Strong CP and DD Levels: A Mirror No-Go, and the Boundary-Condition Status of θ̄
强CP与DD层级：一条镜像no-go，与θ̄的边界条件地位
The genuine target of the strong CP problem is the rephasing-invariant combination $\bar\theta = \theta + \arg\det(M_u M_d)$, experimentally bounded by $\bar\theta < 10^{-10}$. The established responses — axion/Peccei-Quinn, Nelson-Barr, and parity-symmetric constructions — all operate inside the Standard Model framework.
A tempting alternative runs as follows: if a theory possesses an exact parity symmetry, then $\theta = -\theta$ and therefore $\theta = 0$. In frameworks that possess a two-sided structure, this is often expressed as "the two sides are exact mirror images of one another, hence parity is exact."
The principal result of this paper is a no-go: that route does not work. If two sectors are causally disconnected and parity exchanges them, then parity has an empty fixed-point set. It constrains the combined system, not either sector individually. What follows is at most a cross-side pairing $\theta_L = -\theta_R$; it is not $\theta_L = 0$. Moreover, even the retreat to the pairing relation is not delivered: the two ledger entries composing $\bar\theta$ each migrate under chiral basis changes, so their cross-side transformation law must be established separately and has not been. The no-go is not specific to Self-as-an-End (SAE); it applies to any construction that realises parity as a map between two non-communicating sectors.
The positive result of this paper is a decoupling. SAE's constitutive ordering assigns mass construction to the 3DD level and weak mixing to the 2DD level. The Yukawa coupling is accordingly not a fundamental object but a low-energy projective superposition of these two levels, so the CKM complex phase does not flow into $\det(M)$: $\bar\theta$ and CKM CP violation are mutually unentangled. This removes half of the tension that makes the Standard Model version of the problem sharp — the half that asks why one and the same Yukawa object yields an $\mathcal{O}(1)$ CKM phase alongside a $\bar\theta$ below $10^{-10}$. It does not remove the other half, namely why $\bar\theta$ is itself anomalously small.
This paper therefore registers $\bar\theta$ as a boundary parameter: a dimensionless, anomalously small input that SAE does not currently explain. SAE does not resolve strong CP naturalness, neither predicts nor excludes the axion, and offers no independent numerical prediction for the neutron electric dipole moment. Experiment merely constrains this boundary parameter.
Keywords: strong CP problem, θ̄, DD levels, mirror no-go, Yukawa emergence, boundary conditions
§1 The Correct Target of the Strong CP Problem
§1.1 θ̄, not bare θ
The QCD Lagrangian admits a topological term:
$$\mathcal{L}_\theta = \frac{\theta}{32\pi^2} g^2 F^a_{\mu\nu} \tilde{F}^{a,\mu\nu}$$
This term violates P and CP. A nonzero θ endows the neutron with an electric dipole moment $d_n \propto \theta$. The experimental bound $d_n < 1.8 \times 10^{-26}$ e·cm (90% CL) translates into $\bar\theta < 10^{-10}$.
The target of the problem is not the bare θ but the rephasing-invariant combination
$$\bar{\theta} = \theta + \arg\det(M_u M_d)$$
where $M_u, M_d$ are the quark mass matrices. Chiral rotations shuttle phase between θ and $\arg\det(M)$, leaving $\bar\theta$ fixed.
This distinction carries weight throughout the present paper and is therefore stated first as a working discipline:
$$\boxed{\text{The individual values of }\theta_{\rm QCD}\text{ and }\arg\det M_q\ \textbf{depend on the choice of chiral basis};\ \text{the sole physical observable is }\bar\theta.}$$
Wherever this paper writes "$\theta = 0$" or "$\arg\det M = 0$", the statement is to be read as concerning a ledger entry in a specified basis, not as an independent physical claim. An argument that addresses only one of the two entries, without supplying the behaviour of the other, has not yet addressed the strong CP problem.
§1.2 Responses within the Standard Model
Axion / Peccei-Quinn. A global $U(1)_{\rm PQ}$ symmetry and its accompanying Goldstone boson (the axion) drive θ dynamically to the CP-conserving point. The mechanism introduces a new particle and a new symmetry. No axion has been found; searches continue (ADMX, IAXO, and others).
Massless up quark. If $m_u = 0$, θ can be rotated away by a chiral rotation. Precision lattice QCD has excluded $m_u = 0$.
Nelson-Barr. CP is an exact symmetry at high energies, spontaneously broken through a specific Yukawa/mediator structure so that $\bar\theta = 0$ at tree level while the CKM phase remains nonzero. The mechanism faces the technical difficulty of radiative regeneration: loop corrections tend to reignite $\bar\theta$.
Parity-symmetric constructions. Parity invariance sets the bare θ to zero, while $\arg\det(M)$ is controlled by Hermitian Yukawa structures or equivalent devices. These face the same radiative-stability difficulty.
What the four share. All operate within the Standard Model framework: they accept that the Yukawa coupling is a fundamental object and that $M = Yv$, then arrange parameters or introduce new particles inside that framework.
§1.3 What this paper delivers
This paper does not propose a new technical solution. It delivers three things.
A no-go (§3). If parity is realised as a mirror between two non-communicating sectors, it cannot yield $\bar\theta = 0$ for either sector individually. The exclusion holds for an entire class of approaches and does not depend on model details.
A decoupling (§4). SAE's level structure prevents the CKM complex phase from flowing into $\det(M)$. This dismantles half of the tension that makes the Standard Model version of the problem sharp.
A registration (§5). $\bar\theta$ is listed as a boundary parameter. SAE currently has no account of why it is anomalously small, and this paper says so explicitly rather than implying otherwise by omission.
§2 DD Levels and Parity
§2.1 The symmetry map of the chisel-construct sequence
The DD level structure:
| Level | Physical object | Characteristic quantity | Key operation |
|---|---|---|---|
| 1DD | Charge, marking | $E$ | Mark without construct (OR) |
| 2DD | Chirality, weak force | $p = E/c$ | Additive path (OR) |
| 3DD | Colour, strong force | $m = E/c^2$ | Multiplicative path (OR) |
| 4DD | Gravity, block structure | $m/c = E/c^3$ | AND closure |
The 2DD chiral splitting is the first explicit symmetry breaking in the chisel-construct sequence: it distinguishes L from R. That the weak force couples only to left-handed fields is the defining property of the 2DD level.
3DD is constructed atop 2DD. The L side and the R side each independently generate a 3DD colour structure.
§2.2 The exact condition under which parity gives θ = 0
Under P the θ-term flips sign: $\theta \to -\theta$. The standard conclusion drawn from this must have its condition stated precisely:
$$\boxed{\textbf{If P is an exact symmetry \textit{of a given theory}}\ \text{— i.e. P maps that theory's Lagrangian back to \textbf{itself} — then }\theta = -\theta,\text{ hence }\theta = 0.}$$
"P maps theory A to a different theory B" does not yield this conclusion, however exactly A and B mirror one another. The first is a symmetry (an automorphism); the second is a relation (a map between two objects). They resemble each other in form but differ in inferential power: invariance under an automorphism constrains the couplings of the theory itself, whereas invariance under a relation constrains only the combined system.
This distinction is the entire content of §3.
§2.3 The L-R 3DD mirror relation in SAE
Proposition S1 (L-R 3DD mirror relation). The 3DD structures produced on the two sides by the DD chisel-construct sequence are exact mirror images of one another.
Argument.
(a) There is only one kind of chisel (negation), and it does not distinguish direction. Every step of the chisel-construct sequence applies the same operation to the L side and the R side.
(b) Binary-label lemma. The 2DD chiral splitting is a 1-bit scalar operation (L or R). It transmits exactly one bit of information: which side is which. It transmits no orientation, no phase, and no other internal structure.
(c) The two sides each independently execute the spatial three-axis splitting. By (a) and (b), the resulting 3DD structures are exact mirror images of one another at the level of the whole.
(d) The remainder-conservation principle of ZFCρ provides independent mathematical support.
Scope of the binary-label lemma. What the lemma establishes is that the label space is binary: $\chi$ itself carries exactly one binary label bit. It does not follow that the actions, backgrounds, and couplings of the two sectors are exact mirrors of one another. Even with a single bit of labelling, one may still have $g_L \neq g_R$, $V_L \neq V_R$, $\mu_L \neq \mu_R$. The lemma therefore cannot by itself answer whether the two sides are exact mirrors. Its canonical home is the Physical-Quantity Ladder paper (.21227329), where it functions as a definition of label capacity and object type; the present paper cites it only.
The precise type of S1. S1 is a proposition about a relation between the two sides. It is not a proposition about a symmetry of either side:
| Statement | Type | Status |
|---|---|---|
| L-3DD and R-3DD are mirror images of one another | a relation between two sides | Near-theorem |
| The 3DD effective description on our side is invariant under P | a symmetry of one side | Does not follow from S1 |
A necessary clarification. S1 requires no axis pairing at the 4DD level. We do not claim that axis 1 on the L side equals axis 1 on the R side (the Generation Paper, G1, argues that no such canonical identification exists). The claim is only that L-3DD as a whole and R-3DD as a whole mirror one another — and it is precisely this "as a whole, of one another" that prevents S1 from delivering any constraint on a single side.
Level. Near-theorem (supported along two independent lines: ZFCρ and the DD axiom chain).
§2.4 Bidirectional information isolation between DD levels
The Generation Paper's G5 (Isospin Globality) establishes a parallel argument: colour (3DD) collapses to a colour singlet under cross-block projection, while isospin (2DD) survives intact. The reason is that isospin originates in the deeper, shared 2DD layer, and 3DD processes cannot destroy it.
The argument of §4 below runs in the opposite direction: 3DD structure does not inherit 2DD breaking. The 2DD chiral splitting introduces an L-R asymmetry — the origin of CP violation — but that asymmetry is encapsulated within 2DD and is not transmitted to 3DD.
Taken together, the two directions constitute a general principle:
$$\boxed{\textbf{Information isolation between DD levels is bidirectional.}}$$
Lower-level structure is not destroyed by higher-level processes (G5); higher-level structure does not inherit lower-level breaking (§4).
This principle is independent of the mirror relation of §2.3 and is upstream of §4.
§3 The Mirror No-Go
§3.1 Proposition N1
Three relevant facts about SAE's dual-4DD structure are already in print:
| Fact | Source |
|---|---|
| The dual 4DDs are causally disconnected (opposite time arrows); constraints cannot be imposed locally across a causal boundary — only global bookkeeping is available | Paper I, §5.4 |
| Our observable universe is one side of one dual-4DD pair within the left-handed 3DD branch | Prequel, §4.4 |
| 4DD light cones belonging to different branches or axes do not overlap; causal structures are independent. Right-handed gauge topology is isolated in the opposite chiral branch | Prequel, §5.1 and §7, Prediction 4 |
It follows that parity P maps our side's 3DD to the opposite side's 3DD, and the opposite side is causally inaccessible.
Proposition N1 (mirror no-go). Let $S_L$ and $S_R$ be two causally disconnected 3DD sectors, let P exchange them, and let P flip the sign of θ. Then what follows is at most the cross-side pairing relation
$$\theta_L = -\theta_R$$
and not $\theta_L = 0$.
Proof. The fixed-point set of P is empty: P exchanges two disjoint sectors. A transformation carrying $\mathcal{L}_L$ into $\mathcal{L}_R$ constrains, by its invariance, the combined system $\mathcal{L}_L \oplus \mathcal{L}_R$, not $\mathcal{L}_L$. Obtaining $\theta_L = 0$ requires a parity operation mapping $S_L$ into itself — that is, $S_L$ must be P-symmetric on its own. This is strictly stronger than "$S_L$ and $S_R$ mirror one another," and S1 does not supply it. □
Level. Theorem (conditional on causal disconnection, a structural fact of SAE).
§3.2 A second layer: even the global paired zero is not delivered
Suppose one grants $\theta_{{\rm QCD},L} = -\theta_{{\rm QCD},R}$. It still does not follow that
$$\bar\theta_L = -\bar\theta_R$$
because $\bar\theta = \theta_{\rm QCD} + \arg\det M_q$, and the transformation law of the quark mass matrices and of their determinant phase under the cross-side map has not been given. By the discipline of §1.1, $\theta_{\rm QCD}$ and $\arg\det M_q$ each migrate under chiral basis changes; their cross-side correspondence must be established separately and cannot be inherited from the transformation law of the bare θ alone.
$$\boxed{\text{Current status: the single-side zero is refuted; the global paired zero is undelivered.}}$$
§3.3 The scope of N1, and what it does not exclude
N1 is not specific to SAE. It applies to any construction that realises parity as a map between two non-communicating sectors. Any such construction seeking to resolve the local strong CP problem must separately specify how parity acts within the sector in question. The proof invokes a single purely structural fact — that the fixed-point set of P is empty — and depends on no model detail.
N1 does not close off every single-sided strong-CP mechanism. It proves none of the following:
- that the QCD sector cannot possess some other exact CP symmetry;
- that a UV completion cannot enforce $\bar\theta = 0$;
- that a Nelson-Barr-type mass-matrix mechanism is impossible;
- that the axion does not exist.
It refutes exactly one route: the causally disconnected mirror.
Falsification condition for N1 (internal to the framework; no experiment required). If it can be shown that SAE's structure admits a parity realisation mapping our side into itself — either because the two sides are not causally disconnected after all, or because some other parity operation carries L-3DD back to L-3DD — then N1 fails. This condition is tied directly to Paper I §5.4 and to Prequel §4.4 and §5.1: should the reading of "causal disconnection" at those three places be revised, the principal result of this paper must be re-examined.
§4 Yukawa Emergence: Decoupling CKM CP from θ̄
> This section depends neither on the mirror relation of §2.3 nor on anything in §3. It rests on the constitutive ordering of the Generation Paper (G7).
§4.1 Constitutive ordering
In the Standard Model the Yukawa matrix $Y$ discharges two functions at once: mass generation ($M = Yv$) and weak mixing (CKM arises from the mismatch in diagonalising $Y_u$ and $Y_d$). $Y$ is complex, so its complex phase enters $M$ and CKM simultaneously and the two cannot be separated. This is the root of the strong CP problem: one cannot retain CKM CP violation while setting $\arg\det(M) = 0$ without introducing additional machinery.
Yukawa emergence principle. Under SAE's constitutive ordering, the Yukawa coupling is not a fundamental object but a projective superposition, in the low-energy effective theory, of a 3DD mass structure and a 2DD mixing structure.
SAE's constitutive ordering (Generation Paper, G7) separates the two functions:
- Mass is constructed independently at the 3DD level. Topological barrier heights determine natural mass; colour-field redistribution determines actual mass. The definition of mass does not depend on the existence of the 2DD weak force — switch off the weak force, and quarks still have mass.
- Weak mixing is supplied independently at the 2DD level. The 2DD weak force reads out physical mass eigenstates already constructed by 3DD, generating inter-generational transition amplitudes. The CKM matrix is the output of that readout, not a by-product of diagonalising the mass matrix.
- The 2DD chiral splitting distinguishes L from R, permitting complex structure — CP violation — to reside in the weak vertex. This is the origin of the CKM CP violation observed in experiment.
- A low-energy experimenter observes mass and weak force together (a W boson simultaneously changes flavour and transfers momentum) and describes both with a single Yukawa matrix $Y$. $Y$ is complex because it superposes 3DD mass information with 2DD complex mixing information in one object. But $Y$ is not fundamental; it is a projective superposition of two distinct DD levels.
Hence, in SAE: $M$ comes from 3DD; $V$ comes from 2DD (a unitary matrix that may carry a complex phase). $V$ does not participate in the construction of $M$, and $V$ does not appear in the expression for $\det(M_u M_d)$.
Level. Structural argument, dependent on the G7 constitutive ordering.
§4.2 What the decoupling delivers, and what it does not
$$\boxed{\textbf{Delivered:}\ \text{CKM CP and }\bar\theta\text{ are decoupled.}}$$
One sharp form of the Standard Model's strong CP problem asks: the CKM phase is $\mathcal{O}(1)$ ($\delta_{\rm CP} \approx 1.14$ rad, $\sin\delta_{\rm CP} \approx 0.91$), so why is the co-originating $\bar\theta$ below $10^{-10}$? In SAE the co-origination does not hold — the two arise at different DD levels and are mutually unentangled. The question "why so disparate in magnitude given a common origin" therefore does not arise.
$$\boxed{\textbf{Not delivered:}\ \textbf{why }\bar\theta\textbf{ is small.}}$$
The decoupling shows only that the CKM phase does not flow into $\bar\theta$. It says nothing about why $\bar\theta$ itself takes an anomalously small value. $\arg\det M$ has an independent origin (§4.3), and SAE currently leaves it undetermined.
Two statements not to be conflated:
- "$\bar\theta$ is unrelated to CKM CP" — claimed by this paper
- "$\bar\theta$ is therefore zero or tiny" — not claimed by this paper
§4.3 The origin of arg det M
In SAE the quark mass matrix is constructed in two steps (Generation Paper, G3 and G6):
Step one: topological barriers give natural mass. Mass differences across generations arise from the topological distance between the observer block and the source block. The natural-mass matrix $D_{\rm nat}$ is real, positive, and diagonal, with $\det(D_{\rm nat}) > 0$.
Step two: the 3DD colour-field redistribution operator $R$ acts on $D_{\rm nat}$, giving $M = R \cdot D_{\rm nat}$.
Concerning $R$, SAE's structure supplies one constraint:
$\det R > 0$. $R$ deforms continuously from the identity: the chisel-construct sequence proceeds stepwise, and colour-field redistribution is not a single jump but a continuous evolution from "no redistribution" ($R = I$) to the actual redistribution strength. Since $\det(I) = 1 > 0$ and $\det R$ varies continuously along the deformation, $\det R$ cannot change sign without passing through zero. But $\det R = 0$ would drive some mass eigenvalue to zero — the oriented volume of the 3DD colour space collapsing, 3DD degenerating to 2DD — which contradicts the premise that 3DD has completed its construction, and is in any case excluded a posteriori (lattice QCD excludes $m_u = 0$). And $\det R < 0$ would require an inversion of 3DD, a reversal of spatial orientation; but 3DD is built inside a subspace whose chiral direction has already been anchored by 2DD, and possesses no degree of freedom with which to execute a global odd-parity reversal. Hence $\det R > 0$.
But $\det R > 0$ does not give $\arg\det M = 0$. The former says $R$ does not reverse orientation; the latter requires $R$ to be real.
$$\boxed{\text{The entire origin of }\arg\det M\text{ resides in the \textbf{phase structure} of the colour-field redistribution operator }R,\text{ which SAE currently leaves undetermined.}}$$
This is a concrete, computable object — considerably more concrete than "prove that some symmetry holds." It is the item on this paper's open list most likely to be settled first (§7.2).
§4.4 On radiative stability
Parity and Nelson-Barr constructions face a radiative-stability difficulty: a tree-level $\bar\theta = 0$ may be reignited by loop corrections. The reason is that they operate inside the Standard Model EFT framework — mass and weak mixing derive from a single Yukawa object, so loops can shuttle phase between them.
SAE's situation differs. The separation of $M$ from $V$ is not a tree-level approximation but a defining property of the DD level structure. DD levels are not a perturbative expansion of an effective theory: 3DD is not "tree-level 3DD plus loop corrections," any more than 1DD is "tree-level charge plus corrections toward 2DD." Levels are ontological structure, and the notion of "a quantum correction breaking a level assignment" has no referent.
What this argument protects is the decoupling of §4.2, not any zero. It shows that loops cannot transport the CKM phase into $\det(M)$, because the two belong to distinct DD levels rather than being two faces of one Yukawa object. This paper claims no tree-level $\bar\theta = 0$, and there is accordingly nothing for loops to reignite.
What SAE still owes is a systematic projection rule carrying stratified DD objects down to low-energy observables. §4.1 supplies a fragment of that projection, not the theory of it. This is an open problem for SAE's physics series as a whole, not for this paper alone (§7.2).
§5 The Status of θ̄
§5.1 A boundary parameter
Combining §3 and §4, SAE's honest statement about $\bar\theta$ is:
$$\boxed{\bar\theta\text{ is a dimensionless, anomalously small \textbf{boundary parameter} that SAE does not currently explain.}}$$
Four consequences, stated individually:
- SAE does not resolve strong CP naturalness. Why $\bar\theta < 10^{-10}$ when its natural value would be $\mathcal{O}(1)$ — SAE has no mechanism.
- SAE neither predicts nor excludes the axion. The Peccei-Quinn motivation is in SAE neither removed nor established.
- SAE offers no independent numerical prediction for the neutron electric dipole moment. Experiment merely constrains this boundary parameter.
- Neutron EDM experiments do not adjudicate SAE's strong-CP content. Whether or not a signal is found, SAE has placed no bet here.
§5.2 Its place in the boundary-condition table
The boundary-condition table of the Four Forces series requires an additional column. $\bar\theta$ differs in type from the existing entries:
| Column | Entry | Type |
|---|---|---|
| Unit / scale inputs | $\hbar$, $G_*$, $m_e$ | Scales; $\mathcal{O}(1)$ in natural units |
| Dimensionless coupling | $\alpha_{\rm em}$ | Dimensionless, $\sim 10^{-2}$ |
| Constrained CP angle | $\bar\theta$ | A dimensionless angle; natural value $\mathcal{O}(1)$, measured $< 10^{-10}$; and not a known exact value but a free parameter with an upper bound |
$$\boxed{\bar\theta\text{ is the only "unnaturally small" entry in that table.}}$$
To place a quantity in the boundary-condition table means "this framework does not derive it." It does not mean "this framework has explained why it takes that value." For $\hbar$, $G_*$, and $m_e$, failing to derive a scale is unremarkable — General Relativity likewise does not predict the numerical value of $G$. For $\bar\theta$, what goes underived is an anomalously small dimensionless number, and that is a heavier debt, to be recorded as such.
§5.3 Comparison with the cosmological constant
SAE has something to say about the cosmological constant's $10^{-122}$ and nothing to say about $\bar\theta$'s $10^{-10}$. The two stand differently, but neither is closed:
| $\Lambda$ | $\bar\theta$ | |
|---|---|---|
| Structural candidate | Yes (frequency difference of the dual topological clocks) | None |
| No-go audit | Yes (Cosmology Paper VII) | N1 of this paper |
| Realisation-level target relation | Yes ($\mathcal{N}^2\lambda k = 8$) | None |
| Parameter-free action-level realisation | Not closed | Not closed |
$\Lambda$ has a structural candidate and a no-go audit, though its realisation level remains unclosed; $\bar\theta$ does not even have a corresponding structural candidate. The two should not be described as standing at the same level.
§6 Comparison and Falsifiability
§6.1 Three classes of approach
| Axion/PQ | Parity/NB | SAE | |
|---|---|---|---|
| Framework | SM + new symmetry/particle | SM + parity/special structure | DD level ontology |
| Suppressing bare θ | Dynamical (axion field) | Parity symmetry | No mechanism |
| Suppressing $\arg\det M$ | Handled by the same axion field | Hermitian Yukawa, etc. | No mechanism |
| Origin of CKM CP | Complex Yukawa | Complex Yukawa (soft parity breaking) | Independent 2DD weak-vertex readout |
| CKM CP vs $\bar\theta$ | Co-originating | Co-originating | Decoupled (§4) |
| Value of $\bar\theta$ | Dynamically → 0 | 0 at tree level; loops may reignite | Boundary parameter, unexplained |
| Requires new particles | Yes (axion) | Possibly | No position taken |
| Radiative stability | Automatic | Difficult (loop regeneration) | Not applicable (DD levels are not an EFT expansion; see §4.4) |
One point to read off the table. SAE's entries in the first two rows are blanks, not "an alternative mechanism." This paper does not compete with axion/PQ or Nelson-Barr as a solution. It differs from them only in the fifth row.
§6.2 Falsifiability status
This paper places no independent bet on experiment. $\bar\theta$ is a boundary parameter (§5.1); neither neutron EDM measurements nor axion searches adjudicate it.
This paper does carry one internal falsification point: the falsification condition for N1 (§3.3) — should SAE's structure admit a parity realisation mapping our side into itself, N1 fails.
A consistency observation (not an anti-prediction). All confirmed CP violation (in the K, B, and D meson systems) has been observed in weak-mediated processes, consistent with the picture of §4. But the Standard Model gives the same description, and the two are indistinguishable here; the observation therefore does not constitute an anti-prediction.
§6.3 Hard anti-predictions of the Four Forces series
| Anti-prediction | Source | Decisive experiment | Window | Level |
|---|---|---|---|---|
| No GUT proton decay | Paper VI | Hyper-K | ~2030s | Hard |
| No magnetic monopoles | Paper VI | Monopole searches | Ongoing | Hard |
| No additional gauge bosons | Paper VI | LHC/FCC | ~2030s–2040s | Hard |
$$\boxed{\text{Three hard anti-predictions, all from Paper VI, all experimentally decidable.}}$$
This paper contributes no hard anti-prediction. The three from Paper VI rest on the structural judgment that Spin(10) is a classification group rather than a gauge group, and bear no dependency on the mirror argument of the present paper.
§7 Discussion
§7.1 Dependency chain
| Dependency | Source | Level | Use here |
|---|---|---|---|
| Dual 4DDs causally disconnected; only global bookkeeping across the boundary | Paper I, §5.4 | Structural rule | Antecedent of N1 |
| Our universe is one side of a dual-4DD pair in the left-handed branch | Prequel, §4.4 | Structural location | Antecedent of N1 |
| 4DD light cones of different branches/axes do not overlap | Prequel, §5.1 | Structural fact | Antecedent of N1 |
| Constitutive ordering: mass precedes weak mixing | Generation Paper, G7 | Structural argument | Sole support of §4 |
| Mass from 3DD topological barriers | Generation Paper, G3 | Structural argument | §4.3 |
| 3DD colour-field redistribution preserves doublet total mass | Generation Paper, G6 | Conditional theorem | §4.3 |
| Isospin globality (parallel to G5) | Generation Paper, G5 | Theorem | §2.4 |
| L-R 3DD mirror relation (S1) | Chisel-construct sequence + ZFCρ | Near-theorem | §2.3; not used for single-side constraint |
This paper does not depend on: L-R 4DD axis pairing (G1); the no-mixing assumption for OR/AND; Mass-Channel Proportionality; any continuous parameter or boundary condition; any single-side parity symmetry.
§7.2 Open problems
1. The anomalous smallness of $\bar\theta$ is unexplained by SAE. This is the largest blank left by the present paper. Listing $\bar\theta$ as a boundary parameter (§5) does not explain its smallness; it states plainly that it is not derived.
2. The phase structure of the colour-field redistribution operator $R$. §4.3 shows that the entire origin of $\arg\det M$ resides here, and that SAE leaves it undetermined. This is a concrete, computable object and the item on this list most likely to be settled first.
3. Formalising the Yukawa emergence principle. §4.1 is this paper's only positive result. Its formalisation — proving that under the G7 constitutive ordering no mechanism returns the CKM phase to $\det(M)$ — is a load-bearing outstanding task, not optional future work.
4. Other routes to a single-sided strong-CP mechanism. N1 refutes only the causally disconnected mirror. The following remain open: whether the QCD sector possesses some other exact CP symmetry; whether a UV completion enforces $\bar\theta = 0$; whether a Nelson-Barr-type mass-matrix mechanism can be realised in SAE. The acceptance object is $\arg\det M_q$, the same as in item 2.
5. A quantitative theory of cross-level leakage. Information isolation between DD levels is very strong but not absolute; cross-level leakage exists in principle. Standard Model calculations give the CKM contribution to $d_n$ as roughly $10^{-32}$ e·cm, which may serve as an a posteriori calibration of the leakage scale — but that is a Standard Model result, not an SAE prediction. SAE currently has no theory giving the leakage magnitude from first principles. Whether the leakage rate can be computed from a "distance" between DD levels — by analogy with the topological distances that set mass hierarchies in the Generation Paper — is a meaningful open question.
6. A complete projection rule from DD to low-energy observables. §4.1 supplies a fragment. Establishing the systematic rule matters for SAE's entire physics series and is a prerequisite for item 3.
7. Electroweak baryogenesis. If CKM is the sole source of CP violation, the sufficiency of baryogenesis requires independent argument. This does not affect the present paper's treatment of $\bar\theta$.
§7.3 The value of a negative result
The principal result of this paper, N1, is negative. Two reasons make it worth publishing on its own.
First, its scope exceeds SAE. N1 states that any construction realising parity as a mirror between two non-communicating sectors cannot thereby resolve the local strong CP problem on either side. This excludes an entire class of approaches, and the reason for the exclusion is purely structural — the fixed-point set of parity is empty — with no dependence on model detail. Frameworks employing two-sided, mirror, parallel-universe, or CPT-paired structures, and wishing to suppress θ by parity, must first answer N1.
Second, within SAE it applies an existing rule in a new place. The rule of Paper I §5.4 — that constraints cannot be imposed locally across a causal boundary — had governed the two-sided bookkeeping of $\Lambda$, $B$, and $E$. This paper shows that it governs $\bar\theta$ as well, a quantity previously treated as suppressible by mirror symmetry alone.
$$\boxed{\text{A correct negative result is sounder than an overstated positive one.}}$$
As for the strong CP problem itself: within SAE it remains a problem. SAE's level separation dismantles half of the tension in the Standard Model version — the two quantities do not co-originate, so no explanation is owed for "why so disparate given a common origin." The other half stands, and SAE has no mechanism for it.
Proposition Status Table
| Proposition | Status | Depends on | Testable |
|---|---|---|---|
| L-R 3DD mirror relation (S1, §2.3) | Near-theorem | Chisel-construct sequence + ZFCρ | Indirectly |
| Binary-label lemma (§2.3) | Lemma (canonical home: Physical-Quantity Ladder paper) | 2DD splitting | Indirectly |
| Mirror no-go (N1, §3.1) | Theorem (conditional on causal disconnection) | Paper I §5.4 + Prequel §4.4/§5.1 | Internal (§3.3) |
| Global paired zero undelivered (§3.2) | Status statement | Absence of a cross-side transformation law for mass matrices | — |
| Yukawa emergence / CKM–θ̄ decoupling (§4.1–4.2) | Structural argument (principal positive result) | Generation Paper G7 | Indirectly |
| $\det R > 0$ (§4.3) | Conditional proposition | Continuous deformation + 3DD closure + no massless quarks | Indirectly |
| $\arg\det M$ undetermined (§4.3) | Open | — | — |
| Bidirectional DD information isolation (§2.4) | General principle | G5 + §4 | Indirectly |
| $\bar\theta$ as a boundary parameter (§5.1) | Registered | — | Experiment constrains but does not adjudicate |
Acknowledgments
This revision arose from a three-series consistency audit of SAE (July 2026), carried out collaboratively by four artificial intelligence systems in a division of labour.
Zilu (Claude) performed the horizontal cross-series audit of object identity, identified the conflict between §3.1, step 3 of v1 and the rule stated in Paper I §5.4, located the point of failure, and recognised that v1 §3.2 shared the same defect while v1 §3.3 was independent of both and could be retained.
Gongxihua (ChatGPT) performed derivational review and vertical reconciliation, confirmed the above diagnosis, added the second layer (that even the global paired zero is undelivered), and insisted that the audit object be $\bar\theta$ rather than the bare θ throughout.
Zixia (Gemini) contributed geometric and narrative review; Zigong (Grok) contributed adversarial stress-testing and independent reconstruction.
Their contributions were confined to auxiliary checking, verification, and criticism. All intellectual decisions, status assignments, and final adjudications were made by the author, and do not constitute authorship of the scientific claims in this paper.
The object discipline and evidential discipline followed here are set out in Construction Constitution for the Three-Series Upgrade, v1.5.
Author Responsibility Statement
The principal conclusion of v1 of this paper is withdrawn by the present v2. The grounds for the withdrawal, its scope, and the results that survive it are set out item by item in the v2 upgrade note below. The author no longer holds any claim withdrawn from v1; responsibility for every claim retained or newly introduced in this v2 rests with the author.
This paper follows the principle "correct the error at once, and mark the correction": v1 is preserved in full in the Zenodo version history, neither deleted nor obscured.
v2 Upgrade Note
> This section is the complete record of changes from v1 to v2, provided for readers of v1 and for subsequent work citing this paper. The body does not compare versions passage by passage; readers of v2 alone may skip this section.
v1 was titled "The Hierarchical Dissolution of Strong CP." It claimed that SAE's L-R 3DD mirror symmetry pins the bare θ to zero, that the quark mass matrices constructed at 3DD take real positive determinants, and that consequently the 3DD contribution to $\bar\theta$ vanishes. v2 withdraws that claim, retitles the paper, and instead delivers a no-go, a weaker but independent structural result, and the registration of $\bar\theta$ as a boundary parameter.
What changed
| v1 section | v1 content | v2 disposition |
|---|---|---|
| Title | The Hierarchical Dissolution of Strong CP | Retitled (no dissolution is claimed) |
| §3.1 | Conditional theorem: bare θ = 0 | Withdrawn; step 3 of the proof does not hold. Rewritten as the no-go of v2 §3 |
| §3.2 | Conditional proposition: $\arg\det M = 0$ | Withdrawn; its "3DD mirror forbids a preferred orientation" shares the defect of §3.1 step 3. The continuous-deformation argument for $\det R > 0$ does not depend on the mirror and is retained (v2 §4.3) |
| §3.3 | Yukawa emergence (supporting argument) | Retained and promoted to the paper's principal positive result (v2 §4). It rests on Generation Paper G7, not on S1 |
| §3.4 | Main conclusion $\bar\theta_{\rm 3DD} = 0$ | Withdrawn (both summands withdrawn) |
| §3.5 | Cross-level leakage | Withdrawn (its antecedent was §3.4). The general principle of leakage moves to v2 §7.2 as an open problem |
| §2.2 | "If a theory possesses exact P symmetry" | Qualified: P must map the theory into itself (v2 §2.2) |
| §2.3(b) | The 1-bit argument | Renamed the binary-label lemma, with its scope narrowed; canonical home relocated to the Physical-Quantity Ladder paper, cited here only |
| §4.3, Predictions 1 and 2 | 3DD contribution to θ̄ vanishes; no strong-CP axion | Both withdrawn |
| §4.3, Prediction 3 | CP violation confined to the weak sector | Retained, downgraded to a consistency observation (indistinguishable from the Standard Model; not an anti-prediction) |
| §4.4 | Five hard anti-predictions plus one structural expectation | Reduced to three (all from Paper VI, unaffected); this paper's contribution goes to zero |
| §5.3 | "Strong CP is not a problem but an SM artifact" | Reclassification withdrawn; the weaker claim is retained: the level structure decouples CKM CP from $\bar\theta$ |
| — | — | New: Proposition N1 (mirror no-go); registration of $\bar\theta$ as a boundary parameter |
Why it changed
Trigger. A three-series consistency audit of SAE (July 2026). The audit performed a cross-series check of object identity across the Four Forces, Mass, and Cosmology series, and found that the proof in §3.1 of this paper's v1 conflicts with a rule stated in Paper I §5.4. The object discipline and evidential discipline established by that audit are set out in Construction Constitution for the Three-Series Upgrade, v1.5, §1.1.
The diagnosis, in two layers.
First layer: the single-side zero does not hold. Proposition S1 states that L-3DD and R-3DD are mirror images of one another. Parity P, however, maps our side to the opposite side, not to itself. Paper I §5.4 states explicitly that constraints cannot be imposed locally across a causal boundary; only global bookkeeping is available. S1 therefore yields at most a cross-side pairing, never a single-side zero. The same diagnosis applies to v1 §3.2's "the 3DD mirror forbids a preferred orientation" — our side is the left-handed branch, and is therefore itself an orientation.
Second layer: even the global paired zero is not delivered. Granting a cross-side sign flip for the bare θ does not yield one for $\bar\theta$: the transformation law of the quark mass matrices and their determinant phase under the cross-side map has not been given. That v1 did not hold its audit object fixed at $\bar\theta$ throughout is why this layer went unnoticed; v2 states it as a discipline in §1.1.
Status changes
| Proposition | v1 | v2 |
|---|---|---|
| bare θ = 0 | Conditional theorem | Withdrawn |
| $\arg\det(M_u M_d) = 0$ | Conditional proposition | Withdrawn |
| $\bar\theta_{\rm 3DD} = 0$ | Conditional conclusion | Withdrawn |
| Cross-level leakage ≪ experimental precision | Structural expectation | Withdrawn |
| No strong-CP-motivated axion | Hard anti-prediction | Withdrawn |
| L-R 3DD mirror (S1) | Near-theorem | Retained, re-typed: a relation between the two sides, not a symmetry of either |
| Yukawa emergence | Structural argument (supporting) | Structural argument (principal positive result) |
| Mirror no-go (N1) | — | New; theorem (conditional on causal disconnection) |
| $\bar\theta$ as boundary parameter | — | Newly registered |
Downstream effects
| Affected | Content |
|---|---|
| Paper VI (.19426067) | Unaffected. Its three hard anti-predictions rest on Spin(10) being a classification group rather than a gauge group, with no dependency on the mirror argument |
| Generation Paper (.19394500) | G3/G6/G7 unaffected. This paper's reliance on G7 in fact increases, G7 becoming the sole support of the only positive result |
| Paper I (.19342106) | §7.1 here cites its §5.4 as an antecedent of N1. Separately: v1 cited "nDD → SU(n)" as a theorem of Paper I; that theorem's argument carries an independent erratum (Paper II §2.2: removing the overall phase yields $PU(n)$), and v2 has removed the citation |
| Physical-Quantity Ladder paper (.21227329) | The binary-label lemma relocates there as its canonical home; this paper cites it only |
| Public statement of the series | The Four Forces series' hard anti-predictions go from five to three, all from Paper VI |
References
Strong CP and θ̄
[1] S. Weinberg, The Quantum Theory of Fields, Vol. 2: Modern Applications, Cambridge University Press (1996), Ch. 23.6.
[2] R.D. Peccei, "The Strong CP Problem and Axions," Lect. Notes Phys. 741, 3–17 (2008), arXiv:hep-ph/0607268.
[3] M. Dine, "The Strong CP Problem," SCIPP preprint, https://scipp.ucsc.edu/~dine/solutions_of_strong_cp.pdf
Axion / Peccei-Quinn
[4] R.D. Peccei, H.R. Quinn, "CP Conservation in the Presence of Instantons," Phys. Rev. Lett. 38, 1440 (1977).
[5] S. Weinberg, "A New Light Boson?", Phys. Rev. Lett. 40, 223 (1978).
[6] F. Wilczek, "Problem of Strong P and T Invariance in the Presence of Instantons," Phys. Rev. Lett. 40, 279 (1978).
Nelson-Barr / parity constructions
[7] A.E. Nelson, "Naturally Weak CP Violation," Phys. Lett. B 136, 387 (1984).
[8] S.M. Barr, "Solving the Strong CP Problem without the Peccei-Quinn Symmetry," Phys. Rev. Lett. 53, 329 (1984).
[9] J. de Vries, P. Draper, H.H. Patel, "Do Minimal Parity Solutions to the Strong CP Problem Work?", arXiv:2109.01630.
[10] A. Valenti, L. Vecchi, "The CKM Phase and θ̄ in Nelson-Barr Models," JHEP 07, 203 (2022), arXiv:2112.09122.
Neutron electric dipole moment
[11] C. Abel et al. (nEDM Collaboration), "Measurement of the Permanent Electric Dipole Moment of the Neutron," Phys. Rev. Lett. 124, 081803 (2020). [Current best bound: $|d_n| < 1.8 \times 10^{-26}$ e·cm]
[12] n2EDM Collaboration at PSI, https://www.psi.ch/en/nedm [Next generation; baseline sensitivity $\sim10^{-27}$ e·cm, upgrade target $\sim10^{-28}$ e·cm]
Standard Model CKM contribution to $d_n$
[13] I.B. Khriplovich, A.R. Zhitnitsky, "What Is the Value of the Neutron Electric Dipole Moment in the Kobayashi-Maskawa Model?", Phys. Lett. B 109, 490 (1982).
[14] N. Yamanaka et al., "Standard model contribution to the electric dipole moment of the neutron, deuteron, and helion," arXiv:1512.03013.
SAE series (all by the present author, published on Zenodo)
> Concept DOIs throughout (these always resolve to the latest version). Load-bearing citations additionally specify the section.
[15] H. Qin, "Self-as-an-End: the Chisel–Construct–Remainder Cycle" (SAE P1–P3), DOI: 10.5281/zenodo.18528813, .18666645, .18727327.
[16] H. Qin, "Four Forces Prequel: DD Splitting and $\alpha_G = \alpha_{\rm em}^{65/4}$," DOI: 10.5281/zenodo.19341042. [Cited here: §4.4, §5.1, §7 Prediction 4]
[17] H. Qin, "Four Forces Paper I: the nDD → SU(n) Correspondence," DOI: 10.5281/zenodo.19342106. [Cited here: §5.4]
[18] H. Qin, "Four Forces Paper III: $\sin^2\theta_W = 3/13$," DOI: 10.5281/zenodo.19379412.
[19] H. Qin, "Generation Paper: Topological Origin and Mass Structure of the Three Fermion Generations," DOI: 10.5281/zenodo.19394500. [Cited here: G3, G5, G6, G7]
[20] H. Qin, "Four Forces Paper VI: Spin(10) Classification and Three Hard Anti-Predictions," DOI: 10.5281/zenodo.19426067. [Cited here: §6.3]
[21] H. Qin, "Four Forces Paper VII: the Three-Layer Correction Structure," DOI: 10.5281/zenodo.19433220.
[22] H. Qin, "Fourfold Nesting of the Physical-Quantity Ladder," DOI: 10.5281/zenodo.21227329. [Canonical home of the binary-label lemma]