The Articulation of L₃: Formal Systems, Computability, and the Arithmetical Sets
L₃ 的阐述：形式系统、可计算性与算术实数

1. Introduction and Position

1.1 Where this paper stands in the series

Paper 1 gives the layer-articulation schema for $L_1$ through $L_5$ and, in its § 6.1, a four-step table for $L_3$ in parallel-display form. Papers 3 and 4 raise $L_1$ and $L_2$ respectively to monograph treatment. This paper does the same for $L_3$.

What this paper fills is the position deferred by Paper 4 § 8.4. That section hands two rule-layer questions to the $L_3$ paper: whether the three Kontsevich–Zagier transformation rules generate all of $\ker(\mathrm{ev})$, and whether equality of general period expressions is decidable. They are received here at § 3.2, inner step 1.

1.2 What this paper delivers

Positive deliverables:

1. the four-main-step articulation of $L_3$ (§§ 3–8), with main step 4 spanning three sections as required by the registration in Paper 3 § 6.1;
2. two T1 theorems (§ 7.1, § 7.2);
3. identification of the criterion required for independence of the two remainders — closure-essential independence — together with counterfactual evidence that the two failure directions are not identical (§ 9); this does not include a proof that any existing closure meets the criterion;
4. a minimal typed organisation of $\mathfrak{C}_T$ (§ 10);
5. eleven upstream errata (§ 11).

Negative deliverables — which this paper regards as no lighter than any of the above:

1. the existing re-zeroing equation $\deg^{0'}(0')=0$ does not exhibit the complete remainder 2 as a non-deletable input, and therefore cannot establish essential arity 2 (§ 10.3);
2. Paper 4 § 13.2's original falsification condition is empty under the adjudicative reading, and by Corollary 7.2 this emptiness extends to all effectively axiomatized extensions, conservative or not (§ 11.2).

Both are affirmative verdicts about negative content, with proofs. That "this paper has not established a mathematical $L_4$" is by contrast an absence, not a verdict, and is therefore not listed here; it belongs to the scope declaration and the open problems (§§ 13, 14, 15.6).

1.3 A methodological discipline that runs through the paper

One earlier failure in this series deserves to be stated in the introduction. A single term denoted two different sets in two places, with no one calling a halt: "the Gödel remainder" meant $\mathrm{Ind}_T$ (degree $0'$) in one passage and $U_T$ (degree $0^{(\omega)}$) in another. An entire chain of judgements built on that equivocation collapsed.

The paper therefore imposes a one-term-one-object discipline (§ 12.3): each object has exactly one definition, and four formulations are prohibited outright while three others may be used only with stated conditions. This table is not rhetorical caution; it is a condition of the paper's standing.

Paired with it is a translatability discipline (§ 12.4): any claim intended to support a mathematical conclusion must first be rendered as a statement of pure mathematics containing no SAE terminology. What cannot be so rendered is one of two things — an SAE-level identification, which is then tiered T3 and may not serve as a premise in a mathematical argument, or else confusion.

1.4 What is not undertaken

See the closing paragraph of the abstract. Two further points require explicit statement.

First, this paper does not argue that $L_4$ cannot close. That claim and "this paper has not established $L_4$" are different claims; only the latter is made.

Second, this paper does not explain why the facts recorded in § 13 amount to a failure of closure. Readings of that kind fall under the jurisdiction of other series and are outside the scope here.

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2. Firewalls and Registrations

This paper inherits the four firewalls of Paper 4 § 2 and grounds each in the material of $L_3$; two further firewalls are erected for the present purpose.

2.1 Substrate and articulation mode

The substrate of $L_3$ is $\mathbb{R}$ (entered by way of arithmetical definability); its articulation mode is $\mathfrak{C}_T$.

The capture core is not the substrate. $\mathrm{ARITH}$ is countable, as is its real-axis counterpart $\mathbb{R}_{\rm arith}$, whereas $\mathbb{R}$ has the cardinality of the continuum. No claim is made that the core exhausts the substrate of $L_3$ — as Paper 4 makes no such claim for $\mathcal{P}$ at $L_2$.

The capture core is also not the articulation apparatus. The apparatus $\mathfrak{C}_T$ is many-sorted, and evaluation is defined only on its formula sort:

$$\mathrm{ev}_{\rm set}:\ \mathrm{Form}_1(L_A)\longrightarrow\mathrm{ARITH}.$$

Details in § 6.4.

2.2 Values, formulas, and operators

Three kinds of object occur here and must be kept apart.

Kind	Instances
Values	members of $\mathbb{R}_{\rm arith}$; $\Omega_U$
Formulas and sentences	members of $\mathrm{Sent}_L$; $\mathrm{Prov}_T(x)$; $\mathrm{Tr}_{\Sigma_n}(x)$
Operators	$C_T^\vdash$; $C_{\mathbb{N}}^\vDash$; the jump; the diagonal lemma

One easily conflated triple. $\mathrm{Con}(T)$ is a single $\Pi^0_1$ sentence; local reflection $\mathrm{Rfn}(T)$ is a sentential schema; uniform reflection $\mathrm{RFN}_\Gamma(T)$ is a schema with a numerical quantifier inside. The three may not be used interchangeably. $\mathrm{Con}(T)$ is one instance of local reflection, taken at $\varphi=(0{=}1)$.

2.3 Three sorts

Following Paper 4 § 2.3: expression objects · value image · expression equivalence relation may not be run together into a single statement. At $L_3$ this lands as follows.

Sort	Content at $L_3$
Expression / language objects	$\mathrm{Sent}_L$; unary arithmetical formulas
Value objects	$\mathrm{ARITH}$; arithmetical sets; oracle sets
Relations	$C_T^\vdash$; $C_{\mathbb{N}}^\vDash$; $\le_T$; $\equiv_T$

One point to be held firmly. The equivalence kernel $\sim$ is an object of the relation sort, not an element of the expression sort. An earlier draft in this series collapsed that distinction once, with a type error as the result.

Quotient objects are flagged separately. A Turing degree is a member of $\mathcal{P}(\omega)/{\equiv_T}$ — a quotient of the value sort by the relation sort — and inhabits no single sort. An individual set has a degree; "the degree space as a whole" does not.

2.4 Formula complexity, degree of a defined set, and reducibility

These three must be kept in separate columns, or $\Sigma_n$, $0^{(n)}$, and Turing degree will start to substitute for one another.

Column	Typical objects
Formula complexity	$\Sigma^0_n$, $\Pi^0_n$, $\Delta^0_n$
Degree of the defined set	$0^{(n)}$, $0^{(\omega)}$
Reducibility between sets	$\le_m$, $\le_T$

One verified consequence. There is no level-by-level identification $\mathrm{RFN}_{\Sigma_n}(T)\leftrightarrow 0^{(n)}$.

For fixed $n$, if $T+\mathrm{RFN}_{\Sigma_n}(T)$ is a consistent c.e. extension of $Q$, then its theorem set remains c.e.-many-one complete, of Turing degree $0'$, and does not climb to $0''$ as $n$ increases; whereas its proof-theoretic strength does increase with $n$.

The consistency hypothesis cannot be dropped: an inconsistent extension proves every sentence, its theorem set is decidable, and its degree is $0$.

Turing degree therefore does not measure the proof-theoretic strength of these reflection theories.

2.5 The category of main step 4

Following Paper 3 § 6.1: main step 4 is a remainder-ledger structure, not a within-layer sub-fractal; its internal organisation runs capture core → unclaimed ledger → trigger for the next layer.

Main step 4 here spans §§ 6, 7, and 8, while main steps 1 through 3 occupy one section each. The asymmetry of the layout is itself that registration.

2.6 Two firewalls erected here

Eligibility of a capture core

A candidate collection qualifies as a capture core only if it meets four conditions (§ 6.5): evaluation precedes the choice of coding; the core is the image of a semantic quotient; it is covariant under legitimate recoding; and its eligibility does not rest on a cardinality argument.

What this excludes. Formal theories, provability relations, Turing degrees, and provability algebras can all be coded as sets of natural numbers and thence as reals. But

$$\text{codable as reals}\ \ne\ \text{naturally carried by a set of reals}.$$

If codability alone sufficed, any countable mathematical structure could be forced back into a "real-number core", and carrier migration would cease to be checkable.

Five criteria for cross-layer regularities

This paper establishes no new cross-layer regularity. Any proposed regularity must satisfy all five criteria of § 12.5: type pre-registration; pre-registration of the coincidence/separation relation; a threshold mechanism; a negative control; and an out-of-sample test.

Three observations are accordingly recorded here as facts without being raised to regularities: the difference between self-supplied and separately posited sub-marks across the three layers (§ 4.2); the fact that the two transition signatures stand in different relations to their respective layers (§ 6.3); and $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$ (§ 5.2 and its interface with § 9).

2.7 Table against misreading

Easily read as	What is actually claimed
The paper proves that mathematics ends at $L_3$	It records that no $L_4$ has been established, and states explicitly that it has not proved $L_4$ cannot be established
$\mathbb{R}_{\rm arith}$ is the capture core of $L_3$	The core is $\mathrm{ARITH}$; $\mathbb{R}_{\rm arith}$ is its real-axis counterpart, of a different type
The Gödel remainder is "true but unprovable"	It is $T$-independence, witnessed by $\mathrm{Ind}_T$; "true but unprovable" is $U_T$, of degree $0^{(\omega)}$
The two remainders and the behaviour collapse at $0'$	$0'$ covers only $K$, $\Omega_U$, and $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$
The paper proves the two remainders independent	It identifies the criterion and gives evidence of non-identity; it does not prove that any existing closure instantiates it
Theorem A establishes the migration prediction	Theorem A cuts along effective / non-effective; the migration prediction cuts along object / rule. Two different axes

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3. Main Step 1: Marking a Meta-Judgemental Interface Without Constructing It

3.1 Main step 1 and inevitability 1

Paper 3 § 1.2 establishes that inevitability 1 corresponds to main step 1, and takes the form of marking without constructing: a mark is fixed as a public token and its operational standing is settled, while its construction as an object is not yet given. At $L_1$ what is so marked is the unit "1"; at $L_2$ it is $i$ (Paper 4 § 3).

At $L_3$ what is marked without construction is a meta-judgemental interface: the standing from which one may speak, from outside a formal system, about the relation between that system and a sentence.

Registration. This main step belongs to the cross-layer threshold category. What it articulates is not an unfolding internal to $L_3$ but the $L_2\to L_3$ transition itself; the input to its first inner step comes from the layer below.

3.2 Inner steps

Inner step 1: the $L_2$ remainder resists absorption at the object layer

Paper 4 § 8.4 has already identified two rule-layer questions at $L_2$: whether the three Kontsevich–Zagier transformation rules generate all of $\ker(\mathrm{ev})$ (derivational completeness), and whether equality of general period expressions is decidable (decision procedure). The subject of both is the rule system itself.

The structure here becomes visible only against $L_1\to L_2$. Paper 4 § 3.2, inner step 1, records that $x^2+1=0$ is posed by $L_1$'s own reverse questioning and that $L_1$'s substrate does not contain the answer: what is missing is an object, and supplying one object supplies it. That is an outside-the-substrate remainder, and it forces a passage between layers.

The situation at $L_2$ is different: nothing is missing as an object. Adjoining any new number to $\mathsf{Int}_{\rm alg}$ will not make "are the three rules complete?" or "is equality decidable?" answerable — these questions do not ask what some value is, but what that system of rules can do. What resists is not the absence of an object but the fact that the question cannot be posed: within the articulation available at $L_2$ there is no position from which "that rule system" can become, as a whole, something spoken about.

The input to $L_2\to L_3$ is therefore not an object awaiting placement but a standing awaiting placement.

〔This paper does not claim that these two rule-layer questions are the two remainders of $L_3$. Their relation is treated separately in §§ 8 and 9.〕

Inner step 2: intersubjective propagation

"Is this proof correct?" and "is this proposition true?" have circulated in mathematical practice for a very long time, and prior to formalisation.

The mode of propagation matches what Paper 4 § 3.2, inner step 2, records for $i$: Cardano refused to grant reality to square roots of negatives while continuing to compute with them, and Bombelli then supplied rules by which they could be passed along — propagation precedes acknowledgement. So here: meta-judgement was long since publicly occupied, but occupied informally.

The difference lies in the type of what propagates. $i$ is a token passed along in computation; meta-judgement is an action jointly performed.

Inner step 3: public binding of a symbol

Cost. So long as "$T$ supports $\varphi$" can only be said informally, that relation cannot itself become the subject of a mathematical statement. Every discussion of inference must re-negotiate its reference.

Binding. What is marked is a single two-faced meta-judgemental interface

$$\Gamma\mathrel{\triangleright_\epsilon}\varphi,\qquad \epsilon\in\{\mathrm{syn},\mathrm{sem}\},$$

whose two polarised realisations are

$$\triangleright_{\mathrm{syn}}=\vdash_T,\qquad \triangleright_{\mathrm{sem}}=\vDash_{\mathbb{N}}.$$

Historically these were bound by the Frege–Hilbert line (the turnstile) and the Tarski line (the satisfaction sign) respectively, roughly half a century apart, and out of two different practices — proof theory and model theory.

Two faces of the binding:

• Handling capacity. Each support relation becomes an object about which mathematical statements can be made. "$T$ proves $\varphi$" and "$\varphi$ is true in $\mathbb{N}$" can from here be asserted and denied separately.
• Remainder. The relation between the two has not yet been internalised. $\vdash_T$ is defined by finitary combinatorial relations on strings; $\vDash_{\mathbb{N}}$ by structures and satisfaction. The two can of course be compared in an external metatheory — soundness, $\Sigma_1$-completeness, and Gödel's theorem are all such comparisons; what cannot yet be done is to carry out that comparison inside the object language by a single internal predicate. Arithmetisation of the provability relation waits for main step 2, inner step 3; uniform internal evaluation of complete truth never arrives (§ 7.1).

Limitation. That the interface polarises into two faces is a type requirement of the trajectory chosen here, not a general theorem about meta-judgement as such. A trajectory that studies only proof apparatus, or only interpretive structure, or that adopts a realizability semantics, is under no compulsion to have exactly two realisations. No necessity beyond this trajectory is claimed.

Notational discipline. Throughout this paper we write $\vdash_T$ and $\vDash_{\mathbb{N}}$, never an unsubscripted $\vdash$ or $\vDash$. The reason: for classical first-order logic with semantic consequence over all structures, the completeness theorem gives $\Gamma\vdash\varphi\iff\Gamma\vDash\varphi$ — between the unsubscripted symbols there is no gap at all. The gap this paper treats is produced by a designated theory and a designated model, and the notation must carry that designation.

Inner step 4: what is constructed, and what remains

Constructed: the two polarisations are available.

Remainder: their relation has not been internalised — comparable externally, not yet effected within the object language by a single internal predicate.

The sharpness of "marked but not constructed" here lies in this: $\mathrm{Prov}$ has long been marked as a concept — people have been asking whether $T$ can prove $\varphi$ for a very long time — while it has not been constructed as a formula of the object language. That waits for arithmetisation (§ 4.2, inner step 3).

The contrast with $L_2$ is worth stating. $i$ is marked but not constructed because its construction as an object — ordered pair, quotient ring, geometric point — has not yet been given. The meta-judgemental interface is marked but not constructed because the two sides of the interface do not yet inhabit a common domain, so that "the interface itself" is not yet an object one can speak about.

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4. Main Step 2: The Additive Path — Provability Closure

4.1 Main step 2 and inevitability 2

Main step 2 articulates the phase of being: from what main step 1 has marked, a positive iterable entity is produced.

At $L_3$ that entity is proof, and what iteration yields is the theorem set.

4.2 Inner steps

Inner step 1: sub-marking the derivation step

Unfolding from the $\vdash_T$ side, the derivation step is sub-marked as the iterable unit.

The contrast with $L_2$ matters. Paper 4 § 4.2, inner step 1, records that the sub-mark at $L_2$ — a direction orthogonal to the real axis — is not derived from within the real line: nothing on the real line designates an orthogonal direction, and that direction is posited in order to house $i$. Historically this step was realised by the geometric representations of Wessel, Argand, and Gauss, roughly two centuries after Euler's notational binding.

At $L_3$ the situation is otherwise: the derivation step comes with the definition of $\vdash_T$ and needs no separate positing. A system of derivation is a set of licences from premises to conclusion, and "applying one licence" is already the iterable unit. In this respect $L_3$'s main step 2, inner step 1, is closer to the $L_1$ case recorded in Paper 4, where the direction of iteration is supplied by the mark itself.

The threefold difference — self-supplied at $L_1$, separately posited at $L_2$, self-supplied at $L_3$ — is recorded as fact. No cross-layer regularity is claimed from it.

Inner step 2: iterating derivation

A proof is a finite sequence of derivation steps; a theorem is one that has a proof. Iteration yields

$$\mathrm{Th}(T)=\{\varphi: T\vdash_T\varphi\}.$$

What this iteration articulates is how far one can get, not what is correct. That is the source of this main step's remainder (inner step 4).

Inner step 3: binding a proof sequence into a single token

Cost. If every proof must be written as a finite sequence of derivation steps, then statements about proofs must be made at the level of sequences. The assertion "there is a proof of $\varphi$" cannot be expressed as an arithmetical condition, and hence cannot become a formula of the object language.

Binding. Gödel numbering $\#$ binds the finite proof sequence into a single token — a natural number. Consequently

$$\mathrm{Prf}_T(p,x)\quad\text{("$p$ codes a $T$-proof whose last line has code $x$")}$$

becomes an arithmetical relation — primitive recursive for recursively axiomatized $T$, and for c.e. axiomatized $T$ reducible to an equally effective form by Craig's reaxiomatisation. Hence

$$\mathrm{Prov}_T(x)=\exists p\,\mathrm{Prf}_T(p,x)$$

is a $\Sigma^0_1$ formula.

Two faces:

• Handling capacity. Statements about proofs become statements about natural numbers. The syntactic side of the interface marked at inner step 1 acquires a representation inside the object language; the diagonal construction of main step 4 becomes possible from here.
• Remainder. The token conceals that it is a sequence. When the internal structure of a proof is needed it must be decoded, and decoding is not an operation the token carries with it.

This occupies the same position as Paper 4 § 4.2, inner step 3, where an ordered pair is bound into a single symbol; here a finite sequence is bound into a single symbol. Both are cost-management bindings across an iteration.

No claim is made that $\#$ and $2\pi i$ occupy parallel positions. In Paper 4, $2\pi i$ first appears at main step 3, as the generator of the deck transformation group, and recurs at main step 4, inner step 2, as residue normalisation; $\#$ first appears here, at main step 2. The positions differ, and nothing may be inferred by analogy. Whether $\#$ also serves at main step 4 as the exchange-law articulation unit assigned to it by Paper 1 § 4.5 is left to § 8.

Inner step 4: what is constructed, and what remains

Constructed. With $T$ consistent, c.e. axiomatized, $T\supseteq Q$, and proof-checking effective,

$$\mathrm{Th}(T)\ \text{is a}\ \Sigma^0_1\ \text{set}.$$

Further, $(P_T,N_T)$ is an effectively inseparable pair, so $\mathrm{Th}(T)$ is creative, many-one complete for $\Sigma^0_1$, and of Turing degree $0'$.

Remainder. Write

$$\mathrm{Dec}_T=\{\varphi: T\vdash_T\varphi\ \text{or}\ T\vdash_T\neg\varphi\},\qquad \mathrm{Ind}_T=\mathrm{Sent}_L\setminus\mathrm{Dec}_T.$$

By Gödel–Rosser, $\mathrm{Ind}_T\ne\varnothing$ under the above hypotheses. $\mathrm{Ind}_T$ is a $\Pi^0_1$ set; by the effective inseparability of $(P_T,N_T)$ it is $\Pi^0_1$-many-one complete, of Turing degree $0'$.

Three ways of stating this remainder incorrectly:

1. $\mathrm{Ind}_T$ is not "the true but unprovable". If $\varphi$ is independent of $T$, then $\varphi$ and $\neg\varphi$ are both in $\mathrm{Ind}_T$; calling one of them true requires fixing the standard model, and no model has been used anywhere in this main step.
2. The complement of $\mathrm{Th}(T)$ is not $\mathrm{Ind}_T$. The former contains all of $N_T$ — the refutable sentences — and the latter does not.
3. "True but unprovable" is a different object. It is $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$, whose definition requires the standard model, and which for sound c.e. $T$ satisfies $U_T\equiv_T\mathrm{Th}(\mathbb{N})$, of Turing degree $0^{(\omega)}$. The remainder taken here is $\mathrm{Ind}_T$, not $U_T$; the reason is given in § 9.

The boundary of the additive path is therefore purely syntactic: iteration can enumerate all theorems but cannot enumerate the complement of what is adjudicated — a failure statable without the concept of truth.

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5. Main Step 3: The Multiplicative Path — Relativisation and the Degree Profile

5.1 Main step 3 and inevitability 3

Main step 3 articulates the phase that is neither being nor non-being: not a new positive entity, but a memory binding on what is already there — recording complexity rather than adding content.

At $L_3$ what is remembered is: how much incomputability a set requires in order to be computed.

5.2 Inner steps

Inner step 1: sub-marking "relative to what"

The existence of $\mathrm{Ind}_T$ makes a new question askable: if a computation may consult an external set, can more be decided?

That question forces the sub-mark "relative to what" — the oracle. Formally, $B\le_T A$ records that $B$ is computable with $A$ as oracle.

This occupies the same position as Paper 4 § 5.2.1, where the remainder of the additive path (rotation is not given by iterating translation) forces the sub-mark of the multiplicative path; here the additive path's remainder $\mathrm{Ind}_T$ forces relativisation.

Inner step 2: composing reductions

$\le_T$ is a preorder. For any $A,B$ the join

$$A\oplus B=\{2n:n\in A\}\cup\{2n+1:n\in B\}$$

gives their least common upper bound. Composition, in this sense, is always available.

Inner step 3: binding an equivalence class into a single token

Cost. Working always at the level of sets, the fact that "$A$ and $B$ require the same amount of incomputability" cannot be stated in a single assertion — each comparison must exhibit reductions in both directions afresh.

Binding. The degree $[A]_T$, as an equivalence class under $\equiv_T$, binds infinitely many sets into a single token.

Two faces:

• Handling capacity. A degree records how much incomputability is needed to compute a member of it. That is exactly what main step 3 requires as memory content: no new sets, but a record of the complexity history of the ones already there.
• Remainder. The token conceals differences within the class. Two sets of the same degree may differ in every other respect, and passing from a degree back to a set is not an operation the token carries.

A discipline of type. A degree is a quotient object. An individual set has a degree; "the degree space as a whole" does not — it is not something reductions can compare. Every statement in this paper involving degrees keeps apart the degree of a set, reducibility between sets, and the structure of the degree space.

Inner step 4: what is constructed, and what remains

Constructed. The degree profile used here is the ideal of arithmetical degrees

$$\mathrm{AR}=\mathrm{DegProf}^{\rm arith}=\{d:\exists n\ d\le 0^{(n)}\}.$$

It has two properties:

• downward closure: if $e\le d$ and $d\in\mathrm{AR}$ then $e\in\mathrm{AR}$. In particular the degree $0$, of the computable sets, lies in it. (We write $e$ rather than $d'$ here; the prime is reserved throughout for the jump of $d$.)
• closure under the jump: $d\le 0^{(n)}\Rightarrow d'\le 0^{(n+1)}$.

Relativisation may therefore be iterated without bound inside this structure: $0,0',0'',\ldots$ all lie in it.

Remainder. $\mathrm{AR}$ has no greatest element, and no least upper bound in $\mathcal{D}$.

Three statements must be kept apart; conflating any one of them produces an error.

1. $\mathrm{AR}$ has no greatest element: $0^{(n)}<_T 0^{(n+1)}$ for every $n$.
2. $\mathrm{AR}$ has upper bounds in $\mathcal{D}$: for instance $0^{(\omega)}=\bigoplus_{n<\omega}0^{(n)}$.
3. $\mathrm{AR}$ has no least upper bound in $\mathcal{D}$.

Statement 3 follows from Spector's exact pair theorem. A countably generated degree ideal $I$ has an exact pair $(a,b)$ with $I=\{d: d\le a\ \text{and}\ d\le b\}$. If $I$ had a least upper bound $c$, then $c\le a$ and $c\le b$, whence $c\in I$ — so $c$ would be the greatest element of $I$, contradicting statement 1. $\square$

Hence one may not write "its supremum is $0^{(\omega)}$." The tempting bad argument runs: any upper bound $u$ satisfies $u\ge_T 0^{(n)}$ for all $n$, therefore $u\ge_T\bigoplus_n 0^{(n)}$. That step fails — it supplies a family of individual reductions, not the single uniform procedure needed to compute the direct sum.

What $0^{(\omega)}$ actually is. For the fixed effective sequence $0,0',0'',\ldots$, say that a degree $u$ computes the sequence uniformly if, for some (equivalently, any) representative $X$ of $u$, there is a uniformly computable index function $f$ with $\Phi^X_{f(n)}=0^{(n)}$ for all $n$. Under this definition $0^{(\omega)}$ is the least uniform upper bound of the sequence: any uniform upper bound computes the direct sum, and $0^{(\omega)}$ is itself a uniform upper bound.

This term must be used with that definition attached, and must not be conflated with "the least upper bound of $\mathrm{AR}$" — which does not exist (Spector).

A uniform upper bound of the ideal does exist, however, and $0^{(\omega)}$ is one. $\mathrm{ARITH}$ is countable, the formulas being countable; enumerating by formula, $\mathrm{Th}(\mathbb{N})$ computes each $A_\varphi$ uniformly — given $\varphi$ and $n$, ask whether $\varphi(\bar n)$ holds in $\mathbb{N}$, by a single procedure — and $\mathrm{Th}(\mathbb{N})\equiv_T 0^{(\omega)}$. So $0^{(\omega)}$ is a uniform upper bound of $\mathrm{AR}$, and the least one, since any uniform upper bound can decide sentences and hence compute $\mathrm{Th}(\mathbb{N})$.

Three things must therefore be distinguished: the least upper bound of $\mathrm{AR}$ does not exist; the least enumeration-uniform upper bound of $\mathrm{AR}$ exists and is $0^{(\omega)}$; and uniformity is always relative to a fixed enumeration, so that changing the enumeration changes what is meant.

A terminological firewall. The notion of "enumeration-uniform upper bound" used here is defined relative to a fixed effective enumeration and is not the standard notion of a uniform upper bound on an ideal found in the literature — the latter is otherwise defined, and its class has no least member. Every use of the term in this paper carries the definition above.

Finally, $0^{(\omega)}\not\le 0^{(n)}$ for any $n$, so $0^{(\omega)}\notin\mathrm{AR}$.

The remainder of the multiplicative path is thus neither a particular jump nor a supremum lying outside the profile, but the fact that the structure has no least upper bound at all.

One inference that may not be drawn. It does not follow that leaving $\mathrm{AR}$ must pass through $0^{(\omega)}$. The exact pair supplies the counterexample: let $(a,b)$ be an exact pair for $\mathrm{AR}$; both are upper bounds; if both computed $0^{(\omega)}$ then $0^{(\omega)}\le_T a,b$, whence $0^{(\omega)}\in\mathrm{AR}$ by the exact pair property — a contradiction. So at least one of $a,b$ is an upper bound of $\mathrm{AR}$ that does not compute $0^{(\omega)}$. More weakly still: any non-arithmetical degree already lies outside the ideal, and leaving $\mathrm{AR}$ requires no command of the whole tower.

The standing of $0^{(\omega)}$ is therefore only this: a canonical uniform direct sum of the finite jump tower, not an object that must be passed through in leaving the ideal.

The correct form of the second-order reverse question. Relative computation permits taking some set $A$ as oracle; but there is no $A$-algorithm that uniformly decides, for all $A$-relative programs, whether they halt. That uniform self-interrogation produces the strictly higher jump $A'$. This is the source of $A<_T A'$ — it comes from self-application, and has nothing to do with lattice-theoretic properties.

〔This paper does not claim that the failure of the Turing degrees to form a lattice — the existence of pairs without infima — bears any generative relation to the remainder here. They are two different matters.〕

An interface with § 9. $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$: the canonical tower-boundary marker selected here, $0^{(\omega)}$, has the same degree as the witness $\mathrm{Th}(\mathbb{N})$ of the second remainder discussed in § 9.

One equivocation to guard against. The remainder of this main step is the structural fact that $\mathrm{AR}$ has no greatest element and no least upper bound; structural facts do not have Turing degrees. What has a degree is the marker $0^{(\omega)}$ selected to stand for the boundary. The two may not be used interchangeably.

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6. Main Step 4(a): The Capture Core

6.0 How main step 4 is organised

Main step 4 is the closure and remainder registration of $L_3$. By the registration in Paper 3 § 6.1 its category is a remainder-ledger structure, not a within-layer sub-fractal; internally it runs capture core → unclaimed ledger → trigger for the next layer, not a re-enactment of the four phases.

It spans three sections: the capture core here (§ 6), the effectivity boundary in § 7, and closure with the remainder ledger in § 8. Three sections against one apiece for main steps 1 through 3 — and that asymmetry of layout is itself the registration, as in Paper 4 § 6.0.

§ 6 and § 7 stand as core and boundary, not as two disjoint bins. The relation to Paper 4's §§ 6 and 7 — "regular core $\hookrightarrow$ irregular enhancement" — is one of functional position, not of content. There, what is added are exponential factors, directions, and Stokes data: the same objects with more structure. Here, what § 7 supplies are two theorems about how far the core can reach — the boundary structure of the core. The difference is recorded as such and is not to be described as sameness of form.

6.1 Carrier migration

Paper 4 § 6.1 establishes that the carrier of the capture range changed type between $L_1$ and $L_2$.

Layer	Capture criterion	Capture core	Carrier type
$L_1$	what equation is satisfied	$\mathbb{Q}^{\rm alg}_{\mathbb{R}}$	scalar field
$L_2$	what representation is available	$\mathcal{P}$	value image of an expression class
$L_3$	in what language a thing is definable	$\mathrm{ARITH}$ (arithmetical subsets)	quotient image of a semantic evaluation

The three slots do not share an ambient type: $\mathbb{Q}^{\rm alg}_{\mathbb{R}}\subseteq\mathbb{R}$, $\mathcal{P}\subseteq\mathbb{C}$, $\mathrm{ARITH}\subseteq\mathcal{P}(\mathbb{N})$. One may not write "all three slots are subsets of the reals" — the period ring is not a subset of $\mathbb{R}$.

What may be said is weaker: each of the three layers retains a real-axis counterpart, and these counterparts form the strictly increasing chain of § 6.3. The full value domains differ; only the counterparts are comparable.

Two things therefore change across the migration: the manner in which the core is constituted (algebraic at $L_1$, representational at $L_2$, semantic-evaluational at $L_3$), and the ambient type itself.

6.2 The capture core

$$\mathrm{ARITH}=\{A\subseteq\mathbb{N}: A\ \text{is arithmetically definable}\},\qquad \mathbb{R}_{\rm arith}=\{x\in\mathbb{R}: \deg(x)\ \text{is an arithmetical degree}\}.$$

Their relation requires two maps and may not be written with one symbol:

$$\mathrm{Form}_1(L_A)\ \xrightarrow{\ \mathrm{ev}_{\rm set}\ }\ \mathrm{ARITH}\ \xrightarrow{\ j\ }\ \mathcal{C}\subseteq\mathbb{R},$$

where $\mathrm{ev}_{\rm set}(\varphi)=A_\varphi$ and $j$ is some fixed coding, say the ternary Cantor coding.

• $\mathrm{ARITH}$ is the capture core; it is the quotient image of $\mathrm{ev}_{\rm set}$.
• $\mathbb{R}_{\rm arith}$ is not the image of $j$: $\mathrm{im}(j)\subseteq\mathcal{C}$, the Cantor set, whereas $\mathbb{R}_{\rm arith}$ contains members outside it — $1/2$, for instance, is computable and hence in $\mathbb{R}_{\rm arith}$, while its ternary expansion uses the digit $1$. So $\mathrm{im}(j\circ\mathrm{ev}_{\rm set})\subsetneq\mathbb{R}_{\rm arith}$.
• "Real-axis counterpart" is therefore a T3 designation only. No canonical surjection $\mathrm{ARITH}\twoheadrightarrow\mathbb{R}_{\rm arith}$ is supplied here. $\mathbb{R}_{\rm arith}$ is defined by its own degree characterisation, and is connected to $\mathrm{ARITH}$ by Post's theorem, not by $j$.

How the core is constituted. Take a unary arithmetical formula $\varphi(n)$ and set

$$A_\varphi=\{n\in\mathbb{N}:\mathbb{N}\models\varphi(n)\}.$$

This evaluation is total in the external semantics, not partial: every unary formula defines a set of natural numbers, and there is no prior need to decide "whether it defines a real". That is precisely why $\mathrm{ARITH}$ is taken as the core rather than some collection of "reals uniquely specified by a formula" — the latter would first require freezing a coding of reals, an equality on codes, and a sense for "uniquely", and this paper does not freeze them.

The difficulty lies not in the evaluation but in the equivalence kernel:

$$\varphi\sim\psi\iff\mathbb{N}\models\forall n\,(\varphi(n)\leftrightarrow\psi(n)),$$

and this relation is Turing equivalent to complete arithmetical truth — given an arithmetical sentence $\sigma$, comparing the constant formula $\sigma$ with a validity reduces arithmetical truth to $\sim$.

The three layers therefore compare as follows.

Layer	Syntactic object	Evaluation	Character
$L_1$	minimal polynomial + isolating interval	select the root	total, computable, equivalence decidable
$L_2$	admissible absolutely convergent period expression	integrate	total on the declared domain; equivalence open
$L_3$	unary arithmetical formula	standard truth set, then coded	externally total; equivalence kernel not internalisable

The syntactic object in the $L_1$ row must carry root-selection data: a minimal polynomial alone does not determine a real algebraic number, since $x^2-2$ has two real roots. A canonical representation must be a pair (irreducible polynomial, rational isolating interval), or a Thom encoding.

6.3 The numerical chain

$$\mathbb{Q}^{\rm alg}_{\mathbb{R}}\ \subsetneq\ \mathcal{P}\cap\mathbb{R}\ \subsetneq\ \mathbb{R}_{\rm comp}\ \subsetneq\ \mathbb{R}_{\rm arith}.$$

The second strict inclusion is witnessed by Yoshinaga's computable non-period reals. The third is witnessed by $\Omega_U$: for a fixed optimal universal prefix-free machine $U$, $\Omega_U$ is left-c.e. and hence $\Sigma^0_2$-definable, while its Turing degree is $0'$ and it is therefore not computable.

One difference of position to be recorded. $\Omega_U$ is the signature of the $L_2\to L_3$ transition, and (as a real) it lies inside $\mathbb{R}_{\rm arith}$, its standard bit set lying in $\mathrm{ARITH}$; whereas $e$ is the signature of $L_1\to L_2$, and whether $e\in\mathcal{P}$ remains open, Paper 4 § 7.1 stating explicitly that nothing is made to rest on it. The two signatures stand in different relations to their respective layers, and no inference is drawn here from that fact.

What Paper 4 calls "definable reals" is not the $\mathbb{R}_{\rm arith}$ of this paper. What that paper establishes is a broad cardinality statement — for any fixed countable formal language, with no parameters or with parameters drawn from a fixed countable source, the definable reals are countable — with no language specified. Taking the first-order language of arithmetic here is a narrowing and specification, and counts as new material.

6.4 Core and apparatus

The capture core is not the articulation apparatus. The apparatus used here is

$$\mathfrak{C}_T=\bigl(\mathrm{Sent}_L,\ C_T^\vdash,\ C_{\mathbb{N}}^\vDash,\ \#,\ \mathrm{DegProf}^{\rm arith}\bigr),$$

where $C_T^\vdash$ and $C_{\mathbb{N}}^\vDash$ are the syntactic consequence closure and the semantic consequence closure over the standard model respectively.

The apparatus is many-sorted, and evaluation is defined on only one of its sorts:

$$\mathrm{ev}_{\rm set}:\ \mathrm{Form}_1(L_A)\longrightarrow\mathrm{ARITH}.$$

$\mathrm{ev}_{\rm set}$ is not defined on the closure operators $C_T^\vdash$, $C_{\mathbb{N}}^\vDash$, nor on the degree ideal $\mathrm{DegProf}^{\rm arith}$. Writing "$\mathfrak{C}_T\to\mathbb{R}_{\rm arith}$" is an abbreviation and not a well-defined map; it is not used anywhere in this paper.

This stands in the same position as the $L_2$ case: there $\mathsf{Int}_{\rm alg}$ is the apparatus, $\mathcal{P}=\mathrm{im}(\mathrm{ev})$ the value core, and $\ker(\mathrm{ev})$ the equivalence kernel, with Paper 4 § 2.3 forbidding the three to be run together. That firewall is retained here.

The two paths land here, but not through a common evaluation. Each component relates to the core in its own way:

Component	Relation to the core $\mathrm{ARITH}$
$\mathrm{Form}_1(L_A)$	evaluated by $\mathrm{ev}_{\rm set}$; its quotient image is the core
$C_{\mathbb{N}}^\vDash$	determines the equivalence kernel of $\mathrm{ev}_{\rm set}$. But its value at the empty premise, $\mathrm{Th}(\mathbb{N})$, is not a member of the core (Tarski) — the syntactic closure falls inside the core while the semantic closure falls outside it, and that asymmetry is exactly what makes Theorem A and the two remainders of § 9 possible
$C_T^\vdash$	its value at the empty premise, $\mathrm{Th}(T)$, is a member of the core (being $\Sigma^0_1$, hence arithmetical); so is the remainder $\mathrm{Ind}_T$ derived from it ($\Pi^0_1$)
$\mathrm{DegProf}^{\rm arith}$	is the degree spectrum of the core, by Post's theorem
$\#$	makes the relations of $C_T^\vdash$ arithmetical — which, together with the effective axiomatisability of $T$ and the effectiveness of proof-checking, supports the third row above (not by $\#$ alone)

So the products of main steps 2 and 3 both relate to the core, but in different ways — one as membership, the other as degree spectrum. No separate pairing is therefore needed for their difference of type; but it may not be said that the two land in a common value image through a common evaluation.

6.5 A criterion: semantic quotient image

$\mathrm{ARITH}$ qualifies as a capture core not because its members can be coded as reals, but because it satisfies the following four conditions.

1. Evaluation precedes the choice of coding. $\mathrm{ev}_{\rm set}$ is prescribed by the capture problem of the layer; one does not first inject objects arbitrarily into some set and then call the injection an evaluation.
2. The core is the image of a semantic quotient: $\{\text{unary arithmetical formulas}\}/{\sim}\ \cong\ \{\text{arithmetical subsets of}\ \mathbb{N}\}$. Equality in the core corresponds exactly to "representing the same object" in the capture problem at hand.
3. Covariance under legitimate recoding. A recoding $r:E\to E'$ is legitimate when (i) $r$ is computable; (ii) it has a computable inverse on its image, or forms a computable bi-interpretation; (iii) it preserves semantic equivalence, $e_1\sim e_2\iff r(e_1)\sim' r(e_2)$; and (iv) there is a designated computable isomorphism $u$ making the evaluation square commute, $u\circ\mathrm{ev}=\mathrm{ev}'\circ r$. Changing the standard Gödel coding, or the syntactic normal form of formulas, is legitimate; an arbitrary set bijection is not.
4. Eligibility does not rest on cardinality. "$E$ is countable, hence embeds in $\mathbb{R}$" confers no eligibility.

Condition 3 has an independent support: by Post's theorem,

$$A\in\mathrm{ARITH}\iff\exists n\ A\le_T 0^{(n)},$$

a characterisation making no reference to syntax at all. Changing the coding does not change it — and that is exactly what "legitimate recoding" designates: one that leaves this characterisation intact.

The criterion is erected because it excludes a class of invalid argument. Formal theories, provability relations, Turing degrees, and provability algebras can all be coded as sets of natural numbers and thence as reals; but

$$\text{codable as reals}\ \ne\ \text{naturally carried by a set of reals}.$$

If codability alone sufficed, any countable mathematical structure could be forced back into a "real-number core", and carrier migration would cease to be checkable.

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7. Main Step 4(b): The Effectivity Boundary

7.0 What this section adds

§ 6 gives the capture core and the manner of its constitution. This section asks: how far can an effective, sound articulation apparatus reach?

The two results of this section are the T1 deliverables of the paper. Their hypotheses are frozen first.

Hypotheses (none dispensable). The formal systems studied in this paper have:

• a recursive language;
• a c.e. axiom set, or a c.e. theorem set;
• an effective proof-checking relation.

"Objects have finite descriptions", "the expression class is countable", and "the theory is effectively axiomatized" are properties of different strength. Paper 4 works with finite expressions and enumerable rules, and this does not by itself yield the hypotheses above; they are therefore frozen here explicitly and not inherited by default.

7.1 Theorem A: obstruction to full closure

Fix the closure target. Let $L_A$ be the first-order language of arithmetic, and

$$C_T^\vdash(\Gamma)=\{\varphi\in\mathrm{Sent}_A: T+\Gamma\vdash\varphi\},\qquad C_{\mathbb{N}}^\vDash(\Gamma)=\{\varphi\in\mathrm{Sent}_A:\mathbb{N}\models\Gamma\Rightarrow\mathbb{N}\models\varphi\}.$$

Lemma 7.1. $C_T^\vdash=C_{\mathbb{N}}^\vDash\iff\mathrm{Th}(T)=\mathrm{Th}(\mathbb{N})$.

Proof. Necessity: take $\Gamma=\varnothing$. Sufficiency: if $\mathbb{N}\models\Gamma$, then $\Gamma\subseteq\mathrm{Th}(\mathbb{N})=\mathrm{Th}(T)$ and both sides yield $\mathrm{Th}(\mathbb{N})$. If $\mathbb{N}\not\models\Gamma$, some $\gamma\in\Gamma$ is false, so $\neg\gamma\in\mathrm{Th}(\mathbb{N})=\mathrm{Th}(T)$, whence $T+\Gamma$ is inconsistent and both sides are the set of all sentences. $\square$

Theorem A. There is no effectively axiomatized theory $T$ with $C_T^\vdash=C_{\mathbb{N}}^\vDash$.

Proof. By Lemma 7.1, $C_T^\vdash=C_{\mathbb{N}}^\vDash$ entails $\mathrm{Th}(T)=\mathrm{Th}(\mathbb{N})$. Effective axiomatisability entails that $\mathrm{Th}(T)$ is c.e., so $\mathrm{Th}(\mathbb{N})$ would be c.e., contradicting Tarski (or, equally, $\mathrm{Th}(\mathbb{N})\equiv_T 0^{(\omega)}$). $\square$

Soundness is not a hypothesis of this theorem. The proof nowhere uses it: unsound effective theories are excluded as well. Soundness is needed only for one half of the corollary below.

Corollary 7.2 (also for language extensions). Let $S$ be a c.e. theory in a recursive language $L'\supseteq L_A$, with an effective proof calculus and a computable embedding of the arithmetical language. Write $\mathrm{Th}_A(S)=\{\varphi\in\mathrm{Sent}_A: S\vdash\varphi\}$. Then $\mathrm{Th}_A(S)$ is c.e.; if in addition $S$ is arithmetically sound and $\mathrm{Th}_A(S)=\mathrm{Th}(\mathbb{N})$, then complete arithmetical truth would be c.e., a contradiction. Hence

$$S\ \text{effectively axiomatized and arithmetically sound}\ \Longrightarrow\ \mathrm{Th}_A(S)\subsetneq\mathrm{Th}(\mathbb{N}).$$

Soundness is what gives $\subseteq$; the failure of equality does not need it.

The scope of the corollary must be stated precisely. It holds for any apparatus that remains effectively axiomatized — whatever it changes: objects, language, axioms, or rules. In particular it holds for the standard recursively axiomatizable large cardinal theories (ZFC together with some fixed large cardinal axiom or recursive axiom schema); but one may not assert it unconditionally of "all conceivable large cardinal theories", since a non-c.e. totality of large cardinal axioms can be stipulated.

The characterisation of the unique satisfier must be relativised to a fixed language. In the fixed language of arithmetic, the unique deductively closed theory satisfying $C_T^\vdash=C_{\mathbb{N}}^\vDash$ is $\mathrm{Th}(\mathbb{N})$ itself; in extended languages it is not unique — for instance the arithmetical reduct of $\mathrm{Th}(\mathbb{N})+\forall x\,P(x)$ is again $\mathrm{Th}(\mathbb{N})$ — and such theories are likewise non-effective.

7.2 Theorem B: threshold for partial closure

For an effectively recognisable class $\Gamma$ of sentences, write $\mathrm{True}_\Gamma=\{\varphi\in\Gamma:\mathbb{N}\models\varphi\}$. Let $T$ be a c.e. theory that is

• $\Gamma$-sound: $T\vdash\varphi\in\Gamma\Rightarrow\mathbb{N}\models\varphi$;
• $\Gamma$-complete: $\mathbb{N}\models\varphi\in\Gamma\Rightarrow T\vdash\varphi$.

Then $\mathrm{True}_\Gamma=\mathrm{Th}(T)\cap\Gamma$, and the right-hand side is c.e. Hence:

Theorem B. If an effective, $\Gamma$-sound and $\Gamma$-complete theory exists, then $\mathrm{True}_\Gamma$ is c.e.

The converse also holds: if $\mathrm{True}_\Gamma$ is itself c.e., adjoining those sentences to a sound base theory yields a c.e. $\Gamma$-complete theory.

The threshold therefore lies not at a level index but at whether $\mathrm{True}_\Gamma$ is c.e.

Fragment	Can an effective sound theory cover it completely?
$\Delta^0_0$	yes
$\Sigma^0_1$	yes
$\Pi^0_1$	no
$\Sigma^0_n$ ($n\ge 2$)	no
$\Pi^0_n$ ($n\ge 1$)	no

The threshold is asymmetric: along $\Sigma$ it breaks between $\Sigma_1$ and $\Sigma_2$; along $\Pi$ it is in force from $\Pi_1$. There is no single threshold statable without specifying a direction.

The mechanism on the $\Sigma_1$ side: a true $\Sigma_1$ sentence $\exists x\,\delta(x)$ has a concrete witness $m$, and $Q$ can verify the bounded computation $\delta(\bar m)$ and then infer $\exists x\,\delta(x)$. So any $T\supseteq Q$ inherits this completeness direction. Equality with standard truth needs $\Sigma_1$-soundness in addition — mere consistency does not suffice, since a consistent extension of $Q$ may still prove a false $\Sigma_1$ sentence.

There is no middle ground. Below the threshold, the base theory has already closed the fragment, and an object extension accomplishes nothing new; above it, no effective sound theory can close it. There is no level of the standard arithmetical hierarchy that is both not yet closed and possibly closable by an effective object extension.

7.3 $0^{(\omega)}$ marked but not entered

Main step 3, inner step 4 has established that the ideal $\mathrm{AR}$ has no greatest element and no least upper bound in $\mathcal{D}$, while $0^{(\omega)}$ is the least uniform upper bound of the finite jump tower and $0^{(\omega)}\notin\mathrm{AR}$. Moreover $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$.

The situation of $L_3$ can then be put in one sentence:

> $0^{(\omega)}$ is pointed at from within $L_3$ and not taken into the degree profile of $L_3$.

It is pointed at because the multiplicative path can write down $0,0',0'',\ldots$ and see that the chain has no cap; it is not taken in because taking it in requires leaving arithmetical definability — which is precisely the criterion defining $\mathrm{ARITH}$.

This occupies the same functional position as Paper 4 § 7.5, "$\mathcal{P}^e$ marked but not entered": in both cases the layer can point at a boundary object it cannot enter. But the objects are of different type — $\mathcal{P}^e$ is the value image of an extension, $0^{(\omega)}$ is a degree — and no sameness of form is claimed.

It should also be added that $0^{(\omega)}$ is the canonical external marker this paper selects for the finite jump tower, and not a boundary degree of $\mathrm{AR}$; the ideal has none.

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8. Main Step 4(c): Closure and the Remainder Ledger

8.1 Coarse closure: diagonalisation

The behaviour of the $L_2\to L_3$ transition is diagonalisation / universal evaluation.

Applied to the two polarisations bound at main step 1, inner step 3, the diagonal lemma yields one witness apiece:

Applied to	Yields
$\neg\mathrm{Prov}_T(x)$	the Gödel sentence
$\neg\mathrm{True}(x)$	the liar, and thence Tarski's undefinability theorem

One construction, two predicates, two results. The first requires a standard provability predicate over a sufficient arithmetical base, and its independence needs a consistency hypothesis besides; the second takes the form: assume a complete truth predicate satisfying all arithmetical T-biconditionals, diagonalise, and derive a contradiction.

Behaviour and its products must be kept apart. The behaviour is diagonalisation; its products include the halting set $K$ and, relative to a fixed optimal universal prefix-free machine $U$, the number $\Omega_U$. Wherever this paper says "the behaviour" it means diagonalisation / universal evaluation and never $K$ or $\Omega_U$.

8.2 Two degrees, two roles

$$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}(\mathbb{N})]_T=0'.$$

The equation holds, but its content must be stated precisely: it records that two products of the behaviour ($K$ and $\Omega_U$) and the first non-trivial fragment of the truth hierarchy coincide in the degree quotient. $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ is a definable, effectively enumerable, successfully captured fragment — not a member of any remainder.

The two degrees therefore have distinct roles.

Degree	Role
$0'$	collapse point — the degree of the behaviour's products (and also of $\mathrm{Th}(T)$ and $\mathrm{Ind}_T$)
$0^{(\omega)}$	least uniform upper bound of the finite jump tower; $\notin\mathrm{AR}$

Hence $0'$ is not a boundary of the degree profile — $\mathrm{AR}$ is closed under the jump and $0'$ is simply one of its members. Nor does $\mathrm{AR}$ have a "boundary degree" at all: it has no greatest element and no least upper bound in $\mathcal{D}$ (§ 5.2). The closure equation $\deg^{0'}(0')=0$ of ZFCρ Paper II § 6.3 records a normalisation at the collapse point.

〔In the relative Turing degrees that equation is close to definitional: given $0'$ as oracle, $0'$ is of course computable. It is registered as such here, and is not treated as evidence for an independent mathematical mechanism.〕

8.3 The remainder ledger

Following the three-class format of Paper 4 § 8.2:

Class	$L_2$	$L_3$
First — outside the substrate	the $\mathbb{H}$ direction (judged impressible, not forced)	none identified 〔T3〕
Second — on the substrate, uncaptured by the mode (a syntactic boundary)	the exponential direction	undetermined
Third — non-computable, non-definable	inherited from $L_1$, still uncaptured at $L_2$	$\mathbb{R}\setminus\mathbb{R}_{\rm arith}$

First class: no forced outside-the-substrate remainder has been identified 〔registered T3, not a T1 fact〕. In the trajectory chosen here — $\mathbb{R}$ as substrate, arithmetical formulas as mode — reverse questioning does not force a new kind of number. This is consistent with what Paper 4 § 8.2 establishes, namely that algebraic closure eliminates the mechanism producing forced outside-the-substrate remainders.

This is not written as "the first class is empty": whether it is depends on the choice of substrate, the choice of mode, and the sense given to "forced", all three of which are registrations of this trajectory rather than mathematical facts. Its caution matches that of the second class.

Second class undetermined. The second class at $L_2$ is a syntactic boundary — $\mathsf{Int}_{\rm alg}$ does not admit exponential factors. The corresponding question at $L_3$ is whether the syntax of first-order arithmetical formulas has an analogous "does not admit". One candidate is set quantification, which the hyperarithmetical requires and the arithmetical language lacks; but the candidate is unverified and may coincide with the third class. It is registered here as open and left unfilled.

Cardinality and naming of the third class. $\mathbb{R}_{\rm arith}$ is countable, so its complement has the cardinality of the continuum. That complement may be called the non-arithmetically-definable reals, but not "the undefinable reals" — in stronger languages many of its members remain definable.

8.4 The $L_3\to L_4$ trigger

The remainder of main step 3, inner step 4 is that the ideal of arithmetical degrees has no greatest element and no least upper bound. One continuation route considered in this paper is to iterate the jump along transfinite indices, forming the family $0^{(\alpha)}$ — and it should be noted that this is a selected route, not the only way of leaving the ideal (§ 5.2).

Transfinite iteration requires a system of ordinal notations. Then:

1. Kleene's $\mathcal{O}$ is $\Pi^1_1$-complete; the admissible notations are not effectively enumerable.
2. The consequences of a progression depend not only on the order type but on the particular computable notation — the intensional character of progressions. Turing 1939, restated in modern terms with local reflection: for every true $\Pi_1$ sentence $\varphi$ there exists a computable notation for $\omega+1$ such that the progression proves $\varphi$; but there is no fixed notation for which a single theory proves all true $\Pi_1$ sentences. The quantifier order is $\forall\varphi\,\exists\dot\alpha$, not $\exists\dot\alpha\,\forall\varphi$.
3. Paths attaining arithmetical completeness exist — Feferman 1962 constructs one using full uniform reflection — but such a path cannot be $\Sigma^1_1$-definable. Moreover, along a $\Pi^1_1$ path satisfying the natural conditions of Feferman and Spector, some true $\Pi^0_1$ sentence is not derivable.

The trigger therefore takes the form: the path-selection function is not definable within arithmetic; complete paths exist but cannot be specified at low complexity.

This trigger is not of the same form as the $L_2\to L_3$ trigger. Paper 4 § 8.4 describes that one as passing "from the level of values to the level of rules": the pressure moves from what a value is to what a rule system can do. The trigger here is "from effectively reachable to requiring external supply of a choice": it is not that the level of the question has changed, but that what is needed to continue cannot be supplied by the articulation itself. The difference is recorded as such and is not to be described as sameness of form.

Three reflection operators must be kept apart. $\mathrm{Con}(T)=\neg\mathrm{Prov}_T(\ulcorner 0=1\urcorner)$ is a single $\Pi^0_1$ sentence; local reflection $\mathrm{Rfn}(T)$ is a sentential schema; uniform reflection $\mathrm{RFN}_\Gamma(T)$ is a schema with an inner numerical quantifier. $\mathrm{Con}(T)$ is one instance of local reflection, taken at $\varphi=(0{=}1)$, and is far weaker than either full local reflection or uniform reflection.

The three facts (1), (2), (3) above come from three different settings: (1) concerns notation systems themselves and is independent of what is iterated; (2) is restated with local reflection; (3) uses full uniform reflection in its first half (Feferman 1962) and, in its second, addresses general recursive progressions of theories satisfying axiomatisation, monotonicity, and effectiveness conditions (Feferman–Spector), classified not by reflection operator.

This paper therefore does not assign each of the three facts to one of the three reflection operators; taken together they characterise several limitations on transfinite progressions, and the paper states them once for all three operators without choosing among them.

〔Limitation: the full list of "natural conditions" in Feferman–Spector has not been checked item by item here; the strict statement should read "covers progressions satisfying the conditions of that paper".〕

Further material on $L_4$ is treated in § 13.

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9. Independence of the Two Remainders

9.1 What is to be shown, and where it bears weight

ZFCρ Paper II § 7.1 establishes that $L_0$ has a single remainder, has a behaviour, and yet has no closure equation; closure requires two independent directions of failure, so that the behaviour can bind them into a loop returning to zero.

The two remainders of this paper are:

	Fact	Witness
Remainder 1	there exist $T$-independent sentences	$\mathrm{Ind}_T$ ($\Pi^0_1$-m-complete, degree $0'$)
Remainder 2	there is no single complete internal truth predicate	$\mathrm{Th}(\mathbb{N})$ (degree $0^{(\omega)}$)

Weight rests on the facts; the witnesses serve only as numerical witnesses — the same arrangement as Paper 4 § 7.5, where the closure remainder of $L_2$ is "the exponential map and the exponential-integral direction" and $e$ serves only as witness.

One thing must be settled before anything else in this section: § 7.1 does not say how strong "independent" has to be. This paper will not proceed on an undefined word.

9.2 Three candidate criteria, all rejected

(a) No shared definitional resources. Too weak. One can contrive two non-overlapping vocabularies for the same object; conversely, two genuinely independent quantities may share a great deal of background apparatus.

(b) No bridge that does not route through the behaviour. Too strong. Two genuinely different structures in mathematics typically still admit comparison maps, partial conservativity, low-complexity zones of agreement, one-way reductions, or common invariants. To deny independence on the strength of any bridge whatever would make independence almost unsatisfiable. § 9.5 will in fact exhibit such a bridge here.

(c) Mutual Turing incomparability of the witnesses. The type is wrong, and the $L_1\to L_2$ prototype itself fails the test: $i$ and $\pi$ are both computable, $\deg_T i=\deg_T\pi=0$, each computes the other. More fundamentally, if remainder 2 is stated as "no complete unified truth predicate exists in the base language", it is an undefinability phenomenon and carries no Turing degree of its own — a degree appears only once a witness set has been selected, and the answer then depends on that selection. Criterion (c) is thus not representation-invariant.

9.3 The criterion adopted: non-redundancy in the closure role

Let the two failure directions be $F_1,F_2$ and let the closure behaviour be $B:F_1\times F_2\to Z$. This paper adopts:

$$\boxed{B\ \text{does not factor through either projection.}}$$

That is, there is no $b_1$ depending on the first argument alone with $B=b_1\circ\pi_1$, and no $b_2$ with $B=b_2\circ\pi_2$.

Informally: delete either direction and the closure task changes; the two failures are not one failure phrased twice. But partial bridges, relations of strength, and one-way computability reductions are all permitted.

The criterion is derived from the mechanism of § 7.1 itself: if the claim is that two items are needed in order to close, what has to be shown is that binarity does not reduce to unarity — not that the two items stand in no mathematical relation whatever.

Call this closure-essential independence.

9.4 Definitional resources: established, but the wording must be narrowed

The principal defining relation of $\mathrm{Ind}_T$ is $\vdash_T$: its definition contains no occurrence of $\mathbb{N}$ and no occurrence of truth. The principal defining relation of $\mathrm{Th}(\mathbb{N})$ is $\vDash_{\mathbb{N}}$: its definition contains no derivation system.

This is also the second reason for taking $\mathrm{Ind}_T$ rather than $U_T$: the definition of $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$ requires $\mathrm{Th}(\mathbb{N})$, and the property fails immediately.

But "no shared definitional resources" overstates it. The two still share a language of arithmetic, a stock of sentences, the propositional connectives; and once computability enters, a syntactic coding scheme as well. The accurate statement is:

> Their principal defining relations differ, but the underlying syntactic carrier is not separate.

9.5 Two bridges, of different standing

Bridge one ($\Sigma_1$-completeness): a genuine pre-behaviour partial bridge.

$Q$ proves every true $\Sigma_1$ sentence (mechanism in § 7.2), so for $T\supseteq Q$:

$$\mathrm{Th}_{\Sigma_1}(\mathbb{N})\subseteq\mathrm{Th}(T)\quad\Longrightarrow\quad \mathrm{Th}_{\Sigma_1}(\mathbb{N})\cap\mathrm{Ind}_T=\varnothing.$$

Semantic information here constrains a purely syntactic object, and the proof runs through representability of computation — the effective verification of a finite witness — and not through the diagonal lemma. It is therefore a bridge that does not route through the behaviour.

But it is only partial: it neither reconstructs complete truth from $\mathrm{Ind}_T$ nor reconstructs the whole independence set from the truth gap. It should be called a zone of semantic–syntactic agreement at the $\Sigma_1$ level, and not "the two remainders have been connected".

Bridge two (the one-way reduction): still routes through arithmetisation.

$\mathrm{Ind}_T$ is $\Pi^0_1$, hence $\le_T 0'\le_T 0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$; the converse fails. So $\mathrm{Th}(\mathbb{N})$ determines $\mathrm{Ind}_T$ and not conversely.

This is not a pre-behaviour bridge, however. The moment one writes $\mathrm{Ind}_T\le_T\mathrm{Th}(\mathbb{N})$, both sides have been presented as sets of natural numbers — formulas coded as numbers, proofs coded as numbers, a machine querying those codes. That requires arithmetisation $\#$. The reduction refutes only a stronger claim ("even after arithmetisation, and short of the diagonal lemma, there is no bridge at all"), not the claim that there is no bridge before arithmetisation.

And the reduction is in any case to be expected: complete arithmetical truth is a very strong oracle, computing every arithmetical set; $\mathrm{Ind}_T$ is an arithmetical set, so its being computed is structurally inevitable. What this reflects is the informational strength of the oracle, not whether the two failure roles are redundant.

9.6 Counterfactual evidence that the two failure directions are not identical

What this subsection does and does not establish. The counterfactuals below show that the two failure directions are not one failure phrased two ways; they do not show that any existing closure takes both as non-deletable inputs — for which see § 9.7 and § 10.3, where the matter is left unresolved.

Each direction can be varied on its own:

Counterfactual	Direction of remainder 1	Direction of remainder 2
$\mathrm{Th}(\mathbb{N})$ (complete but non-effective)	$\mathrm{Ind}_{\mathrm{Th}(\mathbb{N})}=\varnothing$ — vanishes	arithmetical truth is still not definable by a single arithmetical formula — boundary stands
$\mathsf{CT}^-[\mathrm{PA}]$	still an effective system, hence still incomplete	truth of the base arithmetical language is compositionally expressed in an extended language, conservatively over PA — partially repaired

Hence

$$\text{incompleteness of the provability closure}\ \ne\ \text{failure of internal definability of complete truth}.$$

9.7 What this section does not supply

The above is the identification of a necessary condition, not a proof that any closure instantiates it.

Specifically: the collapse in § 8.2 involves two products of the behaviour and the first non-trivial fragment of the truth hierarchy, and does not contain the complete remainder 2. A typed argument is therefore still owed:

> Why must diagonalising / relativising closure take both Gödel incompleteness and Tarski undefinability as non-deletable inputs?

This is not a question about the definition of independence; it is that the closure map has not been fully written down. It is registered as open (§ 15).

Should that argument never be given, the $L_2\to L_3$ transition should be called a coupled re-zeroing of two boundaries rather than a closure of two independent remainders. This fallback is registered in advance, so that the stronger description cannot later be retained on the ground that nothing has yet refuted it.

9.8 Conclusion of this section

> The principal defining relations of the Gödel and Tarski directions are $\vdash_T$ and $\vDash_{\mathbb{N}}$ respectively, but the two are not absolutely insulated: the true $\Sigma_1$ sentences form a pre-behaviour partial bridge, and complete truth computes the syntactic independence set one way. This paper therefore does not claim independence in the sense of having no bridge, or of Turing incomparability. The weaker condition it requires is that both failure directions be non-deletable in the closure task. Whether that closure-essential independence is fully realised by the existing $0'$ re-zeroing structure remains to be shown.

§ 10.3 goes further: the existing $0'$ re-zeroing equation does not exhibit the complete remainder 2 as a non-deletable input, and therefore that equation by itself cannot show that it instantiates the criterion. This section and § 10.3 do not cancel each other: this section shows that the two failure directions genuinely differ; § 10.3 shows that the existing equation does not exhibit dependence on both inputs — which is not the claim that the existing closure does not in fact use both. Together they constitute the debt registered in § 9.7.

9.9 A narrowing of one upstream formulation

ZFCρ Paper II § 3.4 states that the exponential map is "the sole source" of the relation between $i$ and $\pi$. As a claim about mathematics in general this is too strong; there are at least two bridges that do not go through the complex exponential.

1. Rotational geometry. Once the complex plane is in place, multiplication by $i$ is a quarter turn, and a quarter turn in radian measure is $\pi/2$. This bridge uses only complex multiplication and Euclidean circle geometry.
2. Contour integration. Take $z(t)=\cos t+i\sin t$; then $dz=i\,z(t)\,dt$, so $\oint_{|z|=1}dz/z=\int_0^{2\pi}i\,dt=2\pi i$. The computation uses no complex exponential.

Moreover Paper 4 already registers the deck-generator identity and the residue-normalisation identity of $2\pi i$ as two presentations of a single statement — so "sole source" is unsupported from within the series as well.

The weaker statement must also be written with care. One may not write "there is no nonzero $P\in\mathbb{Q}[X,Y]$ with $P(i,\pi)=0$": that has an elementary counterexample, $P(X,Y)=X^2+1$, for which $P(i,\pi)=i^2+1=0$.

The correct statement is: $\pi$ is transcendental over $\mathbb{Q}(i)$. Equivalently, if $P\in\mathbb{Q}[X,Y]$ and $P(i,\pi)=0$, then $P$ lies in the ideal $(X^2+1)$ — there is no nontrivial additional algebraic relation involving $\pi$; that $i$ satisfies $X^2+1=0$ is of course not in question.

But "$\pi$ is transcendental over $\mathbb{Q}(i)$" does not amount to "there is no geometric, analytic, or topological bridge".

The passage should accordingly read: the exponential map is the canonical binding source, within this trajectory, of the relation between $i$ and $\pi$. See § 11.4 for the erratum registration.

One dividend: $L_1\to L_2$ likewise has pre-behaviour partial bridges. What § 9.5 reports is therefore not peculiar to $L_3$; the two transitions are on a par in this respect.

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10. A Minimal Typed Organisation of $\mathfrak{C}_T$

10.0 What this section delivers, and what it does not

Delivered: a minimal typed organisation of $\mathfrak{C}_T$ — two tables. The first gives the type and internal operations of each component (§ 10.1); the second gives the cross-component interfaces (§ 10.2).

Not delivered: the closure map owed by § 9.7.

And this section will show that the nature of that debt differs from its earlier registration (§ 10.3). It was previously registered as "the closure map has not been written down". The organisational work here shows that the more accurate statement is: the existing re-zeroing equation does not exhibit the complete remainder 2 as a non-deletable input, and therefore cannot by itself establish closure-essential independence. Whether a closure equation meeting the criterion exists, this paper neither affirms nor denies.

Attribution: everything in the two tables is classical. What is done here is the organisation — as in Paper 4 § 9.4.

The word "minimal" is to be held to. What is delivered is at the level of a skeleton. It makes $\mathfrak{C}_T$ more than a list of five items; it does not make it a completed closure apparatus.

10.1 Table I: types and internal operations of the components

Component	Type	Internal operations
$\mathrm{Sent}_L$	decidable set of codes	connectives; substitution
$C_T^\vdash$	closure operator on $\mathcal{P}(\mathrm{Sent}_L)$	extensive, monotone, idempotent
$C_{\mathbb{N}}^\vDash$	as above	extensive, monotone, idempotent
$\#$	coding $\mathrm{Form}\to\mathbb{N}$	compatible with substitution and proof-checking
$\mathrm{DegProf}^{\rm arith}$	downward-closed set of arithmetical degrees	join $\oplus$; closed under the jump

On the idempotence in rows two and three. Both are closure operators, so $C(C(x))=C(x)$. This matters in the comparison with $L_4$: the reflection progressions of § 13.2 do not give a closure operator but a re-basing. In a standard progression each of whose stages is consistent, effectively axiomatized, and sufficiently strong, and in which the reflection principle adjoined at each stage implies at least the consistency of that stage, successor stages extend strictly:

$$T_\alpha\subsetneq T_{\alpha+1}.$$

〔The qualification cannot be dropped: inconsistent theories, and non-effective theories already containing the relevant reflection, are counterexamples.〕

Idempotence and strict ascent are two different operator behaviours, and they belong to two different layers.

On the last row. $\mathrm{DegProf}^{\rm arith}$ is closed under the jump and has no greatest element; it has no least upper bound in $\mathcal{D}$ either (§ 5.2). It is a downward-closed set, not a tower — the degree $0$ of the computable sets belongs to it, so it fills out the arithmetical degrees from $0$ upward rather than climbing from $0'$.

10.2 Table II: cross-component interfaces

Interface	Map	Statement
$\mathrm{Form}_1\to$ value	$\mathrm{ev}_{\rm set}$	$\varphi(n)\mapsto A_\varphi=\{n:\mathbb{N}\models\varphi(n)\}$; quotient image is $\mathrm{ARITH}$
$\mathrm{Sent}\to$ arithmetic	$\#$	makes $\mathrm{Prf}_T$ an arithmetical relation and $\mathrm{Prov}_T$ a $\Sigma^0_1$ formula
value $\leftrightarrow$ degree	Post	$A\in\mathrm{ARITH}\iff\exists n\ A\le_T 0^{(n)}$ — a characterisation free of syntax
$C_{\mathbb{N}}^\vDash\leftrightarrow$ degree	Post's hierarchy	$[\mathrm{Th}_{\Sigma_n}(\mathbb{N})]_T=0^{(n)}$ for $n\ge 1$
$C_T^\vdash\leftrightarrow C_{\mathbb{N}}^\vDash$	(no uniform internal map)	see below

The last row is the substance of the table, and each of its clauses must be stated separately.

• A zone of agreement. For $T\supseteq Q$, $\mathrm{Th}_{\Sigma_1}(\mathbb{N})\subseteq\mathrm{Th}(T)$ — the two agree at the $\Sigma_1$ level (§ 9.5, bridge one).
• One-way inclusion, under a condition. For sound $T$, $C_T^\vdash\subseteq C_{\mathbb{N}}^\vDash$; without soundness this fails.
• Non-equality. By Theorem A, $C_T^\vdash\ne C_{\mathbb{N}}^\vDash$ for any effectively axiomatized $T$.
• Two distinct objects must be kept apart. For sound $T$, the actual set difference at the empty premise is $C_{\mathbb{N}}^\vDash(\varnothing)\setminus C_T^\vdash(\varnothing)=U_T$; whereas $\mathrm{Ind}_T$ is not that difference — it contains both sides of each independent pair. $U_T$ is the semantic gap under a soundness hypothesis; $\mathrm{Ind}_T$ is the object that witnesses the incompleteness of $T$ purely syntactically and without depending on soundness. This paper takes the latter.
• No uniform internal map. By Tarski, there is no single arithmetical formula in the base arithmetical language effecting standard truth evaluation uniformly. Tarski forbids neither external maps, nor metatheoretic comparison, nor a truth predicate in an extended language.

This row occupies the same position as its counterpart at $L_2$. Paper 4 § 2.3 establishes $R_{\rm KZ}\subseteq\ker(\mathrm{ev})$, with equality being the Kontsevich–Zagier conjecture. The two layers correspond thus:

	Derivable	True	Their relation
$L_2$	$R_{\rm KZ}$	$\ker(\mathrm{ev})$	open (KZ)
$L_3$	$C_T^\vdash$	$C_{\mathbb{N}}^\vDash$	provably unequal (Theorem A)

This reading is not imposed by the present paper: in both cases what is compared is some derivation system against some true relation, and in both cases the objects of comparison are supplied by the diagram structure of § 2.3 within the respective layer.

〔One overreach to avoid: the two relations differ — one open, one provably unequal. No cross-layer regularity is claimed from this (§ 12.5, criterion 3: a threshold mechanism must not be defined using material from the target layer).〕

10.3 A finding: the strength of the negative result

The criterion of § 9.3 requires a closure behaviour $B:F_1\times F_2\to Z$ not factoring through either projection. The organisational work of this section brings one thing into view.

First, an argument that cannot be used.

A natural thought is that the candidate behaviours at $L_3$ — the diagonal lemma, the jump, universal evaluation — are all unary, and so none can carry $B$. That argument is unusable, for two independent reasons.

1. Arity is not a representation-invariant property. Any binary computable function can be recast as unary by a pairing function, and any unary function as a binary one ignoring its second argument. Universal evaluation is in any case written $U(e,x)$; the join $A\oplus B$ used in this very paper is binary. So surface arity cannot decide whether two failure directions are non-redundant in a closure.
2. Binary fixed-point constructions are classical. Smullyan's strong double recursion theorem yields recursive functions $t_1(y_1,y_2)$, $t_2(y_1,y_2)$.

So one may not say "no binary construction is available". But neither may one say that such constructions satisfy the criterion of § 9.3 — what these theorems deliver is simultaneity (simultaneous fixed points), not essential dependence: if the given r.e. relation $M_1$ simply ignores $y_2$, the resulting solution factors through the first projection perfectly well. From "simultaneous fixed points exist" it does not follow that "both inputs are non-deletable".

The obstacle must therefore be located afresh (see third point below).

Second, the existing re-zeroing equation does not exhibit the second direction as a non-deletable input.

$$\deg^{0'}(0')=0,$$

where $0'=\deg(K)=\deg(\mathrm{Th}(T))=\deg(\mathrm{Ind}_T)=\deg(\Omega_U)$; whereas $\mathrm{Th}(\mathbb{N})$, the witness of remainder 2, has degree $0^{(\omega)}$ and does not appear in the equation.

One overreading to guard against: it does not follow that "all the sources of $0'$ lie on the syntactic side". The degree quotient forgets provenance; the same $0'$ is realised jointly by $K$, $\Omega_U$, $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$, $\mathrm{Th}(T)$, $\mathrm{Ind}_T$ and other heterogeneous objects, and "syntactic side" cannot be read off a degree. What does stand is only: the witness of the complete remainder 2 is not in the equation.

Third, locating the obstacle: two steps, neither completed.

Step one — the inputs are not yet a well-defined pair. The content of remainder 2 is precisely that "there is no complete $\mathrm{True}(x)$ formula in the base arithmetical language". One cannot therefore define remainder 2 that way and, in the same breath and without extending the language, feed $\neg\mathrm{True}(x)$ to the strong double recursion theorem as an ordinary formula. Working instead in an extended truth language requires specifying afresh: the new language; which layer of language the truth predicate evaluates; the theory used and its compositional axioms; and the arithmetical or modal interpretation of the fixed-point theorem in that setting. This paper has not completed that step.

Step two — even with well-defined inputs, simultaneity is insufficient. By the first point above, one would still have to show that the resulting object depends on both inputs non-deletably, and the classical theorems do not deliver this.

This paper therefore neither claims that the obstacle lies in the type of the input slots, nor that substitution would suffice. Both go beyond what it can show.

Fourth, the exact strength of what is established here.

The one statement that stands is:

> The equation $\deg^{0'}(0')=0$ does not exhibit the complete remainder 2 as a non-deletable input, and therefore cannot establish that the existing closure has essential arity 2.

This is a verdict of insufficient evidence, and not a theorem that the equation "has been shown to factor through $\pi_1$" — the latter would require a $B$ already defined, and this paper defines no $B$.

Neither of the two available ways forward is taken here:

• find a binary closure action for which neither direction is deletable;
• or revise the requirement of "two independent failure directions" so as to permit unary closure together with one unclosed remainder.

The second would bear on ZFCρ Paper II § 7.1's treatment of $L_0$ — which argues precisely from "only one remainder, hence no closure" — and is not to be altered lightly.

Registration: this problem moves here from § 9.7, more precisely stated but still unresolved. See § 15.2.

10.4 Two functional differences from $L_2$

Point	$L_2$ (Paper 4 § 9)	$L_3$ (this section)
Types of the components	group, functional, pairing, representation, operator family, scalar invariant — types not uniform, so § 9.1 there needs a separate paragraph for $G_\infty$	code set, two closure operators, a coding map, a degree profile — more uniform, with only $\mathrm{DegProf}$ a quotient structure
What is not delivered	a unified categorical encapsulation of the six data types; the obstacle is translation between the language of chains 1–2 and that of chains 3–4	a binary re-zeroing equation taking the two failures as inputs; neither of two steps completed (§ 10.3)

The two steps owed at $L_3$: first, make both inputs well defined in an explicit setting of language and theory; second, show that the resulting construction depends essentially on both. Whether existing binary tools can complete these two steps is not adjudicated here.

The two obstacles should not be described as being of the same kind. At $L_2$ the tools (the RH formalism) exist and the map has not been built — a docking problem. At $L_3$ it is not even settled whether the second input can be made well defined in some explicit setting. No similarity of type is claimed.

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11. Relations to Other Papers and Series

11.0 The two roles of this section

Besides registering how this paper connects to its upstream, this section carries an errata function: in the course of articulating $L_3$, eleven passages across three upstream papers were found to require change.

Order of publication. The errata are given here first; the upstream papers will each issue new versions subsequently.

Citation discipline arising from this. Where this paper cites upstream, it cites the published version and notes the corresponding erratum item at the point of citation. Readers should not assume that the upstream has been updated in step with this paper.

Tiering of the errata. Each item below is tiered by the three-way scheme of Paper 4 § 12.1: correction of error (the original statement is mathematically untenable); clarification (the original reads through, but the wording admits a misreading); substantive revision (the role of the original conclusion changes).

11.1 Relation to Paper 3 ($L_1$)

Paper 3 supplies three formats used here: the division of labour between four main steps and four inner steps; the registration of main step 4 as a remainder-ledger structure rather than a within-layer sub-fractal (§ 6.1 there); and the three-way classification of remainders (§ 6 there).

One detail to be supplied, not constituting an erratum. The $L_1$ row of the three-layer table in § 6.2 above takes "minimal polynomial plus isolating interval" as the syntactic object. A minimal polynomial alone does not determine a real algebraic number ($x^2-2$ has two real roots). The corresponding passage in Paper 3 does not develop this, since its articulation does not depend on uniqueness of canonical representation; the present paper must supply it because it compares the evaluation behaviour of three layers.

11.2 Relation to Paper 4 ($L_2$)

This paper inherits from Paper 4 the four firewalls (§ 2), carrier migration (§ 6.1), the three-way classification of remainders (§ 8.2), the organising principle of main step 4 (§ 6.0), and the claim tiers and prediction discipline (§ 12).

What this paper fills is the position deferred by Paper 4 § 8.4: that section hands the two rule-layer questions of $L_2$ to the $L_3$ paper, and they are received at § 3.2, inner step 1 above.

Erratum 6 · Paper 4 § 13.2 (the migration prediction)

The original states a migration prediction — that after object-domain extension is exhausted, pressure passes into the space of representations and rules — and supplies a falsification condition: the prediction fails if the principal closure action of $L_3$ can be accomplished solely by adjoining new kinds of object to the object language, without changing rules of equivalence, proof, or decidability. The accompanying table registers conservative class-theoretic extension (NBG) as an instance that "can trigger".

Three things are withdrawn.

(a) The registration that NBG can trigger falsification. 〔correction of error〕

Let $S$ conservatively extend $T$ over the old language $L$, and let $G$ be an $L$-sentence with $T\nvdash G$ and $T\nvdash\neg G$. If $S\vdash G$ then $T\vdash G$ by conservativity — a contradiction; similarly for $\neg G$. Hence a conservative object extension cannot adjudicate any sentence undecidable in the old language.

(b) The narrowing of "the only falsification route is conservative object extension" — and its strengthening. 〔correction of error〕

By (a) the conservative branch is empty. And by Corollary 7.2, the emptiness extends to all effective extensions, conservative or not:

Type of extension	Why it does not trigger
conservative (NBG-type)	adjudicates no old-language independent sentence, by (a)
non-conservative but still effective (large cardinals, forcing axioms)	adjudicates some, but never attains full closure (Corollary 7.2)

Hence, once the closure target is made precise as $C_T^\vdash=C_{\mathbb{N}}^\vDash$ by (c) below, the falsification condition is empty for every effectively axiomatized extension. This gives the word "structurally" in the discipline added by Erratum 7 ("a falsification condition subsequently shown to be structurally unsatisfiable") a theorem behind it, rather than a combinatorial observation.

But one still may not write "the original condition is empty under every reading" — what has been shown is emptiness under the adjudicative reading; the expressive reading is a separate matter.

(c) The standing of the original falsification condition. 〔substantive revision〕

The original condition does not define the closure target. It describes the extension mechanism in detail while never saying what "the principal closure action of $L_3$" refers to. The NBG row therefore establishes only that $S/T$ is conservative and adds a language of classes; it does not establish that $S$ accomplishes the principal closure of $L_3$.

What this paper does instead. Once the closure target is made precise as $C_T^\vdash=C_{\mathbb{N}}^\vDash$, what results is not a new falsification condition but Theorem A of § 7.1. That theorem does not establish the original object–rule migration prediction: Theorem A cuts along effective / non-effective, the migration prediction along object / rule — two different axes. Indeed $\mathrm{Th}(\mathbb{N})$ attains full closure while being no effective rule system, and it need not be described as "migration into rules" either — it simply places complete semantic truth directly into the axiom set.

(d) Downgrading the migration statement. 〔substantive revision〕

The original migration statement is downgraded to a T3 explanatory proposal, and no longer claims to possess an operationalised falsification condition.

What happened must be stated accurately. One may not write "the prediction was subsequently proved to be a theorem" — that would conceal a change of object. The accurate registration is a conjunction of three:

$$\text{withdraw the operationalisation of the old prediction}\ +\ \text{add a theorem}\ +\ \text{downgrade the migration reading}.$$

One further term must be unpacked 〔clarification〕: "rule" in the original condition covers at least five axes — R0 the logical calculus · R1 language/signature · R2 non-logical axioms · R3 consequences in the old language · R4 algorithmic structure. The original wording reads like R0 while the table in fact applies R3. Expressive power of the metalanguage belongs to R1, not R2. Any successor condition must be written as a condition vector and may not use the bare word "rule" again.

Erratum 7 · Paper 4 § 12.5 (prediction discipline)

Three passages in that section depend directly on § 13.2 being a prediction with an operationalised falsification condition:

Passage	What the original says	After the erratum
mode of registration	§ 13.2 is registered by "state the prediction, supply an operationalised falsification condition, indicate known counterexample candidates"	the premise fails
the counterexample instance	what can genuinely trigger falsification is conservative object extension, instantiated by conservative class-theoretic extension	exactly what Erratum 6(a) withdraws
bearer of testability	the testability of § 13.2 is borne chiefly by its operationalised falsification condition	the bearer fails

〔correction of error + substantive revision〕

One case must also be added 〔addition〕. § 12.5 writes: this series makes predictions; predictions can be wrong; when falsified we say so. But what has occurred here is not falsification — the prediction was not overturned by a counterexample; its falsification condition was found to be empty. That case is unprovided for. Proposed addition:

> If the falsification condition of a prediction is subsequently shown to be structurally unsatisfiable, that condition lapses, and the prediction must be downgraded to an explanatory proposal or rebuilt; it may not be retained on the ground that it has not yet been falsified.

The reason: a discipline that provides only for "own up when wrong" is silent in the face of "the condition could never have been triggered" — and the latter is the more concealed form of immunisation. The addition therefore strengthens the section rather than weakening it.

Also: § 12.5 states that large cardinal and forcing extensions remain legitimate objects of stress-testing but do not constitute falsification routes under the current condition — a sentence that must be rewritten once the condition is withdrawn.

Treatment of the priority claim. § 12.5 states that the version in question "carries the public priority of § 13.2". After the downgrade this must be stated in two parts: the observation has priority (that pressure appears to pass into rule space was indeed published before the $L_3$ paper), while the predictive claim is withdrawn (its operationalisation does not stand). Failing to separate them yields "retain the priority and withdraw the content".

11.3 Relation to Paper 1 (Architecture)

Paper 1 § 6.1 supplies the four-step parallel-display table for $L_3$, which this paper raises to monograph treatment; Paper 1 § 4.5 assigns the exchange-law articulation unit of each layer.

Erratum 1 · Paper 1 § 6.1, the boundary marker of $L_3$-2 〔substantive revision〕

The original records the boundary marker of $L_3$-2 as "the true but unprovable".

The problem. That phrase denotes $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$, whose definition requires the standard model and whose Turing degree, for sound c.e. $T$, is $0^{(\omega)}$. But the remainder of the additive path at $L_3$ is the purely syntactic $\mathrm{Ind}_T$, of degree $0'$, whose definition uses no model at any point.

Read instead: the boundary marker of $L_3$-2 is the $T$-independent sentences ($\mathrm{Ind}_T$). Where "true but unprovable" is genuinely needed, it must be identified as $U_T$ and accompanied by the soundness of $T$.

Erratum 2 · Paper 1 § 6.1, the placement of $L_3$-3 / $L_3$-4 〔substantive revision〕

The original places relative Turing degrees, the structure of the degree space, and fine-grained formulas of the form $\deg^A(\cdot)$ at $L_3$-3.

The division adopted here:

Remaining at $L_3$-3	Passing to $L_3$-4
the reducibility preorder $\le_T$	diagonal self-interrogation
the equivalence $\equiv_T$	the jump $A\mapsto A'$ as re-zeroing mechanism
the degree quotient and join	relativisation with $A'$ as oracle
the arithmetical degree profile	the re-zeroing equation $\deg^{0'}(0')=0$

That is: the jump is produced as a remainder at main step 3, inner step 4, and received as a re-zeroing mechanism at main step 4. $0'$ carries a double role — boundary marker of main step 3 and re-zeroing point of main step 4.

Note: the exchange-law articulation unit $\#$ does not move on this account.

Errata 3–5 · Three pre-v2 assignments

Paper 1 was published before SAE Foundation v2, so its $L_4$-related assignments were made against the mixed framework of v1. Paper 1 is not at fault, and neither is v2 — v2 § 1.4 declares its redrawing of scope transparently; what was missing is a registration of the downstream effect of that redrawing upon already published papers of the mathematics series. This paper supplies it.

#	Passage	Paper 1's assignment	v2's redrawing	Tier
3	Paper 1 § 4.5	the articulation unit of $L_4$ is $c^2$	$c$ is the spatialisation transition signature of $L_2\leftrightarrow L_3$; its later appearances within layers are extended use of an established spacetime metric, not a fresh signature at each layer	clarification
4	Paper 1 § 6.2	$L_4$-1 marks $ct$ and $G$ as handles (i.e. treats $G$ as a remainder)	$G_N$ is not a remainder; it is a response signature	clarification
5	Paper 1 § 6 as a whole	a single ladder $L_1$–$L_5$, changing domain at $L_4$	two independent ladders, each running $L_0$–$L_5$, isomorphic in structure and independent in content	substantive revision

Item 5 bears weight. On the single-ladder reading one can say "the ladder has not stopped, it has merely become physics", which would weaken the provisional closing of § 14; on the two-ladder reading the $L_3\to L_4$ step of the physical-quantity ladder is not a continuation of the mathematical $L_3$. This paper adopts v2's two-ladder reading explicitly and registers Paper 1 § 6's single-ladder narration as pre-v2.

Consequence: the exchange-law units of Paper 1 § 4.5 are redistributed, leaving three on the mathematical side ($\log$, $2\pi i$, $\#$), with a corresponding reduction in evidential strength — see § 14.3.

11.4 Relation to ZFCρ Paper II

That paper's § 6 supplies the $L_2\to L_3$ transition structure — two remainders plus one behaviour, yielding closure — which this paper inherits. §§ 8 and 9 above fill in and revise that structure.

Erratum 8 · ZFCρ Paper II § 6.1, remainder 2 〔correction of error / substantive replacement〕

The original records remainder 2 as Tarski (the boundary of expression), with the remainder given as "the class of $\Sigma_1$ truth predicates".

The disambiguation. Read as the family of stratified truth predicates having no unifying member, the phrase is Tarski's theorem itself, and the reading is correct. Read as the set of true $\Sigma_1$ sentences $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$, it fails: for each $n\ge 1$ there is a partial truth predicate $\mathrm{Tr}_{\Sigma_n}$ of complexity $\Sigma_n$ with PA proving the corresponding T-biconditionals formula by formula, so $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ is definable and effectively enumerable and cannot carry undefinability.

The former reading is taken, but this must be acknowledged as replacement rather than disambiguation: the original explicitly writes $\Sigma_1$, and construing that as the full stratified family is too charitable. The item is therefore tiered as correction / substantive replacement, not as clarification.

Precise statement: there is no single complete truth predicate within the base language realising all levels uniformly. (In an external metatheory there is an effective procedure producing, for each $n$, the code of the $n$-th partial truth predicate; what does not exist is a unifier for all levels inside the base language.)

Erratum 9 · ZFCρ Paper II § 6.2, the reading of the collapse 〔substantive revision〕

The equation is unchanged:

$$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}(\mathbb{N})]_T=0'.$$

Its reading must change. The original reads it as "the two remainders and the behaviour collapse together". That does not stand: complete arithmetical truth has degree $0^{(\omega)}$ and cannot enter this single point; and $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ is a successfully captured fragment, not a member of any remainder.

Read instead: two products of the universal-evaluation behaviour ($K$ and $\Omega_U$) coincide in the degree quotient with the first non-trivial fragment of the truth hierarchy.

So although the equation is untouched, this remains a substantive revision — the role of the conclusion has changed, and it may not be registered as disambiguation.

Erratum 10 · A note to the table of ZFCρ Paper II § 6.3 〔addition〕

That table lists the collapse point of $L_2\to L_3$ as $0'$, the closure equation as $\deg^{0'}(0')=0$, and the character of the re-zeroing as asymmetric. All three stand.

Two notes are to be added:

1. $0'$ is the collapse point — the degree of the behaviour's products — and not the boundary of the degree profile. The profile given in § 5.2 above is the ideal $\mathrm{AR}$ of arithmetical degrees, closed under the jump, so $0'$ is merely one of its members. And $\mathrm{AR}$ has no least upper bound in $\mathcal{D}$ (Spector's exact pair theorem), so one may not say "the supremum of the profile is $0^{(\omega)}$" — $0^{(\omega)}$ is the least uniform upper bound of the finite jump tower, not the supremum of $\mathrm{AR}$.
2. The re-zeroing equation does not exhibit the complete remainder 2 as a non-deletable input, and therefore cannot establish "closure of two independent remainders" (§ 10.3 above).

Erratum 11 · ZFCρ Paper II § 3.4, "the sole source" 〔correction of error〕

The original states that the exponential map is the sole source of the relation between $i$ and $\pi$.

As a claim about mathematics in general this is too strong; see the two counterexample bridges in § 9.9 above (rotational geometry, contour integration), together with Paper 4's own registration of the double identity of $2\pi i$.

Read instead: the exponential map is the canonical binding source within this trajectory of the relation between $i$ and $\pi$. The weaker mathematical statement that may be retained is: $\pi$ is transcendental over $\mathbb{Q}(i)$ (§ 9.9; the formulation "there is no nonzero $P$ with $P(i,\pi)=0$" has the elementary counterexample $P=X^2+1$).

11.5 Relations to other series

SAE Foundation v2

This paper adopts the two-ladder reading of v2 § 2.1 (see § 14.2). v2 has already registered the mathematical $L_3\to L_4$ row as "awaiting future articulation"; what this paper supplies is "not yet established" — stronger than "awaiting articulation" and weaker than "cannot be established".

v2 itself requires no revision. Its § 2.1 lists only the behaviour and signature in the mathematical column, and neither closure equation nor two remainders, so the four ZFCρ errata above do not land on it. When v3 is prepared, the mathematical column should cite this paper.

Form and Flow, Paper 1

That paper's § 3.6 hands Gödel, Tarski, and Turing jointly to SAE-Mathematics as a "family of remainder-persistence phenomena", listing a unified articulation as future work.

A partial response. In this paper the three occupy the following positions:

	Position
Gödel	remainder 1 (incompleteness of the provability closure)
Tarski	remainder 2 (complete truth not uniformly definable in the base language)
Turing	a product of the behaviour (universal evaluation produces the halting set $K$)

That is: two remainders plus one product of the behaviour, not three remainders.

The response is partial, with two limits. First, this paper does not claim that the three "collapse at $0'$" (see Erratum 9). Second, the question of what makes the three a family is not answered: this paper gives only their placement within the articulation of $L_3$, not a general form for the family across domains. The debt remains partly unsettled.

The ZFCρ line

Paper 4 § 13.1 hands the full composition law for $\omega_\rho^{(2)}$ to "the $L_2\to L_3$ interface paper". This paper occupies that position but does not discharge the debt. Its present state and obstacle are given in § 15.1.

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12. Claim Tiers, Jurisdiction, and Disciplines

12.1 Three tiers

Following Paper 4 § 12.1:

Tier	Content
T1	mathematical fact, with a source or a proof; where cited rather than reproved, the source is named
T2	a reframing of an existing result; the mathematical content is not new, the organisation is
T3	an SAE-internal identification; without external endorsement, and not usable as a premise in a mathematical argument

The T1 deliverables of this paper are Theorem A (§ 7.1) and Theorem B (§ 7.2); their hypotheses are frozen explicitly at § 7.0.

One boundary to be held strictly. Identifying a necessary condition and proving an instance are two different things. § 9 identifies the criterion required for independence of the two remainders (closure-essential independence) and supplies counterfactual separation as evidence, but does not prove that the existing closure structure has essential arity 2 for the two remainders (§ 9.7). The former may not be written as the latter.

12.2 Explicit freezing of hypotheses

The formal systems studied in this paper have: a recursive language · a c.e. axiom set or theorem set · an effective proof-checking relation.

"Objects have finite descriptions", "the expression class is countable", and "the theory is effectively axiomatized" are properties of different strength. Paper 4 works with finite expressions and enumerable rules, and this does not by itself yield the above; the hypotheses are frozen here explicitly and not inherited by default.

12.3 One term, one object

The reference of the following terms is fixed. Any reading at variance with this is not what this paper claims.

Term	Denotes only	Never denotes
the Gödel remainder	the fact that $T$-independent sentences exist; witness $\mathrm{Ind}_T$	$U_T$; some particular $G$
the Tarski remainder	the fact that there is no single complete internal truth predicate; witness $\mathrm{Th}(\mathbb{N})$	$\mathrm{Th}_{\Sigma_1}(\mathbb{N})$; $\mathrm{Tr}_{\Sigma_n}$
$\mathrm{Th}(T)$	the deductive closure of $T$	$\mathrm{Th}(\mathbb{N})$
the behaviour	diagonalisation / universal evaluation	its products $K$, $\Omega_U$
the collapse	$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}]_T=0'$	any statement containing remainder 1
reflection	one of $\mathrm{Con}(T)$ / $\mathrm{Rfn}(T)$ / $\mathrm{RFN}_\Gamma(T)$, specified	the three used interchangeably

Four prohibited formulations:

1. "the complement of $\mathrm{Th}(T)$ is the true but unprovable" — the complement contains sentences false and unprovable, and, depending on coding conventions, possibly non-sentence codes;
2. "the first level of the Tarski remainder is $\mathrm{Th}_{\Sigma_1}$" — the latter is a definable, effectively enumerable captured fragment;
3. "the two remainders and the behaviour collapse together" — the $0'$ equation covers only $K$, $\Omega_U$, $\mathrm{Th}_{\Sigma_1}$;
4. "Tarski forbids adjoining a truth predicate" — a truth predicate may be added as a new primitive to an extended language ($\mathsf{CT}^-$ is even conservative over PA); what is forbidden is defining complete truth within the base language, or demanding an unrestricted T-schema for an untyped self-applicable predicate.

Three usable only under stated conditions:

1. "true but unprovable" may be used, but only for $U_T$, and each use must state that $T$ is sound;
2. $\vdash$ / $\vDash$ may appear unsubscripted in the statement of theorems where ordinary first-order consequence has been fixed; where this paper compares $T$-provability with truth in the standard model it must write $\vdash_T$ and $\vDash_{\mathbb{N}}$;
3. $\Omega_U$ requires a fixed optimal universal prefix-free machine $U$; "the canonical value of $\Omega$" and "a machine-independent $\Omega_U$" may not be written.

12.4 Three columns, and translatability

Three columns. Complexity of a formula, degree of a defined set, and reducibility between sets are to be written separately; otherwise $\Sigma_n$, $0^{(n)}$, and Turing degree will substitute for one another.

Translatability. Any claim intended to support a mathematical conclusion must first be rendered as a statement of pure mathematics containing no SAE terminology. What cannot be so rendered is one of two things — an SAE-level identification, which is then tiered T3 and may not serve as a premise in a mathematical argument, or else confusion.

12.5 Five criteria for cross-layer regularities

This paper establishes no new cross-layer regularity. Any candidate must satisfy all five:

1. Type pre-registration — before looking at the material of the several layers, fix what type $A_L$ and $B_L$ are (number, expression, relation, algorithm, theory, map). If the types keep changing between layers, "coincidence" and "separation" have no fixed predicate.
2. Pre-registration of the coincidence/separation relation — "coincide" must be specified in advance as one of: identity of objects; isomorphism; one object in two roles; one mapping to the other; or merely witnessed by the same example. These are of five different strengths.
3. A threshold mechanism — there must be a mechanism, not defined using material from the target layer, explaining why the change occurs at layer $k$. "It looks as though they separate here" is not a mechanism.
4. A negative control — when the regularity is stated, one must also state what phenomenon would count as its failure, rather than being redescribed as some other kind of separation.
5. An out-of-sample test — at least one layer or independent mathematical example that did not participate in constructing the regularity.

Three observations recorded here as facts and not raised to regularities: the self-supplied / separately posited difference among the sub-marks of the three layers (§ 4.2); the two transition signatures standing in different relations to their respective layers (§ 6.3); and $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$ (§ 5.2 and its interface with § 9).

12.6 Dual positioning

One object may carry both a source coordinate (which boundary question it answers) and a development coordinate (in which layer it receives full articulation). Their coexistence is not a failure of disambiguation, and this paper does not shift indices in one direction merely to make them agree.

Example: Gödel and Tarski are triggering objects at the $L_2\to L_3$ boundary, while their full theoretical content is developed at $L_3$.

12.7 Eligibility of a capture core

A candidate collection qualifies as a capture core only under the four conditions of § 6.5 (evaluation precedes coding · image of a semantic quotient · covariance under legitimate recoding · no cardinality argument).

The criterion excludes a class of invalid argument: codability as reals is not the same as being naturally carried by a set of reals. If codability alone sufficed, any countable mathematical structure could be forced back into a "real-number core", and carrier migration would cease to be checkable.

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13. $L_4$: An Open Problem

13.1 Scope of this section

This section does three things only:

1. states that reflection and the jump iterate without bound;
2. states that the totality of recursive ordinal notations is not effectively decidable;
3. registers that this paper has not established a mathematical $L_4$.

It does none of the following four, in each case for want of material:

• it does not claim that the "outer" remainder of $L_4$ is empty;
• it does not claim that no totality containing the whole reflection progression exists;
• it does not claim that the mathematical ladder has been shown to terminate;
• it does not explain why the facts above should constitute a failure of closure — readings of that kind fall outside this paper's scope.

13.2 The shape of the front part of $L_4$

Continuing from the remainder of $L_3$, what is marked is a transfinite index: the degree profile of $L_3$ has no greatest element, and the continuation route considered here iterates along $0^{(\alpha)}$ (a selected route, not the only one; see § 5.2).

The public binding of that index is a system of ordinal notations — $\varepsilon_0$, $\Gamma_0$, $\psi(\Omega_\omega)$, and Kleene's $\mathcal{O}$. Notation systems exist and serve: along a path of notations, the progression

$$T_0=T,\qquad T_{\alpha+1}=T_\alpha+R(T_\alpha),\qquad T_\lambda=\bigcup_{\alpha<\lambda}T_\alpha$$

is well defined, where $R$ is one of $\mathrm{Con}$, $\mathrm{Rfn}$, or $\mathrm{RFN}_\Gamma$ (their differences at § 8.4).

Where "marked but not constructed" sits here. Everyone says "the $\alpha$-th iteration", while from inside the system no canonical $\alpha$-index can be constructed.

13.3 Three T1 facts

(i) Kleene's $\mathcal{O}$ is $\Pi^1_1$-complete; the admissible notations are not effectively enumerable.

It must be registered alongside that undecidable is not indeterminate. With a standard coding of recursive ordinal notations fixed, $\mathcal{O}$ is a set of natural numbers with a determinate extension; $\omega_1^{\rm CK}$ is a determinate countable ordinal; the class of hyperarithmetical sets is determinate. This paper draws from (i) no conclusion whatever about "no totality exists".

(ii) The consequences of a progression depend not only on the order type but on the particular computable notation (the intensional character of progressions). Turing 1939, restated with local reflection: for every true $\Pi_1$ sentence $\varphi$ there exists a computable notation for $\omega+1$ such that the progression proves $\varphi$; but there is no fixed notation for which one theory proves all true $\Pi_1$ sentences.

The quantifier order is $\forall\varphi\,\exists\dot\alpha$, not $\exists\dot\alpha\,\forall\varphi$.

(iii) Paths attaining arithmetical completeness exist — Feferman 1962 constructs one with full uniform reflection — but such a path cannot be $\Sigma^1_1$-definable. Moreover, along a $\Pi^1_1$ path satisfying the natural conditions of Feferman and Spector, some true $\Pi^0_1$ sentence is not derivable.

Hence "that completeness is empty" is too strong: complete paths exist; they simply cannot be captured uniformly by a low-complexity definition.

13.4 What this paper has not supplied

From (i), (ii), (iii) one may state: the path-selection function is not definable within arithmetic; complete paths exist but cannot be specified at low complexity.

But it does not follow that $L_4$ cannot close, for two reasons.

First, every layer depends on choices. $L_1$ depends on the coefficient base $\mathbb{Q}$, $L_2$ on the expression class $\mathsf{Int}_{\rm alg}$, $L_3$ on the theory $T$, the coding, and the universal machine $U$. To demand of $L_4$ that it depend on neither path nor notation system is to erect a standard stricter than the one the lower layers meet. Advancing a standard that only the candidate under judgement fails is what Paper 4 § 12.4 forbids.

The reasonable standard can only be: within the trajectory chosen, is there canonicity in an appropriate sense of equivalence?

Second, one candidate is known and not excluded. The question "which recursive relations are genuinely well-orderings?" is posed naturally by the very discussion of well-ordering and reflection, and it points to $\mathcal{O}$ or an equivalent hyperjump object — an object that is determinate, lies outside the hyperarithmetical hierarchy, and is named by the present question. This paper has not shown why that object should fail to meet the standard for an "outer" remainder, and therefore may not register the outer column as empty.

13.5 A checklist for current candidates for a mathematical $L_4$

The whole of this list is tiered T3. It is not a necessary or sufficient condition in the sense of an external mathematical theorem; the "minimum deliverables" below are this paper's present working definition and reviewing threshold, not a mathematical constraint on future candidates. Its purpose is to make the open problem takeable, not to erect a barrier.

Minimum deliverables (absent any of which a candidate does not yet amount to a layer):

1. a clear type for the object or apparatus;
2. an evaluation or universal property that is not cheap coding (the criteria of § 6.5 may be consulted, not required);
3. an explicit interface with $L_3$;
4. a criterion showing that it is not an iteration internal to $L_3$.

Stress tests for current candidates (all T3; future candidates are not required to satisfy them simultaneously):

• canonicity of the path — required to be of the same strength of parameter dependence as $L_1$–$L_3$, and not required to be parameter-free;
• a new capture image;
• multiple failure directions;
• some mechanism of closure or re-basing — with no advance requirement that a tower terminate; re-zeroing, a limit closure, or a universal object might equally carry closure.

This paper therefore does not write "once condition 4 is supplied, a mathematical $L_4$ is established" — an earlier draft did, and that was too strong.

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14. The Provisional Closing of the Series

14.1 What has been established, and what has not

The mathematical ladder of this series, as of this paper:

Layer	Capture core	Exchange-law articulation unit	Transition closure
$L_1$	$\mathbb{Q}^{\rm alg}_{\mathbb{R}}$	$\log$	$L_1\to L_2$: $e^{i\pi}+1=0$
$L_2$	$\mathcal{P}$	$2\pi i$	$L_2\to L_3$: collapse point and re-zeroing equation registered; closure of two independent remainders not shown (§§ 8.2, 9.7, 10.3)
$L_3$	$\mathrm{ARITH}$	$\#$ (undetermined; § 15.4)	—
$L_4$	not established	—	—

The mathematical main line of this series therefore closes provisionally at $L_3$. This paper has neither identified nor established a mathematical $L_4$.

Two sentences must be kept apart:

• may be written: within the sequence of capture criteria given so far, no candidate for a mathematical $L_4$ has yet been identified;
• may not be written: mathematics ends at $L_3$, or a mathematical $L_4$ cannot exist.

The difference is between this series stopping here and no successor existing mathematically. Only the former is claimed.

14.2 Relation to the physical-quantity ladder

By SAE Foundation v2 § 2.1, the mathematical ladder and the physical-quantity ladder are two independent ladders, isomorphic in structure and independent in content:

	Mathematical ladder	Physical-quantity ladder
$L_0\to L_1$	single remainder, no closure	single remainder, no closure
$L_1\to L_2$	dualisation: $e^{i\pi}+1=0$	dualisation: $[\hat x,\hat p]=i\hbar$
$L_2\to L_3$	diagonalisation (signature $\Omega_U$)	spatialisation: $E=mc^2$ (signature $c$)
$L_3\to L_4$	this paper: not established	causalisation: Schwarzschild (response signature $G_N$)
$L_4\to L_5$	—	irreversibilisation: $S=k_B\ln W$ (signature $k_B$)

This reading bears directly on the claim of this paper. On a single-ladder reading one could say "the ladder has not stopped; it has merely become physics". On the two-ladder reading, the $L_3\to L_4$ step of the physical-quantity ladder is not a continuation slot of the mathematical $L_3$ but a step on another ladder. The mathematical ladder closes provisionally here — and v2 § 2.1 has already registered the mathematical $L_3\to L_4$ row as "awaiting future articulation", where what this paper supplies is "not yet established": stronger than "awaiting articulation", weaker than "cannot be established".

14.3 A reduction in evidential strength

Paper 1 § 4.5 assigns the exchange-law articulation units of the layers as $\log$, $2\pi i$, $\#$, $c^2$, $k_B$, and takes "all five layers have one" as evidence that the cross-layer prediction has passed its test.

After the two ladders are separated, $c^2$ and $k_B$ belong to the physical-quantity ladder, and three remain on the mathematical side.

The consistency of the redistribution must be stated accurately. Of the three on the mathematical side, $\log$ and $2\pi i$ are settled, while whether $\#$ also serves as the exchange-law articulation unit at $L_3$ is not determined by this paper (§ 15.4). One may therefore say "two settled, the third undetermined", and may not say "exactly three layers have a unit" — the latter counts an undetermined item as settled, in breach of the boundary drawn at § 12.1 between identifying a necessary condition and proving an instance.

And this consistency constitutes no new evidence: the three items are precisely the three used in constructing the pattern, so criterion 5 of § 12.5 (out-of-sample test) is not met. The accurate statement is: after the ladders are separated, the existing items on the mathematical side remain compatible with the original pattern; that compatibility is not a fresh test.

Evidential strength therefore falls twice over: five layers of evidence become three (the third undetermined), and the three lie on one ladder, so the evidence is no longer cross-domain.

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15. Open Problems

15.1 The unified categorical encapsulation of $\omega_\rho^{(2)}$ — unsettled, and the assignment needs re-examination

The exact content of the debt

Paper 1 § 11.1 defers the composition law for the six components of $\omega_\rho^{(2)}$ to a later paper. Paper 4 § 9 pays part of it.

Delivered (Paper 4 §§ 9.1, 9.2): the minimal typed composition skeleton — two tables and four chains.

The six components and their types:

Component	Type
$\mathrm{Div}$	free abelian group $\mathrm{Div}(X)$
$\mathrm{Res}$	local linear functional
$\mathrm{Per}$	bilinear pairing $H_1\times H^1_{\rm dR}\to\mathbb{C}$
$\mathrm{Mon}$	representation of $\pi_1$
$\mathrm{Stokes}$	direction-ordered family of operators
$G_\infty$	scalar asymptotic invariant (order $\rho$ and type $\sigma$)

The four cross-component interfaces:

Chain	Map	Language
1↔2	$d\log$: $\mathrm{Res}_p(df/f)=\mathrm{ord}_p(f)$	classical complex analysis
2↔3	small-loop restriction: $\oint_{\gamma_p}\omega=2\pi i\,\mathrm{Res}_p(\omega)$	classical complex analysis
3↔4	(wild) Riemann–Hilbert	RH
within 4	Ramis density; formal-layer semidirect product $G^{\rm formal}=\mathcal{T}\rtimes\overline{\langle\hat\gamma\rangle}^{\rm Zar}$	RH

Not delivered (Paper 4 § 9.3): a unified categorical encapsulation of the six data types — organising all six as a single categorical object with all four chains as morphisms within it.

The exact location of the obstacle: not the absence of categorical tools (the Riemann–Hilbert correspondence and its wild extension are fully formalised) but that the six components have not been mapped onto RH data. Specifically: chains 3 and 4 have entered the RH language, chains 1 and 2 remain in classical complex analysis, and no translation between the two has been supplied.

This paper does not discharge it, and the assignment needs re-examination

Paper 4 § 13.1 defers the debt to "the $L_2\to L_3$ interface paper", on the ground that the encapsulation has a meta-level character (Paper 4 § 8.4) which, by the trigger verdict of Paper 1 § 5.5, belongs to the rule layer; and that by Paper 1 § 11.4 an adequate engagement with categorical structuralism must be handled at $L_2$ and $L_3$ both.

This paper is that interface paper, and does not discharge the debt. It also holds that the ground of the deferral needs re-examination:

> The chain of inference runs: meta-level character → rule layer → $L_3$. But the rule-layer content of $L_3$ is formal systems, computability, and decidability, whereas what the encapsulation lacks is a translation between the language of divisors and residues and that of Riemann–Hilbert data. Both may be called "about rules", but they are not the same kind of rule. The middle step of that inference runs two different senses of "rule layer" together.

The registration here is therefore: the debt is unsettled; and its natural home is more likely a monograph on the $L_2$ side treating the RH translation than the $L_3$ paper. This paper does not itself reassign it — reassignment must be made by whoever commands that material — but records the reasoning here, so that the debt does not continue to be passed between layers on a ground already in question.

By Paper 4 § 12.4: a debt is registered as a debt and is not converted into a finding. This paper neither claims discharge nor diagnoses its inability as an identification.

What would count as discharge

A unified typed container in which the six kinds of data and the four interfaces are composable within a single formal language, with chains 1–2 and chains 3–4 stated at the same level in that language.

No advance requirement that it be an ordinary category with the six components as objects: a pairing, a functional, a representation, an operator family, and a scalar invariant are not naturally objects of one kind; forcing them all into Set would trivialise the requirement. The real apparatus may be a many-sorted diagram, a fibration, a bicategory, or some other typed construction.

15.2 Construction of a binary closure equation

> Owed: a binary closure equation in which both Gödel incompleteness and Tarski undefinability are non-deletable inputs. > > Obstacle (neither step completed): (i) $\neg\mathrm{True}(x)$ is not a formula of the base arithmetical language — that is precisely the content of remainder 2 — so without extending the language it is not a well-defined input to a binary construction; (ii) the classical binary fixed-point theorems deliver simultaneity, not essential dependence, so even with well-defined inputs one must show separately that both are non-deletable. > > Three ways forward: > > 1. Attempt an existing binary construction — Smullyan's strong double recursion theorem supplies a candidate binary form, but does not automatically satisfy § 9.3 (which requires not merely two inputs but non-factorisation through either projection, and simultaneous fixed-point theorems do not deliver that); and the second input must first become a well-defined formula in an explicit extended truth language; > 2. Change the type of operator — some meta-level operations are naturally binary, such as the oracle universal evaluation functional $\Phi:(e,X)\mapsto\Phi_e^X$, where the program code on the syntactic side and the oracle on the semantic side occupy one slot each; or the binary modality $A\triangleright B$ of interpretability logic, which settles syntactic extension and semantic strength together; > 3. Revise the requirement of "two independent failure directions" — but this would bear on ZFCρ Paper II § 7.1's treatment of $L_0$, and is not to be altered lightly. > > A further point: what matters is informational dependence, not syntactic arity. A construction of the form $F_1,F_2\mapsto F_1\oplus F_2\mapsto\Phi(F_1\oplus F_2)$ may have a unary final operator while its input is canonically coupled from two directions; such a form can equally be tested for whether it can be completed relative to either single input. > > If this is never resolved: the $L_2\to L_3$ transition should be called a coupled re-zeroing of two boundaries rather than a closure of two independent remainders (§ 9.7).

〔Two formulations that cannot be used: "the candidate behaviours are all unary" (arity is not representation-invariant); "the obstacle is the type of the input slots" (beyond what can be shown). The course of their withdrawal is given in Appendix A.1.〕

〔The statement of this item has been narrowed several times; the process is recorded in Appendix A.1, and the sole surviving claim is the one stated above.〕

15.3 What the second class of remainder is at $L_3$

The second class at $L_2$ (Paper 4 § 8.2) is a syntactic boundary — $\mathsf{Int}_{\rm alg}$ does not admit exponential factors.

The corresponding question at $L_3$ is whether the syntax of first-order arithmetical formulas has an analogous "does not admit". One candidate is set quantification, which the hyperarithmetical requires and the arithmetical language lacks. The candidate is unverified and may coincide with the third class.

§ 8.3 registers the cell as undetermined and does not fill it.

15.4 Whether $\#$ also serves as the exchange-law articulation unit

Paper 1 § 4.5 assigns the exchange-law articulation unit of $L_3$ to Gödel numbering $\#$.

§ 4.2, inner step 3 above gives the first appearance of $\#$ (at main step 2, as the binder of proof sequences), but does not determine whether it appears again at main step 4 and there serves as the exchange-law unit.

That question depends on how the diagonal construction of § 8.1 is written. This paper does not answer it.

One thing from which nothing may be inferred: in Paper 4, $2\pi i$ first appears at main step 3 and recurs at main step 4, inner step 2, whereas $\#$ first appears at main step 2. The positions differ, and no analogy may be drawn.

15.5 Whether $L_3$ has a side branch

Paper 4 § 10 judges Cayley–Dickson to be a cost branch of $L_2$, on the criterion that the branch is impressible rather than forced — it is not forced by the layer's reverse questioning but adopted in order to pay some cost (there, commutativity).

The corresponding criterion at $L_3$ would be: a branch is a cost branch if it is not forced by the reverse questioning of $L_3$ but adopted in order to pay some cost of $L_3$ (the law of excluded middle in classical logic, say, or the expressive limits of a first-order language).

Four candidates, none currently classifiable:

Candidate	Why it cannot yet be classified
intuitionistic logic	it changes the interface marked at main step 1 itself (the derivation rules of $\vdash_T$), rather than opening a branch under a fixed interface
second-order arithmetic	the strength spectrum of its subsystems straddles $L_3$ and the hyperarithmetical; classification would first require settling whether it falls within the layer or has already left it
nonstandard models	it changes the domain on the $\vDash$ side, whereas $\vDash_{\mathbb{N}}$ has been fixed here to the standard model; loosening that fixture changes a premise rather than opening a branch
type theory	correspondence with classical first-order arithmetic goes through a translation whose own standing is undetermined

This paper makes no adjudication, and does not designate one merely to fill a slot.

Nor may "$L_3$ has no cost branch" be inferred by default from the absence of a survey — that conclusion, if it is to hold, would require excluding each of the four candidates on the criterion above and arguing that the list is exhaustive, and this paper does neither.

15.6 A mathematical $L_4$

This paper has not established a mathematical $L_4$, and has not shown that one cannot be established.

This is an absence, not a verdict, and it is therefore not listed among the negative deliverables of § 1.2; it belongs to the scope declaration (§ 14) and to this section.

§ 13.5 gives a checklist for current candidates (all T3) to make the problem takeable.

15.7 The debt of Form and Flow, Paper 1 § 3.6 — partly unsettled

That paper hands Gödel, Tarski, and Turing jointly to this series as a "family of remainder-persistence phenomena".

§ 11.5 above gives the placement of the three within the articulation of $L_3$ — two remainders plus a product of the behaviour. But no cross-domain unifying mechanism is supplied: what makes the three a family is not answered here; only their position within this layer is.

The debt remains partly unsettled.

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Appendix A · Version and Review Provenance

This paper was subject to four-party internal review (Claude, ChatGPT, Gemini, Grok, one seat each), with the full draft passing through four rounds (v1 → v4), the ChatGPT seat returning Major Revision on v1, Not Passed on v2, and conditional sign-off on v3; a further round of formatting followed.

"Review" here denotes model-assisted internal structural review organised by the author, and is not equivalent to journal peer review.

A.1 Three corrections bearing directly on the anti-immunisation discipline of § 12

First, a mathematical error. The outline and v1 claimed that the supremum of the ideal $\mathrm{AR}$ of arithmetical degrees is $0^{(\omega)}$, with the proof: any upper bound $u$ satisfies $u\ge_T 0^{(n)}$ for all $n$, hence $u\ge_T\bigoplus_n 0^{(n)}$. That step does not hold — individual reductions do not supply the uniform procedure needed to compute the direct sum; and by Spector's exact pair theorem a countable degree ideal has no least upper bound at all. The error ran through six passages (§§ 5.2, 7.3, 8.2, 8.4, 10.1, 13.2) and was corrected throughout once identified in review (§ 5.2 now states three things separately: no greatest element, upper bounds exist, no least upper bound).

Second, an assertion of completion that had not been verified. The revision record of v2 stated that the six passages above had been corrected along with § 5.2, whereas in fact only two had been. A paper that carries an anti-immunisation declaration cannot let its own revision record assert unverified completion, or the declaration has nothing to stand on. From v3 onward, a whole-text scan for proscribed formulations was run after each batch of revisions, and the scan results submitted with the draft.

Third, two instances of withdrawing one overclaim and installing another. The location of the obstacle in § 10.3 passed through three drafts. The first claimed that "the candidate closure actions at $L_3$ are all unary" — untenable, since arity is not representation-invariant (a pairing function changes arity; universal evaluation is written $U(e,x)$). v2 then claimed that "the obstacle is that the input slots of the available binary constructions are formula slots rather than failure-direction slots" — likewise beyond what could be shown. From v3 the statement became two steps, neither completed, with the explicit note that this paper claims neither that the obstacle lies in the type of the input slots nor that substitution would suffice.

A further instance ran in the opposite direction: v3 § 5.2, in the course of narrowing the standing of $0^{(\omega)}$, wrote that "no enumeration-uniform upper bound of the ideal exists"; that sentence is false — $0^{(\omega)}$ is precisely the least such. So the common failure across the three rounds was not "claiming more than the material supports" but asserting without verifying, in either direction.

A.2 Other load-bearing decisions withdrawn or downgraded in review

All are registered at the relevant places in the body; they are listed here for traceability.

• The capture core was changed from $\mathbb{R}_{\rm arith}$ to $\mathrm{ARITH}$, with $\mathbb{R}_{\rm arith}$ downgraded to a real-axis counterpart (§§ 6.1, 6.2) — the real-number coding image of $\mathrm{ev}_{\rm set}$ being strictly smaller than $\mathbb{R}_{\rm arith}$;
• "All three slots are subsets of the reals" withdrawn (§ 6.1) — the period ring is not a subset of $\mathbb{R}$;
• The arithmetical-soundness hypothesis removed from Theorem A (§ 7.1) — it is redundant, and the theorem is stronger without it;
• $\mathrm{ev}_3$ split into $\mathrm{ev}_{\rm set}$ and the coding $j$ (§§ 6.2, 6.4) — the original notation carried two senses;
• The deliverable of § 9 downgraded — from "evidence that $L_3$ satisfies the independence criterion" to "counterfactual evidence that the two failure directions are not identical";
• The negative result of § 10.3 downgraded — from "the existing re-zeroing equation does not instantiate the criterion" to "has not been shown to instantiate it";
• § 13.5 changed from "sufficiency standards" to "a checklist for current candidates", wholly T3;
• The first class of remainder in § 8.3 changed from "empty" to "none identified", and re-tiered T3;
• The formulation about $P(i,\pi)$ corrected (§ 9.9) — the original "there is no nonzero $P$ with $P(i,\pi)=0$" has the elementary counterexample $P=X^2+1$, and is replaced by "$\pi$ is transcendental over $\mathbb{Q}(i)$";
• Paper 1 § 6's single-ladder narration registered as pre-v2 (§ 11.3), the two-ladder reading of Foundation v2 being adopted here.

A.3 The function this appendix serves

This appendix serves traceability of provenance; it does not serve to make the paper credible. The internal evidence for the disciplines of § 12 is the body itself: negative results registered alongside positive deliverables (§ 1.2); independence of the two remainders claimed only as far as non-identity, and not as an instance (§ 9.8); the hypotheses of Theorem A frozen item by item and not one item more (§ 7.0); eleven upstream errata each tiered explicitly (§ 11). This appendix only supplies the traceability of the version chain.

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References

Note: each entry carries a line stating what it supplies to this paper.

Incompleteness and undefinability

1. K. Gödel, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik 37 (1930). — the completeness theorem; the basis for the notational discipline of § 3.2 (there is no gap between an unsubscripted $\vdash$ and $\vDash$).
2. K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38 (1931). — the first incompleteness theorem; used at § 4.2 inner step 4 and § 8.1.
3. J. B. Rosser, Extensions of some theorems of Gödel and Church, Journal of Symbolic Logic 1 (1936). — lowers the hypothesis for independence from $\omega$-consistency to plain consistency; the choice of $\mathrm{Ind}_T$ over $U_T$ depends on this form.
4. A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica 1 (1936). 〔Polish original 1933〕 — undefinability of truth; the semantic-side load-bearer for remainder 2 and Theorem A.

Recursion theory

1. E. L. Post, Recursively enumerable sets of positive integers and their decision problems, Bulletin of the AMS 50 (1944), 284–316. — creative sets and effective inseparability; used at § 4.2 inner step 4. 〔This paper does not contain the arithmetical hierarchy theorem; for that see item 6〕
2. S. C. Kleene, Introduction to Metamathematics, North-Holland (1952), § 57. — $A$ arithmetical $\iff\exists n\,A\le_T 0^{(n)}$; criterion 3 of § 6.5 and § 10.2 rest on this. 〔That book attributes the connection between hierarchy and jump to Post's 1948 abstract〕
3. J. Myhill, Creative sets, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 1(2) (1955), 97–108. DOI 10.1002/malq.19550010205. — creative $\iff$ c.e.-many-one complete; yields the $\Pi^0_1$-completeness of $\mathrm{Ind}_T$ at § 4.2 inner step 4.
4. C. Spector, On degrees of recursive unsolvability, Annals of Mathematics, 2nd ser., 64 (1956), 581–592. DOI 10.2307/1969604. — the exact pair theorem; § 5.2 inner step 4 uses it to show that $\mathrm{AR}$ has no least upper bound, and thence to refute "leaving $\mathrm{AR}$ requires passing through $0^{(\omega)}$".
5. S. C. Kleene, On notation for ordinal numbers, Journal of Symbolic Logic 3 (1938). — establishment of the notation system $\mathcal{O}$.
6. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw–Hill (1967), §§ 11.7–11.8. — $\Pi^1_1$-completeness of $\mathcal{O}$ and the hyperarithmetical hierarchy; § 13.3 (i) rests on this.
7. R. M. Smullyan, Recursion Theory for Metamathematics, Oxford Logic Guides 22, Oxford University Press (1993), ch. IX. — symmetric and double recursion theorems; § 10.3 uses the strong double recursion theorem as a ready-made binary construction while stating explicitly that it does not by itself deliver essential dependence.
8. W. Craig, On axiomatizability within a system, Journal of Symbolic Logic 18(1) (1953), 30–32. — c.e. axiomatisability can be replaced by recursive axiomatisability; used at § 4.2 inner step 3 to give $\mathrm{Prf}_T$ an equally effective form for c.e. axiomatized $T$.

Algorithmic information theory

1. G. J. Chaitin, A theory of program size formally identical to information theory, Journal of the ACM 22 (1975). — construction of $\Omega$; § 6.3 uses it as witness for $\mathbb{R}_{\rm comp}\subsetneq\mathbb{R}_{\rm arith}$.
2. A. Kučera and T. A. Slaman, Randomness and recursive enumerability, SIAM Journal on Computing 31(1) (2001), 199–211. DOI 10.1137/S0097539799357441. — every left-c.e. ML-random real is Turing complete; this paper's $\deg(\Omega_U)=0'$ follows.

Transfinite progressions and reflection principles

1. A. M. Turing, Systems of logic based on ordinals, Proceedings of the London Mathematical Society 45 (1939). — ordinal logics; §§ 8.4 and 13.3 (ii) use its sentence-by-sentence $\Pi_1$-completeness for local reflection progressions (quantifier order $\forall\varphi\,\exists\dot\alpha$).
2. S. Feferman, Transfinite recursive progressions of axiomatic theories, Journal of Symbolic Logic 27 (1962). — arithmetical completeness attained with full uniform reflection; used at § 13.3 (iii).
3. S. Feferman and C. Spector, Incompleteness along paths in progressions of theories, Journal of Symbolic Logic 27 (1962). — incompleteness along $\Pi^1_1$ paths; addressed to general recursive progressions of theories and not classified by reflection operator (whence § 8.4 states the three facts for all three operators at once).
4. G. Kreisel and A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 14 (1968). — $\mathrm{PA}+\mathrm{RFN}(\mathrm{PA})\equiv\mathrm{PA}+\mathrm{TI}(\varepsilon_0)$.
5. L. D. Beklemishev, Iterated local reflection versus iterated consistency, Annals of Pure and Applied Logic 75(1–2) (1995), 25–48. DOI 10.1016/0168-0072(95)00007-4. — $\alpha$ iterations of local reflection have the same $\Pi^0_1$ consequences as $\omega^\alpha$ iterations of consistency; this paper accordingly does not claim that local reflection fails to connect with ordinal analysis.
6. L. D. Beklemishev, Provability algebras and proof-theoretic ordinals I, Annals of Pure and Applied Logic 128(1–3) (2004), 103–123. DOI 10.1016/j.apal.2003.11.030. — ordinal analysis of iterated reflection and conservativity spectra; § 2.4 uses it to explain why reflection strength does not correspond to Turing degree.

Provability logic

1. R. M. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics 25 (1976). — arithmetical completeness of GL (not a fixed-point theorem); used at § 10.1 in connection with Magari algebras.
2. The fixed-point theorem of D. de Jongh and G. Sambin, in G. Boolos, The Logic of Provability, Cambridge University Press (1993), ch. 8. — the standard single-variable GL fixed-point theorem; used at § 10.1 where GL is mentioned.
3. R. Magari, The diagonalizable algebras, Bollettino dell'Unione Matematica Italiana, ser. 4, 12, Supplemento fasc. 3 (1975), 117–125. — Magari algebras; mentioned at § 10.1.

Theories of truth

1. H. Kotlarski, S. Krajewski and A. H. Lachlan, Construction of satisfaction classes for nonstandard models, Canadian Mathematical Bulletin 24(3) (1981), 283–293. DOI 10.4153/CMB-1981-045-3. — conservativity of $\mathsf{CT}^-$ over PA is a standard consequence of this construction (not a theorem stated there directly); used at § 9.6 and in Erratum 6 of § 11.2.

Periods and transcendence

1. M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited — 2001 and Beyond, Springer (2001), 771–808. — the definition of the period ring $\mathcal{P}$ and the three transformation rules; used at § 3.2 inner step 1 and § 6.2.
2. M. Yoshinaga, Periods and elementary real numbers, arXiv:0805.0349 (2008). — a computable non-period real; used at § 6.3 as witness for $\mathcal{P}\cap\mathbb{R}\subsetneq\mathbb{R}_{\rm comp}$.
3. F. Lindemann, Über die Zahl $\pi$, Mathematische Annalen 20 (1882), 213–225. — transcendence of $\pi$; the sole external load-bearing theorem of § 9.9 (whence $\pi$ is transcendental over $\mathbb{Q}(i)$).

This series

1. Han Qin, SAE Mathematics, Paper 1: Architecture. DOI 10.5281/zenodo.20153791. — the layer-articulation schema; its § 6.1 four-step table for $L_3$ is what this paper raises to monograph treatment, and its § 4.5 assigns the exchange-law units of the layers.
2. Han Qin, SAE Mathematics, Paper 2. DOI 10.5281/zenodo.20199082. — the multidimensionality of $L_0$ and three closure paths; the full-exactification trajectory of this paper is one of them.
3. Han Qin, SAE Mathematics, Paper 3: $L_1$. DOI 10.5281/zenodo.20266351. — the four-main-step / four-inner-step format used here, and the category registration of main step 4.
4. Han Qin, SAE Mathematics, Paper 4: $L_2$. DOI 10.5281/zenodo.21466722. — the direct upstream; what this paper fills is the position deferred by its § 8.4.
5. Han Qin, ZFCρ Paper II. DOI 10.5281/zenodo.18927658. — the $L_2\to L_3$ transition structure (two remainders plus one behaviour, yielding closure); §§ 8 and 9 above fill in and revise it.
6. Han Qin, SAE Foundation v2. DOI 10.5281/zenodo.20322460. — the two-ladder reading, adopted at § 14.2.
7. Han Qin, SAE Form and Flow, Paper 1. DOI 10.5281/zenodo.20066986. — its § 3.6 hands Gödel, Tarski, and Turing jointly to this series; § 11.5 gives a partial response.

1. 引言与定位

1.1 本文在系列中的位置

Paper 1 给出 $L_1$ 至 $L_5$ 的层articulation图式, 并在 § 6.1 以并行展示的形式给出 $L_3$ 的四步表. Paper 3 与 Paper 4 分别把 $L_1$ 与 $L_2$ 抬为专题阐述. 本文对 $L_3$ 作同一件事.

本文所填的是 Paper 4 § 8.4 所推的位置. 该节把 $L_2$ 的两个规则层问题 (三条 Kontsevich–Zagier 变换规则是否生成全部 $\ker(\mathrm{ev})$; 一般周期表达式的等同性是否可判定) 推给 $L_3$ 论文. 本文 § 3.2 内层 1 接收之.

1.2 本文交付什么

正面交付:

1. $L_3$ 的四主步骤阐述 (§ 3–8), 其中主步骤 4 按 Paper 3 § 6.1 的登记横跨三节;
2. 两条 T1 定理 (§ 7.1、§ 7.2);
3. 两项余项独立性所需判据 (closure-essential independence) 的识别, 与两种失败方向非同一的反事实证据 (§ 9) —— 不含"现有闭合满足该判据"的证明;
4. $\mathfrak{C}_T$ 的最小 typed 组织 (§ 10);
5. 十一条上游勘误 (§ 11).

负面交付 (本文认为其分量不低于上述任何一项):

1. 现有归零式 $\deg^{0'}(0')=0$ 未把完整余项 2 展示为不可删除的输入, 故不能证明现有闭合具有 essential arity 2 (§ 10.3);
2. Paper 4 § 13.2 的原证伪条件在裁决性读法下为空, 且由推论 7.2可扩至一切有效公理化扩张, 不分保守与否 (§ 11.2).

两项皆是对否定性内容的肯定判定, 有证明. 而"本文未能建立数学 $L_4$"是一项缺席, 不是判定, 故不列于此 —— 它属辖区声明与开放问题 (§ 13、§ 14、§ 15.6).

1.3 一条贯穿全文的方法纪律

本系列此前的一次错误值得写在引言: 一个词在两处指了两个不同的集合而中途无人喊停 —— "Gödel 余项"曾在一处指 $\mathrm{Ind}_T$ (度 $0'$)、在另一处指 $U_T$ (度 $0^{(\omega)}$), 由此建起的一整条判断链随之崩塌.

本文因此立一词一物纪律 (§ 12.3): 每个对象只有一个定义, 且列出四条禁用的说法与三处须带条件使用者. 该表不是修辞上的谨慎, 而是本文得以成立的条件.

与之配套的是可翻译性纪律 (§ 12.4): 凡拟支撑数学结论的主张, 须先翻成不含任何 SAE 术语的纯数学陈述; 翻不动者只有两种情形 —— 它是 SAE 层面的识别 (则标 T3 且不得作数学论证的前提), 或它是混乱的.

1.4 本文不承担者

见摘要末段. 另须明写两条:

其一, 本文不论证"$L_4$ 不可闭合". 该主张与"本文未能建立 $L_4$"是两回事; 本文只主张后者.

其二, 本文不解释 § 13 所陈述的事实为何构成不闭合的理由. 相关读法涉及本系列之外的辖区, 不在本文范围.

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2. 防火墙与登记声明

本文继承 Paper 4 § 2 的四条防火墙, 并按 $L_3$ 的材料各作一次落地. 另立两条本文所需者.

2.1 基底与阐述模式

$L_3$ 的基底是 $\mathbb{R}$ (经算术可定义性进入); 其阐述模式是 $\mathfrak{C}_T$.

捕获核心不等于基底: $\mathrm{ARITH}$ 至多可数, 其实轴对应物 $\mathbb{R}_{\rm arith}$ 亦然, 而 $\mathbb{R}$ 有连续统基数. 本文不主张核心穷尽 $L_3$ 的基底 —— 与 Paper 4 不主张 $\mathcal{P}$ 穷尽 $L_2$ 基底同.

捕获核心亦不等于阐述装置: 装置 $\mathfrak{C}_T$ 是多排序的, 求值只定义在其公式排序上, 写作 $\mathrm{ev}_{\rm set}:\mathrm{Form}_1(L_A)\to\mathrm{ARITH}$. 详 § 6.4.

2.2 数值、公式与算子

本文所涉三类对象须分开:

类	例
数值	$\mathbb{R}_{\rm arith}$ 的成员; $\Omega_U$
公式／句子	$\mathrm{Sent}_L$ 的成员; $\mathrm{Prov}_T(x)$; $\mathrm{Tr}_{\Sigma_n}(x)$
算子	$C_T^\vdash$; $C_{\mathbb{N}}^\vDash$; jump; 对角引理

一处易混: $\mathrm{Con}(T)$ 是单个 $\Pi^0_1$ 句子, 局部反思 $\mathrm{Rfn}(T)$ 是语句 schema, 一致反思 $\mathrm{RFN}_\Gamma(T)$ 是带数字量词的 schema. 三者不得互称; $\mathrm{Con}(T)$ 只是局部反思在 $\varphi=(0{=}1)$ 处的一个实例.

2.3 三个排序

承 Paper 4 § 2.3: 表达式对象 · 取值像 · 表达式等价关系三者不得并作一句. $L_3$ 处的落地是:

排序	$L_3$ 的内容
表达式／语言对象层	$\mathrm{Sent}_L$; 一元算术公式
取值对象层	$\mathbb{R}_{\rm arith}$; 算术集; 神谕集
关系层	$C_T^\vdash$; $C_{\mathbb{N}}^\vDash$; $\le_T$; $\equiv_T$

须特别守住的一处: 等价核 $\sim$ 是关系层的对象, 不是表达式层的元素. 本系列此前曾把该区分压掉一次而致类型错误.

商对象须另标: Turing 度是 $\mathcal{P}(\omega)/{\equiv_T}$ 的成员, 属"取值层按关系层取商", 不住任一单一排序. 单个集合有度; "整个度空间"没有度.

2.4 公式复杂度、集合的度与归约关系

三者须分栏, 否则 $\Sigma_n$、$0^{(n)}$ 与 Turing 度会互相换义:

栏	典型对象
公式复杂度	$\Sigma^0_n$、$\Pi^0_n$、$\Delta^0_n$
所定义集合的度	$0^{(n)}$、$0^{(\omega)}$
集合之间的归约关系	$\le_m$、$\le_T$

一处已核的后果: 不存在 $\mathrm{RFN}_{\Sigma_n}(T)\leftrightarrow 0^{(n)}$ 的逐级同一对应.

对固定的 $n$, 若 $T+\mathrm{RFN}_{\Sigma_n}(T)$ 是一致的 c.e. $Q$-扩张, 则其定理集仍为 c.e.-many-one 完全集, Turing 度为 $0'$, 不随 $n$ 升为 $0''$; 而其证明论强度确随 $n$ 增长.

〔一致性条件不可省: 若该扩张不一致, 则它证出全部句子, 定理集可判定, 度为 $0$.〕

故 Turing 度不能测量这些反思理论的证明论强度.

2.5 主步骤 4 的类别登记

按 Paper 3 § 6.1: 主步骤 4 的类别是余项账本结构, 不是层内 sub-fractal; 其内部按"捕获核心 → 未捕获账本 → 下一层触发条件"组织.

本文主步骤 4 横跨 § 6、§ 7、§ 8 三节, 而主步骤 1 至 3 各占一节. 该版面不对称本身即上述登记.

2.6 两条本文新立者

捕获核心的资格判据

一个候选集合有资格称为捕获核心, 须满足四条 (详 § 6.5): 评价先于编码选择 · 核心是语义商的像 · 对合法重编码协变 · 不靠基数论证.

该判据排除的是: 形式理论、证明关系、Turing 度与可证性代数皆可编码成自然数集再编码成实数; 但可编码成实数不等于自然承载体是实数子集. 若仅凭可编码性即认定核心, 承载体迁移将变得不可检验.

跨层规律的五条判据

本文不新立跨层规律. 拟立者须同时满足类型预注册、关系预注册、阈值机制、负面控制与样本外测试五条 (详 § 12.5).

本文因此如实登记而不上升为规律者三处: 三层 sub-mark 的自带／另设差别 (§ 4.2); 两个跃迁签名与各自层的关系不同型 (§ 6.3); $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$ (§ 5.2 与 § 9 的接口).

2.7 读者防误解表

易读成	实为
本文证明了数学止于 $L_3$	本文登记"尚未建立 $L_4$", 且明写未能证明其不可建立
$\mathbb{R}_{\rm arith}$ 是 $L_3$ 的捕获核心	核心是 $\mathrm{ARITH}$ (算术子集之集); $\mathbb{R}_{\rm arith}$ 是其实轴对应物, 二者类型不同
Gödel 余项是"真而不可证"	是 $T$-独立语句 $\mathrm{Ind}_T$; "真而不可证"是 $U_T$, 度 $0^{(\omega)}$
两余项与行为在 $0'$ 坍缩	$0'$ 只覆盖 $K$、$\Omega_U$ 与 $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$
本文证明了两余项独立	本文识别了所需判据并给出证据; 未证明现有闭合是其实例
定理 A 证明了迁移预测	定理 A 划"有效／非有效"轴, 迁移预测划"对象／规则"轴; 两根不同的轴

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3. 主步骤 1: 标而不构元判断接口

3.1 主步骤 1 与 inevitability 1

Paper 3 § 1.2 立: inevitability 1 对应主步骤 1, 其形式是标而不构 —— 标记被固定为公众 token, 其运算地位被确定, 而它作为对象的构造并未给出. 在 $L_1$, 被标而不构者是单位 "1"; 在 $L_2$, 是 $i$ (Paper 4 § 3).

在 $L_3$, 被标而不构者是元判断接口: 可以站在一个形式系统之外, 就该系统与某语句的关系作出判断.

登记类别: 跨层阈口类. 本主步骤阐述的不是 $L_3$ 内部的展开, 而是 $L_2\to L_3$ 过渡本身; 其内层 1 的输入来自上一层.

3.2 内层步骤

内层 1: $L_2$ 的余项抵抗对象层纳入

Paper 4 § 8.4 已识别 $L_2$ 的两个规则层问题: 三条 Kontsevich–Zagier 变换规则是否生成全部 $\ker(\mathrm{ev})$ (推导完备性), 以及一般周期表达式的等同性是否可判定 (决策程序). 二者的主词是规则系统本身.

此处的结构须与 $L_1\to L_2$ 对照才看得清. Paper 4 § 3.2 内层 1 记: $x^2+1=0$ 由 $L_1$ 自己的追问提出, 而 $L_1$ 的基底不含答案 —— 缺的是一个对象, 补一个对象即补上. 这是基底外余项, 迫使跨层.

$L_2$ 此处不同: 不缺对象. 向 $\mathsf{Int}_{\rm alg}$ 加入任何新的数, 都不会使"三条规则是否完备"或"等同性是否可判定"变得可答 —— 这两个问题不问某个值是什么, 而问那套规则能做到什么. 所抵抗的不是某个对象的缺席, 而是问不出来: 在 $L_2$ 的阐述内, 没有一个位置可以让"该规则系统"整体成为被谈论者.

故 $L_2\to L_3$ 的输入不是一个待安置的对象, 而是一个待安置的站位.

〔本文不主张这两个规则层问题即是 $L_3$ 的两项余项. 二者的关系在 § 8 与 § 9 另行处理.〕

内层 2: 跨主体传播

"这个证明对不对"与"这个命题是不是真的", 在数学实践中长期流通, 且先于形式化.

其传播方式与 Paper 4 § 3.2 内层 2 所记的 $i$ 同格: Cardano 拒绝承认负数平方根的实在性而仍在计算中使用它们, Bombelli 随后给出其运算规则 —— 传播先于承认. 元判断在此处亦然: 一个使用者判定某推理有效, 另一个使用者可以接着往下推, 而无须先就"有效"一词的确切含义达成一致.

区别在于所传播者的类型. $i$ 是一个在运算中被传递的记号; 元判断是一种被共同执行的动作. 它被公众占用已久, 但占得非形式.

内层 3: 符号的公众绑定

代价: 若"$T$ 支持 $\varphi$"始终只能非形式地说, 则该关系本身无法成为数学陈述的主词. 关于推理的每一次讨论都须重新约定所指.

绑定: 所标出者是一个双面的元判断接口

$$\Gamma\mathrel{\triangleright_\epsilon}\varphi,\qquad \epsilon\in\{\mathrm{syn},\mathrm{sem}\},$$

其两种极化实现为

$$\triangleright_{\mathrm{syn}}=\vdash_T,\qquad \triangleright_{\mathrm{sem}}=\vDash_{\mathbb{N}}.$$

历史上这两种实现分别由 Frege–Hilbert 一系的推导线与 Tarski 一系的满足号完成. 二者的绑定时间相隔约半个世纪, 且各自出自不同的实践 (证明论与模型论).

两面:

• 处理能力: 两种支持关系各自成为可作数学陈述的对象. "$T$ 证得 $\varphi$"与"$\varphi$ 在 $\mathbb{N}$ 中为真"从此可以被分别断言、分别否定;
• 余项: 二者之间的关系尚未被内部化. $\vdash_T$ 定义在字符串的有限组合关系上, $\vDash_{\mathbb{N}}$ 定义在结构与满足关系上. 二者在外部元理论中当然可以比较 (soundness、$\Sigma_1$-完全性、Gödel 不完备性皆是这样的比较); 尚不能做的是在原对象语言内由一个统一的内部谓词完成该比较 —— 证明关系的算术化要等主步骤 2 内层 3, 而完整真理的求值则永远不能在原算术语言内统一化 (§ 7.1).

限定: 极化为两面是本文所选 trajectory 的类型要求, 不是元判断概念本身的普遍定理. 只研究证明装置、只研究解释结构, 或采用 realizability 一类语义的 trajectory, 不必被迫恰有两个实现. 本文不主张该 trajectory 之外的必然性.

记号纪律: 本文全篇写 $\vdash_T$ 与 $\vDash_{\mathbb{N}}$, 不写无下标的 $\vdash$ 与 $\vDash$. 理由是: 对经典一阶逻辑与全体结构上的语义后承, 完备性定理给出 $\Gamma\vdash\varphi\iff\Gamma\vDash\varphi$ —— 无下标的两个记号之间没有缝隙. 本文所处理的缝隙由指定理论与指定模型给出, 记号须承载这一指定.

内层 4: 构与余项

构: 两种极化可用.

余项: 二者的关系尚未被内部化 —— 可在外部元理论中比较, 而不能在原对象语言内由一个统一的内部谓词完成 (详内层 3).

标而不构在此处的锋利之处在于: $\mathrm{Prov}$ 作为概念已被标出 —— 人们早已谈论"$T$ 能否证明 $\varphi$" —— 而作为对象语言内的公式尚未被构造. 后者要等算术化 (§ 4.2 内层 3).

这与 $L_2$ 的情形可以对照: $i$ 被标而不构, 是因为它作为对象的构造 (有序对、商环、几何点) 尚未给出; 而元判断接口被标而不构, 是因为接口两侧尚不在同一个论域内, 因而"接口本身"还不是一个可谈论的对象.

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4. 主步骤 2: 加法路径 —— 证明闭包

4.1 主步骤 2 与 inevitability 2

主步骤 2 阐述有相: 从主步骤 1 所标者出发, 给出一个可迭代的正面实体.

在 $L_3$, 该实体是证明, 迭代所得是定理集.

4.2 内层步骤

内层 1: 推导步骤的 sub-mark

从 $\vdash_T$ 一侧展开: 推导步骤作为可迭代单位被 sub-mark.

此处须与 Paper 4 § 4.2 内层 1 对照. 该处记: $L_2$ 的 sub-mark (与实轴正交的方向) 不是从实直线内部导出的 —— 实直线上没有任何结构指定一个正交方向, 该方向是为安置 $i$ 而设的; 历史上这一步以 Wessel、Argand、Gauss 的几何表示实现, 距 Euler 的记号绑定约两个世纪.

$L_3$ 此处不同: 推导步骤由 $\vdash_T$ 的定义自带, 不需另设. 一个推导系统就是一组从前提到结论的许可, 而"施加一次许可"本身即是可迭代的单位. 就此而言, $L_3$ 主步骤 2 内层 1 更近 Paper 4 所记的 $L_1$ 情形 —— 迭代方向由标记自身给出.

三种情形的差别须如实登记: $L_1$ 的方向自带, $L_2$ 的方向另设, $L_3$ 的方向自带. 本文不主张这构成任何跨层规律.

内层 2: 推导的迭代

证明是推导步骤的有限序列; 定理是有证明者. 迭代给出

$$\mathrm{Th}(T)=\{\varphi: T\vdash_T\varphi\}.$$

该迭代阐述的是"能走到哪里", 不是"哪里是对的". 这一点是本主步骤余项的来源 (内层 4).

内层 3: 证明序列绑成单一 token

代价: 若每个证明都写作推导步骤的有限序列, 则关于证明的陈述须始终在序列层面进行. "存在一个 $\varphi$ 的证明"这一断言无法作为算术条件表述, 因而无法成为对象语言内的公式.

绑定: Gödel 数化 $\#$ 把有限证明序列绑成单一 token —— 一个自然数. 于是

$$\mathrm{Prf}_T(p,x)\quad\text{("$p$ 编码一个 $T$-证明, 其末式码为 $x$")}$$

成为算术关系; 对递归公理化的 $T$ 它是 primitive recursive, 对 c.e. 公理化的 $T$ 可经 Craig 重公理化得到同等有效的形式. 由此

$$\mathrm{Prov}_T(x)=\exists p\,\mathrm{Prf}_T(p,x)$$

成为一个 $\Sigma^0_1$ 公式.

两面:

• 处理能力: 关于证明的陈述成为关于自然数的陈述. 内层 1 所标的元判断接口, 其语法侧从此有了在对象语言内的表示; 主步骤 4 的对角构造由此成为可能;
• 余项: token 掩盖了它是序列这一事实. 需要证明的内部结构时须重新解码, 而解码不是 token 自带的操作.

这与 Paper 4 § 4.2 内层 3 的有序对绑定同位: 该处把有序对绑成单一记号, 此处把有限序列绑成单一记号. 两者都是跨迭代的代价管理绑定.

本文不主张 $\#$ 与 $2\pi i$ 的出场格位同型. Paper 4 中 $2\pi i$ 首次出场在主步骤 3 (作为覆叠变换群的生成元), 再现于主步骤 4 内层 2 (作为留数归一); 而 $\#$ 首次出场在本处, 即主步骤 2. 格位不同, 不得据此推论. $\#$ 在主步骤 4 是否再次承担 Paper 1 § 4.5 所派的交换律阐述单位一职, 留待 § 8 处理.

内层 4: 构与余项

构: 在 $T$ 一致、c.e. 公理化、$T\supseteq Q$ 且证明检查有效的前提下,

$$\mathrm{Th}(T)\ \text{是}\ \Sigma^0_1\ \text{集}.$$

进一步, $(P_T,N_T)$ 为有效不可分对, 故 $\mathrm{Th}(T)$ 是 creative 集, 在 many-one 归约下 $\Sigma^0_1$-完全, Turing 度为 $0'$.

余项: 记

$$\mathrm{Dec}_T=\{\varphi: T\vdash_T\varphi\ \text{或}\ T\vdash_T\neg\varphi\},\qquad \mathrm{Ind}_T=\mathrm{Sent}_L\setminus\mathrm{Dec}_T.$$

由 Gödel–Rosser, 在上述前提下 $\mathrm{Ind}_T\ne\varnothing$. 而 $\mathrm{Ind}_T$ 是 $\Pi^0_1$ 集; 由 $(P_T,N_T)$ 的有效不可分性, 它是 $\Pi^0_1$-many-one 完全, Turing 度为 $0'$.

该余项须精确陈述, 三处易误:

1. $\mathrm{Ind}_T$ 不是"不可证的真". 若 $\varphi$ 独立于 $T$, 则 $\varphi$ 与 $\neg\varphi$ 两者皆在 $\mathrm{Ind}_T$ 中; 称其中之一为真须先固定标准模型, 而本主步骤全程未使用模型.
2. $\mathrm{Th}(T)$ 的补集不等于 $\mathrm{Ind}_T$. 前者含 $N_T$ 的全部成员 (可反驳者), 后者不含.
3. "真而不可证"是另一个对象. 它是 $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$, 其定义须用标准模型, 且在 $T$ sound 且 c.e. 时 $U_T\equiv_T\mathrm{Th}(\mathbb{N})$, Turing 度为 $0^{(\omega)}$. 本文所取的余项是 $\mathrm{Ind}_T$, 不是 $U_T$; 理由在 § 9 给出.

故加法路径的边界是纯语法的: 迭代能列出全部定理, 而列不出全部被裁决者的补 —— 这一失败不需要"真"这个概念即可陈述.

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5. 主步骤 3: 乘法路径 —— 相对化与度剖面

5.1 主步骤 3 与 inevitability 3

主步骤 3 阐述非有也非无相: 不是新的正面实体, 而是对已有实体的记忆绑定 —— 记录复杂度而非增加内容.

在 $L_3$, 所记忆者是: 一个集合需要多少不可计算性才能被算出来.

5.2 内层步骤

内层 1: "相对于什么"的 sub-mark

$\mathrm{Ind}_T$ 的存在使一个新的问题成为可问的: 若允许在计算中查询某个外部集合, 能否判定更多?

该问题逼出 sub-mark "相对于什么" —— 神谕. 形式上, $B\le_T A$ 记"以 $A$ 为神谕可计算 $B$".

此处与 Paper 4 § 5.2.1 同格: 该处记加法路径的余项 (旋转不被平移迭代给出) 迫使乘法路径的 sub-mark; 此处是加法路径的余项 ($\mathrm{Ind}_T$) 迫使"相对化"这一 sub-mark.

内层 2: 归约的合成

$\le_T$ 是预序. 对任意 $A,B$, join

$$A\oplus B=\{2n:n\in A\}\cup\{2n+1:n\in B\}$$

给出二者的最小公共上界. 合成在此意义上总是可行的.

内层 3: 等价类绑成单一 token

代价: 若始终在集合层面工作, 则"$A$ 与 $B$ 需要同样多的不可计算性"这一事实无法作为单一陈述表述 —— 每次比较都须重新给出双向归约.

绑定: 度 $[A]_T$ 作为 $\equiv_T$ 的等价类, 把无穷多个集合绑成单一 token.

两面:

• 处理能力: 一个度记录的是"要多少不可计算性才能算出它". 这正是主步骤 3 所要求的记忆内容 —— 不增加新的集合, 而记录已有集合的复杂度史;
• 余项: token 掩盖了类内成员的差异. 同度的两个集合可以在其他一切方面完全不同; 由度回到集合不是 token 自带的操作.

一处类型纪律: 度是商对象. 单个集合有度, 而"整个度空间"没有度 —— 后者不是一个可被归约比较的对象. 本文凡涉度的陈述, 须分清"某集合的度"、"度之间的归约关系"与"度空间的结构"三者.

内层 4: 构与余项

构: 本文所用的度剖面是算术度理想

$$\mathrm{AR}=\mathrm{DegProf}^{\rm arith}=\{d:\exists n\ d\le 0^{(n)}\}.$$

该集有两条性质:

• 下闭: $e\le d$ 且 $d\in\mathrm{AR}$ 则 $e\in\mathrm{AR}$. 特别地, 度 $0$ (可计算集) 在其中. 〔此处用 $e$ 而非 $d'$ —— 后者在本文专指 $d$ 的 jump〕;
• 对 jump 封闭: $d\le 0^{(n)}\Rightarrow d'\le 0^{(n+1)}$.

故相对化在此结构内可以无限迭代而不越界: $0,0',0'',\ldots$ 全部在其中.

余项: 该结构无最大元, 且在全体 Turing 度中无最小上界.

三件事须分开陈述, 混淆其一即致错:

1. $\mathrm{AR}$ 内部无最大元: $0^{(n)}<_T 0^{(n+1)}$ 对一切 $n$;
2. $\mathrm{AR}$ 在 $\mathcal{D}$ 中有上界: 例如 $0^{(\omega)}=\bigoplus_{n<\omega}0^{(n)}$;
3. $\mathrm{AR}$ 在 $\mathcal{D}$ 中无最小上界.

第 3 条由 Spector 的 exact pair 定理得: 可数生成的度理想 $I$ 有 exact pair $(a,b)$, 使 $I=\{d:d\le a\ \text{且}\ d\le b\}$. 若 $I$ 有最小上界 $c$, 则 $c\le a$ 且 $c\le b$, 故 $c\in I$ —— 即 $c$ 是 $I$ 的最大元, 与第 1 条矛盾.

故不可写「其上确界是 $0^{(\omega)}$」. 一个易犯的错误推理是: 任一上界 $u$ 满足 $u\ge_T 0^{(n)}$ 对一切 $n$, 故 $u\ge_T\bigoplus_n 0^{(n)}$. 该步不成立 —— 它只给出一族逐个的归约, 不给出统一计算直和所需的一致程序.

$0^{(\omega)}$ 的正确身份: 对固定的有效序列 $0,0',0'',\ldots$, 称度 $u$ 一致地计算该序列, 若对 $u$ 的某个 (从而任一) 代表 $X$, 存在统一可计算的指标函数 $f$ 使 $\Phi^X_{f(n)}=0^{(n)}$ 对一切 $n$. 〔此处不用字母 $U$ —— 本文 § 6.3、§ 8.1 已把 $U$ 固定为 optimal universal prefix-free machine〕 在此定义下, $0^{(\omega)}=\bigoplus_n 0^{(n)}$ 是该序列的最小一致上界 —— 任何一致上界必计算其直和, 而 $0^{(\omega)}$ 自身即一致上界.

该词须带上述定义使用, 且不得与「$\mathrm{AR}$ 的最小上界」混用 —— 后者不存在 (Spector).

但「理想的一致上界」是存在的, 且 $0^{(\omega)}$ 即是一个: $\mathrm{ARITH}$ 可数 (公式可数); 以公式为枚举, $\mathrm{Th}(\mathbb{N})$ 一致地计算每个 $A_\varphi$ (给定 $\varphi$ 与 $n$, 问 $\varphi(\bar n)$ 在 $\mathbb{N}$ 中是否真 —— 单一程序); 而 $\mathrm{Th}(\mathbb{N})\equiv_T 0^{(\omega)}$. 故 $0^{(\omega)}$ 是 $\mathrm{AR}$ 的一致上界, 且为最小 (任何一致上界皆能判语句真假, 从而计算 $\mathrm{Th}(\mathbb{N})$).

故须区分三者: $\mathrm{AR}$ 的最小上界不存在; $\mathrm{AR}$ 的最小枚举一致上界存在且为 $0^{(\omega)}$; 而一致性是相对于所固定的枚举而言的, 换枚举即换所指.

一条术语防火墙: 本文的「枚举一致上界」是相对于所固定的有效枚举的自定义概念, 不等同于文献中「理想的 uniform upper bound」这一标准概念 —— 后者另有定义, 且其类没有最小成员. 本文凡用该词处皆带上述限定. 而 $0^{(\omega)}\not\le 0^{(n)}$ 对任何 $n$, 故 $0^{(\omega)}\notin\mathrm{AR}$.

故乘法路径的余项不是某一次跳跃, 也不是一个位于剖面之外的上确界, 而是「该结构连一个最小上界都没有」这一事实本身.

一处不可作的推论: 不得由此说"越出 $\mathrm{AR}$ 须经 $0^{(\omega)}$". 反例即由 exact pair 给出: 设 $(a,b)$ 为 $\mathrm{AR}$ 的 exact pair, 二者皆为上界; 若二者都计算 $0^{(\omega)}$, 则 $0^{(\omega)}\le_T a,b$, 由 exact pair 性质得 $0^{(\omega)}\in\mathrm{AR}$, 矛盾. 故 $a,b$ 中至少有一个是 $\mathrm{AR}$ 的上界而不计算 $0^{(\omega)}$. 更弱地说, 任何非算术度都已在理想之外, 离开 $\mathrm{AR}$ 并不要求统摄全塔.

$0^{(\omega)}$ 的地位仅是: 有限 jump 塔的一种规范统一直和, 非离开该理想的必经对象. 这一点须与一个易犯的读法区分: 并非"某个 jump 减不掉"—— jump 在本剖面内可以任意多次施加而始终留在剖面内; 而是"施加任意多次之后所得的极限落在剖面之外".

第二级反向追问的正确形式: 相对计算允许把某集合 $A$ 当作神谕; 但不存在由 $A$ 自身统一决定所有 $A$-相对程序是否停机的 $A$-算法. 该统一自询问产生严格更高的 $A'$. 这是 $A<_T A'$ 的来源 —— 它出自自应用, 与格论性质无关.

〔本文不主张 Turing 度空间的格论失败 (存在无下确界的度对) 与本处余项有生成关系. 二者是两回事.〕

一处与 § 9 的接口: $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$. 即本处选定的规范塔界标记 $0^{(\omega)}$ 与 § 9 所论第二项余项的见证 $\mathrm{Th}(\mathbb{N})$ 同度.

须防一处换义: 本主步骤的余项是"$\mathrm{AR}$ 无最大元且无最小上界"这一结构事实, 而结构事实没有 Turing 度; 有度的是所选定的标记 $0^{(\omega)}$. 二者不得互称. 该同一是两条已知事实的合取, 不是本文的识别; 其结构意义在 § 9 讨论, 本处不作推论 —— 度相同不蕴含在阐述中占同一位置.

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6. 主步骤 4(a): 捕获核心

6.0 主步骤 4 的组织

主步骤 4 是 $L_3$ 的闭合与余项登记. 按 Paper 3 § 6.1 的登记声明, 它的类别是余项账本结构, 不是层内 sub-fractal; 其内部按"捕获核心 → 未捕获账本 → 下一层触发条件"组织, 不按四相重演组织.

主步骤 4 横跨三节: 本节 (§ 6) 阐述捕获核心, § 7 阐述有效性边界, § 8 阐述闭合与余项账本. 三节各占一节而主步骤 1 至 3 各占一节, 这一版面不对称本身即上述登记 —— 与 Paper 4 § 6.0 同.

§ 6 与 § 7 的关系是核心与边界, 不是互斥分箱. 与 Paper 4 § 6/§ 7 的"正则核心 $\hookrightarrow$ 不规则增强"是功能对位, 不是内容类比: 该处所加者是指数因子、方向与 Stokes 数据, 即同类对象的增强; 本文 § 7 所给者是该核心能到哪里的两条定理, 即核心的边界结构. 两处的差别须如实登记, 不得说成同型.

6.1 承载体迁移

Paper 4 § 6.1 立: 捕获范围的承载体在 $L_1\to L_2$ 之间换了类型.

层	捕获判据	捕获范围	承载体类型
$L_1$	满足什么方程	$\mathbb{Q}^{\rm alg}_{\mathbb{R}}$	标量域
$L_2$	有什么表示	$\mathcal{P}$	表达式类的取值像
$L_3$	在什么语言里可定义	$\mathrm{ARITH}$ (算术子集之集)	语义评价的商像

三格的环境类型并不相同: $\mathbb{Q}^{\rm alg}_{\mathbb{R}}\subseteq\mathbb{R}$, $\mathcal{P}\subseteq\mathbb{C}$, $\mathrm{ARITH}\subseteq\mathcal{P}(\mathbb{N})$. 不可写"三格皆为实数的子集" —— 周期环不是 $\mathbb{R}$ 的子集.

可成立的是较弱的一句: 三层各保留一个实轴对应物, 而这些对应物构成 § 6.3 的严格递增链. 完整取值域不同, 只有对应物可比.

变的因此有两件: 核心被构成的方式 ($L_1$ 代数的、$L_2$ 表示的、$L_3$ 语义评价的), 以及环境类型本身.

6.2 捕获核心

$$\mathrm{ARITH}=\{A\subseteq\mathbb{N}:\ A\ \text{是一阶算术可定义的}\},\qquad \mathbb{R}_{\rm arith}=\{x\in\mathbb{R}:\ \deg(x)\ \text{是算术度}\}.$$

二者的关系须用两个映射写清, 不得共用一个记号:

$$\mathrm{Form}_1(L_A)\ \xrightarrow{\ \mathrm{ev}_{\rm set}\ }\ \mathrm{ARITH}\ \xrightarrow{\ j\ }\ \mathcal{C}\subseteq\mathbb{R},$$

其中 $\mathrm{ev}_{\rm set}(\varphi)=A_\varphi$, 而 $j$ 是某个固定编码 (如三进制 Cantor 编码).

• $\mathrm{ARITH}$ 是捕获核心; 它是 $\mathrm{ev}_{\rm set}$ 的商像.
• $\mathbb{R}_{\rm arith}$ 不是 $j$ 的像: $\mathrm{im}(j)\subseteq\mathcal{C}$ (Cantor 集), 而 $\mathbb{R}_{\rm arith}$ 含 Cantor 集外的成员 —— 如 $1/2$ 可计算故在 $\mathbb{R}_{\rm arith}$ 中, 而其三进制展开用到数字 $1$. 故 $\mathrm{im}(j\circ\mathrm{ev}_{\rm set})\subsetneq\mathbb{R}_{\rm arith}$.
• 故「实轴对应物」一词只作 T3 称呼: 本文未给出一个规范的满射 $\mathrm{ARITH}\twoheadrightarrow\mathbb{R}_{\rm arith}$. $\mathbb{R}_{\rm arith}$ 由其自身的度刻画定义 (见上), 与 $\mathrm{ARITH}$ 经 Post 定理相关联, 而非经 $j$ 相关联.

构成方式: 取一元算术公式 $\varphi(n)$, 令

$$A_\varphi=\{n\in\mathbb{N}:\mathbb{N}\models\varphi(n)\},$$

再以固定的无歧义编码 (如 $\sum_{n\in A_\varphi}2\cdot 3^{-(n+1)}$) 把 $A_\varphi$ 表为实数.

该求值在外部语义中是全的, 不是偏的: 每个一元公式都定义一个自然数集合, 不需先判定"它是否定义了一个实数" —— 这正是取 $\mathrm{ARITH}$ 而非某个"由公式唯一指定的实数"之集作核心的理由. 后者须先冻结实数代码、代码判等与"唯一"的所指, 而本文不作该冻结.

难处不在求值, 在等价核:

$$\varphi\sim\psi\iff\mathbb{N}\models\forall n\,(\varphi(n)\leftrightarrow\psi(n)),$$

而该等价关系与完整算术真理 Turing 等价 —— 给定算术句 $\sigma$, 把常值公式 $\sigma$ 与恒真公式比较即把算术真理归约到 $\sim$.

故三层的求值性状如下:

层	语法对象	求值	性状
$L_1$	极小多项式 + 隔离区间	选定实根	全、可计算、等价可判
$L_2$	合法的绝对收敛周期表达式	积分取值	在声明域上全; 一般等价未决
$L_3$	一元算术公式	标准真集, 再作实数编码	外部全; 等价核不能算术内部化

$L_1$ 那一行的语法对象须带根选择数据: 极小多项式单独不足以唯一确定一个实代数数 —— $x^2-2$ 有两个实根. 典范表示须为 (不可约多项式, 有理隔离区间), 或 Thom 编码.

6.3 数值链

$$\mathbb{Q}^{\rm alg}_{\mathbb{R}}\subsetneq\mathcal{P}\cap\mathbb{R}\subsetneq\mathbb{R}_{\rm comp}\subsetneq\mathbb{R}_{\rm arith}.$$

第二个严格包含由 Yoshinaga 的可计算非周期实数见证. 第三个由 $\Omega_U$ 见证: 对固定的 optimal universal prefix-free machine $U$, $\Omega_U$ 左 c.e. 故 $\Sigma^0_2$-可定义, 而其 Turing 度为 $0'$, 故不可计算.

一处须登记的位置差别: $\Omega_U$ 是 $L_2\to L_3$ 的跃迁签名, 而它 (作为实数) 落在 $\mathbb{R}_{\rm arith}$ 之内, 其标准位集属 $\mathrm{ARITH}$; 而 $e$ 是 $L_1\to L_2$ 的跃迁签名, $e\in\mathcal{P}$ 是否成立至今开放, Paper 4 § 7.1 明写不以之承重. 两个签名与各自层的关系不同型, 本文不据此作任何推论.

Paper 4 所说的"可定义实数"与本文的 $\mathbb{R}_{\rm arith}$ 不是同一个对象. 该处挣得的是一条宽的基数陈述 (对任何固定可数形式语言, 无参数或只允许固定可数参数源时, 可定义实数至多可数), 未指定语言. 本文取一阶算术语言, 是一次收窄与具体化, 属新加材料.

6.4 核心与装置

捕获核心不等于阐述装置. 本文所用的装置是

$$\mathfrak{C}_T=\bigl(\mathrm{Sent}_L,\ C_T^\vdash,\ C_{\mathbb{N}}^\vDash,\ \#,\ \mathrm{DegProf}^{\rm arith}\bigr),$$

其中 $C_T^\vdash$ 与 $C_{\mathbb{N}}^\vDash$ 分别是句法后承闭包与标准模型上的语义后承闭包. 装置是多排序的, 求值只定义在其中一个排序上:

$$\mathrm{ev}_{\rm set}:\ \mathrm{Form}_1(L_A)\ \longrightarrow\ \mathrm{ARITH}.$$

$\mathrm{ev}_{\rm set}$ 不定义在闭包算子 $C_T^\vdash$、$C_{\mathbb{N}}^\vDash$ 或度理想 $\mathrm{DegProf}^{\rm arith}$ 上. 写作"$\mathfrak{C}_T\to\mathbb{R}_{\rm arith}$"是简写, 不是良定义的映射; 本文全篇不用该记法.

这与 $L_2$ 的情形同位: 该处 $\mathsf{Int}_{\rm alg}$ 是装置, $\mathcal{P}=\mathrm{im}(\mathrm{ev})$ 是取值核心, $\ker(\mathrm{ev})$ 是等价核; Paper 4 § 2.3 明令三者不得并作一句. 本文沿用该防火墙.

两条路径在此归位, 但不经同一求值. 各分量与核心的关系逐条如下:

分量	与核心 $\mathrm{ARITH}$ 的关系
$\mathrm{Form}_1(L_A)$	经 $\mathrm{ev}_{\rm set}$ 求值, 其商像即核心
$C_{\mathbb{N}}^\vDash$	决定 $\mathrm{ev}_{\rm set}$ 的等价核. 但其在空前提处的取值 $\mathrm{Th}(\mathbb{N})$ 不是核心的成员 (Tarski) —— 句法闭包落在核心内而语义闭包落在核心外, 这条不对称正是定理 A 与 § 9 两项余项之所以可能的原因
$C_T^\vdash$	其在空前提处的取值 $\mathrm{Th}(T)$ 是核心的成员 ($\Sigma^0_1$ 故算术); 由其导出的余项 $\mathrm{Ind}_T$ 亦是 ($\Pi^0_1$)
$\mathrm{DegProf}^{\rm arith}$	是核心的度谱 (由 Post: $A$ 算术 $\iff\exists n\ A\le_T 0^{(n)}$)
$\#$	使 $C_T^\vdash$ 的关系成为算术的 —— 与 $T$ 的有效公理化、证明检查的有效性合起来, 支撑上表第三行 (非单凭 $\#$)

故主步骤 2 与主步骤 3 的产物皆与核心相关, 但相关的方式不同 —— 一为成员, 一为度谱. 本文因此仍不须为二者的"类型不同"另设配对; 但不得说二者经同一求值落到同一取值像 (初稿如此写, 过强).

6.5 一条判据: 语义商像

$\mathrm{ARITH}$ 之所以有资格称为捕获核心, 不因其成员可编码成实数, 而因它满足以下四条:

1. 评价先于编码选择: $\mathrm{ev}_{\rm set}$ 由本层的捕获问题规定, 不是先把对象随意注入某集合再把注入结果叫作取值;
2. 核心是语义商的像: $\{\text{一元算术公式}\}/{\sim}\ \cong\ \{\text{算术子集}\subseteq\mathbb{N}\}$ —— 核心中的相等恰好对应"表示同一对象";
3. 对合法重编码协变: 一个重编码 $r:E\to E'$ 称为合法, 当 (i) $r$ 可计算; (ii) 在其像上有可计算逆, 或形成可计算双解释; (iii) 保持语义等价 ($e_1\sim e_2\iff r(e_1)\sim' r(e_2)$); (iv) 存在指定的可计算同构 $u$ 使求值方图交换 ($u\circ\mathrm{ev}=\mathrm{ev}'\circ r$). 换标准 Gödel 编码、换公式的语法规范形皆合法; 任意集合双射不合法;
4. 不靠基数论证: "$E$ 可数故可嵌入 $\mathbb{R}$"不构成核心资格.

第 3 条另有一条独立支持: 由 Post 的层级定理,

$$A\in\mathrm{ARITH}\iff\exists n\ A\le_T 0^{(n)},$$

这是一条不涉任何语法的刻画. 换编码不改变它 —— 而这正是"合法重编码"一词的所指: 保持该刻画不变者.

该判据须立, 是因为它排除一类无效论证: 形式理论、证明关系、Turing 度与可证性代数皆可编码成自然数集再编码成实数; 但

$$\text{可编码成实数}\ \ne\ \text{自然承载体是实数子集}.$$

若仅凭可编码性即认定核心, 则任何可数数学结构都能被强行装回"实数核心", 承载体迁移将变得不可检验.

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7. 主步骤 4(b): 有效性边界

7.0 本节所增加者

§ 6 给出捕获核心与其构成方式. 本节问: 有效可靠的阐述装置能覆盖到哪里.

本节的两条结论是本文的 T1 交付. 二者的前提须先冻结.

前提 (缺一不可): 本文所研究的形式系统具有

• 递归语言;
• c.e. 公理集或 c.e. 定理集;
• 有效证明检查关系.

"对象有有限描述"、"表达式类可数"与"理论有效公理化"是不同强度的性质. Paper 4 使用有限表达式与可枚举规则, 不自动给出本前提; 故此处显式冻结, 不由上游默推.

7.1 定理 A: 全闭合障碍

先给闭合目标的确切形式. 令 $L_A$ 为一阶算术语言,

$$C_T^\vdash(\Gamma)=\{\varphi\in\mathrm{Sent}_A: T+\Gamma\vdash\varphi\},\qquad C_{\mathbb{N}}^\vDash(\Gamma)=\{\varphi\in\mathrm{Sent}_A:\mathbb{N}\models\Gamma\Rightarrow\mathbb{N}\models\varphi\}.$$

引理 7.1: $C_T^\vdash=C_{\mathbb{N}}^\vDash\iff\mathrm{Th}(T)=\mathrm{Th}(\mathbb{N})$.

必要性取 $\Gamma=\varnothing$. 充分性: 若 $\mathbb{N}\models\Gamma$, 则 $\Gamma\subseteq\mathrm{Th}(\mathbb{N})=\mathrm{Th}(T)$, 两边皆得 $\mathrm{Th}(\mathbb{N})$; 若 $\mathbb{N}\not\models\Gamma$, 则某 $\gamma\in\Gamma$ 为假, 故 $\neg\gamma\in\mathrm{Th}(\mathbb{N})=\mathrm{Th}(T)$, 于是 $T+\Gamma$ 不一致, 两边皆为全体句子. $\square$

定理 A: 不存在有效公理化的理论 $T$ 使 $C_T^\vdash=C_{\mathbb{N}}^\vDash$.

证: 由引理 7.1, $C_T^\vdash=C_{\mathbb{N}}^\vDash$ 蕴含 $\mathrm{Th}(T)=\mathrm{Th}(\mathbb{N})$. 而 $T$ 有效公理化蕴含 $\mathrm{Th}(T)$ 为 c.e., 于是 $\mathrm{Th}(\mathbb{N})$ 为 c.e., 与 Tarski (或与 $\mathrm{Th}(\mathbb{N})\equiv_T 0^{(\omega)}$) 矛盾. $\square$

可靠性不是本定理的前提. 全证不用 soundness —— 不可靠的有效理论同样被排除. 可靠性只在下述推论的 $\subseteq$ 一半需要.

推论 7.2 (对语言扩张亦成立): 设 $S$ 是递归语言 $L'\supseteq L_A$ 中的 c.e. 理论, 证明演算有效, 算术语言的嵌入可计算. 记 $\mathrm{Th}_A(S)=\{\varphi\in\mathrm{Sent}_A: S\vdash\varphi\}$. 则 $\mathrm{Th}_A(S)$ 是 c.e.; 若 $S$ 算术可靠且 $\mathrm{Th}_A(S)=\mathrm{Th}(\mathbb{N})$, 则完整算术真理为 c.e., 矛盾. 故

$$S\ \text{有效公理化且算术可靠}\ \Longrightarrow\ \mathrm{Th}_A(S)\subsetneq\mathrm{Th}(\mathbb{N}).$$

该推论的范围须精确: 它对任何仍然有效公理化的装置成立 —— 不论该装置改变的是对象、语言、公理还是规则. 特别地, 对标准的递归可公理化大基数理论 (ZFC 加某固定大基数公理或递归公理模式) 成立; 但不可无条件断言"所有可想象的大基数理论", 因为可以人为取一个非 c.e. 的大基数公理全集.

唯一满足者的刻画须限固定语言: 在固定算术语言中, 唯一满足 $C_T^\vdash=C_{\mathbb{N}}^\vDash$ 的演绎闭理论是 $\mathrm{Th}(\mathbb{N})$ 本身; 在扩张语言中不唯一 (如 $\mathrm{Th}(\mathbb{N})+\forall x\,P(x)$ 的算术 reduct 仍为 $\mathrm{Th}(\mathbb{N})$), 而这些理论同样非有效.

7.2 定理 B: 分片闭合阈值

对有效可识别的句子类 $\Gamma$, 记 $\mathrm{True}_\Gamma=\{\varphi\in\Gamma:\mathbb{N}\models\varphi\}$. 设 $T$ 为 c.e. 理论, 且

• $\Gamma$-可靠: $T\vdash\varphi\in\Gamma\Rightarrow\mathbb{N}\models\varphi$;
• $\Gamma$-完备: $\mathbb{N}\models\varphi\in\Gamma\Rightarrow T\vdash\varphi$.

则 $\mathrm{True}_\Gamma=\mathrm{Th}(T)\cap\Gamma$, 右侧 c.e. 故

定理 B: 有效、$\Gamma$-可靠且 $\Gamma$-完备的理论存在 $\Longrightarrow\mathrm{True}_\Gamma$ 是 c.e.

反向亦成立: 若 $\mathrm{True}_\Gamma$ 本身 c.e., 则把这些真句加入一个可靠基理论即得 c.e. 的 $\Gamma$-完备理论.

故阈值不在层级序号, 而在"$\mathrm{True}_\Gamma$ 是否 c.e.":

片段	有效可靠理论能否完全覆盖
$\Delta^0_0$	能
$\Sigma^0_1$	能
$\Pi^0_1$	不能
$\Sigma^0_n\ (n\ge 2)$	不能
$\Pi^0_n\ (n\ge 1)$	不能

阈值不对称: 沿 $\Sigma$ 断在 $\Sigma_1\,|\,\Sigma_2$; 沿 $\Pi$ 自 $\Pi_1$ 即起. 无不加方向说明的单一阈值.

$\Sigma_1$ 一侧的机制是: 真 $\Sigma_1$ 句 $\exists x\,\delta(x)$ 有具体见证 $m$, 而 $Q$ 能验证 $\delta(\bar m)$ 这一有限计算, 继而证 $\exists x\,\delta(x)$. 故任何 $T\supseteq Q$ 继承该完备方向. 但与标准真理相等还需 $\Sigma_1$-可靠 —— 单纯一致性不够, 因为一致的 $Q$-扩张仍可能证出假的 $\Sigma_1$ 句.

无中间地带: 低于阈值时基理论已经闭合 (对象扩张什么也没完成); 高于阈值时任何有效可靠理论皆闭不了. 不存在既尚未闭合、又可能由有效对象扩张闭合的标准算术层级.

7.3 $0^{(\omega)}$ 被标出而不被进入

主步骤 3 内层 4 已给出: 算术度理想 $\mathrm{AR}$ 无最大元, 且在 $\mathcal{D}$ 中无最小上界; 而 $0^{(\omega)}$ 是有限 jump 塔的最小一致上界, $0^{(\omega)}\notin\mathrm{AR}$. 又 $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$.

于是 $L_3$ 的处境可以一句话说清:

> $0^{(\omega)}$ 在 $L_3$ 内被指出, 而不被 $L_3$ 的度剖面纳入.

它被指出, 是因为 $L_3$ 的乘法路径能写出 $0,0',0'',\ldots$ 并看出这条链没有封顶; 它不被纳入, 是因为纳入它须离开算术可定义性 —— 而那正是 $\mathrm{ARITH}$ 的判据.

此处与 Paper 4 § 7.5 ("$\mathcal{P}^e$ 被标出而不被进入") 是功能对位: 两处都是本层能指出而不能进入的边界对象. 但两处的对象类型不同 —— $\mathcal{P}^e$ 是一个扩张的取值像, $0^{(\omega)}$ 是一个度. 本文不主张二者同型.

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8. 主步骤 4(c): 闭合与余项账本

8.1 粗闭合: 对角化

$L_2\to L_3$ 的行为是对角化／通用求值.

对角引理施于主步骤 1 内层 3 所绑的两种极化, 各得一个见证:

施于	所得
$\neg\mathrm{Prov}_T(x)$	Gödel 语句
$\neg\mathrm{True}(x)$	说谎者, 由此得 Tarski 不可定义性

同一个构造, 两个谓词, 两个结果. 前者须限定标准可证性谓词与足够的算术基础, 且其独立性另需一致性条件; 后者的形式是: 假设存在满足全部算术 T-biconditional 的完整真谓词, 对角化即得矛盾.

行为与其产物须分开. 行为是对角化; 其产物包括停机集 $K$ 与 (相对固定的 optimal universal prefix-free machine $U$ 的) $\Omega_U$. 本文凡称"行为"处只指对角化／通用求值, 不指 $K$ 或 $\Omega_U$.

8.2 两个度的角色分离

$$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}(\mathbb{N})]_T=0'.$$

该等式成立. 但其内容须精确陈述: 它记的是行为的两个产物 ($K$ 与 $\Omega_U$) 与真理分层的第一个非平凡片段在度商中重合. $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ 是一个可定义、可有效枚举的被成功捕获的片段, 不是任何余项的成员.

由此须把两个度的角色分开:

度	角色
$0'$	坍缩点 —— 行为产物的度 (另: $\mathrm{Th}(T)$ 与 $\mathrm{Ind}_T$ 亦在此度)
$0^{(\omega)}$	有限 jump 塔的最小一致上界; $\notin\mathrm{AR}$

故 $0'$ 不是度剖面的边界 —— 算术度对 jump 封闭, $0'$ 只是其中一个度. 且 $\mathrm{AR}$ 根本没有一个"边界度": 它无最大元, 在 $\mathcal{D}$ 中亦无最小上界 (§ 5.2). ZFCρ Paper II § 6.3 所给的闭合方程 $\deg^{0'}(0')=0$ 记的是坍缩点处的归一化.

〔该方程在相对 Turing 度中近乎定义性: 给定 $0'$ 作神谕, $0'$ 自身当然可计算. 本文按此登记, 不把它当作一项独立数学机制的证据.〕

8.3 余项账本

按 Paper 4 § 8.2 的三类格式:

类	$L_2$	$L_3$
第一类 · 基底外	$\mathbb{H}$ 方向 (判为可施加而非被迫)	尚未识别出 〔T3〕
第二类 · 基底上而模式不捕获 (语法边界)	指数方向	未定
第三类 · 不可计算、不可定义	承自 $L_1$, $L_2$ 仍不捕获	$\mathbb{R}\setminus\mathbb{R}_{\rm arith}$

第一类: 尚未识别出被迫的基底外余项 〔T3 登记, 非 T1 事实〕. 在本文所选的 trajectory 中 ($\mathbb{R}$ 作基底、算术公式作模式), 反向追问未逼出新的数种. 这与 Paper 4 § 8.2 已挣得的"代数闭消除了生产被迫的基底外余项的机制"相容.

该条不写作"第一类为空": 它取决于基底的选择、模式的选择与"逼出"的定义, 三者皆是本 trajectory 的登记, 非数学事实. 其谨慎程度与第二类 (未定) 一致.

第二类未定: $L_2$ 的第二类是一条语法边界 —— $\mathsf{Int}_{\rm alg}$ 不允许指数因子. $L_3$ 的对应须问: 一阶算术公式这一语法有无对应的"不允许"? 一个候选是集合量词 (超算术需要而算术语言不含), 但该候选未验, 且可能与第三类重合. 本文将其登记为开放, 不填.

第三类的基数与命名: $\mathbb{R}_{\rm arith}$ 至多可数, 故其补集有连续统基数. 但该补集只能称非算术可定义实数, 不可称"不可定义实数" —— 在更强的语言中其中许多成员仍可定义.

8.4 $L_3\to L_4$ 触发

主步骤 3 内层 4 的余项是: 算术度理想无最大元, 且无最小上界. 本文所考察的一条延续路线是沿超限指标继续迭代跳跃, 即形成 $0^{(\alpha)}$ 一族 —— 须注意这是一条选定的路线, 不是离开该理想的唯一方式 (§ 5.2).

而超限迭代须有一个序数记号系统. 于是:

1. Kleene 的 $\mathcal{O}$ 是 $\Pi^1_1$-完全的, 合法记号无有效枚举;
2. 理论后果不只依赖序型, 还依赖具体的可计算记号. Turing 1939 (以现代局部反思进程重述): 对每个真 $\Pi_1$ 句 $\varphi$, 存在一个 $\omega+1$ 的可计算记号使该进程证出 $\varphi$; 但不存在一个固定记号使同一理论证出全部真 $\Pi_1$ 句. 量词序是 $\forall\varphi\,\exists\dot\alpha$, 不是 $\exists\dot\alpha\,\forall\varphi$;
3. 达到算术完备性的路径存在, 由 Feferman 1962 以完整一致反思构造; 但这样的路径不能是 $\Sigma^1_1$-可定义的. 且沿满足 Feferman–Spector 自然条件的 $\Pi^1_1$ 路径, 必有真 $\Pi^0_1$ 句不可导出.

故触发的形状是: 路径选取函数在算术内不可定义; 完备路径存在而不可低复杂度指定.

该触发与 $L_2\to L_3$ 的触发不同型. Paper 4 § 8.4 的触发是"值的层面 → 规则的层面": 压力从"某个值是什么"转向"那套规则能做到什么". 本处的触发是"有效可达 → 须外部供给选择": 不是问题的层面变了, 而是继续下去所需的东西不能由阐述自身供给. 两者的差别须如实登记, 不得说成同型.

三个反思算子须分清. $\mathrm{Con}(T)=\neg\mathrm{Prov}_T(\ulcorner 0=1\urcorner)$ 是单个 $\Pi^0_1$ 语句; 局部反思 $\mathrm{Rfn}(T)$ 是语句 schema; 一致反思 $\mathrm{RFN}_\Gamma(T)$ 是带数字量词的一致版本. $\mathrm{Con}(T)$ 只是局部反思在 $\varphi=(0{=}1)$ 处的一个实例, 远弱于完整局部反思或一致反思.

上述三项事实分别来自三种不同的进程设定: (1) 是关于记号系统本身的, 与所迭代的算子无关; (2) 的现代重述用局部反思; (3) 的前半用完整一致反思 (Feferman 1962), 后半针对满足公理化、单调性与有效性条件的一般递归理论进程 (Feferman–Spector), 不按反思算子分类.

故本文不把每项事实逐一归给三种反思算子; 三项合起来刻画超限进程的若干限制, 而本文对三种算子一并陈述, 不择一.

〔限定: Feferman–Spector 的全部"自然条件"本文未逐条复核; 严格陈述应为"覆盖满足该文条件的进程".〕

$L_4$ 的进一步内容在 § 13 处理.

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9. 两项余项的独立性

9.1 本节所欲证者与其承重之处

ZFCρ Paper II § 7.1 立: $L_0$ 只有一个余项, 有行为而没有闭合方程; 闭合需要两个独立的失败方向, 行为才能把它们绑成回零的环.

本文的两项余项是:

	事实	见证
余项 1	存在 $T$-独立语句	$\mathrm{Ind}_T$ ($\Pi^0_1$-m-完全, 度 $0'$)
余项 2	无单一完整的内部真谓词	$\mathrm{Th}(\mathbb{N})$ (度 $0^{(\omega)}$)

承重放在事实上, 见证只作数值见证 —— 与 Paper 4 § 7.5 把 $L_2$ 的闭合余项定为"指数映射与指数积分方向"而 $e$ 只作见证同格.

本节须先澄清一件事: § 7.1 未说明"独立"须有多强. 本文不沿用一个未经界定的词.

9.2 三个候选判据皆不取

(a) 定义资源无共享. 太弱 —— 可以给同一个对象人为安排两套不共享词汇的定义; 反过来, 真正独立的两个量也可能共享大量背景资源.

(b) 无任何不经行为的桥. 太强 —— 数学中两个真正不同的结构通常仍有比较映射、部分保守性、低复杂度一致区、单向归约或公共不变量. 只要存在任何桥即否定独立, 会使"独立"几乎无法满足. 且本文 § 9.5 将给出一座这样的桥.

(c) 两见证互不 Turing 归约. 类型不对, 且 $L_1\to L_2$ 的原型自己就不满足: $i$ 与 $\pi$ 皆可计算, $\deg_T i=\deg_T\pi=0$, 彼此可计算. 更根本地, 余项 2 若表述为"原语言中不存在完整统一真谓词", 它是一个不可定义性现象, 并不天然带 Turing 度 —— 只有选定某个见证集合后才有度, 而结果依赖见证选择. 故 (c) 不具表示不变性.

9.3 所取判据: 闭合角色上的非冗余性

设两个失败方向为 $F_1,F_2$, 闭合行为为 $B:F_1\times F_2\to Z$. 本文取

$$\boxed{B\ \text{不通过任何一个投影单独因子化}}$$

即不存在只依赖第一项的 $b_1$ 使 $B=b_1\circ\pi_1$, 亦不存在只依赖第二项的 $b_2$ 使 $B=b_2\circ\pi_2$.

直观地: 删去任一方向, 闭合任务即改变; 两个失败不是同一个失败换两种措辞. 但允许二者有部分桥、强弱关系或单向计算归约.

该判据由 § 7.1 自身的机制推出: 既然主张"需要两项才能闭合", 真正须证的是二元性不可降为一元性, 而不是两项在一切数学意义上毫无关系.

称之为 closure-essential independence.

9.4 定义资源: 成立, 但须收窄措辞

$\mathrm{Ind}_T$ 的主定义关系是 $\vdash_T$ —— 其定义中不出现 $\mathbb{N}$, 不出现"真". $\mathrm{Th}(\mathbb{N})$ 的主定义关系是 $\vDash_{\mathbb{N}}$ —— 其定义中不出现任何推导系统.

这也是本文取 $\mathrm{Ind}_T$ 而非 $U_T$ 的第二个理由: $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$ 的定义须用 $\mathrm{Th}(\mathbb{N})$, 上述性质当场失效.

但"定义资源互不相涉"略强. 二者仍共享: 同一算术语言、同一批句子、逻辑联结词与否定; 一旦谈及可计算性, 还共享句法编码体系. 准确的陈述是:

> 二者的主定义关系不同, 但底层句法载体并不分离.

9.5 两座桥, 二者的地位不同

桥一 ($\Sigma_1$-完全性): 真正的行为前部分桥.

$Q$ 证一切真 $\Sigma_1$ 句 (机制见 § 7.2), 故对 $T\supseteq Q$:

$$\mathrm{Th}_{\Sigma_1}(\mathbb{N})\subseteq\mathrm{Th}(T)\quad\Longrightarrow\quad \mathrm{Th}_{\Sigma_1}(\mathbb{N})\cap\mathrm{Ind}_T=\varnothing.$$

这是用语义信息约束一个纯语法对象, 且其证明走可计算性表示 (有限见证的有效验证), 不走对角引理. 故它是一座不经行为的桥.

但它只是部分桥: 它既不从 $\mathrm{Ind}_T$ 重构完整真理, 也不从真理缺口重构整个独立集. 应称$\Sigma_1$ 层面的语义—句法一致区, 不应称"两项余项已被连接".

桥二 (单向归约): 仍经算术化.

$\mathrm{Ind}_T$ 是 $\Pi^0_1$ 故 $\le_T 0'\le_T 0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$; 反向不成立. 即 $\mathrm{Th}(\mathbb{N})$ 决定 $\mathrm{Ind}_T$, 反之不然.

但这不是不经行为的桥: 一旦写下 $\mathrm{Ind}_T\le_T\mathrm{Th}(\mathbb{N})$, 两边已被表示成自然数集合 —— 公式编码成数, 证明编码成数, 机器查询这些代码. 这需要算术化 $\#$. 它只反驳一个更强的说法 ("即使完成算术化, 只要不用对角引理二者仍完全无桥"), 不反驳"在算术化之前没有桥".

且该归约本应预期: 完整算术真理是一个极强的神谕, 能计算所有算术集合; $\mathrm{Ind}_T$ 是算术集合, 被它计算是结构上必然的. 这反映的是神谕的信息量, 不是两个失败角色是否冗余.

9.6 两种失败方向非同一的反事实证据

须先说明本小节证到哪里: 下述反事实证明的是两个失败方向不是同一个失败换两种措辞; 它不证明现有闭合以二者为不可删除输入 —— 后者见 § 9.7 与 § 10.3, 尚未获证.

反事实分离: 两个方向可以各自单独变动.

反事实	余项 1 方向	余项 2 方向
$\mathrm{Th}(\mathbb{N})$ (完整但非有效)	$\mathrm{Ind}_{\mathrm{Th}(\mathbb{N})}=\varnothing$ —— 消失	算术真理仍不能由一个算术公式统一定义 —— 边界仍在
$\mathsf{CT}^-[\mathrm{PA}]$	仍是有效系统, 故仍不完全	在扩展语言中组合地表达原算术语言的真理, 且对 PA 保守 —— 部分修复

故

$$\text{证明闭包的不完备}\ \ne\ \text{完整真理的内部可定义性失败}.$$

二者不是同一失败的同义表达.

9.7 本节未能给出的

上述是必要条件的识别, 不是闭合实例的证明.

具体地: § 8.2 的坍缩涉及行为的两个产物与真理分层的第一个非平凡片段, 不含完整的余项 2. 故仍缺一条 typed 论证:

> 对角化／相对化闭合为何必须同时以 Gödel 不完备与 Tarski 不可定义为不可删除的输入?

这不是独立性定义的问题, 而是闭合映射尚未完全写出的问题. 本文将其登记为开放 (§ 15).

若该论证始终不能给出, 则 $L_2\to L_3$ 应称两边界耦合归零, 而不称两独立余项闭合. 本文预先登记此退路, 以免将来以"尚未被反驳"为由继续持有更强的说法.

9.8 本节结论

> Gödel 与 Tarski 两方向的主定义关系分别为 $\vdash_T$ 与 $\vDash_{\mathbb{N}}$, 但它们并非绝对隔离: 真 $\Sigma_1$ 形成一座行为前的部分桥, 完整真理又单向计算句法独立集. 因此本文不主张无桥或 Turing 不可比意义上的独立性. 本文所需的较弱条件是: 两个失败方向在闭合任务中均不可删除. 该 closure-essential independence 是否由现有的 $0'$ 归零结构完整实现, 仍须另证 —— 而 § 10.3 进一步显示: 现有的 $0'$ 归零式未把完整余项 2 展示为不可删除的输入, 故该方程本身不能证明它是该判据的实例. 本节与 § 10.3 不互相抵消: 本节证的是两个失败方向确实不同; § 10.3 证的是现有方程未展示双输入依赖, 不是"现有闭合事实上未使用二者". 二者合起来即 § 9.7 所登记的欠账.

9.9 对上游一处措辞的收窄

ZFCρ Paper II § 3.4 称"指数映射是 $i$ 与 $\pi$ 关系的唯一来源". 作为全数学命题过强 —— 至少有两座不经复指数的桥:

1. 旋转几何: 复平面构造完成后, 乘以 $i$ 是四分之一转, 而弧度制中四分之一转是 $\pi/2$. 该桥只用复乘法与欧氏圆周几何;
2. 围道积分: 取 $z(t)=\cos t+i\sin t$, 则 $dz=i\,z(t)\,dt$, 故 $\oint_{|z|=1}dz/z=\int_0^{2\pi}i\,dt=2\pi i$. 该计算不用复指数.

且 Paper 4 已把 $2\pi i$ 的覆叠生成元身份与留数归一身份登记为同一陈述的两种呈现 —— 从系列内部亦不支持"唯一".

可保留的较弱陈述须精确书写. 不可写"不存在非零 $P\in\mathbb{Q}[X,Y]$ 使 $P(i,\pi)=0$" —— 该陈述有初等反例: 取 $P(X,Y)=X^2+1$, 则 $P(i,\pi)=i^2+1=0$.

正确的陈述是: $\pi$ 在 $\mathbb{Q}(i)$ 上超越. 等价地, 若 $P\in\mathbb{Q}[X,Y]$ 且 $P(i,\pi)=0$, 则 $P$ 必属于理想 $(X^2+1)$ —— 即不存在涉及 $\pi$ 的非平凡额外代数关系; $i$ 自身当然满足 $X^2+1=0$.

但"$\pi$ 在 $\mathbb{Q}(i)$ 上超越"不等于"没有任何几何、解析或拓扑桥".

故该处应改为"本 trajectory 中的规范绑定来源", 不写"唯一来源". 详见 § 11 的勘误登记.

一处红利: $L_1\to L_2$ 亦有行为前的部分桥. 故本节 § 9.5 所报的情形不是 $L_3$ 特有的, 两次跃迁在这一点上同格.

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10. $\mathfrak{C}_T$ 的最小 typed 组织

10.0 本节交付什么、不交付什么

交付: $\mathfrak{C}_T$ 的最小 typed 组织 —— 两张表. 表一给出各分量的类型与内部运算 (§ 10.1), 表二给出跨分量接口 (§ 10.2).

不交付: § 9.7 所欠的闭合映射.

且本节将说明该欠账的性质与先前的登记不同 (§ 10.3). 先前 (§ 9.7、§ 15.2) 把它登记为"闭合映射尚未写出". 本节的组织工作显示: 更准确的陈述是 —— 现有的归零式未把完整的余项 2 展示为不可删除的输入, 故它本身不能证明 closure-essential independence. 至于是否存在满足该判据的闭合式, 本文既不主张有, 亦不主张无.

归属: 两张表的全部内容均为经典结果. 本节所做的是组织 —— 与 Paper 4 § 9.4 同.

"最小"二字须守住: 本节所交付者是骨架级. 它使 $\mathfrak{C}_T$ 不再只是一个五元组的清单, 但不使它成为一个完整的闭合装置.

10.1 表一: 各分量的类型与内部运算

分量	类型	内部运算
$\mathrm{Sent}_L$	可判定的码集	逻辑联结词; 代入
$C_T^\vdash$	$\mathcal{P}(\mathrm{Sent}_L)$ 上的闭包算子	广延、单调、幂等
$C_{\mathbb{N}}^\vDash$	同上	广延、单调、幂等
$\#$	$\mathrm{Form}\to\mathbb{N}$ 的编码	与代入、证明检查相容
$\mathrm{DegProf}^{\rm arith}$	算术度的下闭集	join $\oplus$; 对 jump 封闭

关于第二、三行的幂等性: 二者皆为闭包算子, 故 $C(C(x))=C(x)$. 这一点在与 $L_4$ 的对照中要紧 —— § 13.2 的反思进程给出的不是闭包算子, 而是重新基底. 在每一阶段均一致、有效公理化且足够强, 且所加的反思原则至少蕴含该阶段一致性的标准进程中, 后继阶段严格扩张:

$$T_\alpha\subsetneq T_{\alpha+1}.$$

〔该限定不可省: 不一致的理论, 或已包含相应反思的非有效理论, 皆是反例.〕

幂等与严格上升是两种不同的算子行为, 分属两层.

关于末行: $\mathrm{DegProf}^{\rm arith}$ 对 jump 封闭而无最大元. 它是下闭集而非塔 —— 度 $0$ (可计算集) 在其中, 故它自 $0$ 起铺满全部算术度, 不是自 $0'$ 起跳的序列. 它无最大元, 且在 $\mathcal{D}$ 中无最小上界 (§ 5.2 内层 4).

10.2 表二: 跨分量接口

接口	映射	陈述
$\mathrm{Form}_1\to$ 取值	$\mathrm{ev}_{\rm set}$	$\varphi(n)\mapsto A_\varphi=\{n:\mathbb{N}\models\varphi(n)\}$; 商像为 $\mathrm{ARITH}$
$\mathrm{Sent}\to$ 算术	$\#$	使 $\mathrm{Prf}_T$ 成为算术关系, $\mathrm{Prov}_T$ 成为 $\Sigma^0_1$ 公式
取值 $\leftrightarrow$ 度	Post	$A\in\mathrm{ARITH}\iff\exists n\ A\le_T 0^{(n)}$ —— 一条不涉语法的刻画
$C_{\mathbb{N}}^\vDash\leftrightarrow$ 度	Post 分层	$[\mathrm{Th}_{\Sigma_n}(\mathbb{N})]_T=0^{(n)}$ ($n\ge 1$)
$C_T^\vdash\leftrightarrow C_{\mathbb{N}}^\vDash$	(无统一映射)	见下

末行是本表的实质内容, 须逐条写清:

• 一致区: 对 $T\supseteq Q$, $\mathrm{Th}_{\Sigma_1}(\mathbb{N})\subseteq\mathrm{Th}(T)$ —— 二者在 $\Sigma_1$ 层面一致 (§ 9.5 桥一);
• 单向包含须条件: 对 sound 的 $T$, $C_T^\vdash\subseteq C_{\mathbb{N}}^\vDash$; 无 soundness 则不成立;
• 不相等: 由定理 A, 对任何有效公理化且算术可靠的 $T$, $C_T^\vdash\ne C_{\mathbb{N}}^\vDash$;
• 两个不同的对象须分清: 对 sound 的 $T$, 空前提处的集合差是 $C_{\mathbb{N}}^\vDash(\varnothing)\setminus C_T^\vdash(\varnothing)=U_T$; 而 $\mathrm{Ind}_T$ 不是该集合差 —— 它含每对独立句的两侧. $U_T$ 是 sound 条件下的语义缺口, $\mathrm{Ind}_T$ 是不依赖 soundness、纯句法地见证 $T$ 不完备的对象. 本文取后者;
• 无统一内部映射: 由 Tarski, 不存在原算术语言内的单一算术公式统一实现标准真理的求值. Tarski 不禁止外部映射、元理论比较或扩展语言中的真谓词.

该行与 $L_2$ 的对应位置同格. Paper 4 § 2.3 立: $R_{\rm KZ}\subseteq\ker(\mathrm{ev})$, 而二者是否相等即 Kontsevich–Zagier 猜想. 两层的对应关系是:

	可导出的	真的	二者关系
$L_2$	$R_{\rm KZ}$	$\ker(\mathrm{ev})$	开放 (KZ 猜想)
$L_3$	$C_T^\vdash$	$C_{\mathbb{N}}^\vDash$	可证不等 (定理 A)

这不是本文强加的读法: 两处都是"某个推导系统"与"某个真关系"之间的比较, 而两处的比较对象在各自的层内由 § 2.3 的方图结构自身给出.

〔须防的一处过度: 两处关系不同 —— 一处开放, 一处可证不等. 本文不据此立任何跨层规律 (§ 12.5 判据 3: 阈值机制须不以目标层材料为定义).〕

10.3 一处发现: 现有闭合动作的元数

§ 9.3 所取的判据要求闭合行为 $B:F_1\times F_2\to Z$ 不通过任何一个投影单独因子化. 本节的组织工作使一件事变得可见:

其一, 一条不可用的论证.

一个自然的想法是: $L_3$ 的候选行为 (对角引理、jump、通用求值) 皆是一元的, 故无从承担 $B$. 该论证不可用, 两条独立理由:

1. 元数不是表示不变的性质. 任意二元可计算函数可经配对函数改写为一元, 任意一元函数亦可改写为忽略第二输入的二元函数. 通用求值本来就写作 $U(e,x)$; 本文自己的 join $A\oplus B$ 亦是二元的. 故"表面元数"不能决定两个失败方向在闭合中是否非冗余.
2. 二元不动点构造是经典的. Smullyan 的强双递归定理给出递归函数 $t_1(y_1,y_2)$、$t_2(y_1,y_2)$.

故不得说"无二元构造可用". 但亦不得说它们满足 § 9.3 的判据 —— 这两条定理交付的是同时性 (simultaneous fixed points), 不是本质依赖: 若所给的 r.e. 关系 $M_1$ 根本忽略 $y_2$, 则所得解完全可以经第一投影因子化. 由"存在同时不动点"推不出"两个输入皆不可删除".

障碍须重新指认 (见其三).

其二, 现有的归零式未把第二方向展示为不可删除的输入.

$$\deg^{0'}(0')=0.$$

其中 $0'=\deg(K)=\deg(\mathrm{Th}(T))=\deg(\mathrm{Ind}_T)=\deg(\Omega_U)$; 而余项 2 的见证 $\mathrm{Th}(\mathbb{N})$ 的度是 $0^{(\omega)}$, 不出现在该式中.

须防一处过强的读法: 不可由此说"$0'$ 的来源全部是语法侧" —— 度商遗忘来源, 同一个 $0'$ 由 $K$、$\Omega_U$、$\mathrm{Th}_{\Sigma_1}(\mathbb{N})$、$\mathrm{Th}(T)$、$\mathrm{Ind}_T$ 等异质对象共同实现, 从度本身读不出"语法侧". 可成立的只是: 完整余项 2 的见证不在该式中.

其三, 障碍的指认: 两步皆未完成.

第一步 · 输入尚不是良定义的一对. 余项 2 的内容恰是"原算术语言中不存在完整的 $\mathrm{True}(x)$ 公式". 故不能一边如此定义余项 2, 一边未经语言扩张就把 $\neg\mathrm{True}(x)$ 当作强双递归定理的一个普通公式输入. 若改在扩展的真语言中工作, 须重新指定: 新语言 · 真谓词作用于哪一层语言 · 所用理论及其组合公理 · 不动点定理在该设定下的算术／模态解释. 本文未完成这一步.

第二步 · 即便输入良定义, 同时性亦不足. 由其一之 2, 须另证所得对象对两项输入皆不可删除, 而现成定理不交付这一点.

故本文既不主张"障碍在输入槽的类型", 亦不主张"代入即可". 二者皆超出本文所能证.

其四, 本节所得结论的确切强度.

由其二, 本节唯一可成立者是:

> 现有方程 $\deg^{0'}(0')=0$ 没有把完整余项 2 展示为不可删除的输入, 故它不能证明现有闭合具有 essential arity 2.

这是一项未获证明的判定, 不是"已证明经 $\pi_1$ 因子化"的定理 —— 后者须先有一个已定义的 $B$, 而本文并未定义 $B$.

两条出路, 本文不选:

• 找一个二元的闭合动作, 使两个方向皆不可删除;
• 或修订"两个独立失败方向"这一要求, 使之允许一元闭合加一项未被闭合的余项.

后一条会影响 ZFCρ Paper II § 7.1 对 $L_0$ 的处理 (该处正是以"只有一个余项故不能闭合"作论证), 故不可轻改.

登记: 该问题自 § 9.7 移至此处, 陈述更精确, 但仍未解. 见 § 15.2 的修订.

10.4 与 $L_2$ 的两处功能差异

处	$L_2$ (Paper 4 § 9)	$L_3$ (本节)
六／五分量的类型	群、泛函、配对、表示、算子族、标量不变量 —— 类型不齐, 故 § 9.1 须为 $G_\infty$ 单立一段	码集、两个闭包算子、编码映射、度剖面 —— 类型较齐, 唯 $\mathrm{DegProf}$ 是商结构
未交付者	六类数据的统一范畴封装; 障碍是链 1、2 与链 3、4 的语言翻译	以两项失败为输入的二元归零式; 两步皆未完成 (见 § 10.3)

$L_3$ 处所欠的两步: 先在明确的语言与理论设置中使两个输入良定义, 再证明所得构造对两项输入具有本质依赖. 现有二元工具能否完成这两步, 本文未判定.

两处的障碍不宜说成同型: $L_2$ 处工具 (RH 形式化) 存在而映射未建, 是一个对接问题; $L_3$ 处则连第二个输入是否可在某个明确设置中良定义都未处理. 本文不主张二者类型相近.

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11. 跨论文与跨系列关系

11.0 本节的两重身份

本节除登记本文与上游的承接关系外, 还承担一项勘误功能: 本文在阐述 $L_3$ 的过程中, 发现上游三篇论文共十一处须改.

发布顺序按 Han 裁定: 勘误先在本文给出; 上游各篇随后各自发布新版本, 不覆盖原版本.

由此产生的引用纪律: 本文引用上游时引其已发表版本, 并在引用处就地注明本节的对应勘误条目. 读者不应默认上游已随本文更新.

勘误的分档: 下列各条按 Paper 4 § 12.1 的三档标注 —— 改错 (原陈述在数学上不成立)、澄清 (原陈述可读通但措辞允许误读)、实质修订 (原结论的角色改变).

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11.1 与 Paper 3 ($L_1$) 的关系

Paper 3 提供本文所用的三项格式: 四主步骤与四内层的分工、主步骤 4 作为余项账本结构而非层内 sub-fractal 的登记 (§ 6.1)、以及余项的三分类 (§ 6).

一处须补的细节, 不构成勘误: 本文 § 6.2 的三档表在 $L_1$ 一行取"极小多项式 + 隔离区间"作语法对象. 极小多项式单独不足以唯一确定一个实代数数 ($x^2-2$ 有两个实根). Paper 3 的相关处未展开该细节, 因其阐述不依赖典范表示的唯一性; 本文因须比较三层的求值性状而须补足.

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11.2 与 Paper 4 ($L_2$) 的关系

本文继承 Paper 4 的四条防火墙 (§ 2)、承载体迁移 (§ 6.1)、余项三分类 (§ 8.2)、主步骤 4 的组织原则 (§ 6.0), 以及 claim 三档与预测纪律 (§ 12).

本文所填的是 Paper 4 § 8.4 所推的位置: 该节把 $L_2$ 的规则层两问推给 $L_3$ 论文, 本文 § 3.2 内层 1 接收之.

勘误 6 · Paper 4 § 13.2 (迁移预测)

原文: 给出迁移预测 (对象域扩张耗尽后, 压力转入表示与规则空间), 并给出证伪条件: 若 $L_3$ 的主要闭合动作能仅通过向对象论语言加入新对象种类完成而无需改变规则, 则被证伪. 表中登记保守的类理论扩张 (NBG) 为"能触发"的实例.

三项须撤回:

(a) 「NBG 能触发证伪」的登记. 〔改错〕

设 $S$ 保守扩张 $T$ 于旧语言 $L$, $G$ 为 $L$-句且 $T\nvdash G$, $T\nvdash\neg G$. 若 $S\vdash G$, 由保守性得 $T\vdash G$, 矛盾; 对 $\neg G$ 同理. 故保守对象扩张不可能裁决旧语言中原本不可判定的句子.

(b) 「证伪路径只有保守对象扩张一条」的撤回, 并升级. 〔改错〕

由 (a), 保守那一支为空. 且由推论 7.2, 该空可扩至一切有效扩张, 不分保守与否:

扩张类型	为何不触发
保守 (NBG 型)	裁决不了任何旧语言独立句 (由 (a))
非保守而仍有效 (大基数、强迫公理)	能裁决一些, 但永远达不到全闭合 (推论 7.2)

故一旦按 (c) 把闭合目标精确化为 $C_T^\vdash=C_{\mathbb{N}}^\vDash$, 该证伪条件对一切有效公理化扩张为空. 这使勘误 7 新补的那条纪律 ("证伪条件事后被证明为结构上不可满足") 中的"结构上"三字有定理支撑, 而不只是一条组合观察.

但仍不可写成"原条件在所有读法下皆为空" —— 已证的是裁决性读法下为空; 表达性读法另论.

(c) 原证伪条件的地位. 〔实质修订〕

原条件未定义闭合目标. 它精细描述了扩张机制, 却未给出"$L_3$ 的主要闭合动作"指什么. 故 NBG 那一行只证明了"$S/T$ 保守并增加类语言", 未证明"$S$ 完成 $L_3$ 的主要闭合".

本文的处理: 把闭合目标精确化为 $C_T^\vdash=C_{\mathbb{N}}^\vDash$ 之后, 所得的不是一条新的证伪条件, 而是 § 7.1 的定理 A. 该定理不证明原对象—规则迁移预测 —— 定理 A 划的轴是"有效／非有效", 迁移预测划的轴是"对象／规则", 两根不同的轴. 事实上 $\mathrm{Th}(\mathbb{N})$ 达成全闭合而它并非有效规则系统, 也不必描述为"规则迁移"—— 它只是把完整语义真理直接放入公理集.

(d) 迁移句降档. 〔实质修订〕

原迁移句降为 T3 解释性提案, 不再声称已有操作化的证伪条件.

须说清发生了什么: 不可写"原预测后来被证明为定理"—— 那会掩盖对象的变化. 准确的登记是三件事的合取:

$$\text{撤回旧预测的操作化}\ +\ \text{新增一条定理}\ +\ \text{原迁移解释降档}.$$

另一处词义须拆开 〔澄清〕: 原条件中的"规则"至少含五轴 —— R0 逻辑演算 · R1 语言／签名 · R2 非逻辑公理 · R3 旧语言后果 · R4 算法结构. 原文措辞读起来像 R0, 而表中判据实取 R3. 元语言表达力属 R1, 非 R2. 后续任何相关条件须写成条件向量, 不得再用裸词"规则".

勘误 7 · Paper 4 § 12.5 (预测纪律)

该节三处直接依赖"§ 13.2 是一条带操作化证伪条件的预测":

处	原文所述	勘误后
登记方式	§ 13.2 按"给出预测、给出操作化的证伪条件、指出已知反例候选"登记	前提失效
反例实例	真正能触发证伪者是对旧语言后果保守的对象扩张, 实例为保守的类理论扩张	即勘误 6(a) 所撤回者
可检验性承担者	§ 13.2 的可检验性主要由其操作化的证伪条件承担	承担者失效

〔改错 + 实质修订〕

另须补一条案例 〔增补〕: § 12.5 写"本系列出预测; 预测可能错; 被证伪时认". 但本次发生的不是证伪 —— 预测未被反例推翻, 而是其证伪条件被发现为空. 该情形原节未规定. 拟补:

> 若某预测的证伪条件事后被证明为结构上不可满足, 则该条件失效, 该预测须降档为解释性提案或重建; 不得以"尚未被证伪"为由继续持有该预测.

理由: 一条只规定"错了要认"的纪律, 遇到"条件根本不可能触发"时是无声的 —— 而后者是更隐蔽的免疫形式. 故此补充是对该节的加强, 非削弱.

又: § 12.5 称"大基数扩展与强迫扩展仍是合法的压力测试对象, 但在当前证伪条件下不构成证伪路径"—— 证伪条件撤回后此句须重写.

公开在先凭据的处理: § 12.5 称原版本"承载 § 13.2 的公开优先性". 降档后须分开陈述 —— 观察在先 (压力似乎转入规则空间, 该观察确实先于本文公开), 预测主张撤回 (其操作化不成立). 二者不分开会成为"保留优先权而撤回内容".

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11.3 与 Paper 1 (架构篇) 的关系

Paper 1 § 6.1 给出 $L_3$ 的并行展示四步表, 是本文所抬举者. Paper 1 § 4.5 派各层的交换律阐述单位.

勘误 1 · Paper 1 § 6.1 的 $L_3$-2 边界标记 〔实质修订〕

原文将 $L_3$-2 的边界标记记为"不可证的真".

问题: 该措辞指的是 $U_T=\mathrm{Th}(\mathbb{N})\setminus\mathrm{Th}(T)$, 其定义须用标准模型, 在 $T$ sound 且 c.e. 时 Turing 度为 $0^{(\omega)}$. 而 $L_3$ 加法路径的余项是纯语法的 $\mathrm{Ind}_T$ (度 $0'$), 其定义全程不用模型.

改为: $L_3$-2 的边界标记是 $T$-独立语句 ($\mathrm{Ind}_T$). 若确须使用"真而不可证"一语, 须指明所指为 $U_T$ 并写明 $T$ sound.

勘误 2 · Paper 1 § 6.1 的 $L_3$-3／$L_3$-4 格位 〔实质修订〕

原文把相对 Turing 度、度空间结构及 $\deg^A(\cdot)$ 一类细粒度公式派在 $L_3$-3.

本文的切分:

留在 $L_3$-3	进入 $L_3$-4
可约性预序 $\le_T$	对角自询问
等价关系 $\equiv_T$	jump $A\mapsto A'$ 作为归零机制
度商与 join	以 $A'$ 为神谕的相对化
算术度剖面	归零方程 $\deg^{0'}(0')=0$

即: jump 在主步骤 3 内层 4 作为余项产出, 在主步骤 4 被接收为归零机制. $0'$ 具双重身份 —— 主步骤 3 的边界标记与主步骤 4 的归零点.

注: 交换律阐述单位 $\#$ 不因此移动.

勘误 3–5 · 三处 pre-v2 的分派

Paper 1 发表于 SAE Foundation v2 之前, 故其 $L_4$ 相关分派对着 v1 的混合框架. Paper 1 无过, v2 亦无过 —— v2 § 1.4 已透明声明其重定范围; 缺的是该重定范围对已发表数学系列论文的下游登记. 本文补之.

#	处	Paper 1 原派	v2 的重定范围	档
3	§ 4.5	$L_4$ 的阐述单位是 $c^2$	$c$ 是 $L_2\leftrightarrow L_3$ 的空间化转换签名; 后续层内出现是已建立时空度规的延伸使用, 非每层重新签名	澄清
4	§ 6.2	$L_4$-1 标 $ct$ 与 $G$ 作把手 (即以 $G$ 为余项)	$G_N$ 不是余项, 是响应签名	澄清
5	§ 6 整体	单条阶梯 $L_1$–$L_5$, 在 $L_4$ 处换领域	两条独立阶梯, 各自 $L_0$–$L_5$, 结构同型而内容独立	实质修订

第 5 条承重: 单阶梯读法下可以说"阶梯并未停止, 只是变成了物理", 这会削弱本文 § 14 的暂收声明; 两阶梯读法下, 物理量阶梯的 $L_3\to L_4$ 不是数学阶梯 $L_3$ 的续格. 本文显式采 v2 的两阶梯读法, 并把 Paper 1 § 6 的单阶梯叙述登记为 pre-v2.

连带: § 4.5 的交换律单位由五项重新分配为数学侧三项 ($\log$、$2\pi i$、$\#$), 检验强度随之下调 —— 详本文 § 14.3.

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11.4 与 ZFCρ Paper II 的关系

该文 § 6 给出 $L_2\to L_3$ 的跃迁结构 (两余项 + 一个行为 → 闭合), 是本文所继承者. 本文 § 8、§ 9 是对该结构的填充与修订.

勘误 8 · ZFCρ Paper II § 6.1 的余项 2 〔改错／实质替换〕

原文记余项 2 为 Tarski (表达的边界), 余项为"$\Sigma_1$ 真谓词类".

须消歧: "$\Sigma_1$ 真谓词类"若读作分层真谓词族无统一成员, 即是 Tarski 定理本身, 该读法正确. 若读作真 $\Sigma_1$ 句的集合 $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$, 则不成立 —— 对每个 $n\ge 1$ 存在复杂度为 $\Sigma_n$ 的部分真谓词 $\mathrm{Tr}_{\Sigma_n}$, PA 逐公式证其 T-biconditional; 故 $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ 是可定义且可有效枚举的, 不能承担不可定义性.

取前一读法, 但须承认这是替换而非消歧: 原文明确写 $\Sigma_1$, 把它解释为全层分层真谓词族过于善意. 本文的处理是以正确对象替换原对象, 故分档为改错／实质替换, 不为消歧.

精确陈述: 不存在原语言内的单一完整真谓词统一实现所有层级. (外部元理论中存在有效过程, 给定 $n$ 产生第 $n$ 层部分真谓词的代码; 不存在的是原语言内的全层统一器.)

勘误 9 · ZFCρ Paper II § 6.2 的坍缩解释 〔实质修订〕

等式一字不改:

$$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}(\mathbb{N})]_T=0'.$$

但其解释须改. 原解释为"两余项与行为三者同时坍缩". 该解释不成立: 完整算术真理的度是 $0^{(\omega)}$, 进不了该单点; 而 $\mathrm{Th}_{\Sigma_1}(\mathbb{N})$ 是被成功捕获的片段, 不是余项成员.

改为: 通用求值行为的两个产物 ($K$ 与 $\Omega_U$) 与真理分层的第一个非平凡片段在度商中重合.

故本条虽等式未动, 仍属实质修订 —— 结论的角色变了, 不可登记为消歧.

勘误 10 · ZFCρ Paper II § 6.3 对照表的注 〔增补〕

该表列 $L_2\to L_3$ 的坍缩点为 $0'$、闭合方程为 $\deg^{0'}(0')=0$、归零性质为不对称. 三项保留.

须补两条注:

1. $0'$ 是坍缩点 (行为产物的度), 不是度剖面的边界. 本文 § 5.2 内层 4 给出的度剖面是算术度理想 $\mathrm{AR}$, 它对 jump 封闭, 故 $0'$ 只是其中一个度. 且 $\mathrm{AR}$ 在 $\mathcal{D}$ 中无最小上界 (Spector exact pair), 故不可说"剖面的上确界是 $0^{(\omega)}$" —— $0^{(\omega)}$ 是有限 jump 塔的最小一致上界, 不是 $\mathrm{AR}$ 的上确界;
2. 该归零式未把完整余项 2 展示为不可删除的输入, 故不能证明"两独立余项闭合" (本文 § 10.3).

勘误 11 · ZFCρ Paper II § 3.4 的「唯一来源」 〔改错〕

原文称"指数映射是 $i$ 与 $\pi$ 关系的唯一来源".

作为全数学命题过强. 详见本文 § 9.9 所给的两座反例桥 (旋转几何 · 围道积分), 及 Paper 4 自身对 $2\pi i$ 双重身份的登记.

改为: 指数映射是 $i$ 与 $\pi$ 关系在本 trajectory 中的规范绑定来源. 可保留的较弱数学陈述是: $\pi$ 在 $\mathbb{Q}(i)$ 上超越 (详 § 9.9; 初稿所写"不存在非零 $P$ 使 $P(i,\pi)=0$"有初等反例 $P=X^2+1$).

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11.5 与其他系列的关系

与 SAE Foundation v2

本文采 v2 § 2.1 的两阶梯读法 (详 § 14.2). v2 已把数学 $L_3\to L_4$ 一行登记为"待未来阐述"; 本文所填的是"尚未建立", 强于"待阐述"而弱于"不可建立".

v2 本身不须修订. 其 § 2.1 的数学列只列行为与签名, 未列闭合方程与两余项, 故上述四条 ZFCρ II 勘误不落在它身上. 建议 v3 时把数学列改为引本文.

与形与流 Paper 1

该文 § 3.6 把 Gödel、Tarski、Turing 三者作为"余项持存现象族"一并推给 SAE-Mathematics, 并把统一阐述列为未来方向.

本文的部分回应: 三者在本文中的位置分别是 ——

	位置
Gödel	余项 1 (证明闭包的不完备)
Tarski	余项 2 (完整真理不能在原语言内统一定义)
Turing	行为的产物 (通用求值产生停机集 $K$)

即: 两项余项加一个行为的产物, 而非三项余项.

该回应是部分的, 两处限制: 其一, 本文不主张三者"在 $0'$ 坍缩"(见勘误 9); 其二, 三者何以构成一个"现象族"这一问题, 本文只给出它们在 $L_3$ 阐述中的分位, 未给出跨领域的统一机制. 该债仍部分未清.

与 ZFCρ 线

Paper 4 § 13.1 把 $\omega_\rho^{(2)}$ 的完整合成律推给"$L_2\to L_3$ 接口论文". 本文即该位置, 但未能偿付该债. 其现状与障碍见 § 15.

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12. claim 三档、辖区与纪律

12.1 三档

沿 Paper 4 § 12.1:

档	内容
T1	数学事实. 有出处或有证明; 本文引用而不重证者须标明出处
T2	对既有结果的重新框定. 数学内容不新, 组织方式新
T3	SAE 内部的识别提案. 无外部背书, 不得用作数学论证的前提

本文的 T1 交付是 § 7.1 的定理 A 与 § 7.2 的定理 B; 二者的前提在 § 7.0 显式冻结.

一处须严守的分界: 必要条件的识别与实例的证明是两件事. § 9 识别了两项余项独立性所需的判据 (closure-essential independence) 并给出反事实分离作证据, 但未证明现有闭合结构对两余项具有 essential arity 2 (§ 9.7). 前者不得写成后者.

12.2 前提的显式冻结

本文所研究的形式系统具有: 递归语言 · c.e. 公理集或 c.e. 定理集 · 有效证明检查关系.

"对象有有限描述"、"表达式类可数"与"理论有效公理化"是不同强度的性质. Paper 4 使用有限表达式与可枚举规则, 不自动给出本前提; 故此处显式冻结, 不由上游默推.

12.3 一词一物

本文对以下词的所指作了固定. 凡与此不符的读法, 皆非本文所主张.

词	只指	绝不指
Gödel 余项	事实: 存在 $T$-独立语句. 见证 $\mathrm{Ind}_T$	$U_T$; 某个具体的 $G$
Tarski 余项	事实: 无单一完整的内部真谓词. 见证 $\mathrm{Th}(\mathbb{N})$	$\mathrm{Th}_{\Sigma_1}(\mathbb{N})$; $\mathrm{Tr}_{\Sigma_n}$
$\mathrm{Th}(T)$	$T$ 的演绎闭包	$\mathrm{Th}(\mathbb{N})$
行为	对角化／通用求值	其产物 $K$、$\Omega_U$
坍缩	$[\Omega_U]_T=[K]_T=[\mathrm{Th}_{\Sigma_1}]_T=0'$	任何含余项 1 的陈述
反思	须指明 $\mathrm{Con}(T)$／$\mathrm{Rfn}(T)$／$\mathrm{RFN}_\Gamma(T)$ 之一	三者互称

四条禁用的说法:

1. 「$\mathrm{Th}(T)$ 的补集即不可证的真」—— 补集含假而不可证者;
2. 「Tarski 余项的第一级是 $\mathrm{Th}_{\Sigma_1}$」—— 后者是被成功捕获的片段;
3. 「两余项与行为同时坍缩」—— $0'$ 等式只覆盖 $K$、$\Omega_U$、$\mathrm{Th}_{\Sigma_1}$;
4. 「Tarski 不可加入」—— 真谓词可作新的原始谓词加入扩展语言 ($\mathsf{CT}^-$ 甚至对 PA 保守); 不可的是在原语言内定义完整真理.

三处须带条件使用:

1. 「真而不可证」可用, 但只指 $U_T$, 且每次使用须写明 $T$ sound;
2. $\vdash$／$\vDash$ 在已固定普通一阶后承关系的定理陈述中可无下标; 本文比较 $T$-可证与标准模型真理处必须写 $\vdash_T$ 与 $\vDash_{\mathbb{N}}$;
3. $\Omega_U$ 须先固定一个 optimal universal prefix-free machine $U$; 不得写「规范的 $\Omega$ 数值」或「与机器无关的 $\Omega_U$」.

12.4 三栏与可翻译性

三栏: 「公式的复杂度」「所定义集合的度」「集合之间的归约关系」须分开写. 否则 $\Sigma_n$、$0^{(n)}$ 与 Turing 度会互相换义.

可翻译性: 凡拟用来支撑数学结论的主张, 须先翻成不含任何 SAE 术语的纯数学陈述. 翻不动者只有两种情形 —— 它是 SAE 层面的识别 (则标 T3 且不得用作数学论证的前提), 或它是混乱的.

12.5 跨层规律的五条判据

本文不新立跨层规律. 凡拟立者须同时满足:

1. 类型预注册 —— 在看各层材料之前先定 $A_L$、$B_L$ 各是什么类型 (数／表达式／关系／算法／理论／映射);
2. 合一／分离关系预注册 —— "重合"须预先规定是对象相等、同构、同一对象两角色、一个映到另一个, 还是仅由同一例子见证; 五种强度不同;
3. 阈值机制 —— 须有一个不以目标层材料为定义的机制, 说明为何在第 $k$ 层发生变化. 「这里看起来正好分开」不算机制;
4. 负面控制 —— 立规律时同时写下什么现象出现即判其失败, 而不是被重新描述成另一种分离;
5. 样本外测试 —— 至少一个未参与构造该规律的层或独立数学例子.

本文因此如实登记而不上升为规律者三处: 三层 sub-mark 的自带／另设差别 (§ 4.2 内层 1); 两个跃迁签名与各自层的关系不同型 (§ 6.3); $0^{(\omega)}\equiv_T\mathrm{Th}(\mathbb{N})$ (§ 5.2 内层 4 与 § 9 的接口).

12.6 双重定位

同一对象可同时有来源坐标 (它回答哪一个边界问题) 与展开坐标 (它在哪一层获得完整阐述). 二者并存不是消歧失败; 本文不为求索引统一而单向挪移.

例: Gödel 与 Tarski 是 $L_2\to L_3$ 边界的触发对象, 其完整理论内容在 $L_3$ 展开.

12.7 捕获核心的资格判据

一个候选集合有资格称为捕获核心, 须满足 § 6.5 的四条 (评价先于编码 · 语义商的像 · 对合法重编码协变 · 不靠基数论证).

该判据排除一类无效论证: 可编码成实数不等于自然承载体是实数子集. 若仅凭可编码性即认定核心, 则任何可数数学结构都能被强行装回"实数核心", 承载体迁移将变得不可检验.

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13. $L_4$: 开放问题

13.1 本节的范围

本节只做三件事:

1. 陈述反思与跳跃可无限迭代;
2. 陈述递归序数记号的总体不可有效判定;
3. 登记: 本文尚未建立数学 $L_4$.

本节不做以下四件, 均因材料不足:

• 不主张 $L_4$ 的"外余项"为空;
• 不主张不存在包含整条反思进程的总体;
• 不主张数学阶梯已被证明终止;
• 不解释上述事实为何构成不闭合的理由 —— 相关读法涉及本系列之外的辖区, 不在本文范围.

13.2 $L_4$ 前段的形状

若沿 $L_3$ 的余项继续, 所标的是超限指标: $L_3$ 的度剖面无最大元, 而本文所考察的延续路线是沿 $0^{(\alpha)}$ 迭代 (该路线是选定的, 非唯一, 见 § 5.2).

该指标的公众绑定是序数记号系统 —— $\varepsilon_0$、$\Gamma_0$、$\psi(\Omega_\omega)$, 以及 Kleene 的 $\mathcal{O}$. 记号系统存在且好用: 沿一条记号路径, 进程

$$T_0=T,\qquad T_{\alpha+1}=T_\alpha+R(T_\alpha),\qquad T_\lambda=\bigcup_{\alpha<\lambda}T_\alpha$$

良定义, 其中 $R$ 取 $\mathrm{Con}$、$\mathrm{Rfn}$ 或 $\mathrm{RFN}_\Gamma$ 之一 (三者的区别见 § 8.4).

标而不构在此处的位置: 大家都说"第 $\alpha$ 次迭代", 而从系统内部构不出一条典范的 $\alpha$-指标.

13.3 三条 T1 事实

(i) Kleene 的 $\mathcal{O}$ 是 $\Pi^1_1$-完全的; 合法记号无有效枚举.

须同时登记: 不可判定不等于不确定. 固定一种标准递归序数记号编码后, $\mathcal{O}$ 是一个外延确定的自然数集; $\omega_1^{\rm CK}$ 是一个确定的可数序数; 超算术集类亦确定. 本文不由 (i) 推出任何关于"总体不存在"的陈述.

(ii) 理论后果不只依赖序型, 还依赖具体的可计算记号 (进程的 intensional character). Turing 1939 (以局部反思进程现代重述): 对每个真 $\Pi_1$ 句 $\varphi$, 存在一个 $\omega+1$ 的可计算记号使该进程证出 $\varphi$; 但不存在一个固定记号使同一理论证出全部真 $\Pi_1$ 句.

量词序是 $\forall\varphi\,\exists\dot\alpha$, 不是 $\exists\dot\alpha\,\forall\varphi$.

(iii) 达到算术完备性的路径存在 —— Feferman 1962 以完整一致反思构造之; 但这样的路径不能是 $\Sigma^1_1$-可定义的. 且沿满足 Feferman–Spector 自然条件的 $\Pi^1_1$ 路径, 必有真 $\Pi^0_1$ 句不可导出.

故"该完备性是空的"这一说法过强: 完备路径存在, 只是不能由低复杂度定义统一抓住.

13.4 本文未能给出的

由 (i)(ii)(iii) 可以陈述: 路径选取函数在算术内不可定义; 完备路径存在而不可低复杂度指定.

但由此推不出 $L_4$ 不可闭合. 理由有二:

其一, 各层皆依赖选择. $L_1$ 依赖系数基底 $\mathbb{Q}$, $L_2$ 依赖表达式类 $\mathsf{Int}_{\rm alg}$, $L_3$ 依赖理论 $T$、Gödel 编码与通用机 $U$. 若对 $L_4$ 要求"不依赖选路、不依赖记号系统", 那是一条比低层更严的新标准. 提出一条只有待判对象过不了的标准, 是 Paper 4 § 12.4 所禁者.

合理的标准只能是: 在选定的 trajectory 内, 是否具有在适当等价意义下的典范性.

其二, 已知一个未被排除的候选. "哪些递归关系是真正的良序"这一问题由当前的良序与反思讨论自然提出, 而它指向 $\mathcal{O}$ 或等价的超跳对象 —— 该对象确定、不在超算术层级内、由当前问题指名. 本文未能说明它为何不合"外余项"的标准, 故不得登记外列为空.

13.5 数学 $L_4$ 的当前候选检查表

本表全部标 T3: 它不是外部数学定理意义上的必要／充分条件; 下述"最低交付要求"是本文当前的工作定义与审稿门槛, 非对未来候选的数学约束. 一条塔终止条件本身不会自动交付捕获核心、两项余项与四主步骤. 本表的用途是使该开放问题可被接手, 而非为未来的候选预设门槛.

最低交付要求 (缺一则该候选尚不成其为一个层):

1. 一个清楚的对象／装置类型;
2. 一个非廉价编码的评价或普遍性质 (§ 6.5 的判据可作参考, 不作要求);
3. 与 $L_3$ 的明确接口;
4. 一条判据, 说明它不是 $L_3$ 内部的迭代.

当前候选的压力测试 (皆 T3, 不要求未来候选同时满足):

• 路径的典范性 —— 须与 $L_1$–$L_3$ 的参数依赖同强度, 不要求绝对无参数;
• 新的捕获像;
• 多个失败方向;
• 某种闭合或重新基底机制 —— 不预先要求"塔终止"; 重新归零、极限闭包或普遍对象亦可能承担闭合.

故本文不写"第 4 条一旦给出, 数学 $L_4$ 即建立" —— 初稿如此写, 过强.

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14. 本系列的暂收声明

14.1 已建立与未建立者

本系列的数学阶梯至本文为止:

层	捕获核心	层内交换律阐述单位	跃迁闭合
$L_1$	$\mathbb{Q}^{\rm alg}_{\mathbb{R}}$	$\log$	$L_1\to L_2$: $e^{i\pi}+1=0$
$L_2$	$\mathcal{P}$	$2\pi i$	$L_2\to L_3$: 坍缩点与归零式已登记; 两独立余项闭合未证 (§ 8.2、§ 9.7、§ 10.3)
$L_3$	$\mathrm{ARITH}$	$\#$ (待定, 见 § 15.4)	——
$L_4$	未建立	——	——

故本系列的数学层级主轨暂收于 $L_3$. 本文没有识别或建立数学 $L_4$.

须区别两句:

• 可写: 在目前已给出的捕获判据序列中, 尚未识别出可作为数学 $L_4$ 的候选;
• 不可写: 数学止于 $L_3$, 或数学 $L_4$ 不可能存在.

二者的差别是"当前系列止步"与"数学上不存在后继"—— 本文只主张前者.

14.2 与物理量阶梯的关系

按 SAE Foundation v2 § 2.1, 数学阶梯与物理量阶梯是两条独立的阶梯, 结构同型而内容独立:

	数学阶梯	物理量阶梯
$L_0\to L_1$	单余项不闭合	单余项不闭合
$L_1\to L_2$	对偶化: $e^{i\pi}+1=0$	对偶化: $[\hat x,\hat p]=i\hbar$
$L_2\to L_3$	对角化 (签名 $\Omega_U$)	空间化: $E=mc^2$ (签名 $c$)
$L_3\to L_4$	本文: 未建立	因果化: Schwarzschild (响应签名 $G_N$)
$L_4\to L_5$	——	不可逆化: $S=k_B\ln W$ (签名 $k_B$)

该读法对本文的主张有直接后果: 单阶梯读法下, 可以说"阶梯并未停止, 只是变成了物理"; 两阶梯读法下, 物理量阶梯的 $L_3\to L_4$ 不是数学阶梯 $L_3$ 的续格, 而是另一条阶梯上的一格. 数学阶梯暂收于此, 而 v2 § 2.1 已把数学 $L_3\to L_4$ 一行登记为"待未来阐述" —— 本文所填的是"尚未建立", 强于"待阐述"而弱于"不可建立".

14.3 一处检验强度的下调

Paper 1 § 4.5 派各层的交换律阐述单位为 $\log$、$2\pi i$、$\#$、$c^2$、$k_B$ 五项, 并以"五层皆有"作为该跨层级预测通过检验的证据.

两阶梯拆分之后, $c^2$ 与 $k_B$ 归物理量阶梯, 数学侧只剩三项.

重新分配后的自洽性须如实陈述: 数学侧三项中, $\log$ 与 $2\pi i$ 已定, 而 $L_3$ 的 $\#$ 是否兼任交换律阐述单位本文未确定 (§ 15.4). 故只能说"两层已定、第三层待定", 不可说"有单位的层恰为三层" —— 后者把一个待定项算作已定项, 违反本文 § 12.1 立的"必要条件的识别 $\ne$ 实例的证明"分界.

且该相容性不构成新增检验证据: 这三项正是构造该模式所用的三项, 不满足 § 12.5 判据 5 (样本外测试). 准确陈述是: 拆阶梯后数学侧现有项与原模式仍相容; 该相容性不是新的检验.

检验强度因此双重下调: 五层证据变三层 (且第三层待定), 且三层同属一条阶梯, 不再是跨领域的.

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15. 开放问题

15.1 $\omega_\rho^{(2)}$ 的统一范畴封装 —— 未偿, 且推的地址须重议

债的确切内容

Paper 1 § 11.1 把 $\omega_\rho^{(2)}$ 六分量的合成律挂账给后续论文. Paper 4 § 9 作了部分偿付:

已交付 (Paper 4 § 9.1、§ 9.2): 最小 typed composition 骨架 —— 两张表与四条链.

六分量及其类型:

分量	类型
$\mathrm{Div}$	自由阿贝尔群 $\mathrm{Div}(X)$
$\mathrm{Res}$	局部线性泛函
$\mathrm{Per}$	双线性配对 $H_1\times H^1_{\rm dR}\to\mathbb{C}$
$\mathrm{Mon}$	$\pi_1$ 的表示
$\mathrm{Stokes}$	按方向排序的算子族
$G_\infty$	标量渐近不变量 (阶 $\rho$ 与型 $\sigma$)

四条跨分量接口:

链	映射	所在语言
1↔2	$d\log$: $\mathrm{Res}_p(df/f)=\mathrm{ord}_p(f)$	经典复分析
2↔3	小环路限制: $\oint_{\gamma_p}\omega=2\pi i\,\mathrm{Res}_p(\omega)$	经典复分析
3↔4	(wild) Riemann–Hilbert	RH
4 内部	Ramis 稠密性; 形式层半直积 $G^{\rm formal}=\mathcal{T}\rtimes\overline{\langle\hat\gamma\rangle}^{\rm Zar}$	RH

未交付 (Paper 4 § 9.3): 六类数据的统一范畴封装 —— 即把六者组织为单一范畴对象, 使四条链全部成为该范畴内的态射.

障碍的确切位置: 不是范畴论工具的缺失 (RH 对应及其 wild 扩展的完整形式化已存在), 而是六分量尚未被映到 RH 数据上. 具体地: 链 3 与链 4 已进入 RH 语言, 链 1 与链 2 仍在经典复分析语言中, 二者之间的翻译尚未给出.

本文未偿, 且须重议推的地址

Paper 4 § 13.1 把该债推给"$L_2\to L_3$ 接口论文", 理由是: 该封装具元层级性质 (Paper 4 § 8.4), 而按 Paper 1 § 5.5 的触发判词, 这类陈述属规则层面; 又按 Paper 1 § 11.4, 范畴论结构主义的充分接洽须在 $L_2$ 与 $L_3$ 两处处理.

本文即该接口论文, 但未能偿付. 且本文认为该推的理由须重议:

> 上述推理的链条是: 元层级性质 → 规则层面 → $L_3$. 而 $L_3$ 的规则层内容是形式系统、可计算性与可判定性; 该封装所欠的却是除子—留数语言与 Riemann–Hilbert 语言之间的翻译. 二者虽都可称"关于规则的", 但不是同一种规则. 中间那一步把两个不同的"规则层面"并作了一个.

故本文的登记是: 该债未偿; 且其自然去处更可能是 $L_2$ 一侧一篇处理 RH 翻译的专题, 而非 $L_3$ 论文. 本文不自行改派 —— 改派须由掌握该材料者定 —— 但把上述理由登记在此, 以免该债继续按一条已被质疑的理由在层级之间转手.

按 Paper 4 § 12.4: 欠账登记为欠账, 不得转成发现. 本文既不声称已偿, 亦不把"给不出"诊断为一项识别.

什么算偿付

一种统一的 typed 容器, 使六类数据与四条接口在同一形式语言中可组合, 且链 1、链 2 与链 3、链 4 在该语言中同格表述.

不预先规定它须是一个普通范畴且六分量逐一为对象: 配对、泛函、表示、算子族与标量不变量并非天然同一种 object; 强行都放进 Set 又会使要求平凡化. 真正的装置可能是多排序 diagram、纤维化结构、双范畴或其他 typed construction. 初稿写"一个范畴 $\mathcal{C}$", 过于具体.

15.2 二元闭合式的构造

> 所欠: 一个二元的闭合式, 使 Gödel 不完备与 Tarski 不可定义皆为不可删除的输入. > > 障碍 (两步皆未完成): (i) $\neg\mathrm{True}(x)$ 不是原算术语言的公式 (那正是余项 2 的内容), 故未经语言扩张即不是二元构造的良定义输入; (ii) 现成的二元不动点定理交付的是同时性而非本质依赖, 故即便输入良定义, 亦须另证两项皆不可删除. > > 〔两条不可用的说法: "候选行为皆一元" (元数非表示不变); "障碍在输入槽的类型" (超出所能证). 二者的撤回过程见附录 A.1.〕 > > 三条出路: > > 1. 尝试现成的二元构造 —— 强双递归定理提供了可尝试的二元形式, 但不自动满足 § 9.3 (该判据不只要求"有两个输入", 还要求"不通过任一投影因子化", 而同时不动点定理不交付这一点); 且第二输入须先在明确的扩展真语言中成为良定义公式; > 2. 改换算子类型 —— 元层级天然具备二元结构者, 如神谕通用求值泛函 $\Phi:(e,X)\mapsto\Phi_e^X$ (语法侧的程序码与语义侧的神谕各占一槽), 或解释逻辑的二元模态 $A\triangleright B$ (同时清算语法扩展与语义强度); > 3. 修订"两个独立失败方向"这一要求 —— 但它会影响 ZFCρ Paper II § 7.1 对 $L_0$ 的处理, 不可轻改. > > 另须注意: 真正要紧的是信息依赖而非语法元数. 形如 $F_1,F_2\mapsto F_1\oplus F_2\mapsto\Phi(F_1\oplus F_2)$ 的构造, 其最终算子仍可是一元, 而输入由两个方向规范耦合 —— 该形式亦可检验能否相对任一单个输入完成. > > 若始终不能解决: $L_2\to L_3$ 应称两边界耦合归零, 而不称两独立余项闭合 (§ 9.7).

〔本条的陈述几经收窄, 过程见附录 A.1; 现存的唯一主张见上.〕

15.3 第二类余项在 $L_3$ 是什么

Paper 4 § 8.2 的第二类是一条语法边界 —— $\mathsf{Int}_{\rm alg}$ 不允许指数因子.

$L_3$ 的对应须问: 一阶算术公式这一语法有无对应的"不允许"? 一个候选是集合量词 (超算术需要而算术语言不含). 该候选未验, 且可能与第三类重合.

本文 § 8.3 将该格登记为未定, 不填.

15.4 $\#$ 是否兼任交换律阐述单位

Paper 1 § 4.5 派 $L_3$ 的交换律阐述单位为 Gödel 算术化 $\#$.

本文 § 4.2 内层 3 给出 $\#$ 的首次出场 (主步骤 2, 作为证明序列的绑定器), 但未确定它在主步骤 4 是否再次出场并承担交换律单位一职.

该问题依赖 § 8.1 的对角构造如何书写. 本文未答.

一处不得据以推论者: Paper 4 中 $2\pi i$ 首次出场在主步骤 3, 再现于主步骤 4 内层 2; 而 $\#$ 首次出场在主步骤 2. 格位不同, 不得据此类推.

15.5 $L_3$ 有无侧支

Paper 4 § 10 判 Cayley–Dickson 为 $L_2$ 的代价支, 判据是: 该分支可施加而非被迫 —— 不由本层的反向追问逼出, 而是为付掉某项代价 (此处为交换律) 而被主动施加.

$L_3$ 的同位判据应为: 一个分支是代价支, 当它不由 $L_3$ 的反向追问逼出, 而是为付掉 $L_3$ 的某项代价 (如经典逻辑的排中律、一阶语言的表达力限制) 而被施加.

四个候选目前皆不能归类:

候选	未能归类的理由
直觉主义逻辑	它改的是主步骤 1 所标的接口本身 ($\vdash_T$ 的推导规则), 而非在既定接口下另开分支
二阶算术	其子系统的强度谱跨越 $L_3$ 与超算术两侧; 归类须先定它落在本层内还是已越界
非标准模型	它改的是 $\vDash$ 一侧的论域, 而本文的 $\vDash_{\mathbb{N}}$ 已固定为标准模型; 松开该固定是改前提, 不是开侧支
类型论	与经典一阶算术的对应须经翻译, 而该翻译本身的地位未定

本文不作判定, 亦不为凑格式而指定一个. 且不由未考察默认"$L_3$ 无代价支" —— 该结论若成立, 须按上述判据逐一排除四个候选并论证清单穷尽, 而本文未做.

15.6 数学 $L_4$

本文未能建立数学 $L_4$, 且未能证明其不可建立.

该条是一项缺席, 不是判定, 故不列于 § 1.2 的负面交付 —— 它属辖区声明 (§ 14) 与本节.

§ 13.5 给出当前候选检查表 (全部 T3), 以使该问题可被接手.

15.7 形与流 Paper 1 § 3.6 的债 · 部分未清

该文把 Gödel、Tarski、Turing 三者作为"余项持存现象族"一并推给本系列.

本文 § 11.5 给出三者在 $L_3$ 阐述中的分位 (两项余项加一个行为的产物). 但未给出跨领域的统一机制 —— 三者何以构成一个"现象族", 本文只答了它们在本层的位置, 未答该族的一般形式.

该债部分未清.

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附录 A · 版本与审稿来源记录

本文经四方内部审查 (Claude、ChatGPT、Gemini、Grok 各任一席), 全稿历四轮审查 (v1 → v4), 其中 ChatGPT 席位对 v1 判 Major Revision, 对 v2 判未通过, 对 v3 判 conditional sign-off; 其后另作一轮体例整理.

此处所称 review 为作者组织的模型辅助内部结构审查, 不等同于期刊同行评审.

A.1 与 § 12 反免疫纪律直接相关的三项改正

其一 · 一处数学错. 大纲与 v1 曾主张算术度理想 $\mathrm{AR}$ 的上确界是 $0^{(\omega)}$, 并给出证明: 任一上界 $u$ 满足 $u\ge_T 0^{(n)}$ 对一切 $n$, 故 $u\ge_T\bigoplus_n 0^{(n)}$. 该步不成立 —— 逐个归约不给出统一计算直和所需的一致程序; 而由 Spector 的 exact pair 定理, 可数度理想根本没有最小上界. 该错误穿过 § 5.2、§ 7.3、§ 8.2、§ 8.4、§ 10.1、§ 13.2 六处, 经审稿指出后全部改正 (§ 5.2 现作三分陈述: 无最大元 · 有上界 · 无最小上界).

其二 · 一处断言未经验证的完成. v2 的修订记录声称上述六处已随 § 5.2 一并改净, 而实际只改了两处. 一篇立有反免疫声明的论文, 若其修订记录本身可以断言未经核验的完成, 该声明即无从依托. v3 起, 每批修订之后对全文作禁语扫描, 扫描结果随稿提交.

其三 · 两次"撤回一个过度断言而换上另一个". § 10.3 的障碍指认历三稿: 初稿主张"$L_3$ 的候选闭合动作皆是一元的"—— 该主张不成立, 因元数不是表示不变的性质 (配对函数可改元数; 通用求值本作 $U(e,x)$); v2 改称"障碍在于现成二元构造的输入槽是公式槽而非失败方向槽"—— 该主张同样超出所能证; v3 起改为两步皆未完成, 且明写本文既不主张障碍在输入槽的类型, 亦不主张代入即可.

同类的一次发生在相反方向: v3 § 5.2 为收窄 $0^{(\omega)}$ 的地位而写"相对于本文所固定公式枚举的一致上界不存在", 该句为假 —— $0^{(\omega)}$ 恰是该意义下的最小一致上界. 故三轮中的共同失误不是"主张多于材料", 而是未验证即断言, 方向可正可负.

A.2 其余在审稿过程中被撤回或降档的承重决定

均已在正文相应位置登记, 此处列举以备追溯:

• 捕获核心由 $\mathbb{R}_{\rm arith}$ 改为 $\mathrm{ARITH}$, $\mathbb{R}_{\rm arith}$ 降为实轴对应物 (§ 6.1、§ 6.2) —— 因 $\mathrm{ev}_{\rm set}$ 的实数编码像严格小于 $\mathbb{R}_{\rm arith}$;
• "三格皆为实数的子集"撤回 (§ 6.1) —— 周期环不是 $\mathbb{R}$ 的子集;
• 定理 A 的算术可靠性前提去除 (§ 7.1) —— 该前提冗余, 去除后定理更强;
• $\mathrm{ev}_3$ 拆为 $\mathrm{ev}_{\rm set}$ 与编码 $j$ (§ 6.2、§ 6.4) —— 原记号一符两义;
• § 9 的交付降档 —— 由"$L_3$ 满足独立性判据的证据"降为"两种失败方向非同一的反事实证据";
• § 10.3 的负面结果降档 —— 由"现有归零式未构成实例"降为"尚未被证明构成实例";
• § 13.5 由"充分性标准"改为"当前候选检查表", 全部标 T3;
• § 8.3 第一类余项由"为空"改为"尚未识别出", 并改档为 T3 登记;
• $P(i,\pi)$ 的陈述改正 (§ 9.9) —— 原写"不存在非零 $P$ 使 $P(i,\pi)=0$"有初等反例 $P=X^2+1$, 改为"$\pi$ 在 $\mathbb{Q}(i)$ 上超越";
• Paper 1 § 6 的单阶梯叙述登记为 pre-v2 (§ 11.3), 本文改采 Foundation v2 的两阶梯读法.

A.3 本附录的职能限定

本附录承担的是来源可追踪功能, 不承担"论文因此可信"功能. § 12 各条纪律的内部证据是正文自身: 负面结果与正面交付并列登记 (§ 1.2); 两项余项的独立性只主张到"非同一"而不主张实例 (§ 9.8); 定理 A 的前提逐条冻结且不多加一条 (§ 7.0); 十一条上游勘误逐条标注分档 (§ 11). 本附录只补充版本链的可追溯性.

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参考文献

说明: 各条后附一句该文在本文中的用处.

不完备性与不可定义性

1. K. Gödel, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik 37 (1930). — 完备性定理; 本文 § 3.2 据此立记号纪律 (无下标的 $\vdash$／$\vDash$ 之间没有缝隙).
2. K. Gödel, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik 38 (1931). — 第一不完备性定理; 本文 § 4.2 内层 4 与 § 8.1 用之.
3. J. B. Rosser, Extensions of some theorems of Gödel and Church, Journal of Symbolic Logic 1 (1936). — 把独立性所需条件由 $\omega$-一致降至简单一致; 本文取 $\mathrm{Ind}_T$ 而非 $U_T$ 依赖此形式.
4. A. Tarski, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica 1 (1936). 〔波兰文原版 1933〕 — 真不可定义性; 本文余项 2 与定理 A 的语义侧承重.

递归论

1. E. L. Post, Recursively enumerable sets of positive integers and their decision problems, Bulletin of the AMS 50 (1944), 284–316. — creative 集与有效不可分性; 本文 § 4.2 内层 4 用之. 〔该文不含算术层级定理, 该定理见条目 6〕
2. S. C. Kleene, Introduction to Metamathematics, North-Holland (1952), § 57. — $A$ 算术 $\iff\exists n\,A\le_T 0^{(n)}$; 本文 § 6.5 判据 3 与 § 10.2 承重于此. 〔该书把层级与 jump 的联系归于 Post 1948 摘要〕
3. J. Myhill, Creative sets, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 1(2) (1955), 97–108. DOI 10.1002/malq.19550010205. — creative $\iff$ c.e.-many-one 完全; 本文 § 4.2 内层 4 由此得 $\mathrm{Ind}_T$ 的 $\Pi^0_1$-完全性.
4. C. Spector, On degrees of recursive unsolvability, Annals of Mathematics, 2nd ser., 64 (1956), 581–592. DOI 10.2307/1969604. — exact pair 定理; 本文 § 5.2 内层 4 由此证 $\mathrm{AR}$ 无最小上界, 并由此反驳「越出须经 $0^{(\omega)}$」.
5. S. C. Kleene, On notation for ordinal numbers, Journal of Symbolic Logic 3 (1938). — 记号系统 $\mathcal{O}$ 的建立.
6. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw–Hill (1967), § 11.7–11.8. — $\mathcal{O}$ 的 $\Pi^1_1$-完全性与超算术层级; 本文 § 13.3 (i) 承重于此.
7. R. M. Smullyan, Recursion Theory for Metamathematics, Oxford Logic Guides 22, Oxford University Press (1993), 第 IX 章. — 强双递归定理; 本文 § 10.3 用作二元构造的现成例, 但明写其不自动交付本质依赖.
8. W. Craig, On axiomatizability within a system, Journal of Symbolic Logic 18(1) (1953), 30–32. — c.e. 公理化可改为递归公理化; 本文 § 4.2 内层 3 用之, 以使 $\mathrm{Prf}_T$ 对 c.e. 公理化的 $T$ 亦有同等有效形式.

算法信息论

1. G. J. Chaitin, A theory of program size formally identical to information theory, Journal of the ACM 22 (1975). — $\Omega$ 的构造; 本文 § 6.3 用作 $\mathbb{R}_{\rm comp}\subsetneq\mathbb{R}_{\rm arith}$ 的见证.
2. A. Kučera and T. A. Slaman, Randomness and recursive enumerability, SIAM Journal on Computing 31(1) (2001), 199–211. DOI 10.1137/S0097539799357441. — 左 c.e. 的 ML-随机实数皆 Turing 完全; 本文由此定 $\deg(\Omega_U)=0'$.

超限进程与反思原则

1. A. M. Turing, Systems of logic based on ordinals, Proceedings of the London Mathematical Society 45 (1939). — 序数逻辑; 本文 § 8.4、§ 13.3 (ii) 用其局部反思进程的逐句 $\Pi_1$-完备性 (量词序为 $\forall\varphi\,\exists\dot\alpha$).
2. S. Feferman, Transfinite recursive progressions of axiomatic theories, Journal of Symbolic Logic 27 (1962). — 以完整一致反思达到算术完备性; 本文 § 13.3 (iii) 用之.
3. S. Feferman and C. Spector, Incompleteness along paths in progressions of theories, Journal of Symbolic Logic 27 (1962). — 沿 $\Pi^1_1$ 路径的不完备性; 针对一般递归理论进程, 不按反思算子分类 (本文 § 8.4 据此对三种算子一并陈述).
4. G. Kreisel and A. Lévy, Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 14 (1968). — $\mathrm{PA}+\mathrm{RFN}(\mathrm{PA})\equiv\mathrm{PA}+\mathrm{TI}(\varepsilon_0)$.
5. L. D. Beklemishev, Iterated local reflection versus iterated consistency, Annals of Pure and Applied Logic 75(1–2) (1995), 25–48. DOI 10.1016/0168-0072(95)00007-4. — $\alpha$ 次局部反思迭代与 $\omega^\alpha$ 次一致性迭代有相同 $\Pi^0_1$ 后果; 本文据此不主张局部反思接不上序数分析.
6. L. D. Beklemishev, Provability algebras and proof-theoretic ordinals I, Annals of Pure and Applied Logic 128(1–3) (2004), 103–123. DOI 10.1016/j.apal.2003.11.030. — 迭代反思与保守性谱的序数分析; 本文 § 2.4 据此说明反思强度与 Turing 度不对应.

可证性逻辑

1. R. M. Solovay, Provability interpretations of modal logic, Israel Journal of Mathematics 25 (1976). — GL 的算术完备性 (非不动点定理); 本文 § 10.1 提及 Magari 代数时用之.
2. D. de Jongh and G. Sambin 的不动点定理, 见 G. Boolos, The Logic of Provability, Cambridge University Press (1993), 第 8 章. — 标准的单变量 GL 不动点定理; 本文 § 10.1 提及 GL 时用之.
3. R. Magari, The diagonalizable algebras, Bollettino dell'Unione Matematica Italiana, serie 4, 12, Supplemento fasc. 3 (1975), 117–125. — Magari 代数; 本文 § 10.1 提及.

真理论

1. H. Kotlarski, S. Krajewski and A. H. Lachlan, Construction of satisfaction classes for nonstandard models, Canadian Mathematical Bulletin 24(3) (1981), 283–293. DOI 10.4153/CMB-1981-045-3. — $\mathsf{CT}^-$ 对 PA 的保守性是其标准后果 (非该文的直接定理); 本文 § 9.6 的反事实分离与 § 11.2 勘误 6 用之.

周期与超越性

1. M. Kontsevich and D. Zagier, Periods, in Mathematics Unlimited — 2001 and Beyond, Springer (2001), 771–808. — 周期环 $\mathcal{P}$ 的定义与三条变换规则; 本文 § 3.2 内层 1 与 § 6.2 用之.
2. M. Yoshinaga, Periods and elementary real numbers, arXiv:0805.0349 (2008). — 可计算而非周期的实数; 本文 § 6.3 用作 $\mathcal{P}\cap\mathbb{R}\subsetneq\mathbb{R}_{\rm comp}$ 的见证.
3. F. Lindemann, Über die Zahl $\pi$, Mathematische Annalen 20 (1882), 213–225. — $\pi$ 超越; 本文 § 9.9 的唯一外部承重定理 (由此得 $\pi$ 在 $\mathbb{Q}(i)$ 上超越).

本系列

1. 秦汉, SAE 数学系列 Paper 1: 架构篇, DOI 10.5281/zenodo.20153791. — 层 articulation 图式; 其 § 6.1 的 $L_3$ 四步表是本文所抬举者, § 4.5 派各层交换律阐述单位.
2. 秦汉, SAE 数学系列 Paper 2, DOI 10.5281/zenodo.20199082. — $L_0$ 的多维性与三条闭合路径; 本文的完整精确化 trajectory 属其中一条.
3. 秦汉, SAE 数学系列 Paper 3: $L_1$, DOI 10.5281/zenodo.20266351. — 本文所用的四主步骤／四内层格式与主步骤 4 的类别登记出此.
4. 秦汉, SAE 数学系列 Paper 4: $L_2$, DOI 10.5281/zenodo.21466722. — 直接上游; 本文所填的是其 § 8.4 所推的位置.
5. 秦汉, ZFCρ Paper II, DOI 10.5281/zenodo.18927658. — $L_2\to L_3$ 跃迁结构 (两余项 + 一个行为 → 闭合); 本文 § 8、§ 9 对其填充与修订.
6. 秦汉, SAE Foundation v2, DOI 10.5281/zenodo.20322460. — 两阶梯读法; 本文 § 14.2 采之.
7. 秦汉, SAE 形与流 Paper 1, DOI 10.5281/zenodo.20066986. — 其 § 3.6 把 Gödel、Tarski、Turing 一并推给本系列; 本文 § 11.5 作部分回应.