Angular Information in Discrete Black-Hole Interiors: The Two-Way Split of Transfer-Level Information — One Exact Bit of Spin Sign and a Geometry-Bounded Amplitude
离散黑洞内区的角向信息：转移层信息之二分 —— 一比特旋向与几何所限之幅值

1 Introduction, Scope, and the Dependency Matrix

1.1 Position in the series and the core question

The first four papers of this series set up, in turn, the information-flow skeleton of the discrete interior, nine adjudicated boundary stakes, the entropy of the state layer together with the multiplicity of the occupancy quotient, and the process account of angular transport. The third paper supplies a ruler: the counting entropy of the occupancy quotient, measuring the resolution the state layer can carry. The fourth proves that this ruler fails to measure something: the cumulative angular account is not a function of the state (non-exactness), and occupancy together with its quotient is blind to the spin sign in two named instances. The present paper takes up the natural sequel: how large the blind spot is, why it is blind, and which ruler must replace the old one before the account becomes visible.

The core question, in its adjudicated form: within the pure-occupancy class of discrete interior models, which functions of the angular account history can the output of a release protocol reconstruct, which can it not, and on which layer of the declared observation ladder do the reconstructible targets sit.

Throughout the paper, "information" is pinned to its combinatorial sense — counting, resolution, and reconstruction sufficiency; no bridge from combinatorial information to thermodynamic entropy, von Neumann entropy, or physical radiation entropy is asserted anywhere.

1.2 Main results: the two-way split

The split named in the title is settled layer by layer.

On the spin side: exactly one bit, and no guaranteed leak. On an occupancy-identity library the sign involution $\rho$ acts freely on $\{\ell_\alpha\neq0\}$, the signs split exactly evenly, and the conditional entropy of the spin sign is exactly one bit, no more and no less. Every observation constant on nonzero-sign $\rho$-orbits — occupancy and its quotient among them — is hereditarily blind to this bit: blind on every $\rho$-closed sub-library, and hereditary blindness is characterized exactly by constancy on nonzero-sign orbits. The nonzero-sign fraction $\pi_\times$ admits, for an isolated saturated ring, the closed form $2/(L_C+1)\sim2\varphi^{-C}$ (Lucas numbers), decaying exponentially; after attaching a single degenerate core cell, the eight computed cases $C=3,\dots,10$ place the first two below $2/3$ and the remaining six above $2/3$, with the finite sequence increasing term by term; the general threshold, monotonicity, and limit are not established.

On the amplitude side: a guaranteed leak with closed forms, its capacity legislated by geometry. The occupancy history delivers an amplitude outer box tick by tick; on the single-ring library with $v\le1$ vacancies the fibers of the occupancy observation fall into exactly three types, and the guaranteed leaks of the three targets each admit a closed form: signed account $\log_2\frac{2C+1}{3}$, account amplitude $\log_2\frac{C+1}{2}$, spin sign zero — the amplitude admits a guaranteed leak while the spin sign admits none, and the split holds fiber by fiber. The spectrum of the transfer layer is clamped by the lattice bound $B=2\lfloor M/G_{\rm cyc}\rfloor+1$; with a degenerate core the spectrum is the full segment and the entropy deficit vanishes; the attainability of the second value from the top is decided exactly by a degenerate-connectivity criterion, and that of the third value by retrograde parent adjacency.

The order of the observation ladder is legislated by the library. Each link of the four-layer counting-entropy chain has its own strictness condition: gross over net current holds already on saturated libraries; net current over shell accounts requires, in general, a pair of transfers with distinct net edge currents and equal shell accounts (on a single tangential ring the threshold is at least two vacancies; on multi-shell saturated libraries mixed radial cycles supply the pair at zero vacancies); shell accounts over the total account require a nonzero intersection of the shell-account image with $\ker\alpha$. The target $\boldsymbol\ell$ sits exactly on the third layer of any library containing both witness types; on the spin-clamped library the first three links collapse entirely — positioning and collapse are two faces of the same library dependence.

1.3 Seven standing prohibitions

Kept throughout. First, combinatorial transfer entropy is never called thermodynamic or von Neumann entropy; all bridges are deferred to the interface. Second, no physical claim of information conservation or information loss is made; every curve is a combinatorial counting curve. Third, no physical realization of an observer is asserted; protocols are function classes only. Fourth, all rates are per-tick rates. Fifth, the word "information" is pinned to its combinatorial sense. Sixth, information is never called a conserved quantity; every verb of information location is rewritten as a statement of reconstruction sufficiency. Seventh, no blindness is called universal; every blindness claim carries its observation class, its library, and its target, and every strictness claim carries its witness.

1.4 Tier notation

D is a definition; T1 a complete derivation under declared preconditions; T2 a structural commitment; T3 a speculative deposit; Open an open problem; C1-P a single-source exact computation, C1-R a figure confirmed by independent recomputation. The senses are inherited from Section 1 of the preceding papers.

1.5 Dependency matrix

Unless stated otherwise, the weight convention is the edge-step norm $\alpha=\mathbf 1$, the definition domain is the per-tick class, and the orientation profile is the co-oriented profile.

Result	Occupancy condition	Library	Orientation profile	Domain	Tier
Amplitude outer box (Theorem A)	any	any legal	any	per-tick and segment	T1
Saturated zero leak (Theorem B1)	saturated throughout	occupancy identity	any	segment	T1
Fixed-$v$ global spectrum (Theorem B2)	exactly $v$ vacancies	single tangential ring	as declared	per-tick	T1
Fiber trichotomy and leak closed forms (Theorem B3)	$v\in\{0,1\}$	single tangential ring	as declared	per-tick	T1
One bit of spin sign (Theorem C)	occupancy identity	occupancy identity; explicit $\alpha$	any	per-tick	T1
Entropy chain and bridge (Theorem D, parts one–two)	any	witnesses declared per link; link two requires a pair with distinct net currents and equal shell accounts (two-vacancy threshold on the single tangential ring)	as declared	per-tick	T1
Collapse on the spin-clamped library (Theorem D, part three)	saturated core; occupancy identity	spin-clamped $\mathsf A_1$; no shell of capacity $2$	as declared	per-tick	T1
Ladder positioning (Theorem E)	any	contains both witness types	as declared	per-tick and segment	T1
Lattice entropy bound (Theorem F, part one)	fully saturated core; occupancy identity	complete occupancy-identity library	as declared	per-tick	T1
The geometric-legislation reading (Theorem F, part two)	as above	plus the full preconditions of the spectral-lattice corollaries of [P4]	co-oriented profile only	per-tick	T1
Full segment (Theorem H)	fully saturated; occupancy identity	complete occupancy-identity library; $C_0\le2$; $M>0$	all	per-tick	T1
Second value from the top (Theorem I-a)	fully saturated; occupancy identity	complete occupancy-identity library; $M>0$; Criterion D	all (criterion restated per profile)	per-tick	T1
Third value from the top (Theorem I-b)	fully saturated; occupancy identity	complete occupancy-identity library; all shells non-degenerate	all (criterion referred to the two adjacent profiles)	per-tick	T1
Equality of the two lattice spacings	fully saturated core	full preconditions of the spectral-lattice corollaries of [P4]	co-oriented profile only	per-tick	T1
Packing (Problem G)	fully saturated core	complete occupancy-identity library	as declared	per-tick	Open
Entropy table of three families (Table 1)	saturated core	spin-clamped library	co-oriented	per-tick	C1-P
Statistics chapter	declared	declared measure	as declared	per-tick	computation T1/C1; reading T2

1.6 Certificate suite

The numerical side evidence is carried by two scripts which share a common basis of graph construction, parent maps, and simple-cycle enumeration, and therefore are not claimed to be independent implementations; their load-bearing targets and their assertion sets differ: spectra are computed exactly by bitmask dynamic programming (no sampling), and counts by enumeration of disjoint-cycle packings. All figures are C1-P and will be promoted to C1-R only upon recomputation by a third implementation. Each script reports its own per-section assertion tally; every figure printed in the body must be checked against that tally and never inferred from older runs. The per-section tallies and the hashes of both scripts are recorded in full in Appendix A.

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2 The Definitions Package

2.0 Inheritance without restatement (preconditions copied alongside conclusions)

From the third paper. Definition 3.0 [P3]: the empty transfer is purged; the scope is pure-occupancy move systems. Proposition 3.2 [P3]: for fixed $N$, under edgewise radially non-increasing sequential single-hop dynamics and the declared onion connectivity, the smooth sector is the unique sink strongly connected class. Lemma 3.3 [P3]: with a saturated core, all vacancies outside it, and edgewise radial non-increase, the core occupancy is invariant, all core-touching moves are tangential, and the radial coordinates of core labels are invariant. Note 6.2b [P3]: conditional on the gross move multiset being fully recorded, empty transfers purged, transfers determined by endpoints together with $M$, and no hidden variables outside $M$, one has $\ker\mathcal O_{\rm gross}=\varnothing$; the note itself states that what it establishes is the emptiness of the zero fiber, not injectivity of the observation. Corollary 8.1 [P3]: the occupancy-quotient multiplicity $\binom{C_{\rm skin}}{m}$, the comparison object of this paper.

From the fourth paper. Definition 2.1 [P4] (onion graph: shells, capacities, cells, inward parent sets each nonempty; tangential edges form a ring for $C_j\ge3$, a single edge for $C_j=2$, none for $C_j=1$; a shell is degenerate iff $C_j\le2$); Definition 2.2 [P4] (a synchronous composite is legal iff all four clauses hold: every move runs along an incident edge; at most one departure per source; at most one arrival per target; the source of every moved token is occupied and the final occupancy remains in $\{0,1\}$ cellwise; the occupancy-identity transfers are exactly the disjoint unions of closed chains); Definition 2.5 [P4] (orientation projection $\tau$; tangential steps in degenerate shells project to zero); Definitions 2.8, 2.9, 2.10 [P4] (spin-clamped libraries; the four-layer observation ladder; sequential interfaces and increments); Definition 4.4 [P4] (the sign involution $\rho$: reverse every move; $\rho\theta$ is again legal and $\boldsymbol\ell(\rho\theta)=-\boldsymbol\ell(\theta)$); Lemma 4.5 [P4] (involution count $N_{\rm all}=N_0+2N_+$; $\mathrm{Fix}(\rho)$ is exactly the disjoint unions of transpositions; $\mathrm{Fix}(\rho)\subseteq\{\boldsymbol\ell=\mathbf 0\}$; its defining sentence is read in the corrected, scalar form of the erratum — see Appendix C); Lemma 4.6 [P4] (preconditions: a non-degenerate shell, an occupancy with $v\ge1$ vacancies, synchronous legality, all moves tangential within the shell; then $|\ell_j|\le C_j-v$); Theorem 4.1 [P4] (rigid-rotation quantization; preconditions: a non-degenerate saturated shell, all moves tangential within it, occupancy identity); Proposition 4.3 [P4] ($|\ell_{\mathbf 1}|\le M$ with $M=\sum_{C_j\ge3}C_j$); Propositions 3.1 [P4] and 3.6 [P4], Lemma 3.2 [P4], Theorem 3.3 [P4] with Corollary 3.4 [P4], Lemma 5.1 [P4], Theorem 5.2 [P4], Corollaries 5.3 [P4] and 5.4 [P4], Propositions 6.1 [P4], 6.2 [P4], and 7.3 [P4], and the measure-declaration conventions of Section 8 of [P4], are all cited with their original preconditions, which are copied again in the sections that use them.

For every item above, this paper cites the number and the preconditions and does not restate the proof.

Definition 2.1 (raw library, component quotient, and the nonemptiness clause). Fix an onion graph $G$. The raw library $\Theta_{\rm raw}$ is a finite set of legal synchronous composite transfers with the empty transfer purged. The component quotient $\bar\Theta:=\Theta_{\rm raw}/\!\sim$, where $\sim$ identifies transfers by support and account vector.

Unless explicitly stated otherwise, the observation lattice, reconstruction sufficiency, and counting leaks of this paper are all defined on $\Theta_{\rm raw}$. An observation may descend to $\bar\Theta$ only after it is proved constant on every equivalence class; without that proof it must not be applied on the quotient universe. Quotients serve only to declare statistical universes and are never projected back onto the raw observation lattice.

Nonemptiness clause. Every counting-entropy statement of this paper presumes $\Theta_{\rm raw}\neq\varnothing$; on an empty library the language of entropy does not apply. Empty libraries are realizable (a spin-clamped library on shells all of capacity $\le2$; the case $N=0$), so this clause is not vacuous. [D]

Definition 2.2 (observation functions and counting entropy). An observation is a map $\mathcal O:\Theta_{\rm raw}\to Y$. Under the clause $\Theta_{\rm raw}\neq\varnothing$, $$H(\mathcal O):=\log_2\bigl|\mathcal O(\Theta_{\rm raw})\bigr|\quad(\text{bits}),$$ the logarithmic cardinality of its fiber partition. The zero fiber $\ker\mathcal O:=\mathcal O^{-1}(0)$ applies only where $Y$ has a zero element. The partition and the zero fiber are two different objects: nesting of zero fibers does not entail a strict drop of partition cardinality; the bridge between them is Theorem D, part two. [D]

Definition 2.3 (the observation lattice, the finer–coarser order, and model premise M1). Write $\mathcal O\preceq\mathcal O'$ when there exists $h$ with $\mathcal O=h\circ\mathcal O'$, and say $\mathcal O'$ is finer than $\mathcal O$. This is a preorder whose quotient is isomorphic to the partition lattice; with the coarser-below convention, $\vee$ is common refinement and $\wedge$ common coarsening. The four-layer ladder — the gross move record $\mathcal O_{\rm gross}$, the tangential net edge-current field $J^\omega$, the shell-account vector $\boldsymbol\ell$, and the weighted total account $\ell_\alpha$ — is a chain in this lattice: $$\ell_\alpha\preceq\boldsymbol\ell\preceq J^\omega\preceq\mathcal O_{\rm gross}.$$

The comparability of the occupancy observation $\mathcal O_{\rm occ}:\theta\mapsto(s\theta,t\theta)$ with the layers of the ladder is library-dependent: on a library containing both witness types — same endpoints with different moves, and same moves with different endpoints — the two are incomparable; on restricted libraries the chain may collapse or comparabilities may appear. No universal claim is made.

Model premise M1 (declared by this paper; not a conclusion inherited from the third paper). Suppose the gross move multiset is fully recorded, empty transfers are purged, a transfer is determined by its endpoints together with $M$, and there are no hidden variables outside $M$. Then on $\Theta_{\rm raw}$ $$\mathcal O_{\rm occ}\vee\mathcal O_{\rm gross}\;\simeq\;\mathrm{id}_{\Theta_{\rm raw}},$$ where $\simeq$ is observational equivalence. The clause of Note 6.2b [P3] bearing the same name is a precondition here, not a conclusion; what that note establishes is only $\ker\mathcal O_{\rm gross}=\varnothing$. M1 does not descend automatically to the component quotient. [D]

Definition 2.4 (three definition domains and the history lift). (a) The per-tick class: $\boldsymbol\ell:\Theta_{\rm raw}\to\mathbb Z^{R+1}$ and its weighting $\ell_\alpha$. (b) The schedule-segment class: fix an initial state $x_0$ and a length $T$, and set $$\operatorname{Hist}_T(\Theta_{\rm raw};x_0):=\{\vec\theta=(\theta_1,\dots,\theta_T):\text{legal schedules with matching endpoints}\},$$ with the account history $\vec{\boldsymbol\Lambda}$ given by $\boldsymbol\Lambda(t)=\sum_{u\le t}\boldsymbol\ell(\theta_u)$ and $\boldsymbol\Lambda(0)=\mathbf 0$; for $T=0$ the account history is the empty tuple with entropy zero. (c) The endpoint class: $\boldsymbol\Lambda(T)$ only.

The history lift of per-tick observations: $\mathcal O_{\rm occ}^{[T]}(\vec\theta):=(x_0,\dots,x_T)$, $\mathcal O_{\rm gross}^{[T]}(\vec\theta):=(M(\theta_1),\dots,M(\theta_T))$, and likewise for the other two layers. Whenever the target is an account history, observation and target must be defined on the same $\operatorname{Hist}_T$; leak expressions that straddle definition domains are rejected. [D]

Definition 2.5 (two protocol classes). (a) The occupancy protocol class $\mathcal P_{\rm occ}$: functions defined on occupancy histories or their quotients. (b) The label protocol class $\mathcal P_{\rm lab}$: functions defined on occupancy histories together with shell-membership histories of labels. The peeling protocol of Section 7 of the third paper belongs to (a); its cross-shell inversion discriminant belongs to (b). Protocols are function classes only. [D]

Definition 2.6 (coarsest sufficient layer within a declared class). Given a target $\Phi$ and a pre-declared sub-poset of observations $\mathfrak C$, call $\mathcal O$ sufficient for $\Phi$ when $\Phi\preceq\mathcal O$; call $\mathcal O\in\mathfrak C$ a coarsest sufficient layer of $\mathfrak C$ when it is sufficient and no strictly coarser member of $\mathfrak C$ is.

Caution. In the full observation lattice, $\Phi$ itself is the unique coarsest sufficient observation; this is a tautology and is not counted as a result here. Every occurrence of "coarsest" names its $\mathfrak C$. [D]

Definition 2.7 (worst-output guaranteed counting leak). $$\mathrm{Leak}_{\rm guar}(\mathcal O;\Phi):=\log_2\bigl|\Phi(\Theta_{\rm raw})\bigr|-\max_{y\in\mathcal O(\Theta_{\rm raw})}\log_2\bigl|\Phi(\mathcal O^{-1}(y))\bigr| .$$ This is the minimal Hartley reduction guaranteed for every observation output; it vanishes exactly when some fiber lets $\Phi$ run over its full range.

Note one. The best-fiber and averaged variants are not load-bearing in this paper; the averaged variant requires a declared measure and is not the Shannon mutual information.

Note two, tiers. The formula is D; its non-negativity and the zero criterion are T1. Definitions of measures and mutual information are D, their computation T1 or C1, and only their statistical reading is T2.

Note three, a type caution. $|\Phi(\Theta_{\rm raw})|$ is the cardinality of the global range and $|\Phi(\mathcal O^{-1}(y))|$ that of a fiber; inferring the sign of a leak from growth or shrinkage of the former is rejected. [D]

Definition 2.8 ($\rho$-equivariance and anti-variance). An observation $F:\Theta_{\rm id}\to Y_F$ is $\rho$-equivariant when a range involution $r_F$ is declared with $F\circ\rho=r_F\circ F$; $\rho$-invariant when $r_F=\mathrm{id}$; $\rho$-anti-variant when $Y_F$ is an abelian group with negation and $r_F=-\mathrm{id}$. On the gross move layer $r_F$ is movewise reversal, not negation; on the net edge-current, shell-account, and scalar-account layers $r_F$ is negation. The spin sign is $\sigma_\alpha:=\operatorname{sgn}\ell_\alpha\in\{-,0,+\}$. [D]

Definition 2.9 (scalar-split counts). Fix an explicit positive weight $\alpha$ and set $$\Theta_\alpha^\times:=\{\ell_\alpha\neq0\},\qquad N_0^\alpha:=|\{\ell_\alpha=0\}|,\qquad N_+^\alpha:=|\{\ell_\alpha>0\}| .$$

Caution. This split differs from the split by account vector: $\{\boldsymbol\ell=\mathbf 0\}\subseteq\{\ell_\alpha=0\}$ is strict in general, so $N_0^\alpha\ge N_0$. Exact enumeration on the full default double-parent library of capacities $(1,3,4)$: the scalar split is $1995=521+2\times737$, the vector split is $1995=357+2\times819$, and the two differ; on the radial-free library the two readings coincide at $53=27+2\times13$. The defining sentence of Lemma 4.5 [P4] is read in the scalar form fixed by the erratum (Appendix C); the two splits stand side by side and are never substituted for one another. [D]

Definition 2.10 (orientation profile, the two lattice spacings, the lattice bound, and the entropy deficit).

Orientation profile (an independent input). This paper does not make sequential structure a precondition of orientation. For every non-degenerate shell $j$ an orientation $\omega_j$ of the ring is chosen independently; the prograde successor, the orientation projection $\tau$, the shell accounts, and the total account are all taken relative to the chosen profile $\omega=(\omega_j)$. When the interfaces satisfy the preconditions of Lemma 2.4 [P4] (sequential), the coherent orientation they transport is called the co-oriented profile; the series takes the co-oriented profile as its default, and all published figures of the preceding papers use it.

The two lattice spacings. The graph-level cycle spacing $$G_{\rm cyc}(G,\omega):=\gcd\bigl\{|\ell_{\mathbf 1}(c)|:c\ \text{a legal nonempty simple closed chain of the graph},\ \ell_{\mathbf 1}(c)\neq0\bigr\};$$ the library-level spectral spacing $G_{\rm spec}(\Theta,\omega):=\gcd(\operatorname{Spec}_\Theta(\ell_{\mathbf 1})\setminus\{0\})$, defined only when $\operatorname{Spec}_\Theta\setminus\{0\}\neq\varnothing$; a library with all-zero spectrum is listed separately as a one-point spectrum, its effective lattice bound set to $1$ and its entropy deficit to $0$, with no empty gcd invoked. For every declared library composed of disjoint unions of closed chains, the account of a transfer is a sum of cycle accounts, hence $G_{\rm cyc}\mid G_{\rm spec}$; on the complete occupancy-identity library the two are equal (every simple closed chain is itself a member; T1). The two names are not interchangeable: on the onion of capacities $(3,4)$ the graph-level spacing is one (its cycle accounts include $3$ and $4$), whereas the declared sub-library containing only the simultaneous rotation of both shells and its sign image has spectrum $\{\pm7\}$ and library-level spacing seven. The structural spacing $G^{\deg}$ (Corollary 5.3 [P4]) is defined only within its full preconditions and the co-oriented profile; within that jurisdiction $G^{\deg}=G_{\rm cyc}$, proved in Section 7.

Preconditions: the library is the complete occupancy-identity library, contains at least one non-degenerate shell, and $M>0$. Set the lattice bound $B:=2\lfloor M/G_{\rm cyc}\rfloor+1$ and the entropy deficit $\Delta H:=\log_2 B-\log_2|\operatorname{Spec}(\ell_{\mathbf 1})|\ge0$; for $M=0$ a separate convention applies. On an arbitrary declared library the same formulas are applied with $G_{\rm spec}$ in place of $G_{\rm cyc}$.

Caution one: all four quantities depend on the orientation. $\operatorname{Spec}$, $G_{\rm cyc}$, $B$, and $\Delta H$ all vary with the choice of $\omega$ and are not intrinsic to the geometry; whoever writes any of the four names the profile in force. Witness: capacities $(12,4)$ with primary parent $p(i)=3i$ — under the co-oriented profile the spectrum has fifteen points, $G_{\rm cyc}=2$, $B=17$, $\Delta H=\log_2(17/15)$; under the counter-oriented profile the spectrum has nine points, $G_{\rm cyc}=4$, $B=9$, $\Delta H=0$.

Caution two: the hole count and the entropy deficit are two different objects — a difference of point counts versus a difference of their logarithms; neither substitutes for the other.

Caution three: divisibility completeness does not entail packing tightness. The equality $G^{\deg}=G_{\rm cyc}$ settles the spacing layer only and does not guarantee that the lattice box is filled. Witness: capacities $(7,5)$ with $G_{\rm cyc}=1$ yet $B=25$, a nine-point spectrum, and sixteen holes.

Tiers. The two spacings and $B$, $\Delta H$ as definitions: D; equality $G_{\rm cyc}=G_{\rm spec}$ on the complete library: T1; $G^{\deg}=G_{\rm cyc}$ within the preconditions of Corollary 5.4 [P4]: T1 (Section 7); the witnesses $(12,4)$, $(7,5)$, $(3,4)$: C1-P. Any use of the formulas on an arbitrary declared library with $G_{\rm spec}$ must say so.

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3 Theorems A and B: What Occupancy Discloses on the Amplitude Side

3.1 Theorem A (the amplitude outer box disclosed by the occupancy history)

Statement. For any legal synchronous composite transfer, on any occupancy and any library, shell by shell $$\bigl|\ell_j(\theta)\bigr|\ \le\ \min\bigl\{N_j(s\theta),\ N_j(t\theta)\bigr\},$$ where $N_j$ is the token count of shell $j$. The occupancy history $\vec x$ therefore delivers, tick by tick, the outer box $\mathcal B_t=\prod_j[-b_{j,t},b_{j,t}]$ with $b_{j,t}=\min\{N_j(x_{t-1}),N_j(x_t)\}$, and $\boldsymbol\Lambda(T)$ lies inside the Minkowski sum of these boxes.

Proof. The tangential steps of shell $j$ have their sources occupied in $s\theta$ with at most one departure per source (Definition 2.2 [P4]), so the sources are distinct and number at most $N_j(s\theta)$; their targets are occupied in $t\theta$ with at most one arrival per target, so at most $N_j(t\theta)$; each step contributes $\tau=\pm1$, and the triangle inequality closes the claim. $\square$

Standing. A generalization of Lemma 4.6 [P4]: that lemma confines all moves to a single shell's tangential edges with $v\ge1$; the present theorem admits radial inflow and outflow and multi-shell composition, and still bounds by the occupancy counts at both endpoints. The box is an outer bound, not in general an exact description of the attainable set; its exact form on the single-ring library with $v\le1$ is precisely the fiber trichotomy of Theorem B3.

One-sentence conclusion. Occupancy is not entirely blind; it delivers at least a per-tick amplitude budget. [T1]

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3.2 Theorem B1 (zero leak on the saturated fiber)

Statement. Fix a saturated occupancy $x^\ast$ and a length $T$; let the schedule universe be the occupancy-identity segment $\operatorname{Hist}_T(\Theta_{\rm id};x^\ast)$. Then $\mathcal O_{\rm occ}^{[T]}$ is constant on it, so for every target $\Phi$, $$\mathrm{Leak}_{\rm guar}\bigl(\mathcal O_{\rm occ}^{[T]};\Phi\bigr)=0 .$$

Proof. Occupancy identity forces $x_t=x^\ast$ at every tick, so $\mathcal O_{\rm occ}^{[T]}$ is single-valued; its unique fiber is the whole universe, on which $\Phi$ runs over its full range, and Definition 2.7 gives zero. $\square$

Standing. The jurisdiction of Proposition 6.2 [P4], parts one and two; by Theorem C part one, this blindness is hereditary. [T1]

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3.3 Theorem B2 (the fixed-$v$ global account spectrum; preconditions narrowed)

Statement. A single non-degenerate ring of capacity $C\ge3$; the library consists of all legal single-tick transfers whose source occupancy has exactly $v$ vacancies and whose moves all run along the ring's tangential edges; the empty transfer is purged. The global account spectrum is $$v=0:\ \{0,\pm C\};\qquad 1\le v\le C-2:\ \{0,\pm1,\dots,\pm(C-v)\};\qquad v=C-1:\ \{\pm1\}.$$

Proof. For $v=0$: the ring is full and all moves stay inside it, hence occupancy identity, and Theorem 4.1 [P4] applies.

For $1\le v\le C-2$: the upper bound $|\ell|\le C-v$ is Theorem A with $N=C-v$. Attainability by prefix-block translation: the $v$ vacancies cut the ring into at most $v$ occupied runs; move the prefixes of these runs, $m$ tokens in total, one step forward each — each run's head enters the vacancy behind it, one departure per source and one arrival per target (the head into a vacancy, the rest into cells just vacated) — a legal transfer of account $+m$ for every $0\le m\le C-v$; the negative side is the $\rho$ mirror (Definition 4.4 [P4]). Zero: since $C-v\ge2$ some occupancy contains two adjacent tokens, whose neighbour transposition has account zero.

For $v=C-1$: a single token; the only legal nonempty steps are $\pm1$; account zero would require standing still, which is the empty transfer, already purged. $\square$

Caution. The spectrum is not monotone in the filling: at $C=6$ the eleven points at $v=1$ collapse to the three points at $v=0$, a drop of about $1.87$ bits, while the two points at $v=C-1$ are the collapse at the other end. Any reading of the form "the fuller, the less information" or "the emptier, the more" is rejected. [T1; the full grid $C=3..6$ is C1-P]

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3.4 Theorem B3 (the fiber trichotomy of the occupancy observation, and closed-form leaks)

Preconditions. A single non-degenerate ring of capacity $C\ge3$; the library $\Theta$ consists of all legal single-tick tangential transfers over all source occupancies with $v\in\{0,1\}$; the empty transfer is purged. The observation is $\mathcal O_{\rm occ}(\theta)=(s\theta,t\theta)$; the target is $\Phi=\ell$.

Part one (fiber trichotomy). The nonempty fibers of $\mathcal O_{\rm occ}$ fall into exactly three types.

Saturated type $(s,t)=(\text{full},\text{full})$: fiber account set $\{0,\pm C\}$, three points.

Vacancy-fixed type $s=t$ with one vacancy $h$: fiber account set $\{0\}$, one point — occupancy identity with a vacancy pins the account to zero.

Vacancy-moving type, source vacancy $h$ and target vacancy $h'\neq h$: writing $\ell_A$ for the arc length from $h'$ prograde to $h$, the fiber account set is exactly $$\{\ \ell_A,\ -(C-\ell_A)\ \},\qquad\text{two points},\quad |a_1|+|a_2|=C .$$

Proof. Saturated type: this is $v=0$ of Theorem B2. Vacancy-fixed type: the tokens occupy a path of length $C-1$; any cycle of length $\ge3$ in the permutation would have to run monotonically around the ring, but the path does not close, so the cycles are fixed points and neighbour transpositions only, all of account zero; since $C\ge3$ the path contains two adjacent tokens, so the fiber is nonempty.

Vacancy-moving type, by the edge-flow argument: to each edge $e=(i,i{+}1)$ assign the net flow $f_e\in\{-1,0,1\}$ (at most one token crosses forward from $i$, at most one backward from $i{+}1$, and a simultaneous pair cancels). Occupancy conservation gives $f_{i-1}-f_i=\delta_{i,h}-\delta_{i,h'}$, so $f$ is constant on each of the two arcs, jumping by $+1$ at $h'$ and by $-1$ at $h$: one arc carries $f=c{+}1$, the other $f=c$, and $\{c,c{+}1\}\subseteq\{-1,0,1\}$ forces $c\in\{-1,0\}$. The account is $\sum f_e=cC+\ell_A$: $c=0$ yields $+\ell_A$ and $c=-1$ yields $-(C-\ell_A)$. Both are attained by prefix-block translation along the respective arc. $\square$

Part two (guaranteed leak of the signed account). Take the signed account $\Phi=\ell$ as the target: the global range is $\{0,\pm C\}\cup\{0,\pm1,\dots,\pm(C-1)\}$, in total $2C+1$ points; the largest fiber is the three-point saturated type. Hence $$\mathrm{Leak}_{\rm guar}\bigl(\mathcal O_{\rm occ};\ell\bigr)\;=\;\log_2\frac{2C+1}{3}\;>\;0\qquad(C\ge3).$$ For $C=3..6$ this gives $1.2224$, $1.5850$, $1.8745$, $2.1155$ bits, matching the closed form value by value.

Corollary (the two-way split of amplitude and spin sign). Take the amplitude $|\ell|$ as the target: the global range is $\{0,1,\dots,C\}$, in total $C+1$ points; the amplitude set of the saturated fiber is $\{0,C\}$, two points, that of a vacancy-moving fiber is $\{\ell_A,C-\ell_A\}$, at most two points, and that of a vacancy-fixed fiber is one point — the largest fiber has two points. Take the spin sign $\sigma$ as the target: the global range has three points, and the saturated fiber already contains all of $\{-,0,+\}$. Hence $$\mathrm{Leak}_{\rm guar}\bigl(\mathcal O_{\rm occ};|\ell|\bigr)=\log_2\frac{C+1}{2}\;>\;0,\qquad \mathrm{Leak}_{\rm guar}\bigl(\mathcal O_{\rm occ};\sigma\bigr)=0 .$$ The occupancy observation guarantees a reduction of the uncertainty of the account amplitude, and guarantees no reduction of the uncertainty of the spin sign — the three targets stand side by side ($\ell$ signed, $|\ell|$ amplitude, $\sigma$ sign), and the split is exact fiber by fiber. For $C=3..6$ the amplitude leak is $1.0000$, $1.3219$, $1.5850$, $1.8074$, and the sign leak is identically zero (certificate, section thirteen). [T1; figures C1-P]

Three readings. First, this completes in the positive direction the type caution of Theorem B2: a positive leak is obtained from fiber cardinalities, and here even in closed form. Second, the order of informativeness runs against intuition: a fully unmoved occupancy (the saturated type) is the observation outcome that pins the account least (three candidates); a fixed vacancy pins it most (the account must be zero); a moving vacancy sits in between (two complementary candidates whose absolute values always sum to $C$). Third, the junction with Theorem C: this library is not an occupancy-identity $\rho$-library, so the predicate of hereditary blindness does not apply here; in the language of guaranteed leaks — the occupancy observation already has a positive guaranteed leak on the signed account (the $v=1$ fibers clamp it to two values), while its guaranteed leak on the spin sign remains zero: the two candidates of a vacancy-moving fiber are always one positive and one negative, so the sign stays undecided; the amplitude, separately, is clamped to two points with a closed-form positive leak. The amplitude admits a guaranteed leak and the spin sign admits none — the split reappears inside the fibers.

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3.5 Junction with the main axis

The four results form the quantitative side of protocol blindness: A gives the outer bound (occupancy discloses at least an amplitude budget), B1 the zero point of guaranteed leaks (the occupancy observation is constant on the saturated fiber), B2 the ranges (the fixed-$v$ spectra, with collapse at both ends), and B3 the exact fibers and the closed-form leaks of the three targets. Read together with Theorem C: the guaranteed leak of the occupancy observation on the amplitude has the closed-form positive value $\log_2\frac{C+1}{2}$, and its guaranteed leak on the spin sign is identically zero — the split named in the title holds not only in whole-library counts but fiber by fiber.

B3 compresses the residual uncertainty within a single occupancy fiber into a set of signed candidates; the next section no longer asks how many candidates there are, but why their signs come in pairs under the sign involution, and which protocols can never break the pairing.

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4 Theorem C: One Bit of Spin Sign

The spine of the paper. It rests on Definitions 2.8 and 2.9; on Definition 4.4 [P4], Lemma 4.5 [P4], and Proposition 6.2 [P4]. Certificate: sections one through six of info.py, 53 assertions with zero failures (the script's total for one run, with the per-section tally, is in Appendix A); the assertions for the capacity structure live separately in spec_probe.py.

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4.0 Preconditions and notation

An occupancy-identity library $\Theta\subseteq\Theta_{\rm id}$, empty transfer purged; the sign involution $\rho$ (Definition 4.4 [P4]): reverse every move; $\rho\theta$ is again legal and $\boldsymbol\ell(\rho\theta)=-\boldsymbol\ell(\theta)$. Fix an explicit positive weight $\alpha$ and an orientation profile $\omega$ (co-oriented by default); the spin sign is $\sigma_\alpha:=\operatorname{sgn}\ell_\alpha\in\{-,0,+\}$. Call a library $\Theta$ $\rho$-closed when $\rho\Theta=\Theta$; $\Theta_{\rm id}$ itself is $\rho$-closed (Definition 4.4 [P4]).

Blindness (per library). An observation $F:\Theta\to Y$ is blind to $\sigma_\alpha$ on $\Theta$ when every fiber of $F$ that contains an element with $\sigma_\alpha\neq0$ contains elements of both signs. That is: knowing the value of $F$ never decides the sign of a nonzero spin.

Hereditary blindness. $F$ is hereditarily blind on $\Theta$ when it is blind on every $\rho$-closed sub-library of $\Theta$.

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4.1 Theorem C, part one: the trichotomy of blindness

Theorem C, part one (characterization by nonzero-sign orbits). For a $\rho$-closed library $\Theta$, set $\Theta_\alpha^\times:=\{\theta\in\Theta:\ell_\alpha(\theta)\neq0\}$. The following are equivalent:

(i) $F$ is blind to the nonzero spin sign on every $\rho$-closed sub-library of $\Theta$ (hereditary blindness);

(ii) $F(\rho\theta)=F(\theta)$ for every $\theta\in\Theta_\alpha^\times$ (constancy on nonzero-sign orbits).

Zero-sign orbits are not constrained by this characterization.

Proof, (ii)⟹(i). Let $\Theta'\subseteq\Theta$ be $\rho$-closed and $\theta\in\Theta'$ with $\sigma_\alpha(\theta)\neq0$. Then $\theta\in\Theta_\alpha^\times$, $\rho\theta\in\Theta'$, $F(\rho\theta)=F(\theta)$ with opposite signs, so that fiber contains both signs. $\square$

Proof, (i)⟹(ii). Suppose $\theta_0\in\Theta_\alpha^\times$ with $F(\theta_0)\neq F(\rho\theta_0)$. The two-element library $\{\theta_0,\rho\theta_0\}$ is itself a $\rho$-closed sub-library on which the two fibers of $F$ each carry a single sign — not blind, contradicting (i). $\square$

Two witnesses that zero-sign orbits fall outside the characterization (both in the certificate). First, take two non-degenerate shells of equal capacity and let $\theta$ be a prograde rigid rotation of one paired with a retrograde rigid rotation of the other: $\boldsymbol\ell(\theta)=(C,-C)\neq\mathbf 0$ while $\ell_{\mathbf 1}=0$; let $F$ be constant on all nonzero-sign orbits while $F(\theta)\neq F(\rho\theta)$ — then $F$ is hereditarily blind (every fiber containing a nonzero sign also contains its negated partner) yet is not constant on that zero-sign orbit. Second, the default double-parent library of capacities $(1,3,4)$ contains five hundred and twenty-one transfers of scalar account zero, among them the neutral transfers of shell-account type $(0,+1,-1)$; take the two-element library of any one of them — the blindness clause holds vacuously there while $F$ may separate the two ends. The witness is no isolated case.

(iii) Caution: per-library blindness is weaker than hereditary blindness, and does not pass to sub-libraries. There exist a $\rho$-closed library $\Theta_4$ and an observation $F$ such that $F$ is blind on $\Theta_4$, not blind on a $\rho$-closed sub-library of it, and not orbit-constant.

Witness (explicit; in the certificate). Take two transfers $\theta_1,\theta_2$ of account $+5$ with distinct supports in the $(7,5)$ library, set $\Theta_4=\{\theta_1,\rho\theta_1,\theta_2,\rho\theta_2\}$, and define $$F(\theta_1)=F(\rho\theta_2)=a,\qquad F(\theta_2)=F(\rho\theta_1)=b .$$ The fiber $\{a\}$ contains $+5$ and $-5$, and so does $\{b\}$, so $F$ is blind on $\Theta_4$; restricting to $\{\theta_1,\rho\theta_1\}$, however, $F$ decides the sign. Blindness here is achieved by cross-orbit sign-swapping pairs, without orbitwise constancy. $\square$

Reading. "Blindness" has two senses. Per-library blindness is weak: it can be a coincidence of pairings, lost on passing to a sub-library. Hereditary blindness is strong and stable, and is characterized exactly by constancy on the nonzero-sign orbits — the quantifier "every sub-library" is not decoration but what makes the characterization true; zero-sign orbits sit outside it, since on them the definition of blindness imposes nothing. The occupancy observation $\mathcal O_{\rm occ}$ is constant on all orbits of $\Theta_{\rm id}$ (occupancy identity gives $s(\rho\theta)=t(\rho\theta)=s\theta$), hence in particular on the nonzero-sign orbits, so its blindness is hereditary; what Proposition 6.2 [P4], parts one and two, established was per-library blindness — the present theorem upgrades it to hereditary blindness, and the two proofs now share a single root. [T1]

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4.2 Theorem C, part two: exactly one bit, conditional on $\ell_\alpha\neq0$

Statement. On a $\rho$-closed library $\Theta$, set $\Theta_\alpha^\times:=\{\ell_\alpha\neq0\}$, $N_0^\alpha:=|\{\ell_\alpha=0\}|$, $N_+^\alpha:=|\{\ell_\alpha>0\}|$. Then:

Part one: $\rho$ acts freely on $\Theta_\alpha^\times$ (since $\mathrm{Fix}(\rho)\subseteq\{\boldsymbol\ell=\mathbf 0\}\subseteq\{\ell_\alpha=0\}$, Lemma 4.5 [P4]); the orbits have length two with opposite signs at the two ends, hence $$N_{\rm all}=N_0^\alpha+2N_+^\alpha,\qquad H\bigl(\sigma_\alpha\,\big|\,\Theta_\alpha^\times\bigr)=1\ \text{bit, no more and no less}.$$

Part two: the condition written as $\boldsymbol\ell\neq\mathbf 0$ is false. If the library contains neutral vectors ($\boldsymbol\ell\neq\mathbf 0$ with $\ell_\alpha=0$), set $q:=\bigl|\{\boldsymbol\ell\neq\mathbf 0,\ \ell_\alpha=0\}\bigr|\big/\bigl|\{\boldsymbol\ell\neq\mathbf 0\}\bigr|$, the fraction of neutral transfers among those of nonzero vector account; then under the uniform counting measure on the library, $H_{\rm Sh}^{\rm unif}(\sigma_\alpha\mid\boldsymbol\ell\neq\mathbf 0)=H_b(q)+(1-q)$, which in general is not equal to one — so $\boldsymbol\ell\neq\mathbf 0$ is not the correct condition for extracting a pure spin bit. The two entropies are kept apart: the "one bit" of part one is a counting entropy (a two-point range); putting the uniform counting measure on $\Theta_\alpha^\times$, the sign involution equates the counts of the two signs, so the Shannon entropy is exactly one bit as well — the two entropies coincide under the correct condition and separate under the wrong one.

Numerical certification (certificate, sections three and four). The scalar splits of the three families, with exact even signs: $(1,3,4)$ double-parent, $1995=521+2\times737$; $(7,5)$, $1947=1529+2\times209$; $(4,4,4)$, $6344=3952+2\times1196$. Under the wrong condition the three families give $1.3693$, $1.3776$, $1.5010$ bits ($q=0.1001, 0.5957, 0.4990$), matching $H_b(q)+(1-q)$ family by family. [T1; figures C1-P]

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4.3 Theorem C, part three: the nonzero-sign fraction

Definition (nonzero-sign fraction). $\pi_\times:=|\Theta_\alpha^\times|/N_{\rm all}=2N_+^\alpha/N_{\rm all}$: the fraction of transfers on which the spin sign is, in principle, something to be decided. It does not name the extraction power of any particular protocol (that power is measured separately by the $\mathrm{Leak}$ quantities). Theorem C, part one, says who can never decide the bit (the hereditarily blind class); $\pi_\times$ says how large the nonzero-sign domain is. "Whether the bit can be reconstructed" and "how large its nonzero domain is" are two different questions — the first qualitative, the second quantified by $\pi_\times$.

Part three (a): the single-ring closed form (T1). For a single saturated non-degenerate ring of capacity $C\ge3$, purely tangential occupancy-identity library (purged), $$N_{\rm all}=L_C+1,\qquad N_+^{\mathbf 1}=1,\qquad \pi_\times=\frac{2}{L_C+1},$$ where $L_C$ are the Lucas numbers ($L_3=4$, $L_k=L_{k-1}+L_{k-2}$).

Proof. The transfers of this library are the permutations of the ring mapping each cell to itself or a neighbour. Any cycle of length $\ge3$ moves by $\pm1$ at every step along a simple cycle, hence always in the same direction, hence around the whole ring — so the cycles come in exactly three kinds: fixed points, neighbour transpositions, and the full-ring rigid rotation (which occupies the whole shell). The non-rotation transfers are disjoint unions of neighbour transpositions, counted by the matchings of the cycle graph $C_C$, which number $L_C$; the rotations are two; purging the empty transfer, $N_{\rm all}=L_C+1$. A transposition has account zero ($+1-1$), so the only transfers of nonzero account are the two rotations, and $N_+^{\mathbf 1}=1$. $\square$

Corollary. $L_C=\varphi^C+(-\varphi)^{-C}$, hence $\pi_\times\sim2\varphi^{-C}$ (no factor $\sqrt5$), exponential decay at the golden ratio: on an isolated shell the nonzero-sign domain dies out exponentially in the capacity. Verified exactly for $C=3..10$, with $\pi_\times\varphi^C$ tending to two (certificate, sections one, six, thirteen).

Part three (b): the lift by a core cell (a C1-P observation). Attach a single degenerate core cell ($(1,C)$, all cells sharing the one parent). The full computed list for $C=3,\dots,10$: $0.60870$, $0.65385$, $0.67290$, $0.69194$, $0.70324$, $0.71237$, $0.71882$, $0.72391$ — $C=3,4$ lie below two-thirds, $C=5,\dots,10$ above two-thirds, and this finite sequence increases term by term; the threshold, monotonicity, and limit over all $C$ remain open. Mechanism reading (T2): the snake-cycle family makes transfers of nonzero account available at every displacement $t$, so the interface enlarges the nonzero-sign domain — the same snake cycles that drive the full segment of Theorem H, seen on the counting side. The general threshold and the limit are open (Section 11.3); the limit of $\pi_\times$ goes to the seed bag. [closed form T1; the $(1,C)$ figures C1-P; mechanism T2; limit Open]

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4.4 Junction with the main axis

Whether the spin sign can be reconstructed from a protocol's output now has a three-layer answer. Who never can: the hereditarily blind class, exactly the observations constant on nonzero-sign orbits, occupancy and its quotient among them (part one). What the others get: exactly one bit, conditional on $\ell_\alpha\neq0$, no more and no less (part two). How large the nonzero domain is: the fraction $\pi_\times$ — exponentially vanishing on an isolated shell; after attaching a core cell, of the eight computed cases the first two lie below two-thirds and the remaining six above, with the finite sequence increasing (part three). In one sentence: the undecidability of the spin bit is algebraic (the $\rho$ symmetry); the size of its nonzero domain is geometric (the interface structure).

Protocol corollary (the core question grounded). On a $\rho$-closed occupancy-identity history library, every occupancy protocol $P\in\mathcal P_{\rm occ}$ is constant on nonzero-sign $\rho$-orbits (the occupancy history is $\rho$-invariant), hence hereditarily blind to the spin sign. For label protocols $P\in\mathcal P_{\rm lab}$ no universal claim is made; one checks, protocol by protocol, whether $P(\rho h)=P(h)$ on the nonzero-sign history orbits. For the non-occupancy-identity single-ring endpoint protocols with $v\le1$, Theorem B3 supplies the exact guaranteed leaks of the three targets — signed account, amplitude, and spin sign.

Theorem C gives the algebraic criterion for spin-sign blindness; Theorems D and E turn to comparing the resolution of the observations that are not blind, asking on which libraries the four-layer compression is strict and on which layer the target falls.

Rejection on record. The fraction $q$ bears no criterion relation to $\Delta H$, and neither does $\pi_\times$: the double-parent $(3,4)$ has $\pi_\times=0.726$ with a positive deficit, while $(4,4)$ has $\pi_\times=0.273$ with deficit zero. The counting side and the spectral side have different sources; neither is to be inferred from the other.

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5 Theorem D and Table 1: Strictness of the Ladder

5.1 Theorem D (the four-layer counting-entropy chain, and the library dependence of its strictness)

Part one (the chain). On any declared library, $$H(\mathcal O_{\rm gross})\ \ge\ H(J^\omega)\ \ge\ H(\boldsymbol\ell)\ \ge\ H(\ell_\alpha),$$ immediately, since each layer is a function of the one before (Definition 2.9 [P4]).

Part two (the strictness bridge, stated here). Let $g=h\circ f$. If the library contains $\theta_0,\theta_1$ with $f(\theta_0)\neq f(\theta_1)$ and $g(\theta_0)=g(\theta_1)$, then $h$ is not injective on $\operatorname{im}f$, so $H(g)

Link one, $H(\mathcal O_{\rm gross})>H(J^\omega)$. A same-edge neighbour transposition paired with a purely radial transposition: both have $J^\omega=0$ while their gross move records differ. Available on saturated libraries: in the saturated default double-parent library of $(1,3,4)$ there are three hundred and seventeen transfers with $J^\omega=0$, twenty-seven purely tangential and one hundred and nineteen purely radial (a purely radial transfer has $J^\omega=0$ by definition, since $\tau$ vanishes on radial moves).

Link two, $H(J^\omega)>H(\boldsymbol\ell)$. The general strictness condition: the library contains two transfers with distinct net edge currents and equal shell-account vectors. Multi-shell saturated libraries already supply it: in the default double-parent $(1,3,4)$, the two disjoint radial three-cycles $(1,1)\to(2,0)\to(2,1)\to(1,1)$ and $(1,2)\to(2,3)\to(2,2)\to(1,2)$ contribute one prograde and one retrograde tangential step on shell two, compose legally in one tick, and give $J^\omega\neq0$ with $\boldsymbol\ell=\mathbf 0$ and $v=0$; witnesses of this kind number forty in that saturated library. Pairing with a purely radial transposition ($J^\omega=0$, $\boldsymbol\ell=\mathbf 0$) completes the strictness pair.

The single-ring threshold (an exact sub-model result). On the single tangential ring the pair is unattainable for $v\le1$ and attainable for $v\ge2$; the threshold statement applies to this sub-model only and is not promoted to a general threshold.

Unattainability ($v\le1$). By the edge-flow lemma (the proof of Theorem B3, part one): for $h=h'$, $f\equiv c$ and the account is $cC$, so $\ell=0\Rightarrow c=0\Rightarrow J^\omega=0$; for $h\neq h'$, the account lies in $\{\ell_A,-(C-\ell_A)\}$ and is never zero. In neither case is there an "account zero, $J$ nonzero" transfer. $\square$

Attainability ($v\ge2$). For $C\ge4$, tokens $\{0,1\}$ and vacancies including $\{2,C{-}1\}$: take $1\to2$ (prograde) and $0\to C{-}1$ (retrograde); sources distinct, both targets vacant, legal; account $+1-1=0$ while $J^\omega$ is nonzero on the two edges. The paired same-shell neighbour transposition is available in the same library. $\square$

The witness of Proposition 6.1 [P4], part two — "capacity five, tokens at columns zero and three, vacancies at one, two, four" — is the single-ring instance with $v=3$, reproduced verbatim in the certificate.

Link three, $H(\boldsymbol\ell)>H(\ell_\alpha)$. The general strictness condition: the library contains two shell-account vectors $v_0\neq v_1$ with $\alpha\cdot v_0=\alpha\cdot v_1$; when the library contains a zero-vector transfer, this says $\operatorname{im}\boldsymbol\ell\cap\ker\alpha$ contains a nonzero element. On the spin-clamped (rigid-rotation) sub-library this condition degenerates to a collision of the sums of two distinct unsigned subsets (equivalently, their difference yields a nontrivial signed relation); outside that sub-library, capacity subset sums are not used as a general criterion. Witness: the saturated default double-parent library of $(1,3,4)$ has five hundred and twenty-one transfers of scalar account zero, of which one hundred and sixty-four are neutral (nonzero vector), of shell-account type $(0,+1,-1)$ — a neutrality produced by cancellation of shell accounts along radial snake cycles (capacities $3,4$ admit no nontrivial signed sum equal to zero), i.e. a witness of the nonzero intersection with $\ker\alpha$; pairing with a zero-vector transfer completes the pair.

Part three (a negative result: collapse on the spin-clamped library $\mathsf A_1$ [P4]). Preconditions: saturated core; occupancy identity; the spin-clamped library $\mathsf A_1$ [P4]; purged; the graph has no shell of capacity $2$. By Proposition 3.6 [P4], the nonempty transfers are exactly the disjoint unions of prograde rigid rotations of non-degenerate shells, so the gross record, the net edge current, and the shell-account vector determine one another — the first three links collapse entirely; the last link is the collision question for unsigned subset sums (their difference being the signed relation), strict or not (the three cases of Table 1). Strictness is not a structural property but a property of the library. That link-two witnesses are unattainable inside $\mathsf A_1$ is a direct corollary of the collapse itself: equal shell accounts mean equal subsets mean equal net currents — it is not an exclusion by occupancy conditions (on general libraries the link-two witness can be had at saturation). [T1; the figures of Table 1 are C1-P]

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5.2 Table 1 (the four counting entropies on the spin-clamped library; the three families of [P4])

By the characterization of part three, nonempty transfers correspond bijectively to nonempty subsets of the non-degenerate shells, numbering $2^k-1$; the first three layers all have entropy $\log_2(2^k-1)$. On a general library the last layer counts the distinct values of signed subset sums; the spin-clamped library contains prograde rotations only, so here it reduces to the distinct values of unsigned subset sums.

Family	Non-degenerate capacities	Nonempty transfers	$H_{\rm gross}=H_J=H_{\boldsymbol\ell}$	$H(\ell_{\mathbf 1})$	Last link
Default double-parent $(1,3,4,5,6)$	$3,4,5,6$	$15$	$\log_2 15$	$\log_2 14$	strict; collision $3{+}6=4{+}5=9$
Equal-capacity single-parent uniform tower $(4,4,4)$	$4,4,4$	$7$	$\log_2 7$	$\log_2 3$	strict
Congruent single-parent tower $(12,4)$	$12,4$	$3$	$\log_2 3$	$\log_2 3$	no drop

The three families realize exactly the three cases: drop by collision, drop by repetition, no drop. The certificate lands every figure of every family; C1-P.

Junction with the main axis. The three cases of Table 1 show that the strictness of the last link is decided not by structure but by the arithmetic of the capacities: collision, repetition, and no-drop are all realizable, so the phrase "strictness of the ladder" always names its library — which is why the positioning of Theorem E must declare its comparison class. Theorem D sets the witness conditions for linkwise strictness; Theorem E, in a single library carrying both upper and lower witnesses, wedges the target $\boldsymbol\ell$ between the net current and the total account.

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6 Theorem E: Positioning on the Ladder

6.1 Theorem E (positioning the target on the four-layer ladder)

Declaration. The comparison class $\mathfrak C$ is the four-layer ladder (the caution of Definition 2.6: in the full observation lattice the target itself is the unique coarsest sufficient observation, so "coarsest" is always relative to $\mathfrak C$; the substance of this theorem lies entirely in the two strictness witnesses). The library $\Theta$ consists of all legal single-tick transfers over all occupancies of the $(1,3,4)$ onion (purged).

Statement. $\Theta$ contains both witness types:

First, the upper side ($J^\omega$ strictly finer): the link-two pair — on shell two take tokens $\{0,1\}$, vacancies $\{2,3\}$, one prograde and one retrograde step, paired with a same-shell neighbour transposition; shell accounts both $\mathbf 0$ while the $J^\omega$ differ.

Second, the lower side ($\ell_{\mathbf 1}$ insufficient): the link-three pair — a neutral transfer of the saturated segment (shell accounts $(0,+1,-1)$) paired with a zero-account transfer; scalars both zero while the vectors differ.

Hence within $\mathfrak C$ $$J^\omega\ \succ\ \boldsymbol\ell\ \succ\ \ell_{\mathbf 1}\quad\text{strictly},$$ and $\boldsymbol\ell$ is the coarsest sufficient layer of $\mathfrak C$ for reconstructing itself. The net edge-current field of Proposition 3.1 [P4] is sufficient but not coarsest.

Contrast of definition domains. To reconstruct the full account history $\vec{\boldsymbol\Lambda}$, lift the four layers tick by tick (Definition 2.4) and apply the same positioning; to reconstruct the endpoint $\boldsymbol\Lambda(T)$ only, the sum of the per-tick vector accounts suffices and the per-tick values are not needed.

Caution. The two witnesses now stand in one and the same library (which carries all occupancies), so the positioning holds on a single declared library with no cross-library patchwork; and the collapse on the spin-clamped $\mathsf A_1$ [P4] of part three shows that changing the library can collapse the positioning entirely — positioning and collapse are two faces of the same library dependence. [T1]

Theorem E answers "to which layer must one ascend to reconstruct the target"; Theorem F answers next how many distinct account values that layer can produce at all, for a given graph and orientation.

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6.2 Junction with the main axis

D and E form the structural side of the main axis: when the four-layer order is strict (a property of the library, each link with its own condition — link one already at saturation, link two requiring a pair with distinct net currents and equal shell accounts, link three requiring a nonzero intersection of the shell-account image with $\ker\alpha$), and on which layer the target sits ($\boldsymbol\ell$ exactly on the third layer of any library containing both witness types). Read together with A, B, and C, the answer to the main axis takes shape: what a protocol can reconstruct is bounded by the amplitude outer box and the fiber trichotomy (A, B3); what it can never reconstruct is exactly the hereditarily blind class, the observations constant on nonzero-sign $\rho$-orbits (C); between the two, resolution falls layer by layer, and the strictness of each fall is legislated by the library (D, E).

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7 Theorem F: The Lattice Entropy Bound, and Equality of the Two Spacings

Theorem F (lattice entropy bound and entropy deficit). Full list of preconditions: fully saturated core; occupancy identity; synchronous partial permutation; complete occupancy-identity library; edge-step norm; $M>0$; the declared orientation profile $\omega$.

Part one. From $\operatorname{Spec}\subseteq G_{\rm cyc}\mathbb Z\cap[-M,M]$ of Definition 2.10, $$H(\ell_{\mathbf 1})\ \le\ \log_2 B,\qquad B=2\lfloor M/G_{\rm cyc}\rfloor+1 .$$ Purely combinatorial; no measure enters.

Part two, the reading, which carries two restrictions. If the non-degenerate interfaces additionally satisfy the full preconditions of Corollaries 5.3 [P4] and 5.4 [P4] (house-style clause included) and the co-oriented profile is taken, then $G^{\deg}=G_{\rm cyc}$, and one may write: the geometry of the interfaces legislates the spacing of the upper bound on the information capacity of the transfer layer. What this sentence establishes is the spacing layer only; tightness of the capacity depends separately on packing (Caution three) and does not follow from the equality. Outside the jurisdiction $B$ remains an upper bound, but the reading in terms of "legislation" does not apply.

Part three, two theorems now give the criterion for a vanishing deficit (their proofs in full in Section 8).

Theorem H (full segment under a degenerate initial run). The common preconditions together with $M>0$ and $C_0\le2$ (no house-style clause needed) give $\operatorname{Spec}=[-M,M]\cap\mathbb Z$, hence $G_{\rm cyc}=1$, $B=2M+1$, and $\Delta H=0$.

Theorem I-a (the second value from the top). The common preconditions together with $M>0$ give $M-1\in\operatorname{Spec}$ if and only if Criterion D holds (there exist a non-degenerate shell $e$ and a cell $x$ such that between $(e,x)$ and its prograde successor there is a path whose interior is nonempty and lies entirely in degenerate shells).

Theorem I-b (the third value from the top). When all shells are non-degenerate, $M-2\in\operatorname{Spec}$ if and only if $\exists\,q_b\equiv q_{b-1}-1$ (in parent-set form, relative to the profiles of the two adjacent shells).

Part four, the cross-checks on record (co-oriented profile): $(1,3,4)$ full segment, fifteen points; $(1,3,4,5)$ full segment, twenty-five points; $(1,3,4,5,6)$ verified value by value in the certificate, all with $\Delta H=0$; $(7,5)$ actual spectrum nine points, $B=25$, sixteen holes, $\Delta H\approx1.474$; $(4,4,4)$ actual spectrum seven points, $B=7$, $\Delta H=0$; $(12,4)$ actual spectrum fifteen points, $B=17$, holes exactly at $\pm14$, $\Delta H\approx0.181$; $(2,12,4)$ full segment, thirty-three points, with tail-segment spacing two but global spacing one.

Tiers: the bound and the criterion for $\Delta H=0$ are T1; Theorems H, I-a, I-b are T1; the figures of the finite families are C1-P; general tightness is Open. [T1 for the bound; Open for tightness]

7.1 The two spacings and the proof of their equality

The two spacings (inherited from Definition 2.10, applied to the complete occupancy-identity library). On the complete library every simple closed chain is itself a legal transfer, while every occupancy-identity transfer is a union of vertex-disjoint closed chains; squeezing from both sides, $$G_{\rm cyc}=G_{\rm spec}=\gcd\bigl(\operatorname{Spec}(\ell_{\mathbf 1})\setminus\{0\}\bigr),\qquad \operatorname{Spec}\subseteq G_{\rm cyc}\mathbb Z\cap[-M,M];$$ $B$ and $\Delta H$ follow Definition 2.10.

The structural spacing $G^{\deg}$ (Corollary 5.3 [P4]) is defined only within its preconditions and the co-oriented profile: computing $G_j$ requires the interface to satisfy the preconditions of Theorem 5.2 [P4] (both shells non-degenerate, the primary parent sequential and non-constant), and $G_j$ depends on $\tau$, which depends on the orientation profile. Within that jurisdiction

$$\boxed{G^{\deg}=G_{\rm cyc}}$$

Proof. Corollary 5.4 [P4] gives $\operatorname{Spec}\subseteq G^{\deg}\mathbb Z$, and every cycle account lies in the spectrum, so $G^{\deg}\mid G_{\rm cyc}$. Conversely, the second clause of Theorem 5.2 [P4] states that $G_j$ is exactly the greatest common divisor of the primitive simple-cycle accounts of that shell; those cycles are supported on two shells and remain legal transfers when the whole onion is saturated, so their accounts lie in the spectrum and $G_{\rm cyc}\mid G_j$, whence $G_{\rm cyc}\mid d=\gcd_jG_j$. If some interface has a degenerate lower shell and a non-degenerate upper shell, clause four of Lemma 5.1 [P4] supplies a cycle of account $+1$, so $G_{\rm cyc}=1=G^{\deg}$. Both divide each other and both are positive, hence equal. $\square$

Full list of the definition domain. The equality applies only within the complete preconditions of Corollaries 5.3 [P4] and 5.4 [P4]: fully saturated core; occupancy identity; synchronous partial permutation; complete occupancy-identity library; the non-degenerate shells form a contiguous outer run; the co-oriented profile; every non-degenerate interface satisfies the preconditions of Theorem 5.2 [P4]. Since Theorem H has dropped the house-style clause, many of its models are ones in which $G^{\deg}$ is simply undefined. Furthermore, on a single-shell branch with no qualifying interface, $d=\gcd_jG_j$ would be an empty gcd; that branch is handled directly by the single-shell rigid-rotation spectrum instead, with no empty gcd invoked.

Caution three: divisibility completeness does not entail packing tightness. What the equality proves is that the divisibility spacing supplied by the interface structure is neither too coarse nor too fine — it is already complete; it does not guarantee that the lattice box is filled, so $B$ may still be a non-tight upper bound. Witness: $(7,5)$ with increments $(3,1,1,1,1)$ has $G^{\deg}=G_{\rm cyc}=1$ yet $B=25$, a nine-point spectrum, and sixteen holes. Hence the sentence "the geometry of the interfaces legislates" establishes the spacing layer only; tightness of the capacity depends separately on packing, which is what Problem 9.1 [P4] asks.

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Junction with the main axis. What the lattice bound clamps is the cardinality of the range, a different source from the counting side of Section 4 (as the rejection on record already states); its vanishing criterion is supplied by the two theorems of Section 8, while tightness remains Open (Problem G). Theorem F gives the lattice outer box of admissible account values; Theorems H and I handle, respectively, when that box is a full segment and when the two gaps nearest the amplitude top are open.

8 Theorems H and I: The Full Segment and the Gaps at the Top

The lemmas and theorems of this section are given in full below; the certificate spec_probe.py (SHA-256 e6d5ff4720a759fc551c7fc93a7805e29d3315ba62079d348111af3604cfc7da, four hundred and ninety-four assertions with zero failures) and its per-section tally are in Appendix A. The numerical side evidence goes to Appendix B.

Common Preconditions HI (a citable item). A fixed finite onion graph (Definition 2.1 [P4]); every cell saturated; synchronous composition (the four clauses of Definition 2.2 [P4]); the library is the complete occupancy-identity library after purging; the edge-step norm $\alpha=\mathbf 1$; an orientation profile $\omega$ declared independently for each non-degenerate shell (Definition 2.10, co-oriented by default). Unless stated otherwise, Theorems H, I-a, and I-b all stand under these preconditions; "the common preconditions" below names exactly this item.

8.1 Lemma J: the radial cost inequality

Lemma J. Write $T_j(\theta)$ for the number of tangential moves inside shell $j$, $\nu^{\rm rad}_j(\theta)$ for the number of cells of shell $j$ that emit a radial move, and $\nu^{\rm rad}(\theta):=\sum_{C_j\ge3}\nu^{\rm rad}_j(\theta)$. Then for every non-degenerate shell $$\ell_j(\theta)\ \le\ T_j(\theta)\ \le\ C_j-\nu^{\rm rad}_j(\theta),$$ hence $\ell_{\mathbf 1}(\theta)\le M-\nu^{\rm rad}(\theta)$. Setting $d_j:=(C_j-\nu^{\rm rad}_j)-\ell_j$, $$d_j=\bigl[(C_j-\nu^{\rm rad}_j)-T_j\bigr]+\bigl[T_j-\ell_j\bigr]\ \ge0,$$ the two brackets being the losses due to stationary source cells and to retrograde steps. Equality holds if and only if both levels are simultaneously tight: every non-degenerate shell cell that emits no radial move emits a tangential move, and all tangential moves are prograde. This "if and only if" is true in both directions; the target clause bears only on legality and adds no further condition for equality.

Proof. First level: $\ell_j$ is a sum of $\pm1$ over the $T_j$ tangential steps. Second level: the source cells of tangential steps lie in shell $j$ with at most one departure per source (Definition 2.2 [P4]), so the sources are distinct; a cell emitting a radial move emits no tangential move. For degenerate shells $\ell_j$ vanishes identically. Summing gives the claim. $\square$

Lemma J' (the trichotomy of cost). Let a closed chain $c$ contain at least one radial move. Its inward and outward crossings of each interface are equal in number, so the total number of radial moves is even and at least two. Three cases. First, all shells met by $c$ are non-degenerate: both moves have sources in non-degenerate shells, contributing at least two. Second, $c$ crosses the boundary between a degenerate and a non-degenerate shell: at least one move has its source in a degenerate shell and is therefore not counted in $\nu^{\rm rad}$, so the contribution is at least one, and one is attainable. Third, the support of $c$ lies entirely in degenerate shells: all sources are degenerate cells and the contribution is zero (the radial transposition on capacities $(1,1)$ is an instance; that family has $M=0$ and lies outside the preconditions of all theorems here, so it serves only as illustration). $\square$

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8.2 Theorem H: the full segment under a degenerate initial run

Theorem H. Preconditions: the common preconditions; $M>0$; $C_0\le2$. No house-style clause is imposed. Put $E=\{e_1<\cdots

Corollary H.1. $G_{\rm cyc}=1$, $B=2M+1$, $\Delta H=0$. If in addition all preconditions of Corollaries 5.3 [P4] and 5.4 [P4] are imposed (house-style clause and co-oriented profile included), then $G^{\deg}$ is defined and $G^{\deg}=G_{\rm cyc}=1$.

Corollary H.2. The conclusion does not depend on the choice of auxiliary selector, nor on sequentiality of the interfaces, nor does it require the non-degenerate shells to form a contiguous outer run, and it holds for every orientation profile.

Three lemmas

Lemma H.a (zero-account connectivity of the degenerate initial run). The subgraph induced on shells $0,\dots,d-1$ is connected and every move within it has account zero; hence between any two cells $u\neq v$ of shell $d$ there is a path whose interior is nonempty, lies entirely in shells $0,\dots,d-1$, and has account zero.

Proof. By the definition of $d$, shells $0,\dots,d-1$ are automatically all degenerate; no house-style clause is needed for this. Connectivity by induction on $k$: shell $0$ is connected ($C_0=1$, a single cell; $C_0=2$, a unique edge); assuming the union of shells $0,\dots,k-1$ is connected with $k\le d-1$, shell $k$ is degenerate hence connected in itself, and each of its cells has at least one inward parent in shell $k-1$, so the union stays connected. Accounts: all shells of the initial run are degenerate, so the orientation projection of a tangential step vanishes, and radial steps have account zero. Every cell of shell $d$ has at least one inward parent in shell $d-1$. $\square$

Lemma H.b (snake cycles). For every $j\in E$ and every $t\in\{0,\dots,C_j-1\}$ there is a legal occupancy-identity single cycle whose account is exactly $t$ and whose support lies in shells $0,\dots,j$.

Proof. For $t=0$: take the transposition along any radial edge at interface $d$, that is, of some cell of shell $d$ with its inward parent in shell $d-1$. Since $C_0\le2$ we have $d\ge1$, so interface $d$ exists and carries radial edges; the transposition has account zero and is supported on shells $d-1,d$, and since $j\ge d$ it lies in shells $\le j$.

Now let $t\ge1$. Take a cell $b$ of shell $j$ and set $b'=b+t$ prograde; since $t+1\le C_j$, the cells $b,\dots,b'$ are distinct. Choose an arbitrary auxiliary selector $q_k$ at each level and descend inward along it, obtaining two chains, each descending to shell $d$; the passages through intervening degenerate shells are radial steps as well, of account zero. Since $q$ is a function, once the two chains meet they coincide from the meeting point onward.

Case (a): the two chains first meet at a cell $w$ of some shell $i Take the cycle: ascend from $w$ along the chain of $b$ up to $b$, then $t$ prograde tangential steps to $b'$, then descend along the chain of $b'$ back to $w$; $w$ is the closing point and is not listed twice. The cells are distinct: before the first meeting the two chains are disjoint, and both lie in shells $

Case (b): the two chains do not meet, ending at distinct cells $a\neq a'$ of shell $d$. Close the cycle by the zero-account path of Lemma H.a from $a'$ back to $a$. The cells are distinct and the account is $t$. For $j=d$ the two chains are trivial and case (b) applies directly. $\square$

Lemma H.c (disjoint unions of rigid rotations). The prograde rigid rotation of a whole shell $j\in E$ is a legal occupancy-identity cycle of account $C_j$ (Theorem 4.1 [P4]). The source sets and target sets of rotations of distinct shells are pairwise disjoint; in their union every move still runs along an incident edge, with at most one departure per source and at most one arrival per target, and under full saturation the final occupancy is again one everywhere — so by the four clauses of Definition 2.2 [P4] the rotations of any subset compose into a single legal synchronous transfer. $\square$

Proof of Theorem H

Put $S_r:=\Sigma(\{C_{e_{r+1}},\dots,C_{e_s}\})$ for the set of subset sums (zero included), $M_r:=\sum_{i>r}C_{e_i}$, and $M_0=M$.

Snake cycles live in shells $\le e_r$ and outer-shell rotations in shells $>e_r$; their supports are disjoint, so they compose in one tick, giving $$F_r:=\{0,1,\dots,C_{e_r}-1\}+S_r\ \subseteq\ \operatorname{Spec}\quad(1\le r\le s),\qquad \Sigma(\{C_{e_1},\dots,C_{e_s}\})\ \subseteq\ \operatorname{Spec}.$$

Put $T_r:=\bigl(\bigcup_{i\ge r}F_i\bigr)\cup\Sigma(\{C_{e_r},\dots,C_{e_s}\})$. By downward induction on $r=s,\dots,1$ we show $T_r\supseteq[0,M_{r-1}]$.

Base $r=s$: $T_s=\{0,\dots,C_{e_s}-1\}\cup\{0,C_{e_s}\}=[0,C_{e_s}]=[0,M_{s-1}]$.

Step: assume $T_{r+1}\supseteq[0,M_r]$. Since $T_r\supseteq T_{r+1}$, only $(M_r,\,M_r+C_{e_r}]$ needs covering. Take $k=M_r+m$ with $1\le m\le C_{e_r}$. If $m\le C_{e_r}-1$ then $M_r\in S_r$ (take the full subset), so $k\in F_r$. If $m=C_{e_r}$ then $k\in\Sigma(\{C_{e_r},\dots,C_{e_s}\})$.

Taking $r=1$ gives $[0,M]\subseteq\operatorname{Spec}$; the negative side is the $\rho$ mirror; Proposition 4.3 [P4] closes the bound. $\square$

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8.3 Theorem I-a: the second value from the top

Criterion D. An onion graph satisfies Criterion D when there exist a non-degenerate shell $e$ and a cell $x\in\mathbb Z_{C_e}$ such that between $(e,x)$ and its prograde successor $(e,x{+}1)$ there is a path whose interior is nonempty and lies entirely in degenerate shells.

Theorem I-a. Preconditions: the common preconditions; $M>0$. No house-style clause is imposed. Then $$M-1\in\operatorname{Spec}(\ell_{\mathbf 1})\iff\text{Criterion D holds}.$$

Two facts about permutations (used twice below). First, cut balance: for any set of cells $S$, the number of moves of the permutation from $S$ into $S^c$ equals the number from $S^c$ back into $S$. Second, orbit closure: every moved cell of a finite permutation lies on a unique closed chain.

Proof, necessity. Suppose $\ell_{\mathbf 1}(\theta)=M-1$. By Lemma J, $\nu^{\rm rad}(\theta)\le1$.

Suppose $\nu^{\rm rad}=0$. Fix a non-degenerate shell $e$: none of its cells emits a radial move, so all outgoing edges are tangential within the shell; by cut balance the number of edges entering shell $e$ equals the number leaving, and the entering ones are tangential-within plus radial-in, while tangential-within moves have equal in- and out-counts, so the radial-in count is zero. Hence all moves of shell $e$ are tangential within it and the occupancy is identical, so Theorem 4.1 [P4] gives $\ell_e\in\{0,\pm C_e\}$. Then $M-\ell_{\mathbf 1}=\sum_{e\in E}(1-\epsilon_e)C_e$, each term lying in $\{0,C_e,2C_e\}$ with $C_e\ge3$, so the sum is zero or at least three, never one. Contradiction; hence $\nu^{\rm rad}=1$.

When $\nu^{\rm rad}=1$ the equality case of Lemma J holds, so shell by shell $\ell_j=T_j=C_j-\nu^{\rm rad}_j$: every non-degenerate shell cell that emits no radial move emits one prograde tangential step. Let $(e,x)$ be the cell emitting the radial move. The $C_e-1$ tangential steps of shell $e$ have as sources all cells other than $x$, and their targets are the prograde successors of those sources, covering the whole shell except $(e,x{+}1)$. Now $(e,x{+}1)$ emits a prograde tangential step, so it is not fixed and must receive some move; that move is not tangential, hence radial, and its source is not in a non-degenerate shell (else $\nu^{\rm rad}\ge2$), hence is a degenerate cell.

Now trace forward from $(e,x)$ along $\theta$. By orbit closure the orbit cannot terminate, nor can it merge into a small degenerate cycle disjoint from $(e,x)$. If the image of $(e,x)$ were a cell $y$ of a non-degenerate shell $f$, then $\nu^{\rm rad}_f=0$ would make every cell of shell $f$ receive one prograde tangential step, and $y$ receiving a radial move as well would violate at most one arrival per target; so the image must be a degenerate cell. By the same reasoning the images of the degenerate cells along the chain cannot be cells of a shell $f\neq e$, nor cells of shell $e$ that already receive a tangential step. Hence the first return to the non-degenerate region must land on the unique vacant target $(e,x{+}1)$. The interior of this chain is nonempty and lies entirely in degenerate shells — exactly the path required by Criterion D.

Proof, sufficiency. Take the path $(e,x)\to w_1\to\cdots\to w_k\to(e,x{+}1)$ of Criterion D, with all $w_i$ degenerate. Take the closed chain $$(e,x)\to w_1\to\cdots\to w_k\to(e,x{+}1)\to(e,x{+}2)\to\cdots\to(e,x),$$ give every other non-degenerate shell its whole-shell prograde rigid rotation, and leave the remaining cells fixed. The supports are disjoint, so the composite is legal by Definition 2.2 [P4]. The account is $(C_e-1)+\sum_{f\in E,f\neq e}C_f=M-1$. $\square$

Corollary I.1 (the house-style form). With the preconditions plus the house-style clause, $M-1\in\operatorname{Spec}\iff C_0\le2$.

Proof. When $C_0\le2$ the initial run is nonempty, and by Lemma H.a any two cells of shell $d$ are joined through it with account zero, so D holds. When $C_0\ge3$ the house-style clause makes all shells non-degenerate, so there are no degenerate cells and D fails. $\square$

The house-style clause is load-bearing in Corollary I.1; its isolating witness. Capacities $(3,2,4)$ satisfy the common preconditions and $M>0$, and violate the house-style clause only ($E=\{0,2\}$ is not a contiguous outer run), while $C_0=3>2$ and $M-1=6\in\operatorname{Spec}$; the biconditional of Corollary I.1 is false there. The clause therefore cannot be dropped.

Corollary I.2 (the boundary of the dependence). Criterion D depends in general on the parent map, and its dependence splits into three regions.

Region (a), provable. If there is a degenerate shell $jholds always, independently of the parent map. Proof: take prograde-adjacent cells $x,x{+}1$ of shell $j+1$, whose inward parents $q_x,q_{x+1}$ lie in shell $j$; shell $j$ being degenerate, its cells are either the same cell or joined by the unique tangential edge; the path $x\to q_x\to(q_{x+1})\to x{+}1$ has nonempty interior lying entirely in degenerate shells. $\square$

Region (b), provable. If there is no degenerate shell there is no degenerate cell, and D fails always, independently of the parent map.

Outside those regions, also provable. When all degenerate shells lie in the outer segment (that is, every degenerate shell $j$ has either $j=R$ or $C_{j+1}\le2$), both behaviours occur, each with an analytic witness.

First, an outermost shell of capacity two depends on the parent map. Let $C_R=2$ with $C_{R-1}\ge3$. If the two degenerate cells are given the same parent, then neither pair of prograde neighbours is joined and D fails; if their parents are prograde-adjacent cells, then the two degenerate cells are themselves adjacent and each meets one of the neighbours, so the path $x\to w_1\to w_2\to x{+}1$ exists and D holds. Witnesses: for $(3,4,2)$, $(3,3,2)$, $(4,4,2)$, the outer-shell parents $(0,0)$ give D false and $(0,1)$ give D true.

Second, the narrow clause: when the only degenerate shell is the outermost shell $R$, with $C_R=1$ and a single parent, D fails always. Here the only degenerate cell of the whole graph is that single cell, so the interior of a Criterion D path can only be it; being single-parented, it cannot meet two distinct prograde neighbours of a non-degenerate shell. Witnesses: for $(3,4,1)$, $(4,3,1)$, $(4,4,1)$, $(3,3,1)$ — families whose only degenerate shell is the outermost — D is false for every single-parent graph. The boundary of the narrow clause: if further degenerate shells lie inside an outermost capacity-one shell, this clause does not adjudicate. $(3,2,1)$ is the instance: give the two cells of shell one (capacity two) the prograde-adjacent parents $(0,1)$, and the path $(0,0)\to(1,0)\to(1,1)\to(0,1)$ has an entirely degenerate interior, so D holds and $M-1$ lies in the spectrum; with parents $(0,0)$, D fails. This branch depends on the parent map, and a complete characterization of general degenerate outer segments is an open problem (see Section 11.3).

The restriction of the second case: single parenthood is its precondition. With two parents the single cell acquires two fathers and can bridge; for $(3,4,1)$ and $(4,4,1)$ the double-parent graphs give D true.

Consequently. $C_0\le2$ implies region (a) (take $j=d-1$, so $C_d\ge3$); the house-style clause implies region (a) or region (b). Both are corollaries, not independent sufficient conditions. Instances inside regions (a) and (b) may be listed by capacities alone; outside them the full parent map must be listed.

$M-1$ does not entail a full segment. The three families below have $M-1$ in the spectrum while the full spectrum has holes, so the conclusion of Theorem I-a does not automatically yield a full segment. All three lie outside the regions of Corollary I.2, so their full parent maps are listed (the accompanying default sequential single parent):

Capacities	Interface-one primary parent $p_1$	Interface-two primary parent $p_2$	$M$	Spectrum points	$M-1$
$(3,4,2)$	$(0,0,1,2)$	$(0,3)$	$7$	$13$	in spectrum
$(7,5,2)$	$(0,3,4,5,6)$	$(0,4)$	$12$	$21$	in spectrum
$(4,4,2)$	$(0,1,2,3)$	$(0,3)$	$8$	$13$	in spectrum

By contrast, the floor single parent. For $(3,4,2)$ take $p_1=(0,0,1,2)$, $p_2=(0,2)$; for $(7,5,2)$ take $p_1=(0,1,2,4,5)$, $p_2=(0,2)$; for $(4,4,2)$ take $p_1=(0,1,2,3)$, $p_2=(0,2)$. In all three, $M-1$ is not in the spectrum. Capacities therefore do not determine the conclusion. The three families have $C_0=3,7,4$ respectively, all violating $C_0\le2$, so they are not isolating witnesses for the house-style clause and must not be used to argue that the clause is load-bearing in Theorem H.

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8.4 Theorem I-b: the third value from the top

Theorem I-b. Preconditions: the common preconditions; $M>0$; all shells non-degenerate ($\min_jC_j\ge3$). Then $$M-2\in\operatorname{Spec} \iff \exists\,j,\ b\in\mathbb Z_{C_j},\ q_b\in\mathrm{par}_j(b),\ q_{b-1}\in\mathrm{par}_j(b{-}1):\quad q_b\equiv q_{b-1}-1\ (\mathrm{mod}\ C_{j-1}),$$ where $b-1$ is taken with respect to $\omega_j$ and the $-1$ on the right with respect to $\omega_{j-1}$.

Proof, necessity. All shells being non-degenerate, there are no degenerate cells, so by Lemma J' case one every closed chain containing a radial move contributes at least two. If $\theta$ has no radial move, then as in the proof of Theorem I-a one gets $M-\ell_{\mathbf 1}\in\{0\}\cup[3,\infty)$, never two. So $\theta$ contains radial moves and $\nu^{\rm rad}\ge2$; since $\ell_{\mathbf 1}\le M-\nu^{\rm rad}$, in fact $\nu^{\rm rad}=2$.

By Lemma J, $d_j\ge0$; and $\sum_jd_j=\sum_jC_j-\sum_j\nu^{\rm rad}_j-\sum_j\ell_j=M-2-(M-2)=0$, so $d_j=0$ shell by shell, and the two equality conditions of Lemma J hold simultaneously in every shell.

With $\nu^{\rm rad}=2$, the radial moves come in a pair crossing the same interface (crossing two interfaces would make the total at least four), so exactly one outward and one inward, with sources in shells $j-1$ and $j$ respectively, the other shells each contributing a whole-shell rotation, together supplying $M-C_{j-1}-C_j$. The $C_{j-1}-1$ prograde tangential steps of shell $j-1$ cover the shell; if their path runs from $a$ to $a-1$, then $a-1$ is the radial emitter and $a$ the cell receiving no tangential step; likewise in shell $j$, if the path starts at $b$, the radial emitter is $b-1$. The two radial moves are therefore $(a-1)\to b$ and $(b-1)\to a$, so $a-1\in\mathrm{par}_j(b)$ and $a\in\mathrm{par}_j(b-1)$.

Proof, sufficiency. Put $a:=q_{b-1}$, so that $q_b\equiv a-1$. Take the closed chain: in shell $j-1$ go prograde from $a$ to $a-1$, then radially to $b$; in shell $j$ go prograde from $b$ to $b-1$, then radially back to $a$; give every other shell its whole-shell prograde rigid rotation. The account is $(C_{j-1}-1)+(C_j-1)+(M-C_{j-1}-C_j)=M-2$. $\square$

Corollary I.3 (single parent). When all interfaces are single-parented, the criterion reads $\exists\,j,b:\ p_j(b)-p_j(b-1)\equiv-1\ (\mathrm{mod}\ C_{j-1})$. This form does not depend on sequential lifting, and is invariant under cyclic-shift relabelling that preserves each shell's given ring orientation; its signs remain relative to the chosen orientation profile and are not claimed to be coordinate-independent in any absolute sense.

Corollary I.4 (single parent, sequential, non-constant primary parent). The lift and the increments are unique, and $p_j(b)-p_j(b-1)\equiv g^{(j)}_{b-1}\pmod{C_{j-1}}$ (for $b=0$ via the periodic lift identity $\tilde p_j(C_j)=\tilde p_j(0)+C_{j-1}$), with $0\le g^{(j)}_{b-1}\le C_{j-1}$; since $C_{j-1}\ge3$, the congruence $\equiv-1$ forces $g^{(j)}_{b-1}=C_{j-1}-1$ (the value $g=C_{j-1}$ gives $\equiv0$, and the two are not conflated). The criterion therefore reads: some increment equals $C_{j-1}-1$.

Corollary I.5 (two parents, sequential, non-constant primary parent). If $\mathrm{par}_j(i)=\{p_j(i),p_j(i)+1\}$, the criterion becomes $g^{(j)}_{b-1}+\delta_b-\delta_{b-1}+1\equiv0$ with $\delta\in\{0,1\}$, and so relaxes to $\exists\,i:\ g^{(j)}_i\in\{0,\ C_{j-1}-2,\ C_{j-1}-1\}$.

Sharpness of the jurisdiction: the single-parent clause is load-bearing. The default double-parent increments of capacities $(3,4)$ are $(0,1,1,1)$, none equal to $C_0-1=2$, yet $M-2=5\in\operatorname{Spec}$; the same family with a single parent has $5\notin\operatorname{Spec}$. The families $(4,5)$ and $(5,6)$ behave the same way. The increment form of Corollary I.4 therefore fails under two parents, and one must use Corollary I.5 or the parent-set form of Theorem I-b.

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8.5 A unified reading

Lemma J supplies one piece of bookkeeping: $\ell_{\mathbf 1}\le M-\nu^{\rm rad}$. Here $M$ is the tangential budget allotted by the geometry and $\nu^{\rm rad}$ is the toll paid for crossing shells. What the two theorems share is not a "common parent" but the low-account radial closer: the structure a closed chain needs in order to leave a shell and return at the lowest possible toll. Three levels follow.

Toll zero. Each shell closes within itself, and the spectrum is the set of signed subset sums of the shell rotations.

Toll one. Achievable only by closed chains adjacent to a degenerate shell (Lemma J', case two). The necessary and sufficient condition for cashing it in is exactly Criterion D. This level delivers $M-1$. A degenerate shell need not sit at the core: under the house-style clause it must lie inward, but outside that clause it may lie in the middle or at the outside (the three regions of Corollary I.2).

Toll two. The parent sets must supply a retrograde adjacency difference $q_b\equiv q_{b-1}-1$, that is, two shells joined each by a prograde path missing one edge. This level delivers $M-2$.

A full segment does not follow from the toll inequality alone. Theorem H relies further on the connectivity of the degenerate initial run, which lets the snake cycle for every $t\in\{0,\dots,C_j-1\}$ close, and on the meshing of the snake-cycle family with the subset sums of the outer-shell rotations. The holes cluster at the top, because the values near the top are the most sensitive to the toll; and whether the toll can be paid is legislated jointly by the location of the degenerate region, the structure of the parent sets, and the chosen orientation profile.

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9 Problem G (Packing): The Current Status

Problem G (packing). Inherited from Problem 9.1 [P4]: the complete criterion for $\Delta H=0$, that is, when the information capacity of the transfer layer saturates its geometric upper bound. The sufficient direction is settled (Theorem H); the necessary direction has two exact criteria (Theorems I-a and I-b); the full solution remains Open: $(4,4,4)$ has all shells non-degenerate and yet $\Delta H=0$, so a degenerate initial run is not a necessary condition for the full segment. Until proved, it is called a problem or a conjecture, never a theorem. [Open]

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10 The Two-Way Split under a Declared Measure

This section supports the preceding theorems with exact moments under a declared measure, following the conventions of Section 8 of [P4]: every conclusion stands relative to its measure, no measure is natural, computations are C1-P, and all readings are T2.

Measure declaration. The universe is the complete occupancy-identity library of the family in question (purged), the measure is uniform on the library with equal weight per transfer; the weight norm is $\alpha=\mathbf 1$ and the profile co-oriented. Every exact value is obtained by direct enumeration, never by sampling.

Table 10.1 (the two-way split statistics of three families; uniform transfer measure).

Family	$N_{\rm all}$	$P(\ell_{\mathbf 1}>0)$	$P(\ell_{\mathbf 1}=0)$	$H_{\rm Sh}(\ell_{\mathbf 1})$	$\log_2	\operatorname{Spec}	$	$\pi_\times$
Default double-parent $(1,3,4)$	$1995$	$0.3694$	$0.2612$	$2.8755$	$3.9069$	$0.7388$
Intermediate-band example $(7,5)$	$1947$	$0.1073$	$0.7853$	$1.2899$	$3.1699$	$0.2147$
Equal-capacity single-parent uniform tower $(4,4,4)$	$6344$	$0.1885$	$0.6230$	$1.4566$	$2.8074$	$0.3770$

Three readings, each naming its measure, all at tier T2.

First. Under the uniform library measure, the conditional distribution of the sign satisfies $P(\sigma=+\mid\ell_{\mathbf 1}\neq0)=\tfrac12$ exactly, family by family — the measure-theoretic image of the even split of Theorem C, part two; the equality comes from the free action of the sign involution, not from any separately posited symmetry.

Second. Under the same measure, $H_{\rm Sh}(\ell_{\mathbf 1})<\log_2|\operatorname{Spec}|$ strictly for every family, the gap running from $1.03$ to $1.88$ bits: the counting entropy measures the cardinality of the range, while the Shannon entropy measures the concentration of the declared measure across the whole spectrum. Their difference comes from the overall non-uniformity of the spectral probabilities: the zero account is the principal point of concentration but does not exhaust the gap — holding $P(\ell_{\mathbf 1}=0)$ fixed, the non-uniformity among the nonzero spectral values contributes a further $0.766$, $0.105$, and $0.474$ bits respectively. This gap and the lattice bound of Theorem F (a range-side quantity) are two different objects and never substitute for one another.

Third. Under the same measure, $\pi_\times=P(\ell_{\mathbf 1}\neq0)$, the measure-theoretic image of the nonzero-sign fraction of Theorem C, part three; the values $0.74$, $0.21$, $0.38$ correspond family by family to their degenerate structure (only $(1,3,4)$ has a degenerate core). No claim is made for general graph families.

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11 Prohibitions Discharged, Negative Results, and Open Problems

11.1 Discharging the seven prohibitions

Where the seven prohibitions of Section 1.3 are honoured in the body: the first and fifth, in that "information" is combinatorial throughout (Definitions 2.2 and 2.7); the second and sixth, in that every mention of reconstruction is a sufficiency statement (the form of Theorems A through E); the third, in that protocols are function classes only (Definition 2.5); the fourth, in that all rates are per-tick (Definition 2.4); the seventh, in that every blindness claim carries its triple and every strictness claim its witness (throughout Theorems C and D).

11.2 The negative results in full (rejections are front-stage content)

First. The general strictness condition for link two is that the library contain two transfers with distinct net currents and equal shell accounts; on a single tangential ring the pair is unattainable for $v\le1$ and attainable for $v\ge2$ (the edge-flow argument) — the threshold statement applies to that sub-model only. Multi-shell saturated libraries supply the witness through mixed radial cycles at zero vacancies (forty of them in the double-parent $(1,3,4)$).

Second. At $v=C-1$ the spectrum collapses to $\{\pm1\}$ and zero is unattainable; the reading "the number of spectral points is monotone in the vacancies" is rejected (the caution of Theorem B2).

Third. The neutral fraction $q$ bears no criterion relation to the entropy deficit $\Delta H$: over twenty-one families, those with zero deficit have $q\in[0,0.5375]$ and those with positive deficit have $q\in[0.1277,0.5957]$, and the intervals overlap; counter-witnesses on both sides are $(4,4)$ ($q=0.5375$ with zero deficit) and the double-parent $(3,4)$ ($q=0.1277$ with positive deficit); the apparent correlation is spurious, induced by $G_{\rm cyc}$ entering both. The same holds for the nonzero-sign fraction $\pi_\times$ against $\Delta H$. The reading "the dilution on the spin side and the deficit on the amplitude side have a common source" is rejected; the counting side and the spectral side have different sources.

Fourth. On the spin-clamped library $\mathsf A_1$ [P4] the first three links collapse entirely (Theorem D, part three): strictness is not a structural property but a property of the library.

Fifth. Attainability of $M-1$ does not entail a full segment (the three families of Section 8.3); and "some shell is degenerate" is not the criterion for $M-1$ (seven counter-families; the criterion is D).

Sixth. Divisibility completeness does not entail packing tightness (Caution three of Definition 2.10): $(7,5)$ has $G_{\rm cyc}=1$ with a deficit of sixteen holes.

Seventh. The claim "an outermost shell of capacity one with a single parent makes D always false" is untrue: in $(3,2,1)$ the inner degenerate shell (capacity two) bridges the prograde neighbours, and with parents $(0,1)$ criterion D holds. The clause is narrowed to the only degenerate shell being the outermost with $C_R=1$ and a single parent; the remaining branches depend on the parent map and are moved to the open list (Section 11.3).

11.3 Open problems

First, the necessary direction of the full segment: $(4,4,4)$ has all shells non-degenerate and yet $\Delta H=0$, so a degenerate initial run is not necessary, and the complete criterion is Open (Problem G). Second, the mixed case of Theorem I-b (asking after $M-2$ when degenerate shells are present) is not established. Third, $M-3$ and deeper structure at the top has not been asked. Fourth, the complete characterization outside the regions of Corollary I.2 is not established. What is proved: an outermost shell of capacity two depends on the parent map; the narrow clause (the only degenerate shell being the outermost with $C_R=1$ and a single parent) is always false, with single parenthood as its limit (two parents can flip it to true); multiple degenerate outer shells (as in $(3,2,1)$) depend on the parent map, and the general criterion is Open. Fifth, a general proof of orientation invariance is not established (over all tested families the three theorems hold for every orientation; C1-P). Sixth, for the nonzero-sign fraction $\pi_\times$ on $(1,C)$, the general threshold, monotonicity, and limit are unproved (computed for $C=3..10$: the first two below two-thirds, the remaining six above, and the finite sequence increasing term by term; deposited at T3).

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Appendix A: The Certificate Suite

Two scripts, with different load-bearing targets but sharing a common basis of graph construction, parent maps, and simple-cycle enumeration, and therefore not claimed to be independent implementations (spectra by bitmask dynamic programming, counts by disjoint-cycle packing; their aggregation and assertions differ); each lists capacities and full parent maps family by family; each reports its own per-section tally, against which every figure in the body must be checked; all figures are C1-P and will be promoted to C1-R upon recomputation by a third implementation.

spec_probe.py (capacity structure) SHA-256 e6d5ff4720a759fc551c7fc93a7805e29d3315ba62079d348111af3604cfc7da; one standard run gives 494 assertions with zero failures. Per section: model validation, five checks, 18; Theorem H, 52; $M-1$ does not entail a full segment, 3; Theorem I-a, 128; Theorem I-b, 44; filling the gap, 3; orientation profiles, 73; H without the house-style clause, 82; the three-region dependence, 21; primary parents of the three families, 12; witnesses outside the regions, 18; the two lattice spacings, 40.

info.py (main axis) SHA-256 e849867bf36bb5113dd2bd93e3b6ccd62449d20da23f2392f9e19eb9f35b3d7c; one standard run gives 130 assertions with zero failures. Per section: the single-ring Lucas closed form, 8; $\rho$-closure and sign reversal, 9; scalar split and even signs, 9; the condition for one bit, 6; the trichotomy of blindness, 6; the nonzero-sign fraction $\pi_\times$, 15; the fixed-$v$ spectrum of B2, 18; the fiber trichotomy of B3, 16; the outer box of A and B1, 5; the three-link witnesses of D, 15; Table 1, 4; the positioning of E, 3; the six regression groups of the whole-paper review, 16.

Appendix B: Cross-Checked Figures (principal items)

Model validation (against the published figures of Appendices C and D of [P4]). Default double-parent $(1,3,4)$: 18 undirected edges, 18 transpositions, 624 directed cycles, 642 single components, 1995 legal composites after purging; scalar split $1995=521+2\times737$; vector split $1995=357+2\times819$; radial-free library $53=27+2\times13$ (the two readings coincide); the radial-free spectrum misses $\pm2,\pm5,\pm6$; $(1,2,8)$ full segment to amplitude eight; $(1,3,4,5)$ full segment, twenty-five points; $(7,5)$ actual spectrum $\{0,\pm2,\pm5,\pm7,\pm12\}$ with sixteen holes.

Capacity structure. Theorem H: twenty-six families in two parent modes, fifty-two cases, all full segments; twenty families violating the house-style clause, three parent modes and three orientations, two hundred and ten cases, all full segments; $(2,12,4)$ with tail-segment spacing two but global spacing one, thirty-three points, full segment. Theorem I-a: criterion D agrees in one hundred and twenty-eight checks over thirty-nine families, with seven families such as $(3,4,1)$ flipping between single and double parents. Theorem I-b: forty-four cases in agreement. Orientation profiles: $(7,5)$ nine points co-oriented against twenty-one counter-oriented; $(12,4)$ fifteen points co-oriented ($\Delta H\approx0.181$, holes exactly at $\pm14$) against nine counter-oriented ($\Delta H=0$). $G^{\deg}=G_{\rm cyc}$: thirty cases, no disagreement.

Main axis. Single ring, $C=3..10$: $N_{\rm all}=L_C+1$ and $N_+^{\mathbf 1}=1$ exactly, value by value. $\pi_\times$ on $(1,C)$, full list $C=3..10$: $0.60870$, $0.65385$, $0.67290$, $0.69194$, $0.70324$, $0.71237$, $0.71882$, $0.72391$. Scalar splits and even signs of the three families: $1995=521+2\times737$, $1947=1529+2\times209$, $6344=3952+2\times1196$. Entropies under the wrong condition: $1.3693/1.3776/1.5010$. The three targets of B3 ($C=3..6$): signed $1.2224/1.5850/1.8745/2.1155$; amplitude $1.0000/1.3219/1.5850/1.8074$; spin sign identically zero. Link-two double-three-cycle witnesses in the saturated $(1,3,4)$: forty. $(3,2,1)$: parents $(0,1)$ give D true, parents $(0,0)$ give D false. A declared sub-library of spacing $\{\pm7\}$ against graph-level spacing one. Link one: $J^\omega=0$ in 317 transfers (27 purely tangential, 119 purely radial). Link three: 164 neutral transfers, of shell-account type $(0,+1,-1)$. The witness of Proposition 6.1 [P4], part two ($C=5$, tokens $\{0,3\}$), reproduced verbatim. Table 1: $15/\log_2 14$, $7/\log_2 3$, $3/3$.

Appendix C: Erratum to Lemma 4.5 [P4] (the erratum itself)

This appendix is the erratum, published together with the present paper; the Zenodo record of [P4] carries no separate note and is not versioned.

Object of the erratum. Angular Transport in Discrete Black-Hole Interiors: Process Accounts, Spin-Order Lattices, and the Rigid-Rotation Spectrum, final text bhp4-final-CN.md, SHA-256 66b25180a606e47d39d91f93542629b870ca22429ad970a516fbf55c5838198d. Two places are affected: the defining sentence of Lemma 4.5 in Section 4, and its restatement in Appendix C.

The error. The text of Lemma 4.5 reads "$N_0$ is the number of transfers whose account vector vanishes, and $N_+$ the number with positive account". The published figures it cites, $1995=521+2\times737$ (the full default double-parent library of capacities $(1,3,4)$), are in fact the split by scalar account: exact enumeration gives 357 transfers with vanishing account vector and 521 with vanishing scalar account. Both counting identities hold in their own reading ($1995=357+2\times819=521+2\times737$); what is wrong is only the mismatch between the defining sentence and the figures cited. Moreover, the phrase "positive account" requires an undeclared order under the vector reading, while under the scalar reading the sign of $\ell_{\mathbf 1}$ suffices.

The correction. The defining sentence of Lemma 4.5 should be read as:

> $N_0$ is the number of transfers with vanishing scalar account, that is $|\{\theta:\ell_{\mathbf 1}(\theta)=0\}|$; $N_+$ is the number with $\ell_{\mathbf 1}(\theta)>0$, the two signs being dual under the sign of $\ell_{\mathbf 1}$; the weight norm in this instance is the edge step $\alpha=\mathbf 1$.

The published figures of Appendix C stand unchanged.

Scope of the impact. The conclusions of Lemma 4.5 — the involution property, the counting identity, the characterization of fixed points, the inclusion $\mathrm{Fix}(\rho)\subseteq\{\boldsymbol\ell=\mathbf 0\}$ and its strictness condition — are independent of which reading $N_0$ takes and are unaffected. The radial-free library's $53=27+2\times13$ coincides under both readings and is unaffected. The figures 521 and 737 occur nowhere in that paper except the note to Lemma 4.5 and Appendix C, with no downstream use. The erratum is therefore local and alters the statement or proof of no theorem, proposition, or corollary.

Basis for recomputation. The vector split $1995=357+2\times819$ is a figure newly recorded by this erratum and stands alongside the scalar split; the two splits are landed by two certificates with different load-bearing targets and separate assertions but a shared underlying construction basis (spec_probe.py SHA-256 e6d5ff4720a759fc551c7fc93a7805e29d3315ba62079d348111af3604cfc7da; info.py SHA-256 in Appendix A); neither substitutes for the other, and no claim of implementation independence is made.

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References

[P1] Han Qin. Area-Level Plateaus and Self-Sufficient Windows in the Frozen-Ledger Model: A First-Principles Construction. Zenodo, DOI 10.5281/zenodo.20913514.

[P2] Han Qin. Discrete Black-Hole Interiors: Nine Adjudicated Boundary Stakes. Zenodo, DOI 10.5281/zenodo.21499053.

[P3] Han Qin. Discrete Black-Hole Interiors: State-Layer Entropy and the Multiplicity of the Occupancy Quotient. Zenodo, DOI 10.5281/zenodo.21521760.

[P4] Han Qin. Angular Transport in Discrete Black-Hole Interiors: Process Accounts, Spin-Order Lattices, and the Rigid-Rotation Spectrum. Zenodo, DOI 10.5281/zenodo.21569768.

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