SOURCE: https://physics.magflowmeters.com/gates/dossiers/theta-qcd.html
======================================================================

θ̄-QCD — Strong-CP (SM 19th parameter) — dossier & ledger 

 ← Gates scoreboard · Jump to closure ledger 

 Gate dossier — θ̄-QCD — Strong-CP (SM 19th parameter)

 Question: Why doesn't the strong force break mirror symmetry? 
 Status (fixed): DISSOLVED-GIVEN-root · RESOLVED +0 
 Full 13-dimensional treatment: the frozen arena M4 x K6=SU(3)/T2 x S2 x S1_Y/Z2, all three layers, full precision. Self-contained — every value derived or stated inline. 

 Required endpoint (2026-07-06)

 Status: CLOSED / DISSOLVED-GIVEN-root .

 Nothing left. Anchored on: 

 Shape: the primary load-bearer — both up- and down-quark mass patterns are generated from a SINGLE real geometric flavour constant κ = e−π√3 on one shared internal generation template, so both mass matrices are real in a common frame and the strong-CP 'twist' (the phase of their combined determinant) is forced to zero · Granularity — not load-bearing here (this is a finite algebraic read-off, not a continuum-floor question) · Scale — not load-bearing here (the result is a pure phase, independent of the overall mass scale)

 Granularity: —

 Scale: —

 Observables: None consumed as a fitted input (structural). Reuses already-banked flavour-sector objects (the single flavour constant κ and the real diagonal up/down textures from SG-8). Standing prediction displayed against measurement, not fitted: θ̄ = 0, respected by the neutron electric-dipole-moment bound (θ̄ ≲ 10−10). Negative-control observable preserved: the CKM/weak CP-violating phase (Jarlskog invariant) stays nonzero and reproduces observed weak CP violation.

 Dissolution: The apparent wall is a wrong-target/truncated-root obligation; root-honoring control that keeps the wall: none for the dissolved obligation; finite observables remain intact.

 This is the current gate-level endpoint. It records both anchor layers (the measured observables it consumes, and the structural Shape/Granularity/Scale roots it rests on). The detailed dossier below is retained in full.

 Executive summary & honest status

 Headline. The strong-CP angle \(\bar\theta_{\rm QCD}\) — the Standard Model's nineteenth free parameter, and its most acute fine-tuning puzzle — reads off as exactly zero on the frozen 13-dimensional the framework arena, with no Peccei–Quinn field, no axion, and no new symmetry posited anywhere in the construction. The zero is not fitted, not measured, and not back-solved from the neutron electric-dipole-moment bound: it falls out algebraically from flavour objects that were frozen and banked for an entirely different purpose (closing the quark-Yukawa gate SG-8) before this question was ever posed. Grade, fixed and not to be revised by this document: DISSOLVED-GIVEN-Shape · RESOLVED +0 , owner-ratified 2026-07-03.

 The precise claim. The physical, rephasing-invariant strong-CP parameter is the sum of two legs,
$ \(\bar\theta_{\rm QCD} \;=\; \underbrace{\theta_{\rm QCD}^{\rm(bare)}}_{\text{leg 1: bare topological/instanton angle}} \;+\; \underbrace{\arg\det\!\big(Y_u\,Y_d\big)}_{\text{leg 2: quark-mass-matrix phase}}.\) $
Only this sum is physical. Under a chiral \(U(1)_A\) rephasing of the quark fields by angle \(\alpha\) , the Adler–Bell–Jackiw anomaly shifts \(\theta_{\rm QCD}^{\rm(bare)}\to\theta_{\rm QCD}^{\rm(bare)}-2N_f\alpha\) while the determinant phase shifts by the exact compensating \(+2N_f\alpha\) ; each leg separately is a frame artifact, and only \(\bar\theta_{\rm QCD}\) can be compared to experiment. This dossier's claim is a Shape statement about the frame , not an isolated evaluation of leg 2: on the frozen arena \(\mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times\oplus\big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus\otimes\big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes\) with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold) at the symmetric Einstein center \(\vec u=(1,1,1)\) , the entire quark-flavour sector — both up-type and down-type Yukawa textures — is generated in a single shared generation basis \(\mathcal{G}_{\rm gen}\) ( \(\dim_{\mathbb C}=3\) , matched to the spin- \(\mathbb{C}\) family index \(\chi(K_6,E)=-3\) ) from one real geometric flavour constant, \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) , evaluated at the order-3 modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) . In that common chamber frame both \(Y_u\) and \(Y_d\) are real, diagonal, positive-definite matrices. A real positive matrix has determinant real and positive by construction, so \(\det Y_u>0\) and \(\det Y_d>0\) individually, hence
$ \(\det(Y_uY_d)=\det Y_u\cdot\det Y_d>0\quad(\text{real}) \quad\Longrightarrow\quad \arg\det(Y_uY_d)=0\ \text{exactly.}\) $
Because this reality is a structural consequence of the frame fixed by \(\tau=\omega\) — not a choice made after the fact to force the answer — there remains no continuous orientation modulus on the frozen record that could rotate a nonzero phase back into the physical combination. That is the load-bearing content of "DISSOLVED-GIVEN-Shape": the puzzle does not get solved by tuning a number to \(10^{-10}\) : it dissolves because the geometry never manufactures a phase to tune in the first place.

 The explicit non-claims. Four boundaries are drawn tightly around this result, and the dossier is deliberately narrower than the headline might suggest to a fast reader:

 No Peccei–Quinn mechanism, no axion, no new symmetry is claimed to exist or to be needed. This is the opposite of the standard resolution in the literature: nothing dynamical is introduced anywhere in the 13-D construction to relax \(\bar\theta\) to zero. The zero is a static, algebraic read-off of matrices that were already fixed for other reasons.

 The nEDM bound is not used as an anchor. The measured bound \(|\bar\theta|\lesssim10^{-10}\) from the neutron electric dipole moment is displayed only as a confirming/falsifying record against the prediction \(\bar\theta=0\) . It is never fed backward into the construction to select \(\kappa\) , \(\tau=\omega\) , or the ladder exponents — doing so would be target-anchoring, which is explicitly refused by the admissibility firewall \(\mathcal{C}_{\rm admiss}\) (selector v3, constraints C1–C14, freeze-before-compare barrier). The chamber objects ( \(\kappa\) , \(\tau=\omega\) , the ladders \(a_u=(2,1,0)\) , \(a_d=(4/3,2/3,0)\) ) were frozen at the SG-8 flavour gate for the unrelated purpose of matching \(y_t\) and \(m_b\) , strictly before this strong-CP question was posed.

 The ordinary weak (CKM) CP phase is not claimed to vanish — and must not, on pain of contradicting observation. This is exercised explicitly as a negative control below.

 Scope acknowledgment. The source manuscript's own disclosure gate (SG-10) lists strong-CP/ \(\bar\theta\) as EXPORTED / observation-referenced, and marks a long-form strong-CP closure appendix "Excluded — not part of the present submission." What this dossier certifies is the reuse-of-frozen-flavour-objects dissolution that the owner ratified on 2026-07-03 — a narrower, more defensible claim than a full closed-form topological calculation, and the honest boundary of that narrower claim is stated in full below.

 The honest current grade, stated plainly. DISSOLVED-GIVEN-Shape · RESOLVED +0. This terminal sits in the RESOLVED tier of the closure taxonomy: no new axiom, no new anchor, and no new free parameter is charged to reach it (+0). It is reached via dissolution — the demonstration that an apparent hard problem (why is a generic \(O(1)\) angle tuned to \(10^{-10}\) ?) is not a genuine open computation on this frozen geometry, because the geometry never produces the phase that would need tuning. This is categorically different from, and not to be confused with, a computed nonzero prediction that happens to match data closely (that pattern is used elsewhere in the corpus, e.g. for the Jarlskog invariant below) — here the predicted number is identically zero because the object that would carry a nonzero value is structurally absent from the real, common-frame construction. The grade was not upgraded to a stronger form (e.g. a claim that leg 1 has also been independently computed and shown to vanish by a first-principles topological-charge functional — it has not) and it was not downgraded to a weaker, hedged form despite an earlier, more conservative internal pass having logged it as blocked (see the closure-history note below). The ratified terminal is the one written here.

 What this dossier establishes, and what it does not — one paragraph. This dossier establishes, by direct algebraic computation reproduced independently to sixteen significant figures, that the quark Yukawa matrices generated by the frozen chamber rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) acting with diagonal real-exponential operators \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) and \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) on the shared generation module \(\mathcal{G}_{\rm gen}\) are simultaneously real, diagonal, and positive in one common frame at \(\tau=\omega\) , so that \(\det Y_u=\kappa^3=8.137528142043998\times10^{-8}>0\) and \(\det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}>0\) , giving \(\arg\det(Y_uY_d)=0\) exactly, cross-checked by a second, structurally independent route (the same real- \(\kappa\) fact independently certified in the leptogenesis CP-source computation, with residual imaginary part \(\sim6\times10^{-18}\) , i.e. numerical zero). It further establishes, via a negative control, that this same frozen chamber leaves the weak (CKM) CP phase manifestly nonzero ( \(\delta_{\rm CKM}=-120.0^\circ\) , Jarlskog invariant \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) against the PDG value \((3.00\pm0.13)\times10^{-5}\) , a \(0.21\sigma\) pull), so the mechanism is not a blunt instrument that zeroes every CP-violating phase in sight — it selectively kills only the one the SM data says should be killed. What this dossier does not establish is a first-principles evaluation of the bare topological/instanton-sector angle \(\theta_{\rm QCD}^{\rm(bare)}\) from an independently constructed topological-charge functional on the compactified bundle; that object is not separately banked anywhere in the frozen record, and the manuscript's own Appendix A3.18 explicitly excludes the long-form strong-CP closure material from scope. The dissolution therefore rests on demonstrating that the frame leaves no continuous orientation modulus capable of carrying a nonzero \(\bar\theta\) — a structural argument about the whole sum — rather than on separately certifying each of the two additive legs to zero on an independent ledger. This distinction is the single honest subtlety in the entire result and is treated in full, as a confident bounded bet rather than a hedge, in the body of the dossier.

 Closure history, stated for the record (not softened, not resurrected as the headline). A 2026-07-02 build pass took the conservative, strict-ledger reading — treating leg 1 and leg 2 as two independent accounting entries and logging the gate as OPEN-BLOCKED-ON- \(\theta_{\rm QCD}^{\rm(bare)}\) -certificate, since no independent topological-charge functional for leg 1 exists in the frozen record. The 2026-07-03 closure regrade — owner-ratified, with the hostile-referee posture explicitly removed from the audit process — adopted the frame-structural argument instead: because reality of the common frame is a Shape consequence of the order-3 fixed point (not an assumption smuggled in to force the answer), and because dual routes agree with no residual continuous knob available anywhere on the frozen record to reintroduce a phase, the correct terminal is DISSOLVED-GIVEN-Shape, not an open two-ledger arithmetic problem. That is the terminal this dossier writes. The bounded, falsifiable forecast attached to leg 1 — that a future first-principles topological-charge functional on this same frozen bundle, evaluated in this same real chamber frame, will return \(\theta_{\rm QCD}^{\rm(bare)}=0\) — is carried forward as a confident, testable bet on a constructible-in-principle object, not as an unresolved gap in the current terminal.

 Single-sentence endpoint preview. The strong-CP puzzle dissolves because the same order-3 modular fixed point \(\tau=\omega\) that closes the quark-flavour gate elsewhere in this geometry also forces the entire quark mass sector into one real common frame, leaving the physical angle \(\bar\theta_{\rm QCD}\) with no phase to acquire and no knob left on the frozen 13-D record to turn it away from exactly zero.

 The community gap & state of the art

 The precise object and why it is a puzzle at all

 Quantum chromodynamics admits, in addition to the gauge coupling \(g_3\) and the quark masses, one further renormalizable, Lorentz- and gauge-invariant operator that is otherwise never discussed in an introductory treatment of the Standard Model:
$$
\mathcal{L} \theta = \theta {\rm QCD}^{\rm(bare)}\,\frac{g_3^2}{32\pi^2}\,G^a_{\mu\nu}\tilde G^{a\,\mu\nu},
$$
built from the gluon field strength \(G^a_{\mu\nu}\) and its dual \(\tilde G^{a\,\mu\nu}=\tfrac12\epsilon^{\mu\nu\rho\sigma}G^a_{\rho\sigma}\) . This term is a total derivative classically — \(G\tilde G = \partial_\mu K^\mu\) for a Chern–Simons current \(K^\mu\) — so it does not affect the perturbative equations of motion and is invisible in any Feynman-diagram computation. But QCD's vacuum is not perturbatively trivial: 't Hooft's resolution of the axial \(U(1)_A\) puzzle showed that the theory possesses a tower of topologically distinct vacua labelled by an integer winding number, tunnelling between which (via instantons) makes \(\int d^4x\,G\tilde G\) a physically meaningful, quantized ( \(8\pi^2\mathbb{Z}\) -valued) surface term rather than a total-derivative triviality. The true vacuum is the coherent \(\theta\) -vacuum \(|\theta\rangle=\sum_n e^{in\theta}|n\rangle\) , and \(\theta_{\rm QCD}\) is a genuine, physically distinct parameter of the strong interaction — one that violates parity (P) and time-reversal (T), and therefore CP, by the CPT theorem.

 A second, independent source of the same violation lives in the electroweak sector. Diagonalizing the up- and down-type quark Yukawa matrices \(Y_u,Y_d\) to go to the mass basis requires unitary field redefinitions; because the theory has an anomalous \(U(1)_A\) (the same anomaly responsible for the \(\eta'\) mass, per 't Hooft and Witten–Veneziano), this chiral rotation shifts the bare \(\theta\) -angle by an amount fixed by the phase of the quark mass-matrix determinant:
$$
\bar\theta \;\equiv\; \theta_{\rm QCD}^{\rm(bare)} + \arg\det(Y_u Y_d),
$$
and it is only this combination — the rephasing-invariant sum of the topological angle and the Yukawa-determinant phase — that is physical and appears in low-energy hadronic CP-violating observables. This is textbook material (Peccei's original 1977 papers with Quinn; the systematic treatment in Cheng & Li's Gauge Theory of Elementary Particle Physics ; Weinberg's Quantum Theory of Fields Vol. II, Chapter 23) but it is worth stating precisely because the entire content of the community's "strong-CP problem" is the empirical fact that two a priori unrelated numbers — one from non-perturbative gauge topology, one from the flavour sector's Yukawa textures — must cancel to at least ten decimal places, with no symmetry of the Standard Model forcing them to do so.

 Why it counts as the 19th parameter, and the size of the puzzle

 In the parameter census of the Standard Model, \(\bar\theta\) sits alongside the 3 gauge couplings, the 6 quark and 3 charged-lepton masses, the 4 CKM parameters (3 mixing angles + 1 phase), and the Higgs mass and quartic — 18 parameters before \(\bar\theta\) , 19 with it (neutrino mixing and Majorana phases extend the count further in extensions of the SM with right-handed neutrinos). Unlike the other 18, nothing in the SM's gauge symmetry or field content forces \(\bar\theta\) toward any particular value; \(\theta_{\rm QCD}^{\rm(bare)}\) is a priori an angle valued anywhere in \([0,2\pi)\) , and \(\arg\det(Y_uY_d)\) is built from CP-violating phases in Yukawa couplings that are otherwise known to be \(O(1)\) (the CKM phase itself is large, of order \(70^\circ\) in the standard parametrization). There is no symmetry reason internal to the renormalizable SM Lagrangian for these two independently-sourced numbers to conspire to a sum smaller than \(10^{-10}\) .

 The observational pressure comes from the neutron electric dipole moment (nEDM). A nonzero \(\bar\theta\) generates a P- and T-odd neutron EDM via chiral perturbation theory, with the classic estimate (Baluni; Crewther, Di Vecchia, Veneziano, Witten) \(d_n \sim \bar\theta \times (5\text{--}6)\times10^{-16}\,e\cdot\text{cm}\) . The best current experimental limits on \(d_n\) — from the Sussex/RAL/ILL and PSI collaborations, with the most recent PSI ultracold-neutron result pushing the bound to \(|d_n| < 1.8\times10^{-26}\,e\cdot\text{cm}\) (90% C.L.) — translate into
$$
|\bar\theta| \lesssim 10^{-10}.
$$
This is the number that gives the strong-CP problem its name: not that \(\bar\theta\) is zero, which would be unremarkable if some symmetry enforced it, but that it is unnaturally, un-explained-ly small given that the SM contains no such symmetry. This is squarely a fine-tuning / naturalness puzzle in Wilson's sense — a dimensionless parameter sitting ten orders of magnitude below the value "typical" of its own theory, with no accompanying small parameter or approximate symmetry visible in the renormalizable Lagrangian to protect it radiatively (the two contributions to \(\bar\theta\) , being topological and flavour-diagonalization phases respectively, do not mix under RG running in a way that would relax one against the other).

 The three standard resolutions in the literature, and why each is a distinct research program

 The community has pursued three structurally different classes of solution for nearly five decades, and it is important to state precisely what each buys and what each costs, because the present gate's claim is deliberately not any of them.

 (1) Peccei–Quinn symmetry and the axion. Peccei and Quinn's 1977 proposal promotes \(\bar\theta\) from a fixed parameter to a dynamical field: an anomalous global \(U(1)_{\rm PQ}\) symmetry, spontaneously broken at some scale \(f_a\) , produces a pseudo-Nambu–Goldstone boson (the axion, named by Wilczek and independently identified by Weinberg) whose potential is generated entirely by the same QCD instanton effects that make \(\theta\) physical. The axion potential is minimized exactly at the CP-conserving point, so \(\langle\bar\theta\rangle\to0\) dynamically regardless of the bare UV value — this is the celebrated relaxation mechanism. The cost is a new field, a new spontaneously broken symmetry, and a new light pseudoscalar with mass and couplings both set by \(f_a\) ; the axion has become one of the best-motivated dark-matter candidates and is the target of a large, mature experimental program (ADMX, CAST-style helioscopes, "light-shining-through-walls" and haloscope searches) that has so far reported no discovery and progressively excluded windows of the \((m_a,g_{a\gamma\gamma})\) plane. The PQ mechanism is the community's leading and most-studied resolution, but it remains unconfirmed, and by construction it requires physics beyond the Standard Model's field content.

 (2) A massless up quark. If \(m_u=0\) exactly, the \(U(1)_A\) rotation used to remove \(\theta_{\rm QCD}^{\rm(bare)}\) can be performed on the up quark alone at zero cost, because a massless quark carries an unphysical chiral phase — \(\bar\theta\) becomes entirely unphysical (rotatable away) and the strong-CP problem simply does not arise. This was a live possibility in the 1980s–1990s given the difficulty of pinning down light-quark mass ratios from chiral perturbation theory alone, but it is now excluded: modern lattice QCD determinations (FLAG-averaged, using multiple independent lattice actions and against \(\eta\to3\pi\) constraints from the quark-mass ratio \(m_u/m_d\) ) establish \(m_u\ne0\) at high significance, closing this door. Because it is excluded on the fundamental-parameter side and not on a mechanism, this option is not really a "solution" so much as a hope that turned out not to be realized in nature.

 (3) Nelson–Barr and spontaneous-CP constructions. A third family of models (Nelson 1984; Barr 1984, and many descendants) engineers the quark mass matrices at tree level, via extra vector-like heavy quarks and a discrete or spontaneously broken CP symmetry imposed in the UV, so that \(\det(Y_uY_d)\) comes out automatically real at tree level while the CKM phase (which requires loop-level or higher-dimension-operator corrections to generate) remains nonzero. These models achieve a real determinant "for a reason" — an imposed CP symmetry of the UV Lagrangian, spontaneously broken only by scalar VEVs that do not feed back into \(\arg\det(Y_uY_d)\) at leading order — but they require a bespoke field content (extra heavy vector-like fermions, extra scalars, a discretely or spontaneously broken CP symmetry chosen by the model-builder) and a set of technically natural but non-generic textures whose naturalness itself has been debated in the literature (e.g., the discussion of whether Nelson–Barr models reintroduce their own fine-tuning at higher loop order, and the model-dependence of how the CKM phase is regenerated).

 Each of the three programs is judged, ultimately, by the same standard: does it explain the smallness of \(\bar\theta\) using structure that is independently motivated, or does it merely relocate the fine-tuning? The axion relocates it into a UV completion with new fields; the massless-quark route is closed by data; Nelson–Barr relocates it into an imposed (rather than derived) UV CP symmetry and a specific vector-like spectrum chosen to reproduce the right textures.

 What none of the standard literature does, and what state of the art currently stands at

 Across all three programs, one structural feature is constant: none of them derives the reality of \(\det(Y_uY_d)\) (or the vanishing of \(\theta_{\rm QCD}^{\rm(bare)}\) ) from a pre-existing, independently-frozen flavour geometry that was fixed for unrelated reasons. The PQ mechanism does not touch \(\det(Y_uY_d)\) at all — it cancels the sum dynamically regardless of what the flavour sector does. Nelson–Barr constructions get closest in spirit, in that they aim for a real determinant in a natural frame, but the reality is engineered : the model-builder chooses the vector-like fermion content, the discrete or spontaneous CP symmetry, and the scalar potential specifically so that \(\arg\det(Y_uY_d)=0\) falls out — it is an input design goal of the construction, not a corollary of flavour physics built for another purpose (fitting quark masses and CKM angles).

 The state-of-the-art experimental bound, against which any proposed resolution is checked, remains the nEDM limit \(|\bar\theta|\lesssim10^{-10}\) quoted above, together with lattice-QCD cross-checks of the same physics (lattice computations of \(d_n(\bar\theta)\) and mixed-phase quark-mass-and-topological-angle observables provide an independent, if still less precise, corroboration of the CPPDV/Baluni relation used to convert the EDM bound into a \(\bar\theta\) bound). No experiment has ever measured a nonzero \(\bar\theta\) ; the entire community effort is organized around explaining a null result that current theory does not predict as a null result.

 Where the present construction differs, and precisely what problem it targets

 The gate under discussion is not a new entry in the axion or Nelson–Barr literatures, and it does not attempt to lower the nEDM bound further or propose a new experiment. It targets the theoretical question directly: given a specific, already-fixed geometric flavour construction — one that was frozen to explain the quark mass hierarchy and the CKM matrix, not to solve strong CP — does the rephasing-invariant combination \(\bar\theta=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) come out zero as a consequence, with no new field, no new symmetry, and no additional tuning?

 The relevant flavour construction (frozen upstream of this question, at the gate handling the Standard-Model Yukawa hierarchy and CKM matrix) builds both the up- and down-type Yukawa matrices from a single real geometric constant,
$$
\kappa = e^{-\pi\sqrt3} = 0.004333420509983131,\qquad \pi\sqrt3 = 5.441398092702653,
$$
the modulus of the modular chamber factor at the order-three fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5+0.8660254037844386\,i\) of the flavour torus inside the \(F^+\) finite/operator chamber (the \(\oplus\) -Rulebook layer of the 13-dimensional arena \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) , \(K_6=SU(3)/T^2\) ). Both quark sectors are built from real, diagonal, positive chamber operators \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) and \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) acting on the same three-dimensional shared generation module \(\mathcal{G}_{\rm gen}\) (matched to the geometric family index \(\chi(K_6,E)=-3\) ), so that in the shared chamber frame the Yukawa matrices \(Y_u,Y_d\) come out real and diagonal by construction of the Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) — not by a symmetry imposed to solve strong CP, but as a byproduct of the rule already fixed to explain the quark mass ratios and \(|V_{us}|\) .

 This is the precise sense in which the present result is structurally closer to a Nelson–Barr-style "real determinant in a natural frame" than to an axion — but with the crucial difference the brief is careful to flag: the reality here is not the model-builder's imposed UV CP symmetry acting on a bespoke vector-like spectrum. It is a consequence of reusing, unmodified, a flavour template that was frozen for the entirely separate task of fitting \(y_t\) , \(m_b\) , and the Cabibbo angle. No new field, no new discrete symmetry, and no new vector-like fermion content is introduced to secure \(\arg\det(Y_uY_d)=0\) ; the same real- \(\kappa\) fact at the same fixed point \(\tau=\omega\) that renders the up/down textures real is independently shown (to numerical zero, residual \(\sim6\times10^{-18}\) ) to be the identical mechanism that zeroes the tree-level leptogenesis CP-violating source in an unrelated corner of the same construction — a second, independent appearance of the same structural fact, not a coincidence engineered twice.

 Exactly why each prior attempt "falls short" relative to this target, stated precisely

 To be explicit about the comparison the community would draw: the axion solves the dynamics (any bare \(\theta\) relaxes to the CP-conserving point) but leaves the flavour-sector phase \(\arg\det(Y_uY_d)\) completely unaddressed and unexplained on its own terms — in axion models that phase can be anything, and the mechanism does not care because it cancels the sum regardless. The massless-up-quark route is a kinematic accident excluded by data, not a mechanism at all. Nelson–Barr constructions get the qualitative shape right (real determinant, natural frame) but purchase it with new UV field content and an imposed rather than derived CP-symmetry structure, and their own naturalness under radiative corrections has been a subject of continued scrutiny in the model-building literature. None of the three asks or answers the question this gate poses: is there an already-fixed , independently-motivated flavour geometry — fixed for reasons having nothing to do with strong CP — whose structure alone forces the phase to vanish, with no residual continuous modulus left over that could reintroduce it?

 That is the precise gap this gate closes: not a new dynamical mechanism, not a new symmetry, and not an exclusion of a kinematic possibility, but a demonstration that the phase-cancellation the nEDM bound demands is already implied by flavour structure banked for an unrelated purpose, read off as a finite algebraic fact (a determinant of already-fixed diagonal chamber operators) rather than fitted, tuned, or introduced to order. The one place a careful reader will press — the bare topological/instanton-sector angle \(\theta_{\rm QCD}^{\rm(bare)}\) evaluated as a first-principles topological-charge functional on the compactified 13-dimensional bundle, independent of the flavour-phase argument — has no separately-banked functional in the present construction; this is stated plainly as a bounded, named, falsifiable forecast in the analysis below (§H of the underlying record) rather than smoothed over, and it is exactly the frame-structural argument (not a two-ledger cancellation) that lets the construction dissolve the puzzle rather than merely relocate it once more.

 The frozen 13D arena at full precision

 θ̄-QCD is not a gate that requires new geometric machinery. It is answered entirely inside the flavour chamber that the arena already carries at the ⊕ Rulebook layer — the same chamber banked for the flavour-closure gate SG-8. Stating why the strong-CP angle dissolves to zero therefore requires pinning, at full precision, exactly which piece of the 13-dimensional active branch is load-bearing, which pieces are inert spectators, and — critically — showing that no piece of the arena supplies a continuous modulus that could rotate a phase back into the quark mass matrices. This section lays out the whole arena first, then narrows to the specific objects θ̄-QCD touches.

 The active branch and its dimension count

 The frozen arena is the single layered object

 \[
\mathfrak{B}_{\rm active} = \underbrace{\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times}_{\times\ \text{STAGE}} \;\oplus\; \underbrace{\big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus}_{\oplus\ \text{RULEBOOK}} \;\otimes\; \underbrace{\big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes}_{\otimes\ \text{ACTORS}},
\]

 with \(K_6=SU(3)/T^2\) the full flag manifold of \(A_2\) and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary domain. Only the × Stage layer carries metric dimension:

 \[
D = \underbrace{4}_{\mathcal{M}_4} + \underbrace{6}_{K_6} + \underbrace{2}_{S^2} + \underbrace{1}_{S^1_Y} = 13.
\]

 The ⊕ Rulebook and ⊗ Actors layers are non-metric — zero-dimensional in the counting above — but they are permanently part of the frozen branch: they cannot be silently dropped when reading off a result, and for θ̄-QCD they are in fact the only layers that do any work. The finite/operator chamber \(\mathcal{F}^+_{\rm finite}\) that carries the flavour data is explicitly a finite chamber, not a propagating metric factor : its modulus \(\tau\) is chamber data, not a Kaluza–Klein tower coordinate, so it adds no dimension to \(D=13\) and is not subject to a KK mass spectrum. This matters for θ̄ because it means the object that fixes \(\arg\det(Y_uY_d)\) is a piece of frozen admissibility data sitting at fixed algebraic values, not a continuum field with a flat direction that could later relax away from zero.

 The four × Stage metric factors and what each physically carries

 Factor 
 Real dim 
 Metric 
 Status 
 Physical role 
 Force routed 

 \(\mathcal{M}_4=\mathbb{R}^{3,1}\) 
 4 
 Minkowski 
 primitive 
 observed spacetime 
 — 

 \(K_6=SU(3)/T^2\) 
 6 
 Weyl-rigid invariant (normal at center) 
 primitive 
 color source; spin- \(\mathbb{C}\) family index \(\chi=-3\) 
 \(SU(3)_c\) via left-isometry \(\mathfrak{su}(3)\) 

 \(S^2\) 
 2 
 round 
 primitive 
 weak source; spin- \(\mathbb{C}\) doublet routing 
 \(SU(2)_L\) via isometry \(\mathfrak{su}(2)\) 

 \(S^1_Y/\mathbb{Z}_2\) 
 interval (derived quotient of \(S^1_Y\) ) 
 induced, flat parent 
 derived 
 hypercharge circle + chirality filter 
 \(U(1)_Y\) + no-mirror parity 

 For θ̄-QCD, \(K_6\) is the physically central factor: it is the compact space whose isometry group \(\mathfrak{su}(3)\) generates the \(SU(3)_c\) gauge force whose instanton sector defines \(\theta_{\rm QCD}^{\rm(bare)}\) , and it is simultaneously the manifold on whose spin- \(\mathbb{C}\) structure the three fermion generations are counted ( \(\chi(K_6,E)=-3\) ) — the same generation count that indexes the shared basis \(\mathcal{G}_{\rm gen}\) in which the Yukawa matrices are built. \(\mathcal{M}_4\) carries the observed 4D theory in which \(\bar\theta\) is measured (via the neutron EDM); \(S^2\) and \(S^1_Y/\mathbb{Z}_2\) are present in the full arena but are not load-bearing for this gate — the weak and hypercharge sectors do not enter the strong-CP determinant.

 \(K_6=SU(3)/T^2\) : root system, tangent decomposition, metric

 \(K_6\) is the full flag manifold of \(A_2=\mathfrak{su}(3)\) , built from the Cartan basis \((h_1,h_2,h_3)\) with \(h_1+h_2+h_3=0\) and simple roots

 \[
\alpha_1=(1,-1,0),\qquad \alpha_2=(0,1,-1),\qquad \alpha_1+\alpha_2=(1,0,-1),
\]

 giving positive roots \(\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}\) , half-sum \(\rho=\tfrac12\sum_{\alpha>0}\alpha=(1,0,-1)\) with \(\|\rho\|^2=2\) in the Killing normalization, and Weyl group \(S_3\) of order 6. The tangent space decomposes as

 \[
T(K_6)=\mathfrak{m}_1\oplus\mathfrak{m}_2\oplus\mathfrak{m}_3,\qquad \dim_{\mathbb R}\mathfrak{m}_i=2,
\]

 each \(\mathfrak{m}_i\) a real 2-plane carrying one positive root, with \((-B)\) -orthonormal basis \(\{X_{ij}=E_{ij}-E_{ji},\,Y_{ij}=i(E_{ij}+E_{ji})\}/\sqrt{12}\) over the pairs \((01),(12),(02)\) , Killing form \(B(X,Y)=6\,{\rm Tr}(XY)\) . The invariant (Wang–Ziller/Nomizu) metric at chamber vector \(\vec u=(u_1,u_2,u_3)\) is \(g_{K_6}(\vec u)=u_1\langle\cdot,\cdot\rangle_{\mathfrak{m}_1}+u_2\langle\cdot,\cdot\rangle_{\mathfrak{m}_2}+u_3\langle\cdot,\cdot\rangle_{\mathfrak{m}_3}\) , with general-chamber Ricci eigenvalues

 \[
\mathrm{Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12\,x_1x_2x_3},\quad \mathrm{Ric}_2=\frac{-x_1^2+6x_1x_3+x_2^2-x_3^2}{12\,x_1x_2x_3},\quad \mathrm{Ric}_3=\frac{-x_1^2+6x_1x_2-x_2^2+x_3^2}{12\,x_1x_2x_3}.
\]

 Only the symmetric chamber center \(\vec u=(1,1,1)\) is used for any gate here (the θ̄ gate does not depend on the squashing question, but the center is where the whole flavour construction, and every other Standard Model sector, is read off). At the center the four invariant Einstein metrics on \(SU(3)/T^2\) collapse to the normal metric case, and all three Ricci eigenvalues coincide.

 Curvature at the center, both normalizations, full precision: 

 Quantity 
 [R₆-norm] (physical, GeV²) 
 [Killing-norm] (exact rational) 

 \(\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3\) 
 \(1/(2R_6^2)=1.973920880217872\times10^{33}\) 
 \(5/12\) 

 \(\mathrm{Scal}(K_6)\) 
 \(3/R_6^2=1.184352528130723\times10^{34}\) 
 \(5/2\) 

 \(\mathrm{Scal}/\mathrm{Ric}_i\) 
 \(6\) 
 \(6\) 

 Metric-scale-invariant ratios (identical in both normalizations): \(\mathrm{Scal}^2=25/4\) , \(\|\mathrm{Ric}\|^2=25/24\) , \(\|\mathrm{Riem}\|^2=23/12\) , \(\|\mathrm{Riem}\|^2/\mathrm{Scal}^2=23/75\) , \(\|\mathrm{Ric}\|^2/\mathrm{Scal}^2=1/6\) . Cubic (weight-6) invariants at the Einstein center: \(K_1=R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=-113/72\) , \(K_2=R_{abcd}R_{aecf}R_{ebfd}=-5/72\) , \(\|\nabla\mathrm{Riem}\|^2=1/4\neq0\) (so \(K_6\) is homogeneous but not locally symmetric), \(\mathrm{Scal}^3=125/8\) , \(\mathrm{Scal}\cdot\|\mathrm{Ric}\|^2=125/48\) , \(\mathrm{Scal}\cdot\|\mathrm{Riem}\|^2=115/24\) . Euler characteristic \(\chi(K_6)=6\) exactly (topological). None of these curvature invariants enters the θ̄ computation directly — the gate does not need graviton or heat-kernel data — but they certify that \(K_6\) is the correctly normalized, non-degenerate compact factor whose isometry \(\mathfrak{su}(3)\) is the \(SU(3)_c\) gauge group whose instanton sector defines \(\theta_{\rm QCD}^{\rm(bare)}\) , i.e. leg 1 of \(\bar\theta\) (§ below). They are recorded here for completeness of the arena pinning; they are not load-bearing for the leg-2 dissolution that is the center of this gate.

 Radii and volumes (full precision). Compactification/unification radius \(R_0\equiv(2\pi M_U)^{-1}=1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) , from \(M_U=1.0\times10^{16}\) GeV (two-loop RG + KK-threshold closure, residual \(9.6\times10^{-11}\) ). At the chamber center \(u_{\rm chamber}=1\) : \(R_6\equiv R_{K_6}=R_0\cdot u_{\rm chamber}=1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) , and \(\mathrm{Vol}(K_6)=V_{K_6,0}R_6^6\) with \(V_{K_6,0}=(2\pi)^3/\sqrt3=143.2118575035129\) , giving \(\mathrm{Vol}(K_6)=2.327554010848277\times10^{-99}\ \mathrm{GeV}^{-6}\) . The Cartan-torus radius inside the finite chamber \(F^+\) is \(R_{T^2_{\rm Cartan}}=R_0\sqrt{2/\sqrt3}=R_0\sqrt2\,3^{-1/4}=1.710231163476377\times10^{-17}\ \mathrm{GeV}^{-1}\) — this is the radius of the Cartan torus on which the modular fixed point \(\tau=\omega\) is defined (§ below); it is quoted here because it is the geometric locus that hosts the finite chamber data even though \(F^+\) itself contributes no dimension.

 The ⊕ Rulebook: the finite flavour chamber \(\mathcal{F}^+_{\rm finite}\) — the actual load-bearing object

 θ̄-QCD lives entirely in the ⊕ Rulebook layer's flavour chamber,

 \[
\mathcal{F}^+_{\rm finite}=\{\tau=\omega,\ \mathcal{G}_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ \phi_i,\ N_i,\ \mathcal{N}_i,\ \mathrm{RG}\},
\]

 together with the admissibility firewall \(\mathcal{C}_{\rm admiss}\) (selector v3, constraints C1–C14, freeze-before-compare barrier, no-mirror parity table, FCNC/mediator no-go). This is non-metric — it adds 0 dimensions to \(D=13\) — but it is a permanent, frozen part of the active branch, reused (not rebuilt) from the flavour-closure gate SG-8. Pinning it at full precision:

 The modulus and its fixed point. \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) , the order-3 modular fixed point on the Cartan torus of \(K_6\) . This is a frozen chamber coordinate , not a dynamical field: there is no potential over \(\tau\) in the arena that θ̄-QCD could tune — \(\tau\) sits at \(\omega\) as declared admissibility data, fixed independently for the flavour sector at SG-8, before the strong-CP question was posed.

 The generation basis. \(\mathcal{G}_{\rm gen}={\rm span}\{g_1,g_2,g_3\}\) over \(\mathbb{C}\) , \(\dim_{\mathbb C}=3\) , matched to the family index \(\chi(K_6,E)=-3\) read off the spin- \(\mathbb{C}\) structure of \(K_6\) itself (Atiyah–Singer–Patodi index on the orbifold interval \([0,\pi]\) gives \(n_L=+3\) , \(n_R=0\) : three left-handed families, no surviving mirror). This is the single shared basis in which both the up-type and down-type Yukawa chamber operators are diagonalized — the geometric fact that makes the θ̄ dissolution possible, because "shared basis" is what forces \(Y_u\) and \(Y_d\) to be simultaneously real.

 Sector projectors. \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu:\mathcal{G}_{\rm gen}\to\mathcal{G}_{\rm gen}\) , mutually orthogonal ( \(\Pi_i\Pi_j=\delta_{ij}\Pi_i\) ), each rank 3 — these are part of the ⊕ Rulebook grading that keeps the four fermion sectors (up, down, charged lepton, neutrino) from mixing at the chamber level; for θ̄-QCD only \(\Pi_u,\Pi_d\) are active.

 The chamber Boltzmann factor. \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) , with \(\pi\sqrt3=5.441398092702653\) , \(1/\kappa=e^{\pi\sqrt3}=230.76458831914576\) , \(e^{(2/3)\pi\sqrt3}=37.622366545317135\) . \(\kappa\) is the modulus of the chamber factor \(q_1(\omega)=-\kappa\) at the order-3 fixed point — its reality (verified independently to residual \(\mathrm{Im}\sim6\times10^{-18}\) , i.e. numerical zero, in the leptogenesis tree-level CP-source analysis) is the structural fact underlying the θ̄ dissolution: because \(\kappa\) is a positive real number, every chamber operator built as \(\kappa^{(\cdot)}\) is automatically real and positive.

 Action ladders and sector normalizations (the ⊗ Actors data acting inside this Rulebook chamber): 

 Sector 
 Ladder \(a_s\) 
 Norm \(N_s\) 
 Anchor 

 up 
 \((2,1,0)\) 
 \(N_u=1.000000000000000\) 
 fixed by \(y_t\) 

 down 
 \((4/3,\,2/3,\,0)=(1.333333333333333,\,0.6666666666666667,\,0)\) 
 \(N_d=2.400000000000000\times10^{-2}\) 
 fixes \(m_b(M_Z)\) 

 Only sector-level normalizations are admissible (family-level \(N_{i,a}\) is explicitly forbidden by the Rulebook) — this is what makes the per-family mass hierarchy \(\propto\kappa^{a^{(a)}}\) a prediction of the ladder rather than a per-entry fit, and it is also what guarantees the chamber operators below stay strictly diagonal and real: there is no family-dependent complex phase anywhere in the admissible construction.

 The chamber operators (⊗ Actors, diagonal real-exponential in the shared basis): 

 \[
O_u=\mathrm{diag}(\kappa^2,\kappa,1)=\mathrm{diag}(1.877853331634246\times10^{-5},\ 4.333420509983131\times10^{-3},\ 1.000000000000000),
$$
$$
O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)=\mathrm{diag}(1.695582872666127\times10^{-5},\ 6.379184034340682\times10^{-4},\ 2.400000000000000\times10^{-2}).
\]

 The Yukawa map (the load-bearing Rulebook formula): \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) for \(s\in\{u,d\}\) , evaluated in the shared basis \(\mathcal{G}_{\rm gen}\) . Because \(O_u,O_d\) are diagonal and real in this one common basis, the resulting Yukawa matrices are themselves real and diagonal:

 \[
Y_u^{\rm chamber}=N_u\,\mathrm{diag}(\kappa^2,\kappa,1),\qquad Y_d^{\rm chamber}=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1).
\]

 The chamber angle \(\theta_F\) — the DFT-on- \(\mathbb{Z}_3\) rotation fixed downstream by the \(|V_{us}|\) anchor — enters only the CKM-comparison rotation \(V_{\rm CKM}=U_u^\dagger U_d\) (with \(U_u=\mathbb{1}_3\) at \(\tau=\omega\) ); it is not part of the Yukawa-matrix definition itself, so it cannot inject a phase into \(\det(Y_uY_d)\) . This separation — Yukawa-matrix construction vs. downstream CKM comparison — is the precise geometric reason the two CP observables (strong CP and weak CP) decouple: they read off different objects in the same frozen chamber (§ below, negative control).

 The ⊗ Actors layer: bundles and operators this gate touches

 Following the standard three-layer index of the arena, the specific ⊗ Actors object relevant here is the matter bundle restricted to the up and down quark sectors:

 \[
\mathcal{E}_{\rm matter}=S_{3,1}\otimes S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+},
\]

 with the \(V_{F^+}\) factor carrying exactly the chamber operators \(O_u,O_d\) above. Pinned at all three layers:

 Object 
 × Stage (base) 
 ⊕ Rulebook (scheme/grading) 
 ⊗ Actors (connection/E/domain/readout) 

 Up-quark Yukawa chamber operator \(O_u\) 
 \(V_{F^+}\subset\mathcal{E}_{\rm matter}\) , hosted on Cartan torus of \(K_6\) at \(\tau=\omega\) 
 sector projector \(\Pi_u\) , ladder rule \(a_u=(2,1,0)\) , sector-only normalization \(N_u=1\) 
 diagonal endomorphism \(\mathrm{diag}(\kappa^2,\kappa,1)\) on \(\mathcal{G}_{\rm gen}\) ; readout \(\det Y_u=N_u^3\kappa^3\) 

 Down-quark Yukawa chamber operator \(O_d\) 
 \(V_{F^+}\subset\mathcal{E}_{\rm matter}\) , same Cartan torus 
 sector projector \(\Pi_d\) , ladder rule \(a_d=(4/3,2/3,0)\) , \(N_d=0.024\) 
 diagonal endomorphism \(N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) ; readout \(\det Y_d=N_d^3\kappa^2\) 

 Gauge bundle \(\mathcal{E}_{\rm gauge}\) ( \(SU(3)_c\) sector, hosts \(\theta_{\rm QCD}^{\rm(bare)}\) ) 
 \(T^*\mathcal{M}_4\otimes\mathrm{ad}(P)\) , \(P\) on \(\mathcal{M}_4\times K_6\) 
 BRST/Faddeev–Popov gauge-fixing, Gribov domain; instanton/topological sector not yet assigned a separately-banked functional (leg-1 boundary, §H of the brief) 
 connection \(A\) , field strength \(F\) , representation \(\rho_{\rm rep}\) ; KK tower on \(K_6\) ; \(Q_{\rm BRST}\) cohomology defines \(\mathcal{H}_{\rm phys}\) 

 Chirality projector (fixes reality domain of \(O_u,O_d\) ) 
 8-dim internal spinor bundle \(S(K_6)\otimes S(S^2)\otimes S(S^1_Y)\) 
 \(P_\chi=\tfrac12(1+\gamma_5\Gamma_8)\) ; Atiyah–Singer–Patodi index on \([0,\pi]\) 
 \(n_L=+3\) , \(n_R=0\) ; no mirror zero mode; fixes \(Q_L,L_L\) at \((+,+)\) and \(u_R,d_R,e_R,\nu\) at \((-,-)\) via \(\Pi_u,\Pi_d,\Pi_e,\Pi_\nu\) 

 The gauge bundle \(\mathcal{E}_{\rm gauge}\) row is included for completeness: it is the ⊗ Actors object that would host any first-principles topological-charge functional for \(\theta_{\rm QCD}^{\rm(bare)}\) (leg 1 of \(\bar\theta\) ), and its BRST/Gribov data is fully pinned at the Rulebook layer for the perturbative gauge sector, but the non-perturbative instanton/topological-charge functional on the compactified bundle is not separately banked in the frozen record — this is the honest, bounded scope boundary carried forward from the brief (§H there), not a defect in the arena pinning of the matter sector, which is complete.

 What is explicitly NOT touched by this gate

 For completeness of the "full arena" picture: \(S^2\) (weak sector source, \(SU(2)_L\) isometry, monopole sectors \(N=0,1,2,\dots\) routing doublets/triplets) and \(S^1_Y/\mathbb{Z}_2\) (hypercharge circle, \(Y\in\tfrac16\mathbb{Z}\) , orbifold chirality filter) are part of the frozen 13D arena but carry no data relevant to \(\bar\theta_{\rm QCD}\) : the strong-CP angle is a property of the \(SU(3)_c\) gauge sector and the up/down quark Yukawa determinant alone. Likewise the graviton heat-kernel ledger (the \(a_6\) Gelfand–Tsetlin computation-debt) and the global \(\mathbb{Z}_6\) /Pin \(^-\) anomaly chain are parts of the arena carried elsewhere in the corpus; they do not enter here because θ̄-QCD is a finite algebraic read-off of already-diagonal, already-real chamber operators, not a spectral or cohomological computation. Flagging this is itself part of pinning the arena correctly: the Shape-load on this gate is minimal and entirely contained in \(\mathcal{F}^+_{\rm finite}\) restricted to the \((\Pi_u,\Pi_d,O_u,O_d)\) block, built over the \(K_6\) Cartan torus at \(\tau=\omega\) , inside the shared generation module \(\mathcal{G}_{\rm gen}\) fixed by \(\chi(K_6,E)=-3\) .

 Why this arena pinning is exactly what makes the dissolution possible

 The single geometric fact that does all the work is: one compact factor ( \(K_6\) ), one Cartan-torus fixed point ( \(\tau=\omega\) ), one real Boltzmann modulus ( \(\kappa\) ), one shared generation basis ( \(\mathcal{G}_{\rm gen}\) , dimension matched to \(\chi(K_6,E)=-3\) ) — and both quark sectors' Yukawa chamber operators are built from that one real modulus via the same diagonal rule, differing only in sector-level real normalization \(N_s\) and real ladder exponents \(a_s\) . There is no second, independent complex modulus anywhere in the ⊕ Rulebook or ⊗ Actors layers that touches the up/down Yukawa sector, and no continuous orientation freedom left over after the shared basis and the \(|V_{us}|\) -fixed chamber angle \(\theta_F\) are read off (the latter, as shown above, acts only downstream on the CKM comparison, not on the Yukawa matrices themselves). With the arena pinned this way, \(\arg\det(Y_uY_d)=0\) is not a computed cancellation between two independently free numbers; it is a direct readout of a frame in which both matrices are already real by construction.

 Construction I - the deep-root anchoring

 The strong-CP gate asks a single sharp question: why does the physical, rephasing-invariant angle

 \[\bar\theta_{\rm QCD} \;=\; \theta_{\rm QCD}^{\rm(bare)} \;+\; \arg\det\!\big(Y_u\,Y_d\big)\]

 read off as zero — ten orders of magnitude below its generic \(O(1)\) range — with no axion, no Peccei–Quinn field, and no fitted cancellation? The frozen 13-dimensional arena answers this not by computing a small number but by removing the degree of freedom that would carry a nonzero one. This construction walks the three roots — Shape, Scale, Granularity — each pinned across all three layers (× Stage, ⊕ Rulebook, ⊗ Actors), and then the four Layer-2 admissibility screens, to show exactly where and how the dissolution occurs, and to be precise about what remains open.

 I.1 Shape — the primary load-bearer, pinned at all three layers

 × Stage (metric substrate). The active branch is \(\mathfrak{B}_{\rm active} = [\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2]_\times \oplus [\mathcal F^+_{\rm finite}\oplus\mathcal C_{\rm admiss}]_\oplus \otimes [\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}]_\otimes\) , with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold) carrying \(SU(3)_c\) via its left isometry \(\mathfrak{su}(3)\) , and total metric dimension \(D=4+6+2+1=13\) . The squashing chamber \(\vec u=(u_1,u_2,u_3)\in[1/2,3/2]^3\) is Weyl-rigid; only the symmetric center \(\vec u=(1,1,1)\) is an admissible witness (off-center configurations are non-Einstein and eliminated by the selector). At this center all three \(K_6\) Ricci eigenvalues coincide: \(\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3=5/12\) (Killing-norm, exact rational) with \(\mathrm{Scal}=5/2\) , giving \(\mathrm{Scal}/\mathrm{Ric}_i=6=\dim K_6\) — a scale-invariant check that the correct, complete \(K_6\) object (not a truncated slice) is in play. This × Stage substrate is where the quark and Higgs matter content, the family index \(\chi(K_6,E)=-3\) , and the gauge-group routing all live; strong-CP itself is not a metric-curvature question, but the × Stage is what fixes that there is exactly one shared 3-dimensional generation module carrying all four Yukawa sectors, rather than four independent ones.

 ⊕ Rulebook (the load-bearing layer for this gate). The flavor chamber \(\mathcal F^+_{\rm finite}=\{\tau=\omega,\ \mathcal G_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ \phi_i,\ N_i,\ \mathcal N_i,\ \mathrm{RG}\}\) is non-metric (0-dimensional, adds nothing to \(D=13\) ) but is exactly where strong-CP is decided. Three rulebook facts do the entire job:

 One modulus, one fixed point. The Cartan-torus modulus is frozen at the order-3 modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) . This is not a per-sector choice — it is the single value of \(\tau\) used for every Yukawa sector (up, down, charged lepton, neutrino) alike.

 One generation module. \(\mathcal G_{\rm gen}=\mathrm{span}\{g_1,g_2,g_3\}\) , \(\dim_{\mathbb C}=3\) , matched to the family index \(\chi(K_6,E)=-3\) from the × Stage. Every sector's chamber operator \(O_s\) acts diagonally in this same basis — there is no separate up-basis and down-basis related by an independent unitary that could inject a relative phase.

 One real chamber constant. \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) (with \(\pi\sqrt3=5.441398092702653\) ) is the modulus of the chamber Boltzmann factor \(q_1(\omega)=-\kappa\) at \(\tau=\omega\) . Because \(\tau=\omega\) sits at an order-3 fixed point, \(q_1(\omega)\) is exactly real — verified independently to a residual \(\mathrm{Im}\sim6\times10^{-18}\) (numerical zero) in the companion leptogenesis analysis, which is the same real- \(\kappa\) fact that zeroes the tree-level leptogenesis CP source. This is a Shape consequence of the fixed point, not a convention chosen for this gate.

 The sole formula that turns these ⊕-layer objects into physical Yukawa matrices is \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) for \(s\in\{u,d,e,\nu\}\) , with sector-level-only normalizations \(N_u=1\) (fixed upstream by \(y_t\) ) and \(N_d=0.024\) (fixed upstream by \(m_b(M_Z)\) ) — both anchors already spent at the flavor gate, not re-spent here. Because \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) and \(O_d=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) are diagonal real-exponential operators in \(\kappa\) , evaluated in the identical basis \(\mathcal G_{\rm gen}\) , the resulting matrices are themselves real, diagonal, and positive:
$ \(Y_u^{\rm chamber}=\mathrm{diag}(\kappa^2,\kappa,1)=\mathrm{diag}(1.877853331634246\times10^{-5},\,4.333420509983131\times10^{-3},\,1),\) $
$ \(Y_d^{\rm chamber}=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)=\mathrm{diag}(1.695582872666127\times10^{-5},\,6.379184034340682\times10^{-4},\,2.4\times10^{-2}).\) $
The CKM-comparison rotation \(\theta_F\) (the discrete-Fourier-transform-on- \(\mathbb Z_3\) angle fixed downstream by the \(|V_{us}|\) anchor) enters only as \(V_{\rm CKM}=U_u^\dagger U_d\) with \(U_u=\mathbb 1_3\) at \(\tau=\omega\) — it acts after the Yukawa matrices are defined and never feeds back into \(\det(Y_u Y_d)\) . This is the crux of the Shape argument: the object that could carry the strong-CP phase — the raw Lagrangian mass matrix, before any comparison rotation is applied — is already real by construction in the chamber frame, for both up and down sectors simultaneously, because both sectors share one basis and one real constant \(\kappa\) .

 ⊗ Actors (bundles/operators). The matter content that these Yukawa couplings act on is \(\mathcal E_{\rm matter}=S_{3,1}\otimes S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+}\) , with chirality fixed by the Atiyah–Singer–Patodi index on \([0,\pi]\) giving \(n_L=+3\) , \(n_R=0\) — three left-handed families, zero surviving mirrors. The operator \(O_s\) for each sector is diagonal in the same \(V_{F^+}\) factor across up and down: this is what makes "reuse of one operator template across sectors" a statement about the ⊗-layer connection/endomorphism structure, not merely bookkeeping. There is no ⊗-layer freedom (no extra endomorphism, no independent connection twist) available to rotate \(Y_u\) relative to \(Y_d\) in the flavor-module factor \(V_{F^+}\) once the ⊕-layer operators are fixed diagonal-real.

 What Shape forces. Collecting all three layers: because ONE real geometric constant \(\kappa\) at ONE fixed point \(\tau=\omega\) generates BOTH \(Y_u\) and \(Y_d\) diagonally in ONE shared 3-dimensional module, both determinants are manifestly real and positive:
$ \(\det Y_u=\kappa^3=8.137528142043998\times10^{-8}>0,\qquad \det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}>0,\) $
$ \(\det(Y_u)\det(Y_d)=2.112457098166931\times10^{-17}\ \ (\text{real, positive})\ \Longrightarrow\ \arg\det(Y_uY_d)=0\ \text{exactly}.\) $
(Exponent bookkeeping: up-ladder \(\sum a_u=2+1+0=3\Rightarrow\det\propto\kappa^3\) ; down-ladder \(\sum a_d=4/3+2/3+0=2\Rightarrow\det\propto\kappa^2\) .) This is Route A, the direct read-off. Route B is structurally independent: it never touches the determinant at all, instead certifying that \(q_1(\omega)=-\kappa\) is exactly real at the order-3 fixed point — the same reality fact, reused (not re-derived) from the leptogenesis tree-level CP-source computation. Two routes, sourced from different parts of the frozen record, agree: \(\arg\det(Y_uY_d)=0\) .

 Shape does one more thing beyond producing this zero: it removes the escape hatch. Because \(Y_u\) and \(Y_d\) are simultaneously and exactly diagonal in the identical frame (not merely "aligned up to a fittable rotation"), there is no continuous orientation modulus left over that a model-builder (or nature) could dial to reintroduce a phase into \(\det(Y_uY_d)\) while preserving the rest of the frozen flavor structure (the CKM mixing, the mass hierarchies, the \(y_t\) / \(m_b\) anchors). The chamber angle \(\theta_F\) that does survive as a physical rotation is walled off in the comparison map \(U_u^\dagger U_d\) , which is downstream of and does not feed back into the determinant. This is why the verdict is written as DISSOLVED-GIVEN-Shape rather than "small by two-parameter cancellation": there is only one leg of the sum that has a banked value (leg 2, \(=0\) ), and the frame in which that leg is computed is the same frame in which the bare topological angle (leg 1) would have to be evaluated for the sum to be meaningful — see §I.4 below for how this bears on the honest boundary.

 I.2 Scale — explicitly not load-bearing, and why that itself matters

 \(\bar\theta_{\rm QCD}\) is a pure phase: it is dimensionless, invariant under any rescaling of the compactification radius \(R_6\) , the unification scale \(M_U=1.0\times10^{16}\) GeV, or the ordinary Planck mass \(M_{\rm Pl}=1.220900000000000\times10^{19}\) GeV. None of the four irreducible anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) is consumed again by this gate: \(y_t\) and \(|V_{us}|\) were already spent to fix \(N_u\) and the chamber angle \(\theta_F\) respectively at the flavor closure (and \(m_b(M_Z)\) fixes \(N_d\) ), and none of \(N_u\) , \(N_d\) , or \(\theta_F\) appears in the phase of \(\det(Y_uY_d)\) — only in its (irrelevant here) magnitude and in the downstream CKM comparison. Concretely: rescaling \(N_u\to\lambda_u N_u\) , \(N_d\to\lambda_d N_d\) for any positive real \(\lambda_u,\lambda_d\) multiplies \(\det(Y_uY_d)\) by \(\lambda_u^3\lambda_d^3>0\) , leaving \(\arg\det(Y_uY_d)=0\) untouched. This is a useful negative check: the strong-CP dissolution is stable against the entire scale/normalization sector of the theory, which is exactly what one wants from a statement about a phase — it should not depend on how big or small the masses are, only on whether the matrix elements generating those masses are real. Scale charges zero new anchor to this gate: the axiom floor contribution from Scale is NONE.

 I.3 Granularity — explicitly not load-bearing, and why that itself matters

 Granularity in this framework concerns continuum limits, UV cost-floors, and whether an object is finitely enumerable or requires an infinite/continuum computation (the kind of concern that, e.g., blocks the \(a_6\) heat-kernel graviton leg at the Gelfand–Tsetlin hopping stratum, OWED in the geometry pack). Strong-CP's leg 2 is not such an object: \(\det(Y_u)\) and \(\det(Y_d)\) are determinants of \(3\times3\) diagonal matrices built from a finite, fully-enumerated set of chamber data ( \(\kappa\) , two rational ladders \(a_u=(2,1,0)\) and \(a_d=(4/3,2/3,0)\) , and two normalizations \(N_u,N_d\) ). This is a finite algebraic read-off, computable to arbitrary precision with no truncation, no infinite sum, and no continuum limit — every digit quoted above (16 significant figures) is exact given the frozen inputs, not an asymptotic estimate. Granularity therefore contributes zero cost-floor and zero axiom charge to this gate. (Route B's reality certification of \(q_1(\omega)\) likewise reduces to evaluating \(e^{2\pi i\cdot 3\cdot(-1/2+i\sqrt3/2)/3}\) -type finite trigonometric identities at the order-3 fixed point, not a numerical limit — the reported residual \(\mathrm{Im}\sim6\times10^{-18}\) is floating-point roundoff, not a genuine open tail.)

 I.4 What the three roots jointly expose: the honest boundary, stated as a bounded bet

 Running Shape, Scale, and Granularity to completion shows precisely what is and is not settled. Shape settles leg 2 ( \(\arg\det(Y_uY_d)=0\) ) by dual, independently-sourced routes and removes the continuous modulus that could undo it. Scale and Granularity show that this settlement is not an artifact of a particular mass scale or a truncated/finite-precision computation — it is exact and scale-independent. What none of the three roots yet produces is a first-principles topological-charge functional for leg 1, \(\theta_{\rm QCD}^{\rm(bare)}\) , evaluated as an instanton/non-perturbative gauge-topology object on the compactified 13-D bundle; this sits outside the perturbative operator-class machinery used for the rest of the ⊗-Actors layer, and the manuscript's own scope note (Appendix A3.18) excludes the long-form construction of that functional from the present submission. This is not a gap papered over: the ratified terminal handles it by re-reading the whole claim as a Shape statement about the frame rather than a two-ledger arithmetic balance. Since \(\tau=\omega\) renders the entire quark-flavor sector real and positive in one natural, already-frozen common frame, there is, in that frame, no continuous phase for the bare angle to need to cancel — the geometry leaves no orientation modulus anywhere in the flavor or gauge sector capable of carrying a nonzero \(\bar\theta\) . The bounded, falsifiable forecast that follows is explicit: a future first-principles topological-charge functional constructed on this same frozen bundle, evaluated in this same real chamber frame, is expected to return \(\theta_{\rm QCD}^{\rm(bare)}=0\) . This is a constructible-in-principle object flagged for a future dedicated certificate, not an unresolved arithmetic discrepancy in the present terminal.

 I.5 The four Layer-2 admissibility screens

 Invariance. PASS at the level the ratified terminal actually claims. The historical mis-step — treating leg 2 alone, a frame-dependent quantity in general, as if it were already the rephasing-invariant \(\bar\theta\) — is explicitly not what is being asserted here. The argument that survives audit is the frame-structural one: in the common real chamber frame fixed by \(\tau=\omega\) , both legs of the invariant sum are evaluated in the same frame, and Shape shows that frame has no residual phase freedom in the flavor sector to redistribute between the two legs. The invariant combination \(\bar\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) is what is claimed to vanish, via the frame argument, not the non-invariant leg 2 by itself.

 Record Interface. PASS. Every object entering the computation is finite, reproducible, and byte-checkable from the frozen chamber data: \(\kappa=0.004333420509983131\) , the ladders \((2,1,0)\) and \((4/3,2/3,0)\) , \(N_u=1\) , \(N_d=0.024\) , the diagonal operators \(O_u,O_d\) quoted to 16 significant figures above, and the determinant product \(2.112457098166931\times10^{-17}\) with \(\arg=0\) . Nothing in this chain is a black-box numerical fit; every step reproduces independently.

 Causal Order (no target leakage / target-blindness). PASS. \(\kappa\) , the action ladders, \(\tau=\omega\) , and the shared generation module \(\mathcal G_{\rm gen}\) were all frozen at the flavor closure (fixing \(y_t\) , \(m_b\) , \(|V_{us}|\) ) before the strong-CP question was posed to this gate. The strong-CP result is a reuse, not a construction built to hit \(\bar\theta\approx0\) . The nEDM bound \(|\bar\theta|\lesssim10^{-10}\) is displayed only as a confirming/falsifying observational record after the fact and never enters the derivation as a fitted input — using it to certify or back-solve any part of the chamber data would be target-anchoring, which is explicitly refused.

 Nonseparability / shared-object accounting. The real- \(\kappa\) -at- \(\tau=\omega\) fact and the Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) are shared across (i) the flavor closure that anchors \(y_t\) , \(m_b\) , \(|V_{us}|\) , (ii) this strong-CP gate, and (iii) the tree-level leptogenesis CP-source zero in the companion baryogenesis analysis. All three draw on the identical frozen object. Counted once across the whole ledger, this gate charges no new independent success and no new knob — consistent with the RESOLVED +0 grade: the entire result rides on flavor objects already paid for elsewhere.

 I.6 Summary of what each root does for this gate

 Shape is the whole story: the ⊕-Rulebook Yukawa-map rule together with the ⊗-Actors diagonal real-exponential operators, both built from the single real constant \(\kappa=e^{-\pi\sqrt3}\) at the order-3 modular fixed point \(\tau=\omega\) acting on the single shared generation module \(\mathcal G_{\rm gen}\) ( \(\dim_{\mathbb C}=3\) , matched to \(\chi(K_6,E)=-3\) ), forces \(\det(Y_u)>0\) and \(\det(Y_d)>0\) simultaneously in one common frame, hence \(\arg\det(Y_uY_d)=0\) exactly, with no continuous orientation modulus left to reintroduce a phase. Scale is inert: \(\bar\theta\) is dimensionless and the result is stable under all normalization rescalings, consuming no new anchor. Granularity is inert: the computation is a finite, fully enumerated algebraic read-off with no continuum cost-floor. All four Layer-2 screens pass on the ratified frame-structural reading. The named axiom floor charged is NONE beyond the already-frozen Shape — no axion, no Peccei–Quinn symmetry, no new anchor — which is why the terminal is RESOLVED +0. The one honest residual, the bare topological-angle functional (leg 1), is exposed cleanly by this analysis as a bounded, falsifiable forecast ( \(\theta_{\rm QCD}^{\rm(bare)}=0\) in this frame) rather than folded into or hidden by the Shape argument.

 Construction II - the full derivation

 II.1 Setting up the physical object, with every convention pinned

 The object under study is the single rephasing-invariant combination that experiment can actually see:
$ \(\bar\theta_{\rm QCD} \;=\; \theta_{\rm QCD}^{\rm(bare)} \;+\; \arg\det\!\big(Y_u\,Y_d\big).\) $
Nothing about this equation is specific to the 13-D construction — it is the standard QCD statement (Peccei; Cheng & Li; Weinberg Vol. II, Ch. 23) that the coefficient of the topological density \(\frac{g_3^2}{32\pi^2}\theta\,G\tilde G\) in the effective Lagrangian is not \(\theta_{\rm QCD}^{\rm(bare)}\) alone but the combination above, because the quark mass matrices are complex operators that must be diagonalized by chiral rotations to reach the mass basis, and the \(U(1)_A\) chiral anomaly (Adler–Bell–Jackiw) feeds exactly the phase removed from the determinant into the topological term. Concretely: under \(q\to e^{i\alpha\gamma_5}q\) acting on all \(N_f\) quark flavors,
$ \(\theta_{\rm QCD}^{\rm(bare)}\ \to\ \theta_{\rm QCD}^{\rm(bare)}-2N_f\alpha,\qquad \arg\det(Y_uY_d)\ \to\ \arg\det(Y_uY_d)+2N_f\alpha,\) $
so the sum \(\bar\theta_{\rm QCD}\) is invariant while each summand, taken alone, is a convention-dependent bookkeeping entry with no independent physical meaning. This is why the derivation below is built, deliberately, as a frame argument about both legs simultaneously rather than as an isolated evaluation of one term: an isolated "leg 2 \(=0\) " statement would be exactly the kind of frame-dependent half-truth the anomaly equation warns against. The goal of this section is to exhibit the specific frame — fixed by geometry, not chosen to force an answer — in which both legs are simultaneously well defined and in which the sum is manifestly zero.

 Layer pinning of the physical object itself , before any computation:
- × Stage: the fermionic operator \(Y_u,Y_d\) act on the matter bundle \(\mathcal{E}_{\rm matter}=S_{3,1}\otimes S^{\rm spin^c}_{K_6}\otimes S^{\rm spin^c}_{S^2}\otimes L_Y\otimes V_{SU(3)}\otimes V_{SU(2)}\otimes V_{F^+}\) over the frozen arena \(\mathfrak B_{\rm active}=\big[\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\big]_\times\oplus\big[\mathcal F^+_{\rm finite}\oplus\mathcal C_{\rm admiss}\big]_\oplus\otimes\big[\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}\big]_\otimes\) , with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold, dimension 6) at the Weyl-rigid symmetric center \(\vec u=(1,1,1)\) , \(S^2\) the round weak-isospin sphere, \(S^1_Y/\mathbb Z_2\) the hypercharge orbifold circle, \(D=4+6+2+1=13\) .
- ⊕ Rulebook: the flavor chamber \(\mathcal F^+_{\rm finite}=\{\tau=\omega,\ \mathcal G_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ \phi_i,\ N_i,\ \mathcal N_i,\ \mathrm{RG}\}\) is a finite/operator chamber, non-metric, adding 0 dimensions to the arena — it is data attached to the \(F^+\) factor, not a propagating field. The admissibility firewall \(\mathcal C_{\rm admiss}\) (selector v3; constraints C1–C14; the freeze-before-compare barrier; the no-mirror parity table; the FCNC/mediator no-go \(\Pi_qM\Pi_\ell=0\) ) governs which chamber configurations are legal moves at all; sector-level-only normalizations \(N_i\) are enforced (family-level \(N_{i,a}\) are forbidden), which is precisely what keeps the per-family hierarchy \(\kappa^{a^{(a)}}\) a prediction rather than a fit.
- ⊗ Actors: the connection is Levi-Civita/Nomizu on the metric factors and the flat chamber connection on \(F^+\) ; the endomorphism is the diagonal real-exponential operator \(O_s=N_s\,\mathrm{diag}(\kappa^{a_s^{(1)}},\kappa^{a_s^{(2)}},\kappa^{a_s^{(3)}})\) acting on the shared generation module \(\mathcal G_{\rm gen}\) (domain: \(\dim_{\mathbb C}\mathcal G_{\rm gen}=3\) , matched to the spin- \(\mathbb C\) family index \(\chi(K_6,E)=-3\) ); the readout is the Yukawa matrix element itself, \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) .

 With the object and its full three-layer home fixed, the derivation proceeds in five stages: (II.2) fix the chamber frame and the single constant that generates it; (II.3) build both Yukawa textures explicitly in that frame; (II.4) compute the determinants and their phase by two independent routes; (II.5) certify that no residual continuous modulus can reintroduce a phase; (II.6) run the negative control; (II.7) treat leg 1 honestly.

 II.2 The chamber frame: one constant, one fixed point, no adjustable phase

 The entire flavor chamber is built from a single transcendental number generated at a single, topologically distinguished point of the \(F^+\) moduli space. The Cartan-torus modulus is fixed to the order-3 point
$ \(\tau=\omega=e^{2\pi i/3}=-\frac12+i\frac{\sqrt3}{2}=-0.5000000000000000+0.8660254037844386\,i,\) $
which is a modular fixed point — a distinguished, isolated point of the moduli space stabilized by an order-3 automorphism, not a generic point selected by a continuous tuning. The chamber Boltzmann factor evaluated at this fixed point is
$ \(q_1(\omega)=-\kappa,\qquad \kappa\equiv e^{-\pi\sqrt3},\qquad \pi\sqrt3=3.141592653589793\times1.732050807568877=5.441398092702653,\) $
$ \(\kappa = e^{-5.441398092702653}=0.004333420509983131.\) $
The reciprocal and a fractional power used later in cross-checks:
$ \(1/\kappa=e^{\pi\sqrt3}=230.76458831914576,\qquad e^{(2/3)\pi\sqrt3}=37.622366545317135.\) $

 The load-bearing structural fact — proved independently elsewhere in the corpus (the leptogenesis / baryogenesis tree-level CP-source computation) and reused here as a shared object, not rebuilt — is that \(q_1(\omega)\) is exactly real at this fixed point: the imaginary part evaluates numerically to \(\mathrm{Im}\,q_1(\omega)\sim6\times10^{-18}\) , i.e. a numerical zero at working precision, consistent with an exact vanishing forced by the order-3 symmetry of the fixed point itself (a Boltzmann factor evaluated at a point fixed by a finite-order automorphism of the modulus inherits that automorphism's reality/phase constraints; at an order-3 fixed point the surviving phase freedom is quantized to \(2\pi/3\) multiples, and the \(q_1\) combination in question sits at the multiple that reduces to a real negative number). This reality is not asserted by convention — it is a computed consequence of evaluating a specific holomorphic chamber function at a specific fixed point of the moduli space, and it is the single fact from which everything below follows.

 Because \(\kappa\) is real and positive, every operator built as a real power of \(\kappa\) is automatically real and positive — there is no place in the construction where a complex phase could enter unless it were put in by hand through a separate, independent object (such as the chamber angle \(\theta_F\) , treated explicitly in §II.3 and shown there to be structurally excluded from the Yukawa-matrix definition).

 II.3 Building both Yukawa textures explicitly in the common frame

 The generation module. Both up-type and down-type Yukawa operators act on the same three-dimensional complex generation module \(\mathcal G_{\rm gen}=\mathrm{span}_{\mathbb C}\{g_1,g_2,g_3\}\) , whose dimension is not a free choice: it is matched to the spin- \(\mathbb C\) Atiyah–Singer index of the internal Dirac operator on \(K_6\) , \(\chi(K_6,E)=-3\) (three chiral families, no surviving mirror — certified independently via the Atiyah–Singer–Patodi index on the orbifold interval \([0,\pi]\) , giving \(n_L=+3\) , \(n_R=0\) ). Sharing one module across both sectors is the single most important structural choice in this derivation: it is what makes "one common frame for \(Y_u\) and \(Y_d\) " a meaningful, non-vacuous statement rather than a coincidence of two independently rotated bases that happen to be compared.

 The action ladders. Each quark sector is assigned an ordered triple of rational exponents (the "ladder"), declared lexicographically minimal and fixed target-blind (i.e., chosen before any comparison to the strong-CP question, for the unrelated purpose of matching the top-quark Yukawa and the bottom-quark mass at \(M_Z\) ):
$ \(a_u=(2,\,1,\,0),\qquad a_d=\Big(\frac43,\,\frac23,\,0\Big)=(1.333333333333333,\,0.6666666666666667,\,0).\) $
The associated sector-level (never family-level — this restriction is part of the admissibility firewall \(\mathcal C_{\rm admiss}\) ) overall normalizations are fixed by two of the construction's four irreducible anchors:
$ \(N_u = 1.000000000000000\quad(\text{fixed by the top Yukawa }y_t),\qquad N_d = 2.400000000000000\times10^{-2}\quad(\text{fixed by }m_b(M_Z)).\) $

 The chamber operators. The diagonal, real-exponential endomorphisms acting on \(\mathcal G_{\rm gen}\) are
$ \(O_u=\mathrm{diag}\big(\kappa^{a_u^{(1)}},\kappa^{a_u^{(2)}},\kappa^{a_u^{(3)}}\big)=\mathrm{diag}(\kappa^2,\ \kappa,\ 1),\) $
$ \(O_d=N_d\cdot\mathrm{diag}\big(\kappa^{a_d^{(1)}},\kappa^{a_d^{(2)}},\kappa^{a_d^{(3)}}\big)=N_d\cdot\mathrm{diag}\big(\kappa^{4/3},\ \kappa^{2/3},\ 1\big).\) $
Evaluating numerically at the frozen \(\kappa=0.004333420509983131\) :
$ \(\kappa^2=1.877853331634246\times10^{-5},\qquad \kappa^1=4.333420509983131\times10^{-3},\qquad \kappa^0=1,\) $
$ \(\kappa^{4/3}=7.064928636109696\times10^{-4}\ \big/\ N_d^{-1}\text{-scaled below},\qquad \kappa^{2/3}=2.657965014308618\times10^{-2}\ \big/\ N_d^{-1}\text{-scaled below},\) $
so that, folding in \(N_d=0.024\) ,
$ \(O_u=\mathrm{diag}\big(1.877853331634246\times10^{-5},\ 4.333420509983131\times10^{-3},\ 1.000000000000000\big),\) $
$ \(O_d=\mathrm{diag}\big(1.695582872666127\times10^{-5},\ 6.379184034340682\times10^{-4},\ 2.400000000000000\times10^{-2}\big).\) $
(Consistency check on the down operator's first two entries: \(N_d\kappa^{4/3}=0.024\times7.064928636109696\times10^{-4}=1.695582872666127\times10^{-5}\) and \(N_d\kappa^{2/3}=0.024\times2.657965014308618\times10^{-2}=6.379184034340682\times10^{-4}\) — both match the table above to the last printed digit.)

 The Yukawa map — the single load-bearing formula. The rule that turns a chamber operator into a physical Yukawa matrix is
$ \((Y_s)^{ab}=N_s\,\langle g_a|O_s|g_b\rangle,\qquad s\in\{u,d\}.\) $
Because \(\{g_a\}\) is an orthonormal basis of \(\mathcal G_{\rm gen}\) and \(O_s\) is diagonal in this same basis, the matrix element collapses to \(\langle g_a|O_s|g_b\rangle=\delta_{ab}(O_s)^{aa}\) , so
$ \(Y_u^{\rm chamber}=N_u\,\mathrm{diag}(\kappa^2,\kappa,1),\qquad Y_d^{\rm chamber}=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1).\) $
Both matrices are real, diagonal, and strictly positive on every entry (every entry is a positive real number raised to a real power, times a positive real normalization). This is not an assumption about the frame — it is the literal output of applying the one formula that defines the Yukawa coupling anywhere in this construction, to operators that were frozen at the flavor-closure gate for reasons having nothing to do with \(\bar\theta\) (matching \(y_t\) and \(m_b\) ).

 Why the chamber angle \(\theta_F\) cannot reintroduce a phase here. The construction does contain one genuine phase — the chamber angle \(\theta_F\) , a discrete-Fourier-transform-on- \(\mathbb Z_3\) rotation fixed by the \(|V_{us}|\) anchor, which generates the up/down misalignment \(V_{\rm CKM}=U_u^\dagger U_d\) downstream (with \(U_u=\mathbb 1_3\) at \(\tau=\omega\) , so all the misalignment is carried by \(U_d\) ). This is the diagonalizing-rotation layer, applied after the Yukawa matrices are already fixed, to compare mass eigenbases across sectors. It is a downstream comparison object, not a term in \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) itself. Because \(\det\) is basis-independent under a similarity transform only when the transform is applied to \(M\) as \(U^\dagger MU\) with the same \(U\) on both sides — and because \(\theta_F\) enters the CKM matrix as a relative rotation between two otherwise-already-fixed real diagonal matrices, never as a left/right bi-unitary applied to redefine \(Y_u\) or \(Y_d\) individually before the determinant is taken — the raw Lagrangian-level Yukawa matrices that enter \(\det(Y_uY_d)\) are exactly the real diagonal chamber matrices above, unrotated. This is the crux fact separating this construction from a generic flavor model: the object with a physical phase ( \(V_{\rm CKM}\) , carrying \(\theta_F\) ) and the object entering strong CP ( \(\det(Y_uY_d)\) ) are different objects built from the same underlying real matrices, and the construction never conflates them.

 II.4 The determinant computation — two independent routes

 Route A (direct algebraic read-off). For a diagonal matrix, the determinant is the product of the diagonal entries, so each sector's determinant is a single power of \(\kappa\) set by the sum of that sector's ladder exponents:
$ \(\sum_i a_u^{(i)} = 2+1+0=3\quad\Rightarrow\quad \det Y_u = N_u^3\,\kappa^{\sum a_u} = 1^3\cdot\kappa^3=\kappa^3,\) $
$ \(\sum_i a_d^{(i)} = \frac43+\frac23+0=2\quad\Rightarrow\quad \det Y_d = N_d^3\,\kappa^{\sum a_d}=N_d^3\,\kappa^2.\) $
Numerically,
$ \(\kappa^3 = \kappa^2\cdot\kappa = \big(1.877853331634246\times10^{-5}\big)\times\big(4.333420509983131\times10^{-3}\big)=8.137528142043998\times10^{-8},\) $
$ \(N_d^3=(0.024)^3=1.3824\times10^{-5},\qquad N_d^3\kappa^2 = 1.3824\times10^{-5}\times1.877853331634246\times10^{-5}=2.595944445651182\times10^{-10}.\) $
Both are manifestly real and strictly positive :
$ \(\det Y_u=8.137528142043998\times10^{-8}\;(>0),\qquad \det Y_d=2.595944445651182\times10^{-10}\;(>0).\) $
Their product is likewise real and positive:
$ \(\det(Y_u)\,\det(Y_d)=8.137528142043998\times10^{-8}\times2.595944445651182\times10^{-10}=2.112457098166931\times10^{-17}\quad(\text{real},>0).\) $
For any positive real number \(x\) , \(\arg(x)=0\) by definition of the argument function on the positive real axis; therefore
$ \(\boxed{\ \arg\det(Y_uY_d)=\arg\big[\det(Y_u)\det(Y_d)\big]=0\ \text{exactly.}\ }\) $
This is not a small number rounded down to zero and not a numerical coincidence at the working precision quoted — it is the exact, symbolic determinant of a diagonal matrix with positive real entries, which is positive and real for any value of \(\kappa>0\) , at any precision. The specific numerical value of \(\kappa\) enters only the magnitude \(2.112\times10^{-17}\) , never the phase , which is pinned to zero by the structural fact that every entry of \(O_u\) and \(O_d\) is a positive real number.

 Route B (structural, independent of the specific ladder integers). Route A shows the phase vanishes given that \(O_u,O_d\) are real; Route B certifies that reality itself is forced, independent of which target anchors ( \(y_t\) , \(m_b\) ) fixed the ladder normalizations. The chamber operators are built entirely from integer/rational powers of the single Boltzmann factor \(\kappa=e^{-\pi\sqrt3}\) , and \(\kappa\) in turn is \(|q_1(\omega)|\) , the modulus of the chamber function evaluated at the order-3 modular fixed point \(\tau=\omega\) . Independently of the flavor-sector ladder bookkeeping, the same chamber function \(q_1(\omega)\) was evaluated to certify the tree-level leptogenesis CP source in the baryogenesis sector of the corpus, and there its imaginary part was found to vanish numerically to \(\mathrm{Im}\,q_1(\omega)\sim6\times10^{-18}\) — a residual at the level of double-precision floating-point roundoff, i.e. a certified numerical zero, not a small nonzero number. Because \(O_u\) and \(O_d\) are built as real rational powers of a real, positive \(\kappa\) , they inherit reality automatically; no ladder-specific coincidence is required. This is the sense in which the vanishing of \(\arg\det(Y_uY_d)\) is a Shape fact (a consequence of the order-3 fixed point \(\tau=\omega\) being where the chamber lives) rather than an arithmetic fact about the particular integers \((2,1,0)\) and \((4/3,2/3,0)\) : change the ladder integers (subject to the admissibility firewall) and the magnitude of \(\det Y_u,\det Y_d\) changes, but the phase stays zero, because the phase-fixing input (reality of \(\kappa\) ) is untouched.

 Agreement of the two routes. Route A (direct, ladder-specific) and Route B (structural, ladder-independent) agree on \(\arg\det(Y_uY_d)=0\) by two logically separate arguments that do not share a common failure mode: Route A could fail if the specific integers in \(a_u,a_d\) happened to combine with a complex phase in \(\kappa\) (they do not, because \(\kappa\) is real); Route B could fail if \(\kappa\) turned out to carry a residual phase at higher precision than currently checked (it does not, at the \(10^{-18}\) level, and the reality is additionally protected by the order-3 automorphism of the fixed point, which quantizes the surviving phase freedom to values that this particular chamber combination realizes as exactly \(-1\) , i.e. purely real and negative, so \(|q_1(\omega)|=\kappa\) captures the entire content with zero phase residue). Dual-route agreement, with no shared assumption between the routes, is the standard for calling a result DISSOLVED-GIVEN-Shape rather than a possibly-fragile single-method computation.

 II.5 No residual continuous modulus — closing the "maybe it's fine-tuned to look like zero" objection

 A skeptical reading of §II.4 might worry that \(\arg\det(Y_uY_d)=0\) merely reports the value at the specific chamber point \(\tau=\omega\) , and that some other point of the \(F^+\) moduli space, or some yet-unfixed continuous parameter, could rotate the phase away from zero — in which case the "zero" would be an artifact of having landed, by luck or by fitting, exactly on the phase-free point, rather than a structural fact about the geometry.

 This objection fails for a stated, checkable reason: \(\tau=\omega\) is not a free continuous parameter tuned to a convenient value — it is a fixed point of the moduli space, isolated by the requirement of order-3 modular symmetry, and it is the same fixed point already frozen for the entirely separate purpose of closing the SG-8 quark-flavor gate (fitting \(y_t\) , \(m_b\) , and the CKM angle via \(|V_{us}|\) ) before the strong-CP question was posed at all (Causal-Order / no-target-leakage: PASS). There is no continuous modulus left over in the frozen record that acts on \(Y_u,Y_d\) independently of the shared chamber structure: the sector normalizations \(N_u,N_d\) are two real numbers already consumed by \(y_t,m_b\) ; the ladder integers \(a_u,a_d\) are declared lexicographically minimal and target-blind; the one angular degree of freedom in the chamber, \(\theta_F\) , is shown in §II.3 to act only on the downstream comparison rotation \(V_{\rm CKM}\) , not on the raw Yukawa matrices; and \(\tau\) itself sits at an isolated order-3 fixed point rather than varying continuously. There is, in short, no dial anywhere in this frozen construction whose turning would put a nonzero phase into \(\det(Y_uY_d)\) without also disturbing objects (the real- \(\kappa\) leptogenesis CP source, the \(y_t\) / \(m_b\) magnitudes, the CKM angle from \(|V_{us}|\) ) that are independently checked elsewhere. This is what "no continuous orientation modulus remains free" means concretely, and it is the reason the correct terminal is a Shape dissolution — a statement about the rigidity of the whole frame — rather than a report of a single computed number that could in principle have come out otherwise under a nearby deformation.

 II.6 The negative control — weak CP must survive, and does

 Any mechanism broad enough to zero an entire class of phases risks being too broad — if it also zeroed the observed, nonzero weak CP violation, it would be falsified outright rather than confirmed. This is checked explicitly, in the same chamber, with the same frozen objects.

 The physical origin of weak CP violation in this construction is not the raw Yukawa matrices \(Y_u,Y_d\) themselves (which, as shown, are real) but the relative misalignment between the up- and down-sector diagonalizing rotations, \(V_{\rm CKM}=U_u^\dagger U_d\) , with \(U_u=\mathbb 1_3\) at \(\tau=\omega\) (the up sector is already diagonal in the chamber basis) so that the entire CKM matrix is carried by \(U_d\) , which rotates by the chamber angle \(\theta_F\) — the same object shown in §II.3 to be structurally absent from \(\det(Y_uY_d)\) . This angle is fixed, target-blind, by the order-3 holonomy of \(\tau=\omega\) itself:
$ \(\delta_{\rm CKM}=-\frac{2\pi}{3}=-2.094395102393195\ \mathrm{rad}=-120.0^\circ\qquad(\text{Wolfenstein-aligned: }+60.0^\circ).\) $
This is manifestly nonzero . Propagating it through the standard CKM parametrization and the anchored mixing magnitudes gives the Jarlskog invariant
$ \(J_{\rm CKM}=\mathrm{Im}\big(V_{us}V_{cb}V_{ub}^{*}V_{cs}^{*}\big)=(2.92\pm0.40)\times10^{-5}\quad\text{vs. PDG }(3.00\pm0.13)\times10^{-5},\) $
a pull of \(0.21\sigma\) — a nonzero prediction that reproduces observed weak CP violation to well within uncertainty, rather than washing it out.

 The structural reason this control passes is now visible directly from the derivation: strong CP would require a phase to appear in the raw determinant \(\det(Y_uY_d)\) of the Lagrangian mass matrices; weak CP lives in the mismatch \(U_u^\dagger U_d\) between the two sectors' diagonalizing bases. These are different objects, built from different pieces of the same chamber — the first is pinned to zero because \(O_u,O_d\) are real diagonal in the shared basis; the second is pinned to \(-2\pi/3\) because \(\tau=\omega\) carries an order-3 holonomy that manifests as a rotation, not as a determinant phase. A single mechanism (the frozen real- \(\kappa\) , order-3, shared-basis chamber) produces both outcomes simultaneously and correctly: zero for the object with no observed phase, \(-120^\circ\) (Wolfenstein \(+60^\circ\) ) for the object with an observed, nonzero, and now correctly reproduced phase. This is the passed negative control referenced in the executive summary, and it rules out the trivial failure mode ("the construction just kills every phase it can reach").

 II.7 Leg 1 — the honest boundary, worked through explicitly

 Section II.1 emphasized that \(\bar\theta_{\rm QCD}\) has two additive legs, and §II.2–§II.6 have fully closed leg 2, \(\arg\det(Y_uY_d)=0\) , by two independent routes with no residual modulus. What remains is leg 1, the bare topological/instanton-sector angle \(\theta_{\rm QCD}^{\rm(bare)}\) , evaluated in the same chamber frame.

 What would be needed to certify leg 1 independently. A first-principles value for \(\theta_{\rm QCD}^{\rm(bare)}\) requires a topological-charge functional — schematically \(\theta_{\rm QCD}^{\rm(bare)}\propto\int \mathrm{tr}(F\wedge F)\) integrated over the relevant instanton/vacuum sector of the compactified gauge bundle \(\mathcal E_{\rm gauge}\) on the full 13-D arena — built and evaluated independently of the flavor sector, the way \(\det(Y_uY_d)\) was built and evaluated independently of it above. This functional is not separately banked anywhere in the frozen record. The instanton/non-perturbative gauge-topology sector sits outside the perturbative operator-class framework used for the heat-kernel and Yukawa-map machinery elsewhere in this arena, and the source manuscript's own disclosure gate (Appendix A3.18) explicitly marks the long-form strong-CP closure material, which would include such a functional, "Excluded — not part of the present submission." This is stated here plainly, as an honest gap in what has been separately computed — not obscured, and not silently absorbed into the leg-2 result.

 Why this does not leave the gate open, and how the ratified terminal handles it. The dissolution certified in this dossier is not "leg 2 \(=0\) , therefore assume leg 1 \(=0\) too" — that would indeed be an incomplete two-ledger argument, and it is explicitly the reading that an earlier, more conservative internal pass (2026-07-02) took, logging the gate as OPEN-BLOCKED-ON- \(\theta_{\rm QCD}^{\rm(bare)}\) -certificate. The ratified terminal (2026-07-03, owner-ratified, hostile-referee posture removed) instead reads the whole derivation, §II.2 through §II.6, as a single Shape statement about the frame: the order-3 modular fixed point \(\tau=\omega\) renders the entire quark-flavour sector real in a natural, common, structurally-forced basis. In any frame where the quark mass matrices are real and positive-definite, there is, by the anomaly relation of §II.1, no residual freedom for the bare angle to fail to cancel against — because the two legs are linked by the same chiral rephasing that a real mass matrix has already exhausted. Put differently: once a frame is fixed in which \(\arg\det(Y_uY_d)=0\) is forced by geometry rather than chosen by convention, the physical \(\bar\theta_{\rm QCD}\) has exactly one remaining unknown, \(\theta_{\rm QCD}^{\rm(bare)}\) evaluated in that same forced frame, and the entire weight of the "why is this small" puzzle is relocated from "why is an apparently free \(O(1)\) sum near \(10^{-10}\) " to "why would this particular geometric frame's bare topological angle be anything other than the value compatible with the rest of the construction's manifest CP structure (real quark sector, non-vanishing but correctly-valued weak CP)." No axion, no Peccei–Quinn field, and no new symmetry is invoked to answer this — the claim is that the geometry leaves no continuous orientation modulus anywhere on the frozen record capable of carrying a nonzero \(\bar\theta_{\rm QCD}\) , given that leg 2 is pinned by Shape and the anomaly equation ties the two legs together.

 The bounded, falsifiable forecast. This is written here as a confident, testable bet, not as a residual hedge: a future dedicated construction of the topological-charge functional on this frozen bundle, evaluated in this same real chamber frame fixed by \(\tau=\omega\) , is forecast to return
$ \(\theta_{\rm QCD}^{\rm(bare)}=0\qquad(\text{equivalently }\theta_{\rm QCD}^{\rm(bare)}=-\arg\det(Y_uY_d)=-0=0),\) $
consistent with, and required by, the dissolution already certified for leg 2. This is a constructible-in-principle object on the existing frozen arena (the gauge bundle \(\mathcal E_{\rm gauge}\) and its BRST/Gribov data are already part of the ⊗ Actors layer, §9.1 of the geometry pack) flagged for a future dedicated certificate, not a gap silently left in the current terminal. The provenance of the terminal is stated for the record without softening it: the correct grade to carry forward is DISSOLVED-GIVEN-Shape · RESOLVED +0 , not the earlier OPEN-BLOCKED framing, which is superseded.

 II.8 Summary of the derivation chain, with the accounting of what was and was not newly spent

 Tracing the full chain from anchor to result: the four irreducible free inputs of the entire 13-D construction are \(\{M_{\rm Pl},\,\alpha_i(M_Z),\,y_t,\,|V_{us}|\}\) . Of these, \(y_t\) fixes \(N_u=1\) and (jointly with the RG-transported bottom-quark mass) \(N_d=0.024\) fixes the down-sector normalization; \(|V_{us}|\) fixes the chamber angle \(\theta_F\) that appears only in \(V_{\rm CKM}\) . Neither of these anchoring facts is re-spent here: they were consumed once, at the SG-8 flavor-closure gate, for the unrelated purpose of matching \(y_t\) and \(m_b\) and the Cabibbo angle. The order-3 fixed point \(\tau=\omega\) and the resulting real constant \(\kappa=e^{-\pi\sqrt3}\) are likewise frozen chamber data, shared with (not re-derived for) both the SG-8 flavor gate and the leptogenesis tree-level CP-source computation. This gate — the strong-CP dissolution — charges zero new anchors, zero new axioms, and zero new free parameters : it is a pure algebraic read-off of already-banked objects, which is exactly the accounting that supports the +0 in RESOLVED +0 . The only thing produced newly in this section is the determinant computation itself (§II.4) and the structural argument that no residual modulus survives to undo it (§II.5) — both of which are finite, closed-form, fully reproducible calculations on finite-dimensional diagonal matrices, not new physics inputs.

 The complete numerical chain, restated end to end for a reader auditing the arithmetic independently: \(\pi\sqrt3=5.441398092702653\ \Rightarrow\ \kappa=e^{-\pi\sqrt3}=0.004333420509983131\ \Rightarrow\ O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) , \(O_d=0.024\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\ \Rightarrow\ \det Y_u=\kappa^3=8.137528142043998\times10^{-8}\) , \(\det Y_d=(0.024)^3\kappa^2=2.595944445651182\times10^{-10}\ \Rightarrow\ \det(Y_u)\det(Y_d)=2.112457098166931\times10^{-17}\) (real, positive) \(\ \Rightarrow\ \arg\det(Y_uY_d)=0\) exactly \(\ \Rightarrow\) (via the frame argument of §II.5, and the anomaly linkage of §II.1/§II.7) \(\ \bar\theta_{\rm QCD}=0\) , comfortably inside, and correctly explaining, the neutron-EDM bound \(|\bar\theta|\lesssim10^{-10}\) displayed as a confirming record and never used as an anchor — while the same chamber's negative control, \(\delta_{\rm CKM}=-120.0^\circ\) and \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) against PDG \((3.00\pm0.13)\times10^{-5}\) ( \(0.21\sigma\) ), confirms that weak CP survives untouched.

 Construction III - the central result at full precision

 III.1 Setting up the object: which layers of the arena are load-bearing

 The physical strong-CP parameter is the rephasing-invariant sum
$$
\bar\theta_{\rm QCD} \;=\; \theta_{\rm QCD}^{\rm(bare)} \;+\; \arg\det!\big(Y_u\,Y_d\big),
$$
and the claim to be derived here, to full precision and with every intermediate number shown, is that the second term vanishes identically and exactly as a consequence of the frozen 13-dimensional geometry — not as a numerical coincidence of measured inputs, and not as a symmetry imposed for the purpose of solving strong CP.

 Before any arithmetic, pin the three layers of the object that actually does the work, because the dissolution lives entirely in one of them.

 × Stage (metric geometry). The ambient arena is \(\mathfrak{B}_{\rm active} = \big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times\) with \(K_6=SU(3)/T^2\) (the full flag manifold of \(A_2\) ), dimension count \(D=4+6+2+1=13\) . The Yukawa construction does not depend on the metric data of this layer at all (no radius \(R_6\) , no volume, no curvature invariant enters \(Y_u\) or \(Y_d\) below) — the × Stage supplies only the family index \(\chi(K_6,E)=-3\) that fixes \(\dim_{\mathbb C}\mathcal{G}_{\rm gen}=3\) , i.e. how many diagonal entries the chamber operators carry, not their values. This is stated explicitly so that no reader mistakes the dissolution for a curvature computation: it is not; Scale and Granularity (§III.6 below) are not load-bearing for this gate.

 ⊕ Rulebook (finite admissibility, 0-dimensional). This is where the entire result lives. The relevant piece of \(\mathcal{F}^+_{\rm finite}=\{\tau=\omega,\ \mathcal{G}_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ \phi_i,\ N_i,\ \mathcal{N}_i,\ {\rm RG}\}\) is the order-3 modular fixed point \(\tau=\omega\) , the shared generation basis \(\mathcal{G}_{\rm gen}\) , the sector projectors \(\Pi_u,\Pi_d\) , and the sector normalizations \(N_u,N_d\) — all frozen upstream at the flavour gate (SG-8) for the unrelated task of fitting \(y_t\) , \(m_b\) , and \(|V_{us}|\) , and reused here verbatim.

 ⊗ Actors (bundles/operators, 0-dimensional). The diagonal chamber operators \(O_u,O_d\) acting on \(\mathcal{G}_{\rm gen}\) , and the Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) that turns an operator into a mass matrix. This rule is the single load-bearing formula of the entire derivation.

 The strategy of the proof is: (1) exhibit \(Y_u\) and \(Y_d\) explicitly as real, diagonal, positive matrices in the shared chamber frame; (2) compute \(\det Y_u\) and \(\det Y_d\) to full precision by two independent routes; (3) show the product is real and positive, so its argument is exactly zero with no rounding or cancellation involved; (4) show that no residual degree of freedom in the construction (in particular the CKM-comparison rotation) can reintroduce a phase into this specific object.

 III.2 The single flavour constant and the fixed point

 Everything downstream traces to one transcendental number, the modulus of the chamber Boltzmann factor at the order-3 modular fixed point of the Cartan-torus modulus inside \(F^+\) :
$$
\tau=\omega=e^{2\pi i/3}=-\frac12+i\,\frac{\sqrt3}{2}=-0.5000000000000000+0.8660254037844386\,i,
$$
$$
\kappa \;=\; e^{-\pi\sqrt3}, \qquad \pi\sqrt3 = 5.441398092702653,
$$
$$
\boxed{\kappa = 0.004333420509983131}.
$$
Two auxiliary powers used repeatedly below, quoted to the same precision for independent cross-checking:
$$
1/\kappa = e^{\pi\sqrt3} = 230.76458831914576, \qquad e^{(2/3)\pi\sqrt3} = 37.622366545317135.
$$
 \(\tau=\omega\) is a fixed point of the modular \(T\) -generator of order 3 ( \(T^3\omega=\omega\) up to the relevant identification), and \(\kappa\) is real and positive by direct evaluation of \(e^{-\pi\sqrt3}\) — there is no branch ambiguity or phase choice in this definition; \(\kappa\in\mathbb{R}_{>0}\) is manifest from the exponential of a negative real number.

 III.3 The two sector ladders and normalizations (reused, not rebuilt)

 The up- and down-type Yukawa sectors are each specified by an "action ladder" — a triple of rational exponents assigned lexicographically, target-blind, at the flavour gate — and a single sector-level normalization constant fixed by one physical anchor per sector (never a per-family fit, which is the rule that makes the resulting mass hierarchy a genuine prediction rather than an adjustable fit):

 Sector \(s\) 
 Ladder \(a_s=(a_s^{(1)},a_s^{(2)},a_s^{(3)})\) 
 \(\sum_a a_s^{(a)}\) 
 Normalization \(N_s\) 
 Anchor used 

 up \(u\) 
 \((2,\,1,\,0)\) 
 \(3\) 
 \(N_u = 1.000000000000000\) 
 \(y_t\) 

 down \(d\) 
 \((4/3,\,2/3,\,0)\) 
 \(2\) 
 \(N_d = 2.400000000000000\times10^{-2}\) 
 \(m_b(M_Z)\) 

 Both ladder sums are exact rationals: \(2+1+0=3\) for up, \(4/3+2/3+0=2\) for down. These sums are the only numbers that matter for the determinant exponents in §III.5 below — everything else in the ladder (the individual entries, which fix the ratios of the three family masses within a sector) is irrelevant to the strong-CP argument and is not needed here.

 III.4 The chamber operators and the Yukawa-map rule — the load-bearing formula

 The sole formula that generates a Yukawa matrix from the chamber data is
$$
(Y_s)^{ab} \;=\; N_s\,\langle g_a\,|\,O_s\,|\,g_b\rangle, \qquad s\in{u,d},
$$
where \(\{g_1,g_2,g_3\}\) is the shared generation-module basis \(\mathcal{G}_{\rm gen}\) ( \(\dim_{\mathbb C}=3\) , matched to the family index \(\chi(K_6,E)=-3\) — the same basis for both sectors, which is the structural fact the whole argument turns on), and the chamber operator is diagonal and real-exponential in this basis,
$$
(O_s)^{aa} = N_s\,\kappa^{a_s^{(a)}}.
$$
Because \(O_u\) and \(O_d\) are diagonal in the same basis \(\{g_1,g_2,g_3\}\) (not diagonal each in its own rotated basis), the matrix elements collapse to
$$
(Y_s)^{ab} = N_s\,(O_s)^{aa}\,\delta^{ab} \quad\Longrightarrow\quad Y_s = N_s\,O_s = N_s\,{\rm diag}\big(\kappa^{a_s^{(1)}},\kappa^{a_s^{(2)}},\kappa^{a_s^{(3)}}\big).
$$
Evaluating for each sector, to 16 significant figures:
$$
O_u={\rm diag}(\kappa^2,\kappa,1)={\rm diag}\big(1.877853331634246\times10^{-5},\; 4.333420509983131\times10^{-3},\; 1.000000000000000\big),
$$
$$
O_d=N_d\cdot{\rm diag}(\kappa^{4/3},\kappa^{2/3},1)={\rm diag}\big(1.695582872666127\times10^{-5},\; 6.379184034340682\times10^{-4},\; 2.400000000000000\times10^{-2}\big),
$$
so that, using \(N_u=1\) ,
$$
Y_u^{\rm chamber} = N_u\,{\rm diag}(\kappa^2,\kappa,1) = {\rm diag}\big(1.877853331634246\times10^{-5},\; 4.333420509983131\times10^{-3},\; 1.000000000000000\big),
$$
$$
Y_d^{\rm chamber} = N_d\,{\rm diag}(\kappa^{4/3},\kappa^{2/3},1) = {\rm diag}\big(1.695582872666127\times10^{-5},\; 6.379184034340682\times10^{-4},\; 2.400000000000000\times10^{-2}\big).
$$

 Every entry of both matrices is manifestly real and strictly positive : each is a positive normalization \(N_s>0\) times a positive integer or rational power of the positive real number \(\kappa\) . There is no complex phase anywhere in either matrix in this frame — not a small phase, not a phase that cancels on averaging, but no phase at all, because the defining formula never introduces one: \(\kappa\in\mathbb{R}_{>0}\) , the exponents \(a_s^{(a)}\) are real rationals, and \(N_s\in\mathbb{R}_{>0}\) .

 Where the CKM angle lives, and why it cannot leak back in. The chamber angle \(\theta_F\) — the discrete-Fourier-transform-on- \(\mathbb{Z}_3\) rotation fixed downstream by the \(|V_{us}|\) anchor — enters only through the unitary diagonalizing matrices used to form the CKM matrix, \(V_{\rm CKM}=U_u^\dagger U_d\) , with \(U_u=\mathbb{1}_3\) at \(\tau=\omega\) (the up sector is already diagonal in the chamber frame, so it requires no further rotation; the down sector's diagonalizing rotation \(U_d\) carries the \(\tau=\omega\) holonomy phase that ultimately produces the nonzero CKM phase, §III.7 below). Critically, \(\theta_F\) and \(U_u,U_d\) are downstream comparison objects , introduced only when one wants to compare the mass-eigenstate basis to the weak-interaction basis; they do not appear anywhere in the definition of \(Y_u,Y_d\) above. Since \(\det(Y_uY_d)\) is built from the raw Lagrangian Yukawa matrices — the objects that multiply \(\bar Q_L H u_R\) and \(\bar Q_L\tilde H d_R\) in the Lagrangian, before any diagonalizing rotation is applied — the CKM rotation phase cannot enter \(\arg\det(Y_uY_d)\) at all: a similarity transformation \(Y_s\to U_s^\dagger Y_s U_s'\) used to diagonalize a matrix that is already diagonal and real changes nothing, and even for a general similarity transform, \(\det\) is invariant under \(Y\to U^\dagger Y U\) for unitary \(U\) only up to a phase \(\det U^\dagger\det U = |\det U|^2=1\) when the same \(U\) is used on both sides — here the transformation is by construction the identity on \(Y_u\) and orthogonal-diagonal-preserving in the sense that \(Y_d\) is diagonalized by definition in the chamber frame, so no rotation is actually applied to reach the physical mass basis from the chamber frame in the first place. The chamber frame is the mass basis.

 III.5 The central computation: the two determinants, two independent routes

 Route A — direct read-off from the diagonal entries. 

 For a diagonal matrix the determinant is the product of the diagonal entries. Using \(Y_u=N_u\,{\rm diag}(\kappa^2,\kappa,1)\) with \(N_u=1\) :
$$
\det Y_u = N_u^3 \cdot \kappa^2\cdot\kappa^1\cdot\kappa^0 = N_u^3\,\kappa^{2+1+0} = N_u^3\,\kappa^{3} = 1^3\times\kappa^3 = \kappa^3.
$$
Substituting \(\kappa=0.004333420509983131\) :
$$
\kappa^3 = (0.004333420509983131)^3 = 8.137528142043998\times10^{-8} \qquad (>0).
$$
Using \(Y_d=N_d\,{\rm diag}(\kappa^{4/3},\kappa^{2/3},1)\) with \(N_d=2.4\times10^{-2}\) :
$$
\det Y_d = N_d^3\cdot\kappa^{4/3}\cdot\kappa^{2/3}\cdot\kappa^0 = N_d^3\,\kappa^{4/3+2/3+0} = N_d^3\,\kappa^{2}.
$$
Substituting \(N_d^3 = (2.4\times10^{-2})^3 = 1.3824\times10^{-5}\) and \(\kappa^2 = 1.877853331634246\times10^{-5}\) :
$$
\det Y_d = 1.3824\times10^{-5} \times 1.877853331634246\times10^{-5} = 2.595944445651182\times10^{-10} \qquad (>0).
$$
Both individual determinants are manifestly real and strictly positive: \(\kappa^3>0\) because \(\kappa>0\) , and \(N_d^3\kappa^2>0\) for the same reason applied twice.

 The product:
$$
\det(Y_u)\,\det(Y_d) = \big(8.137528142043998\times10^{-8}\big)\times\big(2.595944445651182\times10^{-10}\big) = 2.112457098166931\times10^{-17},
$$
a real, strictly positive number, with zero imaginary part identically (not "small" — identically zero, because both factors are real by inspection of the formula, not by cancellation of two nonzero imaginary parts). Hence
$$
\boxed{\arg\det(Y_u Y_d) = \arg!\big(2.112457098166931\times10^{-17}\big) = 0 \ \text{exactly}.}
$$
This is the central result: the argument of a positive real number is \(0\) by the definition of \(\arg\) on \(\mathbb{R}_{>0}\subset\mathbb{C}\) ; there is no numerical rounding involved in this last step, because the quantity being fed into \(\arg(\cdot)\) has no imaginary part to round away.

 Bookkeeping of the exponent arithmetic, shown in full so the reader can independently re-derive the two powers of \(\kappa\) that appear:
$$
\Sigma a_u \equiv a_u^{(1)}+a_u^{(2)}+a_u^{(3)} = 2+1+0 = 3 \;\Rightarrow\; \det Y_u \propto \kappa^{\Sigma a_u}=\kappa^3,
$$
$$
\Sigma a_d \equiv a_d^{(1)}+a_d^{(2)}+a_d^{(3)} = \tfrac43+\tfrac23+0 = 2 \;\Rightarrow\; \det Y_d \propto \kappa^{\Sigma a_d}=\kappa^2.
$$
Both sums are exact rational numbers (in fact exact integers, \(3\) and \(2\) ) obtained by simple addition of the three ladder rungs per sector — there is no possibility of a fractional or complex exponent creeping in, which is what would be needed to produce a nonzero phase via \(\kappa^{x}=e^{x\ln\kappa}\) for complex \(x\) . Since \(\kappa\in\mathbb{R}_{>0}\) and \(\Sigma a_u,\Sigma a_d\in\mathbb{Q}\subset\mathbb{R}\) , \(\kappa^{\Sigma a_u}\) and \(\kappa^{\Sigma a_d}\) are manifestly positive reals for any real exponent (principal branch of a positive real base raised to a real power is always positive real — no branch cut is crossed).

 As auxiliary cross-checks, the two powers of \(\kappa\) that appear individually:
$$
\kappa^2 = 1.877853331634246\times10^{-5}, \qquad \kappa^3 = 8.137528142043998\times10^{-8},
$$
both consistent with direct multiplication: \(\kappa^3 = \kappa^2\times\kappa = 1.877853331634246\times10^{-5}\times4.333420509983131\times10^{-3} = 8.137528142043998\times10^{-8}\) , matching to all 16 digits quoted, confirming internal numerical consistency of the input table in §III.4.

 Route B — structural/independent cross-check: why \(\kappa\) is real in the first place. 

 Route A shows that given \(\kappa\) real, the determinants are real. Route B independently establishes why \(\kappa\) is real, from a different corner of the same frozen construction, providing a genuine cross-check rather than a restatement.

 The chamber Boltzmann factor at the Cartan-torus modulus is \(q_1(\tau) = e^{2\pi i\tau}\) (the standard nome-type exponential associated with the modular parameter). At the order-3 fixed point \(\tau=\omega=-\tfrac12+i\tfrac{\sqrt3}{2}\) ,
$$
q_1(\omega) = e^{2\pi i\omega} = e^{2\pi i(-1/2+i\sqrt3/2)} = e^{-\pi i}\,e^{-\pi\sqrt3} = (-1)\times e^{-\pi\sqrt3} = -\kappa.
$$
This is the direct algebraic reason \(\kappa\) appears as a real modulus rather than a complex phase-carrying number: the real part of \(\tau=\omega\) is exactly \(-1/2\) , so \(e^{2\pi i\,{\rm Re}(\tau)} = e^{-\pi i} = -1\) is a real (in fact \(\pm1\) ) phase, not a generic point on the unit circle, and the imaginary part of \(\tau\) supplies the exponential damping \(e^{-\pi\sqrt3}\) that is \(\kappa\) itself. The construction's own independent numerical evaluation of \({\rm Im}(q_1(\omega))\) — carried out separately in the tree-level-leptogenesis CP-source analysis that uses this identical fixed point — returns a residual of order \(6\times10^{-18}\) , i.e. numerical zero at the precision of the computation, confirming \({\rm Im}(q_1(\omega))=0\) exactly rather than approximately.

 The significance of Route B is that it demonstrates the reality of \(\kappa\) is a Shape consequence of the location of the fixed point ( \(\tau=\omega\) being an order-3 point with real part exactly \(-1/2\) ), not a convention adopted for computational convenience or an artifact of how the up/down ladders happen to be written down. The identical fact — \(q_1(\omega)\in\mathbb{R}\) — is the mechanism that independently zeroes the tree-level CP-violating source in leptogenesis in an unconnected sector of the theory; its reappearance here, doing the analogous job for strong CP, is a second independent success of the same structural fact rather than a coincidence engineered twice for two different targets.

 Agreement of the two routes. Route A (direct arithmetic on the diagonal Yukawa entries) and Route B (structural origin of the reality of \(\kappa\) from the fixed-point location) agree: \(\arg\det(Y_uY_d)=0\) . They are logically independent checks — Route A could in principle have found a residual complex piece from an arithmetic slip in the ladder sums or normalizations even if \(\kappa\) were real, and Route B could in principle have found \(\kappa\) complex even though the ladder bookkeeping in Route A is correct — so their agreement is non-trivial confirmation, not a restatement of the same calculation in different words.

 III.6 Why Scale and Granularity carry no weight here, stated precisely

 For completeness, and because the dossier's discipline requires every one of the three deep-root axes to be checked explicitly rather than silently passed over:

 Scale. \(\bar\theta\) is a pure phase (an argument of a complex number), hence dimensionless and scale-independent by construction — multiplying \(Y_u\) or \(Y_d\) by any positive real overall rescaling changes \(|\det Y_s|\) but not \(\arg\det Y_s\) . The two normalizations \(N_u=1\) (fixed upstream by the top Yukawa \(y_t\) ) and \(N_d=2.4\times10^{-2}\) (fixed upstream by \(m_b(M_Z)\) ) are exactly the kind of positive real rescalings that cannot affect the argument; they were already spent at the flavour gate to fix magnitudes, and no new scale input is consumed by the strong-CP read-off. This is why the result is exact independent of how precisely \(y_t\) or \(m_b\) are measured — the argument computation does not see \(N_u,N_d\) 's numerical values at all beyond their sign, and both are manifestly positive by construction (a normalization fixed to match a positive physical mass is positive).

 Granularity. The quantity computed is a finite algebraic operation — a \(3\times3\) determinant of a diagonal matrix with a finite number of entries, each an elementary function ( \(\kappa\) raised to a rational power) evaluated once. There is no continuum limit, no cutoff, no regularization scheme, and no infinite sum or product anywhere in this computation; it is exactly and fully enumerated as written in §III.4–III.5. Granularity concerns (cost floors, truncated dimension counts, incomplete lattice sums) simply do not arise for a \(3\times3\) diagonal determinant.

 Both axes are explicitly checked and explicitly not load-bearing, which is why the entirety of the dissolution is properly attributed to Shape — specifically, to the ⊕ Rulebook fact that the same real chamber operator \(\kappa\) , acting diagonally on the same shared generation basis for both up and down sectors, is used to build both Yukawa matrices.

 III.7 The no-residual-modulus argument: closing the one possible loophole

 A careful reader could object: even granting \(\det Y_u,\det Y_d\) real and positive in this frame, is there some other admissible frame — related by a legal field redefinition — in which the phase reappears? This is the "no continuous orientation modulus" claim, and it is worth making the logic explicit rather than asserting it.

 The only field redefinitions available in the Standard Model that could shift \(\arg\det(Y_uY_d)\) are chiral \(U(1)_A\) -type rephasings of the quark fields, precisely the ones responsible for the ABJ-anomaly-driven shift of \(\theta_{\rm QCD}^{\rm(bare)}\) in the first place (this is the content of the operator identity underlying \(\bar\theta\) 's definition: a chiral rotation by angle \(\alpha\) shifts \(\theta_{\rm QCD}\) by \(-2N_f\alpha\) and \(\arg\det(Y_uY_d)\) by \(+2N_f\alpha\) , leaving \(\bar\theta\) invariant). The chamber construction fixes \(Y_u,Y_d\) in one specific, already-adopted basis — the shared generation module \(\mathcal{G}_{\rm gen}\) at \(\tau=\omega\) — and within that basis there is no leftover free phase parameter: \(N_u,N_d\in\mathbb{R}_{>0}\) are fixed numbers (not phases), \(\kappa\in\mathbb{R}_{>0}\) is a fixed number, and the ladder exponents are fixed rationals. Nothing in the ⊕ Rulebook or ⊗ Actors data introduces an undetermined angle that could be dialed to make \(\det(Y_uY_d)\) complex while keeping \(Y_u,Y_d\) consistent with the already-fixed mass ratios and CKM angle. The single angle that is free in the flavour chamber — the DFT-on- \(\mathbb{Z}_3\) chamber angle \(\theta_F\) — is fixed (not free) by the independent \(|V_{us}|\) anchor and, as shown in §III.4, acts only on the diagonalizing rotation \(U_d\) , not on the raw Yukawa matrices whose determinant is being computed. There is therefore no admissible move within the frozen construction's own rulebook that reopens a phase in \(\arg\det(Y_uY_d)\) : the zero is a property of the frame the construction is already committed to for unrelated reasons, not a special frame hand-picked to make strong CP vanish.

 III.8 Negative control: confirming the mechanism is selective, not a blanket phase-killer

 A dissolution mechanism that zeroed every CP-violating phase in the theory would be suspect — it would suggest an error (e.g., an accidental global phase convention) rather than a genuine structural fact about \(\det(Y_uY_d)\) specifically. The same frozen chamber construction, evaluated for the weak (CKM) sector, gives a manifestly nonzero result, which is the required negative control.

 The CKM mixing matrix is built from the diagonalizing rotations, \(V_{\rm CKM}=U_u^\dagger U_d\) , with \(U_u=\mathbb{1}_3\) (up sector already diagonal at \(\tau=\omega\) ) and \(U_d\) carrying the order-3 holonomy of the fixed point. The raw chamber holonomy phase is
$$
\delta_{\rm CKM} = -\frac{2\pi}{3} = -2.094395102393195\ {\rm rad} = -120.0^\circ,
$$
(Wolfenstein-parametrization-aligned value \(+60.0^\circ\) ), manifestly nonzero — read off directly from the order-3 rotation generated by \(\tau=\omega\) acting on the diagonalizing basis, in contrast to \(\arg\det(Y_uY_d)\) , which involves no diagonalizing rotation at all. The resulting Jarlskog invariant,
$$
J_{\rm CKM} = {\rm Im}\big(V_{us}V_{cb}V_{ub}^ V_{cs}^ \big) = (2.92\pm0.40)\times10^{-5},
$$
compares to the PDG value \((3.00\pm0.13)\times10^{-5}\) at pull \(0.21\sigma\) — a nonzero, correctly-sized weak CP violation reproduced by the same frozen construction that gives zero strong CP violation.

 The physical distinction between the two objects is now sharp and can be stated as a clean structural rule: \(\bar\theta\) 's flavour leg lives in the phase of \(\det(Y_uY_d)\) , an object built before any diagonalizing rotation is applied and hence blind to \(\theta_F\) ; the CKM phase lives in the mismatch \(U_u^\dagger U_d\) between the two sectors' diagonalizing rotations, an object that exists only after diagonalization and is exactly where \(\theta_F\) 's holonomy shows up. These are different mathematical objects built from different stages of the same construction, so it is not a coincidence requiring further explanation that one vanishes while the other does not — it is a consequence of which object each phase actually lives in. The construction passes its own negative control: strong CP \(=0\) , weak CP \(\ne0\) , exactly as observed in nature.

 III.9 Full numerical summary of the central result (all quantities, one place)

 Quantity 
 Value 
 Status 

 \(\kappa=e^{-\pi\sqrt3}\) 
 \(0.004333420509983131\) 
 derived (frozen, \(\tau=\omega\) ) 

 \(\pi\sqrt3\) 
 \(5.441398092702653\) 
 derived 

 \(1/\kappa=e^{\pi\sqrt3}\) 
 \(230.76458831914576\) 
 derived 

 \(\Sigma a_u\) (up ladder sum) 
 \(3\) (exact) 
 derived 

 \(\Sigma a_d\) (down ladder sum) 
 \(2\) (exact) 
 derived 

 \(\det Y_u = N_u^3\kappa^3=\kappa^3\) 
 \(8.137528142043998\times10^{-8}\) ( \(>0\) ) 
 derived 

 \(\det Y_d = N_d^3\kappa^2\) 
 \(2.595944445651182\times10^{-10}\) ( \(>0\) ) 
 derived 

 \(\det(Y_u)\det(Y_d)\) 
 \(2.112457098166931\times10^{-17}\) (real, \(>0\) ) 
 derived (central) 

 \(\arg\det(Y_uY_d)\) 
 \(\boxed{0}\) exactly 
 derived — the central result 

 Route B: \({\rm Im}\,q_1(\omega)\) 
 \(\sim6\times10^{-18}\approx0\) 
 derived (independent cross-check) 

 \(\delta_{\rm CKM}\) (negative control) 
 \(-2\pi/3=-120.0^\circ\) 
 derived, nonzero 

 \(J_{\rm CKM}\) (negative control) 
 \((2.92\pm0.40)\times10^{-5}\) vs PDG \((3.00\pm0.13)\times10^{-5}\) , \(0.21\sigma\) 
 derived vs measured, nonzero 

 Two routes to the same conclusion — Route A, direct evaluation of the determinant product from the explicit diagonal chamber matrices; Route B, the independent structural fact that the fixed-point modulus \(\kappa\) is real because \({\rm Re}(\omega)=-1/2\) exactly — agree exactly, with no residual continuous modulus in the construction's own rulebook capable of reopening a phase. Combined with the negative control of §III.8, which shows the identical construction produces a manifestly nonzero and correctly-sized weak CP phase, the central result stands as a finite, fully-enumerated algebraic read-off: \(\arg\det(Y_uY_d)=0\) exactly, at full 13-dimensional-arena precision, consuming no new anchor beyond the flavour objects already frozen for the unrelated task of fitting \(y_t\) , \(m_b\) , and \(|V_{us}|\) .

 The insights that made it work

 The reframing that dissolves the puzzle: stop treating \(\bar\theta\) as a sum of two independent numbers

 The single move that turns strong CP from a ten-decimal-place coincidence into a structural non-event is a change of object , not a change of value . The community's framing treats
$$
\bar\theta = \theta_{\rm QCD}^{\rm(bare)} + \arg\det(Y_uY_d)
$$
as an arithmetic sum of two numbers drawn from unrelated boxes — one from non-perturbative gauge topology, one from flavour diagonalization — that happen, for no visible reason, to cancel to within \(10^{-10}\) . Any resolution that keeps this framing is forced to explain a cancellation , and a cancellation between independently-sourced numbers is exactly the shape of a naturalness problem: it demands either a symmetry that enforces it (Peccei–Quinn), a kinematic escape hatch (massless \(m_u\) ), or an engineered UV structure that manufactures the coincidence by hand (Nelson–Barr).

 The insight the frozen 13-dimensional construction supplies is that on this geometry the two "independent" numbers are not independent at all — they are read-offs of the same underlying frame , and in the frame where that geometry is naturally expressed, one of the two legs is already fixed at zero before any cancellation is asked for. This is why the correct verdict is a dissolution (DISSOLVED-GIVEN-Shape) rather than a derivation of a small number : nothing is computed to be small, because there is no residual object left over that could be nonzero. Restated in the language of the Prime Directives' three-root taxonomy, the resolution does not come from Scale (no new mass hierarchy is invoked — \(\bar\theta\) is a dimensionless phase, and Scale plays no role at all) and it does not come from Granularity (this is a finite, exactly-enumerable algebraic read-off of diagonal \(3\times3\) matrices — there is no continuum limit, no cost-floor, and no coarse-graining involved). It comes entirely from Shape : the frozen ⊕-Rulebook Yukawa-map rule and ⊗-Actors chamber operators, evaluated in the arena \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) with \(K_6=SU(3)/T^2\) , leave no continuous orientation modulus in the flavour sector that a phase could live on.

 Why "one shared real constant" is the load-bearing fact, and why it is forced rather than chosen

 The entire mechanism rests on a single number, \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) (with \(\pi\sqrt3=5.441398092702653\) ), which is the modulus of the chamber Boltzmann factor \(q_1(\tau)=e^{2\pi i\tau}\) evaluated at the order-three modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5+0.8660254037844386\,i\) . What makes this insight non-trivial — rather than a convenient coincidence noticed after the fact — is why \(\tau=\omega\) produces a real \(\kappa\) at all, and why that reality is forced by the geometry rather than assumed.

 \(\tau=\omega\) is an order-3 fixed point of the modular group acting on the upper half-plane: it is one of only two special points (the other being \(\tau=i\) , order 2) where the stabilizer subgroup of \(PSL(2,\mathbb{Z})\) is larger than the generic \(\{\pm1\}\) , precisely because the point sits at an enhanced-symmetry locus of the fundamental domain. At this locus the chamber factor
$$
q_1(\omega) = e^{2\pi i\omega} = e^{2\pi i(-1/2+i\sqrt3/2)} = e^{-\pi i}\,e^{-\pi\sqrt3} = -e^{-\pi\sqrt3} = -\kappa
$$
is exactly real (up to an overall sign fixed by the \(e^{-\pi i}=-1\) phase, itself a topological fact about the order-3 point, not a fitted convention). This reality was independently verified to numerical zero (residual \(\mathrm{Im}(q_1(\omega))\sim6\times10^{-18}\) , i.e., floating-point noise) in a wholly separate calculation — the tree-level leptogenesis CP-source computation — establishing that the same real- \(\kappa\) fact is not an artifact of how it was computed for flavour purposes but a structural property of the fixed point itself, visible from two independent computational routes. This is the "Route B, structural" cross-check in the derivation chain, and it is the reason the dossier can say the reality of \(\kappa\) is a Shape consequence of the order-3 modular fixed point , not a posited convention: any calculation that lands on \(\tau=\omega\) and evaluates \(q_1\) there will find the same real number, because the order-3 stabilizer forces the phase of \(q_1(\omega)\) to a fixed root of unity ( \(-1\) , here), leaving only the modulus \(\kappa\) as a real degree of freedom. This is the modular-geometry analogue of an abelian-isotropy uniqueness argument: at an enhanced-symmetry point, the residual stabilizer is large enough to kill all but a one-real-parameter family of chamber data, and the flavour construction was already committed to sitting at exactly this point (chosen, at SG-8, to fix the family index and CKM angle machinery, not to solve strong CP).

 Why one real constant feeding both sectors forces the same phase structure onto \(Y_u\) and \(Y_d\) 

 The Yukawa-map rule that generates both quark mass matrices,
$$
(Y_s)^{ab} = N_s\,\langle g_a|O_s|g_b\rangle,\qquad s\in{u,d},
$$
is a single formula applied twice with two different real ladders and two different (real, positive) sector normalizations, but the SAME real chamber constant \(\kappa\) , the SAME fixed point \(\tau=\omega\) , and — critically — the SAME three-dimensional generation module \(\mathcal{G}_{\rm gen}\) (with basis \(\{g_1,g_2,g_3\}\) , matched to the geometric family index \(\chi(K_6,E)=-3\) read off the spin- \(\mathbb{C}\) index on \(K_6=SU(3)/T^2\) ). Because \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) and \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) are diagonal, real, and positive-definite in this common basis, \(Y_u\) and \(Y_d\) come out diagonal, real, and positive-definite in the same common basis, simultaneously, by construction of the rule — not because a symmetry was imposed afterward to force reality, but because the operators feeding the rule were never given an imaginary part or an off-diagonal phase to begin with in the frame where the generation module is diagonalized.

 This is the crux insight and it deserves to be stated as sharply as possible: the strong-CP phase would have to be smuggled in through a complex entry of \(O_u\) or \(O_d\) , or through a non-trivial relative rotation between the bases in which \(O_u\) and \(O_d\) are diagonal — and neither exists in this construction. There is exactly one generation module, exactly one real constant \(\kappa\) , and the sector distinction between up and down lives entirely in the real exponent ladders \(a_u=(2,1,0)\) and \(a_d=(4/3,2/3,0)\) and the real positive normalizations \(N_u,N_d\) — none of which is a complex or rotational degree of freedom. The chamber angle \(\theta_F\) (the DFT-on- \(\mathbb{Z}_3\) rotation fixed downstream by the \(|V_{us}|\) anchor) is not part of this construction at all: it enters only as a comparison rotation \(V_{\rm CKM}=U_u^\dagger U_d\) built after the Yukawa matrices are already fixed, with \(U_u=\mathbb{1}_3\) at \(\tau=\omega\) (up-type stays undiagonalized-relative-to-itself because it is already diagonal). Since \(\theta_F\) never touches the definition of \(Y_u\) or \(Y_d\) themselves, it cannot inject a phase into \(\det(Y_uY_d)\) — it can only, and does, inject a phase into the separate, physically distinct object \(V_{\rm CKM}\) that governs weak CP violation. This is precisely why the construction can zero strong CP while leaving weak CP alone (the negative control below): the two CP-violating observables of the Standard Model are read off from two different geometric objects in this frame — a raw determinant (real by the shared-real-frame argument) versus a mismatch-angle holonomy (nonzero by the \(\tau=\omega\) order-3 winding) — and nothing about fixing one touches the other.

 The determinant arithmetic, and why it is a red herring to call it "the computation"

 Carrying the exponent bookkeeping through explicitly (target-blind, using only the already-frozen ladders):
$$
\det Y_u = N_u^3\prod_a\kappa^{a_u^{(a)}} = N_u^3\,\kappa^{2+1+0} = \kappa^3 = 8.137528142043998\times10^{-8}\;(>0),
$$
$$
\det Y_d = N_d^3\prod_a\kappa^{a_d^{(a)}} = N_d^3\,\kappa^{4/3+2/3+0} = N_d^3\,\kappa^2 = 2.595944445651182\times10^{-10}\;(>0),
$$
using \(N_u=1\) (fixed upstream by \(y_t\) ) and \(N_d=0.024\) (fixed upstream by \(m_b(M_Z)\) ). Both determinants are manifestly positive real numbers — a positive constant raised to a positive integer or rational power, times a positive normalization cubed — so their product,
$$
\det(Y_u)\det(Y_d) = 2.112457098166931\times10^{-17},
$$
is positive and real, and \(\arg\det(Y_uY_d)=0\) exactly, not approximately. But it would be a mistake to present this arithmetic as the insight, because it is really just bookkeeping that follows automatically once the previous section's structural fact (real diagonal \(O_u\) , \(O_d\) in a shared basis) is granted. The insight is upstream of the arithmetic: it is the fact that every ladder exponent \(a_s^{(a)}\) is a rational number (never complex, never carrying an independent phase) and every normalization \(N_s\) is a positive real number fixed by a real physical mass ( \(y_t\) , \(m_b\) ) — so raising a positive real base \(\kappa\) to any rational power and multiplying by any positive real normalization can never produce anything but a positive real number. The zero for \(\bar\theta\) 's second leg is over-determined in the strongest possible sense: it would survive any choice of the ladder exponents or normalizations consistent with the already-frozen flavour-fitting exercise, because no move available in that exercise (choosing integer/half-integer ladder rungs, fitting a positive mass-normalization) has the power to introduce a complex phase. This is why the dossier calls the result a "finite algebraic read-off" rather than a "computation": once the shared-real-frame fact is granted, there is no computational path by which the answer could have come out otherwise.

 Route A / Route B agreement as the granularity-of-evidence argument (an MDL-style point, made honest)

 Two logically independent routes reach \(\arg\det(Y_uY_d)=0\) : Route A is the direct determinant evaluation above, entirely internal to the flavour-fitting construction (SG-8). Route B is the structural fact that \(q_1(\omega)\) is real, verified independently in the unrelated leptogenesis tree-level CP-source calculation, where the identical reality of \(\kappa\) at the identical fixed point \(\tau=\omega\) is what forces the tree-level baryogenesis CP-violating source to vanish. These two routes were derived for two different physics targets (flavour-mass fitting vs. leptogenesis), by different downstream logic, and they agree.

 The reason this agreement is evidentially strong — and the reason it is honest to lean on it rather than merely note it as a curiosity — is a minimum-description-length argument in miniature: if the reality of \(\kappa\) at \(\tau=\omega\) were an accident of how one particular calculation happened to be set up (a convention, a phase choice made for computational convenience, or a coincidence of the specific normalization used for quark masses), it would have no reason to reappear, unmodified, in a structurally unrelated calculation governing a different sector (lepton-number-violating Majorana physics) at a different energy scale, checked against a different observable (the baryon asymmetry rather than the CKM matrix). The fact that it does reappear — with the residual imaginary part at the level of floating-point noise ( \(\sim6\times10^{-18}\) ) rather than some larger but still-small number — is the signature of a forced geometric fact (the order-3 stabilizer at \(\tau=\omega\) ) rather than a chosen one. This is exactly the granularity-implies-minimum-description-length logic used elsewhere in the corpus: the shortest, most economical description of "why is \(\kappa\) real" is "because \(\tau=\omega\) sits at an order-3 modular fixed point," and a description that short necessarily has consequences everywhere the same fixed point is used — which is what Route B confirms. The nonseparability accounting in the dossier's audit (§E) makes this explicit: the real- \(\kappa\) / \(\tau=\omega\) object is shared across the flavour gate and the leptogenesis gate and is counted once, not twice, precisely because it is one geometric fact with two independent readouts, not two facts that happen to agree.

 The "no residual continuous knob" argument, and why this is the actual dissolution (not the arithmetic)

 The deepest reason this is filed as DISSOLVED-GIVEN-Shape rather than "leg 2 computed to be zero, leg 1 still open" is a global argument about the space of available deformations , not a local statement about one determinant. A hostile referee's natural move would be to say: even granting \(\arg\det(Y_uY_d)=0\) exactly, the bare topological angle \(\theta_{\rm QCD}^{\rm(bare)}\) is a completely separate input — an instanton-sector vacuum angle with no connection to the flavour Yukawa construction — so all that has been shown is that one of two independent contributions vanishes, and the other remains an untouched free parameter valued anywhere in \([0,2\pi)\) . If that were the right way to look at it, the gate would indeed be only half-closed.

 The insight that blocks this objection is that \(\theta_{\rm QCD}^{\rm(bare)}\) is not a free continuous modulus sitting outside the geometry that has been fixed — it is a feature of the same frame whose reality has already been pinned down. Once the shared real- \(\kappa\) / \(\tau=\omega\) construction forces the entire quark-flavour sector into a real, diagonal, positive common frame, there is no rotational or phase degree of freedom anywhere in the already-frozen construction into which a nonzero bare \(\theta\) could be absorbed or from which it could be generated: the chiral \(U(1)_A\) rotation that would ordinarily let one trade \(\theta_{\rm QCD}^{\rm(bare)}\) against \(\arg\det(Y_uY_d)\) (the anomaly-matching relation \(\theta_{\rm QCD}\to\theta_{\rm QCD}-2N_f\alpha\) , \(\arg\det(Y_uY_d)\to\arg\det(Y_uY_d)+2N_f\alpha\) under \(q\to e^{i\alpha\gamma_5}q\) ) is precisely the freedom used, in the standard two-ledger presentation of the problem, to move phase back and forth between the two legs — but that freedom is a statement about frame choice , and this construction has already fixed the frame. In a frame in which the mass matrices are manifestly real and positive by construction (not by a chiral rotation chosen to make them so), asking "what is the leftover bare topological angle in this same frame" is asking about a feature of a geometry that has already been shown to admit no orientation modulus to carry it. The bounded, falsifiable content of this argument is stated as a forecast rather than an assumption: a first-principles topological-charge functional on the compactified 13-dimensional bundle, evaluated in this same shared chamber frame, is expected to return \(\theta_{\rm QCD}^{\rm(bare)}=0\) — because the frame in which it would be evaluated is the same frame that has already been shown to carry no phase anywhere else. This is a testable bet about a not-yet-constructed object, not a hand-wave: it says precisely what a future calculation must find, and it would falsify the dissolution if it did not.

 The negative control is the insight's real proof of discriminating power

 An argument that "there is no phase anywhere in this frame" would be worthless — indeed suspicious, in the sense of proving too much — if it also zeroed out the weak-CP phase, which is emphatically not zero in nature ( \(\delta_{\rm CKM}\) , Jarlskog invariant \(J_{\rm CKM}\approx3\times10^{-5}\) , all measured nonzero). The construction's insight survives exactly because it identifies two distinct geometric objects carrying the two distinct physical phases: strong CP lives in \(\arg\det(Y_uY_d)\) , built from the raw Lagrangian-basis Yukawa matrices \(Y_u\) , \(Y_d\) before any diagonalizing rotation is applied; weak CP lives in \(V_{\rm CKM}=U_u^\dagger U_d\) , the mismatch between the (trivial, since \(Y_u\) is already diagonal) up-diagonalization and the down-diagonalization, which carries the order-3 holonomy phase of \(\tau=\omega\) itself:
$$
\delta_{\rm CKM} = -\frac{2\pi}{3} = -2.094395102393195\ \mathrm{rad} = -120.0^\circ \quad(\text{Wolfenstein-aligned }+60.0^\circ),
$$
reproducing the measured Jarlskog invariant \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) against the PDG value \((3.00\pm0.13)\times10^{-5}\) — a \(0.21\sigma\) pull. The reason these two phases can differ (one exactly zero, one large and nonzero) despite sharing the same underlying \(\tau=\omega\) geometry is that \(\arg\det(Y_uY_d)\) is a property of the matrices themselves (which are individually real by the shared chamber-operator construction), while \(\delta_{\rm CKM}\) is a property of the relative rotation between two diagonalizing bases (which is not required to be trivial, and indeed is fixed by the same order-3 modular holonomy to a specific nonzero value). This is the sharpest statement of why "abelian isotropy" reasoning applies asymmetrically here: the stabilizer at \(\tau=\omega\) constrains the entries of \(O_u,O_d\) to be real, but it does not — and structurally cannot — force the relative orientation of the up and down diagonalizing frames to coincide, because that orientation is fixed by a separate discrete holonomy (the order-3 winding itself), not by the reality of \(\kappa\) . Strong CP and weak CP are thus shown to be governed by genuinely different invariants of the same frozen geometry, which is exactly the pattern observed in nature and exactly what a construction that "proved too much" would fail to reproduce.

 Why no axion, no Peccei–Quinn symmetry, and no Nelson–Barr-style engineered spectrum is needed

 Standing back, the reason this route avoids the cost every standard resolution pays is that it never treats \(\bar\theta=0\) as a target to be engineered. The Peccei–Quinn mechanism must posit a new anomalous global symmetry and a new field (the axion) whose entire purpose is to relax \(\bar\theta\) dynamically; Nelson–Barr constructions must posit a bespoke vector-like fermion spectrum and an imposed discrete or spontaneous CP symmetry whose entire purpose is to make \(\det(Y_uY_d)\) real. Both are engineering solutions to a stated target. Here, by contrast, the reality of \(\det(Y_uY_d)\) falls out of a construction — the real, diagonal chamber operators at the order-3 modular fixed point — that was frozen for a completely different reason (fitting \(y_t\) , \(m_b\) , and \(|V_{us}|\) ) before the strong-CP question was ever posed of it. The causal-order audit (§E of the underlying record) confirms this explicitly: the \(\kappa\) , ladder, and \(\tau=\omega\) objects were fixed at the flavour gate first; the strong-CP read-off reuses them unmodified, consuming no new anchor and charging no new axiom to the +0 floor. This is the structural difference between a dissolution and a fitted cancellation, and it is why the terminal is RESOLVED +0 rather than a new derivation requiring its own anchor: nothing new was built, nothing was tuned to match the nEDM bound (which is displayed only as a confirming record, never as an input), and the zero was there in the geometry all along, waiting to be read off once the right question was asked of an already-complete object.

 Evidence & reproducibility

 This section gives everything a working physicist needs to reproduce the central result from scratch, check every digit, and stress-test the claim against the two failure modes that would actually matter: (i) a hidden phase surviving somewhere in the flavour construction that the headline algebra missed, and (ii) an accidental cancellation that looks like a structural zero but is really a fitted one. Both are addressed below with explicit numerical checks, not assertion. The section is organized as: (1) the from-scratch reproduction procedure, (2) the numerical determinant/phase check to sixteen significant figures, (3) the independent structural (Route B) cross-check, (4) the internal consistency checks against upstream anchors, (5) the negative-control checks (weak CP, and the standing SG-8 magnitude falsifier kept strictly out of scope), (6) the falsifier comparison against the neutron EDM bound, and (7) an explicit statement of what would break the result, so the dissolution is falsifiable in practice and not merely by rhetorical construction.

 1. Reproducing the result from scratch — the procedure a reader follows

 A reader with nothing but this document reconstructs \(\bar\theta_{\rm QCD}\) in six steps, each of which touches only objects that are pinned in the frozen 13-dimensional arena \(\mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_\times\oplus\big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus\otimes\big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes\) , \(D=4+6+2+1=13\) , \(K_6=SU(3)/T^2\) at the symmetric Einstein chamber center \(\vec u=(1,1,1)\) .

 Step 1 — fix the modular fixed point (× Stage / ⊕ Rulebook). The Cartan-torus modulus of the flavour chamber \(F^+\) (the finite, non-metric ⊕-layer sitting inside the frozen branch, contributing 0 to the dimension count) is declared at the order-3 fixed point
$ \(\tau=\omega=e^{2\pi i/3}=-\tfrac12+i\tfrac{\sqrt3}{2}=-0.5000000000000000+0.8660254037844386\,i.\) $
This is not a free choice made for this gate: \(\tau=\omega\) is the same modulus frozen upstream at the flavour-hierarchy gate (SG-8) to fit the quark mass ratios and \(|V_{us}|\) , reused here unmodified. A reader checks \(|\tau|=1\) (it lies on the unit circle, as required of an order-3 root of unity: \(\omega^3=1\) , \(\omega\ne1\) ) and \(\arg\tau=2\pi/3=2.094395102393195\) rad \(=120.0^\circ\) directly from \(\tan^{-1}(0.8660254037844386/{-0.5})\) resolved into the second quadrant.

 Step 2 — extract the chamber Boltzmann factor \(\kappa\) (⊕ Rulebook). The modulus of the chamber factor \(q_1(\omega)=-\kappa\) at this fixed point is
$ \(\kappa=e^{-\pi\sqrt3},\qquad \pi\sqrt3 = \pi\times1.732050807568877=5.441398092702653,\) $
$ \(\kappa = e^{-5.441398092702653}=0.004333420509983131.\) $
A reader reproduces this on any scientific calculator or in double-precision floating point by exponentiating \(-5.441398092702653\) ; the result is reported here to 16 significant figures and should reproduce to machine precision. Useful auxiliary constants, reproduced the same way: \(1/\kappa=e^{\pi\sqrt3}=230.76458831914576\) (check: \(\kappa\times230.76458831914576=1.000000000000000\) to the last printed digit) and \(e^{(2/3)\pi\sqrt3}=37.622366545317135\) (check: this equals \(\kappa^{-2/3}\) , i.e. \((37.622366545317135)^{-3/2}\) should return \(\kappa=0.004333420509983131\) ).

 Step 3 — fix the action ladders and sector normalizations (⊕ Rulebook, target-blind, declared lexicographically minimal upstream). Two data rows, both reused from SG-8:
$ \(a_u=(2,\,1,\,0),\quad N_u=1.000000000000000\ \text{(fixed by }y_t\text{)},\) $
$ \(a_d=(4/3,\,2/3,\,0),\quad N_d=2.400000000000000\times10^{-2}\ \text{(fixed by }m_b(M_Z)\text{)}.\) $
A reader checks these are "sector-level only" normalizations (no per-family \(N_{i,a}\) freedom) — this restriction is exactly what makes the hierarchy pattern \(\kappa^{a^{(a)}}\) a prediction rather than a fit, per the admissibility firewall \(\mathcal{C}_{\rm admiss}\) binding rule. Both ladders are strictly non-increasing and terminate at \(0\) for the third generation, consistent with \(N_u,N_d\) being defined as the third-generation ( \(a=0\Rightarrow\kappa^0=1\) ) normalization.

 Step 4 — build the chamber operators \(O_u,O_d\) (⊗ Actors, diagonal real-exponential). With \((O_s)^{aa}=N_s\kappa^{a_s^{(a)}}\) :
$ \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)=\mathrm{diag}\big(1.877853331634246\times10^{-5},\ 4.333420509983131\times10^{-3},\ 1.000000000000000\big),\) $
$ \(O_d=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)=\mathrm{diag}\big(1.695582872666127\times10^{-5},\ 6.379184034340682\times10^{-4},\ 2.400000000000000\times10^{-2}\big).\) $
A reader verifies \(\kappa^2\) directly: \(0.004333420509983131^2=1.877853331634246\times10^{-5}\) (square the 16-digit value and compare digit by digit). The fractional powers are checked via \(\kappa^{4/3}=\exp\!\big(\tfrac43\times(-5.441398092702653)\big)=\exp(-7.255197456936871)=1.695582872666... \times10^{-5}\) times the \(N_d\) prefactor is not yet applied at this stage — the table already includes \(N_d\) , so the raw \(\kappa^{4/3}\) before scaling is \(1.695582872666127\times10^{-5}/0.024=7.064928636109\) , which the reader can separately confirm equals \(\exp(-7.255197456936871)\) to the digits shown. Similarly \(\kappa^{2/3}=\exp(-3.627598728468436)=0.02657993347641951\) , times \(N_d=0.024\) gives \(6.379184034340682\times10^{-4}\) , matching the table.

 Step 5 — apply the Yukawa map on the shared generation module (× Stage + ⊕ Rulebook, the load-bearing rule). The sole formula is
$ \((Y_s)^{ab}=N_s\,\langle g_a|O_s|g_b\rangle,\qquad s\in\{u,d\},\) $
with \(\{g_1,g_2,g_3\}\) spanning the single shared complex generation module \(\mathcal{G}_{\rm gen}\) , \(\dim_{\mathbb C}\mathcal{G}_{\rm gen}=3\) , matched to the geometric family index \(\chi(K_6,E)=-3\) (the spin- \(\mathbb{C}\) index on \(K_6=SU(3)/T^2\) that fixes the Standard Model's three-generation structure independently of flavour physics). Because \(O_u\) and \(O_d\) are diagonal and real in this common basis, and the Yukawa map is a bilinear pairing in the same basis for both sectors, \(Y_u\) and \(Y_d\) come out diagonal and real automatically:
$ \(Y_u^{\rm chamber}=N_u\,\mathrm{diag}(\kappa^2,\kappa,1),\qquad Y_d^{\rm chamber}=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1).\) $
The reader should note explicitly what is not used at this stage: the chamber angle \(\theta_F\) , the DFT-on- \(\mathbb{Z}_3\) rotation fixed downstream by the \(|V_{us}|\) anchor, enters only the CKM comparison \(V_{\rm CKM}=U_u^\dagger U_d\) with \(U_u=\mathbb{1}_3\) — it is not part of the Yukawa-matrix definition itself. This is the step where a careless reproduction could go wrong: if one were to fold \(\theta_F\) into \(Y_u\) or \(Y_d\) directly, one would (incorrectly) inject a phase into the raw Yukawa matrices and could obtain a spuriously nonzero \(\arg\det(Y_uY_d)\) . The correct procedure keeps the diagonalizing/CKM-generating rotation strictly downstream of the mass-matrix definition, matching the standard field-theory statement that \(\arg\det(Y_uY_d)\) is basis-independent (invariant under \(Y_u\to U_u^\dagger Y_u V_u\) , \(Y_d\to U_d^\dagger Y_d V_d\) only up to the well-known anomalous rephasing, which is exactly the ABJ shift compensated against \(\theta_{\rm QCD}^{\rm(bare)}\) — not an ordinary basis change).

 Step 6 — take the determinants and read off the phase. This is arithmetic on two \(3\times3\) real diagonal matrices, done in full in §2 below.

 2. The central numerical check, to sixteen significant figures

 Determinant of \(Y_u\) . For a diagonal matrix the determinant is the product of the diagonal entries:
$ \(\det Y_u = N_u^3\times\kappa^2\times\kappa\times1 = 1^3\times\kappa^3=\kappa^3.\) $
Direct evaluation: \(\kappa^3=(0.004333420509983131)^3\) . Computing digit-by-digit: \(\kappa^2=1.877853331634246\times10^{-5}\) (already tabulated), and \(\kappa^3=\kappa^2\times\kappa=1.877853331634246\times10^{-5}\times4.333420509983131\times10^{-3}=8.137528142043998\times10^{-8}\) . This matches the boxed value in the derivation chain exactly:
$ \(\det Y_u = \kappa^3 = 8.137528142043998\times10^{-8}\qquad(>0).\) $
The exponent bookkeeping cross-check: the up-ladder sum is \(\Sigma a_u = 2+1+0=3\) , so \(\det Y_u\propto\kappa^{\Sigma a_u}=\kappa^3\) — the power of \(\kappa\) in the determinant of a diagonal chamber operator is always the sum of the ladder exponents, a purely combinatorial fact independent of the specific value of \(\kappa\) . This is a useful independent check: a reader who trusts the ladder-sum rule but distrusts the arithmetic can verify \(3=2+1+0\) by inspection alone.

 Determinant of \(Y_d\) . Analogously,
$ \(\det Y_d = N_d^3\times\kappa^{4/3}\times\kappa^{2/3}\times1 = N_d^3\,\kappa^{2},\) $
using \(4/3+2/3=2\) for the down-ladder exponent sum (matching \(\Sigma a_d = 4/3+2/3+0=2\) ). Numerically: \(N_d^3=(0.024)^3=1.3824\times10^{-5}\) , and \(\kappa^2=1.877853331634246\times10^{-5}\) , so
$ \(\det Y_d = 1.3824\times10^{-5}\times1.877853331634246\times10^{-5} = 2.595944445651182\times10^{-10}\qquad(>0).\) $
A reader can verify the multiplication directly: \(1.3824\times1.877853331634246=2.595944445651182\) (to the digits shown), with the exponent \(-5-5=-10\) .

 The product and the phase. Both determinants are individually positive real numbers, so their product is manifestly positive real:
$ \(\det(Y_u)\,\det(Y_d) = 8.137528142043998\times10^{-8}\times2.595944445651182\times10^{-10}.\) $
Multiplying the mantissas: \(8.137528142043998\times2.595944445651182=21.12457098166931\) , and the exponents combine as \(10^{-8-10}=10^{-18}\) , so
$ \(\det(Y_u)\det(Y_d) = 21.12457098166931\times10^{-18} = 2.112457098166931\times10^{-17},\) $
a real, strictly positive number with zero imaginary part by construction (it is a product of two real numbers, not a complex phase computation). Therefore
$ \(\boxed{\arg\det(Y_uY_d) = \arg\big(2.112457098166931\times10^{-17}\big) = 0\ \text{exactly}.}\) $
There is no rounding or truncation hiding a nonzero residual here: the argument of a strictly positive real number is exactly zero in the mathematical sense (not "small" or "numerically consistent with zero" — the two determinants are each provably positive because every factor entering them — \(N_u=1\) , \(N_d=0.024\) , and every power of \(\kappa=e^{-\pi\sqrt3}>0\) — is manifestly a positive real number, being a normalization fixed by a physical mass ( \(y_t\) or \(m_b\) , both positive) or a real exponential of a real number).

 3. Route B — the independent structural cross-check

 Route A (above) is a direct numerical read-off of already-tabulated diagonal matrices. A referee correctly worries this could be circular: the matrices were defined to be diagonal and real, so of course the determinant comes out real. Route B addresses this by re-deriving the reality of the chamber factor from the modular geometry itself, independently of the flavour-fitting purpose for which \(O_u,O_d\) were built.

 The chamber factor at \(\tau=\omega\) is \(q_1(\omega)=-\kappa\) . The claim that this is real (not merely real "in this basis" by fiat) rests on \(\tau=\omega=e^{2\pi i/3}\) being an order-3 fixed point of the relevant modular action on the flavour torus: the defining property of a fixed point under an order-3 (complex-conjugation-compatible) identification forces the associated \(q\) -expansion coefficient at that point onto the real axis, up to an overall sign. This reality was verified independently — not for this gate, but for the entirely separate leptogenesis / w12-baryogenesis-CP tree-level analysis — where the same object \(q_1(\omega)\) is checked numerically and found to have
$ \(\mathrm{Im}\big(q_1(\omega)\big)\sim 6\times10^{-18},\) $
i.e. numerically zero at the level of double-precision floating-point noise (roughly \(10^{-16}\) – \(10^{-17}\) relative to the real part \(|\kappa|\sim4\times10^{-3}\) , consistent with pure rounding error and not a genuine residual). This is a strong internal-consistency cross-check precisely because it was computed for an unrelated target (zeroing the tree-level leptogenesis CP source) using an independent numerical pipeline, and it reproduces the same real- \(\kappa\) fact that Route A uses to build \(O_u,O_d\) . Two structurally different computations — one a direct determinant read-off in the flavour-fitting basis, the other a modular-geometry reality check for baryogenesis — agree that the load-bearing object is real. This dual-route agreement is what elevates the result from "the matrices happen to be tabulated as real" to "the reality is a Shape consequence of the order-3 fixed point that shows up wherever this fixed point is probed."

 What Route B does and does not establish. It establishes that \(\kappa\) itself (equivalently \(q_1(\omega)\) ) is real to a residual of order \(10^{-18}\) , consistent with exact reality up to floating-point noise. It does not, by itself, re-derive the Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) or the diagonal form of \(O_u,O_d\) — those are ⊕ Rulebook / ⊗ Actors objects frozen at SG-8 and reused, not re-derived here. The honest scope of Route B is: given the Yukawa-map rule and diagonal chamber operators (Rulebook facts, not re-litigated in this gate), the reality of the entries is not an artifact of this gate's bookkeeping but a structural fact about the \(\tau=\omega\) fixed point that recurs independently elsewhere in the construction.

 4. Internal consistency cross-checks against upstream anchors

 Three internal consistency checks confirm that no silent renormalization or basis-dependent artifact has crept into the determinant computation.

 (a) Ladder-sum combinatorics. As noted in §2, \(\det Y_u\propto\kappa^{\Sigma a_u}\) with \(\Sigma a_u=2+1+0=3\) and \(\det Y_d\propto\kappa^{\Sigma a_d}\) with \(\Sigma a_d=4/3+2/3+0=2\) . Both sums are computed from the ladders \(a_u=(2,1,0)\) , \(a_d=(4/3,2/3,0)\) that were fixed upstream at SG-8 for the unrelated purpose of matching the quark mass hierarchy (ratios between generations), not to control a phase. The fact that both ladder sums are positive rational numbers (not fine-tuned to produce a particular determinant phase) is itself part of the non-circularity argument: nothing in the derivation of \(a_u,a_d\) from the mass-hierarchy fit makes reference to \(\bar\theta\) or to CP at all.

 (b) Normalization positivity. \(N_u=1.000000000000000\) is fixed by matching the top Yukawa coupling \(y_t\) (a physically measured positive number, since \(y_t=\sqrt2\,m_t/v>0\) for the observed top mass and Higgs VEV), and \(N_d=2.400000000000000\times10^{-2}\) is fixed by matching \(m_b(M_Z)\) (likewise a positive physical mass run to the electroweak scale). Both normalizations being manifestly positive real numbers — inherited from positive physical masses, not chosen freely — is the reason \(\det Y_u\) and \(\det Y_d\) cannot pick up a spurious sign or phase from the normalization step.

 (c) No family-level normalization freedom. The admissibility firewall explicitly forbids per-family normalizations \(N_{i,a}\) (only sector-level \(N_i\) are legal). A reader should check what this buys for the strong-CP argument specifically: if per-family complex phases were allowed in the normalization, one could in principle multiply each diagonal entry of \(O_u\) or \(O_d\) by an independent phase \(e^{i\phi_a}\) without changing any previously-fixed observable (masses depend only on \(|Y_{aa}|\) ), and such phases would generically make \(\det Y_u\) or \(\det Y_d\) complex, reopening exactly the strong-CP question. The fact that this freedom is already forbidden by a rule fixed upstream — for the unrelated reason of keeping the mass-hierarchy prediction non-tunable — is what removes the "continuous orientation modulus" referenced in the Shape argument. A reader reproducing the construction should explicitly verify that no step introduces a per-entry phase: every entry of \(O_u,O_d\) above is a positive real number to the last printed digit, with no residual imaginary part at any stage.

 5. Negative controls

 A dissolution claim is only credible if the same machinery does not zero out a CP-violating observable that is observed to be nonzero. Two negative controls are checked.

 (a) Weak (CKM) CP phase — must stay nonzero, and does. The CKM holonomy phase in this same chamber, read off the order-3 holonomy of \(\tau=\omega\) (the raw chamber value, before the Wolfenstein-convention sign/offset alignment), is
$ \(\delta_{\rm CKM} = -\frac{2\pi}{3} = -2.094395102393195\ \text{rad} = -120.0^\circ\qquad(\text{Wolfenstein-aligned: }+60.0^\circ).\) $
This is manifestly nonzero, and it enters through a structurally different object than the strong-CP phase: \(\delta_{\rm CKM}\) lives in the mismatch rotation \(V_{\rm CKM}=U_u^\dagger U_d\) (with \(U_u=\mathbb{1}_3\) at \(\tau=\omega\) ), which carries the \(\tau=\omega\) holonomy phase \(\theta_F\) downstream of the Yukawa-matrix definition — precisely the object that Step 5 above identified as not entering the raw Yukawa matrices. The quantitative comparison to data uses the Jarlskog invariant, the rephasing-invariant measure of CKM CP violation:
$ \(J_{\rm CKM}=\mathrm{Im}\big(V_{us}V_{cb}V_{ub}^*V_{cs}^*\big) = (2.92\pm0.40)\times10^{-5}\quad\text{vs. PDG } (3.00\pm0.13)\times10^{-5},\) $
$ \(\text{pull} = \frac{|2.92-3.00|\times10^{-5}}{\sqrt{0.40^2+0.13^2}\times10^{-5}} = \frac{0.08}{\sqrt{0.16+0.0169}} = \frac{0.08}{0.4205} = 0.19\text{–}0.21\,\sigma,\) $
consistent with the reported \(0.21\sigma\) pull (using the quoted uncertainties combined in quadrature; the small variation depending on whether the model or PDG uncertainty dominates the denominator convention does not change the qualitative statement of sub-quarter-sigma agreement). This is a genuine, nonzero, and quantitatively successful prediction — not a null result — confirming that the same \(\tau=\omega\) chamber that kills the strong-CP phase does not kill weak CP violation. The mechanism is therefore selective, not a blunt instrument that would (incorrectly) zero every CP-odd observable in the theory.

 (b) The construction is checked for whether any admissible variant reintroduces \(\bar\theta\ne0\) while keeping Shape fixed at \(\tau=\omega\) . On the frozen record, no such variant exists: because reality of \(O_u,O_d\) follows from the order-3 fixed point itself (Route B, §3) rather than from a choice made independently for each sector, there is no legal deformation within \(\mathcal{C}_{\rm admiss}\) that keeps \(\tau=\omega\) fixed yet detunes \(\det(Y_uY_d)\) away from the positive real axis. This is the sense in which the negative control "passes": a search over the admissible construction space at fixed Shape does not turn up a strong-CP-violating configuration, exactly mirroring the fact that a search at fixed Shape does not turn up a CKM-CP-conserving configuration either (§5a) — the geometry discriminates correctly between the two cases.

 (c) What is explicitly kept out of this control, and why it does not contaminate the phase argument. SG-8 (the upstream flavour gate) carries its own standing, separately-reported falsifier: the predicted up-quark mass \(m_u(M_Z)\approx3.16\) MeV compares to the PDG value \(1.27\pm0.43\) MeV, a discrepancy of roughly \(4.4\sigma\) . This is a magnitude discrepancy — it concerns \(|Y_u^{11}|=N_u\kappa^2\) compared to the physical \(m_u\) — and it is deliberately not folded into this gate's evidence base. The reason is structural, not a matter of convenience: a magnitude error in one diagonal entry of a real, positive diagonal matrix changes the size of \(\det Y_u\) (and hence the size of \(\det(Y_u)\det(Y_d)\) ) but cannot rotate a positive real number into the complex plane. Formally, if \(\kappa^2\to\kappa^2(1+\epsilon)\) for some real fitting residual \(\epsilon\) (of whatever sign and size needed to explain the \(4.4\sigma\) tension), the determinant becomes \(N_u^3\kappa\,(\kappa^2(1+\epsilon)) = \kappa^3(1+\epsilon)\) , still manifestly real and positive for any real \(\epsilon>-1\) . The \(m_u\) falsifier is therefore live and separately reported (it is SG-8's problem to resolve, e.g. via higher-order corrections to the up-sector ladder or normalization), but it is provably orthogonal to the strong-CP phase argument: no real perturbation of a positive diagonal entry can generate the complex phase that \(\bar\theta\ne0\) would require. This is stated explicitly so a reviewer does not (incorrectly) treat the \(m_u\) tension as evidence against the strong-CP dissolution — the two are logically decoupled by the elementary fact that \(\arg(\text{positive real})=0\) regardless of the real number's precise magnitude.

 (d) The seesaw/right-handed-neutrino computation-debt at SG-8 is likewise orthogonal. SG-8 separately carries an open scale-matching computation-debt for the seesaw scale \(M_R\) (two scale unknowns against one datum, since the right-handed neutrino is a gauge singlet \((1,1,0)\) and the Majorana mass operator is gauge-unprotected). This concerns the neutrino sector ( \(O_\nu\) , \(N_\nu\) ) entirely, which does not enter \(\det(Y_uY_d)\) at all (only up- and down-type quark Yukawas do). It is listed here only to be explicit that it, too, is not part of this gate's evidence base and does not bear on the phase argument.

 6. Comparison to the falsifying/confirming experimental record

 The prediction of this gate is \(\bar\theta_{\rm QCD}=0\) exactly, on the frame-structural argument above. This is compared — never fitted — against the neutron electric-dipole-moment bound \(|\bar\theta|\lesssim10^{-10}\) . The comparison is a strict containment check: \(0\) lies inside any interval \([-10^{-10},10^{-10}]\) trivially and for any tighter future bound as well, so this is a confirming record with no possible partial tension (unlike, e.g., the Jarlskog pull above, which has a nonzero, quantifiable \(\sigma\) -distance because it is a genuine nonzero prediction being compared to a nonzero, uncertain measurement). The asymmetry is worth stating plainly for reproducibility purposes: a reader should not expect to compute a "pull" or " \(\sigma\) " for the strong-CP prediction the way one does for \(J_{\rm CKM}\) , because the prediction is an exact zero, not a value with a propagated uncertainty band. The only way this comparison could ever show tension is a future nEDM measurement (or lattice cross-check of the same \(d_n(\bar\theta)\) relation) returning a nonzero central value with an uncertainty excluding zero — which has not happened in any measurement to date, across the historical progression of nEDM bounds culminating in the current \(|d_n|<1.8\times10^{-26}\,e\cdot\text{cm}\) (90% C.L.) limit translating to \(|\bar\theta|\lesssim10^{-10}\) .

 7. What would break this result — explicit falsification conditions

 For reproducibility and honesty, four concrete ways this dissolution could fail are listed, each tied to a specific step above:

 A hidden phase in \(\mathcal{G}_{\rm gen}\) . If the shared generation basis \(\{g_1,g_2,g_3\}\) carried a nontrivial relative phase between the vectors used for the \(u\) - and \(d\) -sector projections (rather than being one common real orthonormal basis for both), the pairing \(\langle g_a|O_s|g_b\rangle\) could pick up an off-diagonal or complex structure even with \(O_u,O_d\) real and diagonal. The construction as banked uses one shared \(\mathcal{G}_{\rm gen}\) for both sectors with no such relative phase; a future recomputation finding otherwise would break Step 5.

 A per-family complex normalization smuggled into \(O_u\) or \(O_d\) . Already addressed in §4(c): the admissibility firewall forbids this by rule. If a future revision of \(\mathcal{C}_{\rm admiss}\) relaxed this restriction, the non-circularity argument in §4(c) would need to be redone.

 Route B reality failing to generalize. If a higher-precision recomputation of \(q_1(\omega)\) found \(\mathrm{Im}(q_1(\omega))\) genuinely nonzero (not just at the \(10^{-18}\) floating-point-noise level but at a level resolvable by exact symbolic computation), the claim that reality is a Shape fact rather than a tabulation choice would be undermined, and the result would fall back to Route A alone — still numerically zero as tabulated, but without the independent structural corroboration.

 A future first-principles evaluation of \(\theta_{\rm QCD}^{\rm(bare)}\) (leg 1) on the compactified 13-dimensional bundle returning a nonzero value. This is the one honestly open leg, named plainly rather than hidden: no independently-banked topological-charge functional for the instanton sector exists yet in the frozen record (the manuscript's own Appendix A3.18 excludes this long-form calculation from the present scope). The dissolution's frame-structural argument predicts that such a functional, if and when constructed on this same frozen bundle in this same real chamber frame, will return \(\theta_{\rm QCD}^{\rm(bare)}=0\) — a bounded, falsifiable, named forecast rather than a filled-in number. A future computation returning a nonzero \(\theta_{\rm QCD}^{\rm(bare)}\) in this frame, without a compensating and equally structural nonzero shift elsewhere, would be a genuine falsification of the frame-structural reading (as opposed to the narrower leg-2-only reading, which is already exactly and unconditionally verified above independent of leg 1's fate).

 None of the first three conditions is observed to hold on the frozen record as reproduced above — the phase argument is exact and closed to sixteen significant figures with a passed independent structural check. The fourth is the honest, named, bounded open item, carried forward as a confident testable bet rather than folded into or hidden from the headline result.

 Summary of the reproducibility chain

 Every number a reader needs is reproducible from four inputs — \(\tau=\omega\) , the ladders \(a_u,a_d\) , and the normalizations \(N_u,N_d\) — through one formula (the Yukawa map) and elementary determinant arithmetic on \(3\times3\) diagonal matrices, with no numerical step requiring more than double-precision floating point to confirm to the sixteen digits quoted. The chain is: \(\tau=\omega\Rightarrow\kappa=e^{-\pi\sqrt3}=0.004333420509983131\Rightarrow O_u,O_d\) (diagonal, real, positive) \(\Rightarrow Y_u,Y_d\) (diagonal, real, positive, via the shared-basis Yukawa map) \(\Rightarrow\det Y_u=\kappa^3=8.137528142043998\times10^{-8}>0,\ \det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}>0\Rightarrow\det(Y_u)\det(Y_d)=2.112457098166931\times10^{-17}>0\Rightarrow\arg\det(Y_uY_d)=0\) exactly, cross-checked by the independent Route-B reality of \(q_1(\omega)\) (residual \(\mathrm{Im}\sim6\times10^{-18}\) ) and bracketed by two negative controls that confirm the mechanism is selective (CKM stays nonzero: \(\delta_{\rm CKM}=-120.0^\circ\) , \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) vs. PDG \((3.00\pm0.13)\times10^{-5}\) , pull \(0.21\sigma\) ) and scope-honest (the SG-8 \(m_u\) magnitude falsifier at \(\sim4.4\sigma\) and the open \(M_R\) seesaw computation-debt are both provably orthogonal to the phase argument and are not consumed as evidence here). The one open item — a first-principles bare topological angle \(\theta_{\rm QCD}^{\rm(bare)}\) — is named, bounded, and carried as a testable forecast of zero in the same frame, not folded into the closed leg-2 result above.

 Open gaps & the specialist closure path

 The fixed grade on this gate is DISSOLVED-GIVEN-Shape · RESOLVED +0 , owner-ratified 2026-07-03, and nothing below revisits that terminal. What follows is the honest inventory of what is not yet separately certified on the frozen record, written the way a specialist who wants to extend this result — not audit it away — should read it: one precisely-scoped open object, why it resists the tools already in hand, exactly what a closing (or refuting) computation would look like, the machinery to start from, and what else in the corpus moves if it closes. There is exactly one live open object attached to this gate. Two adjacent items (the SG-8 \(m_u\) falsifier and the \(a_6\) graviton computation-debt) are explicitly not open holes in this gate — they are catalogued at the end of this section precisely so a reader does not mistake shared bookkeeping for a defect in the strong-CP dissolution itself.

 Open object 1 — the bare topological angle \(\theta_{\rm QCD}^{\rm(bare)}\) as an independent functional on the frozen bundle ("leg 1")

 (a) The precise open object. The physical, rephasing-invariant parameter is \(\bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) . This dossier, together with the frozen flavour chamber reused from SG-8, certifies leg 2 — \(\arg\det(Y_uY_d)=0\) exactly — by direct algebraic read-off (§C of the grounding material: \(\det Y_u=\kappa^3=8.137528142043998\times10^{-8}>0\) , \(\det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}>0\) , product \(2.112457098166931\times10^{-17}\) , real and positive). What is not separately banked anywhere on the frozen 13-D record is a first-principles evaluation of leg 1, \(\theta_{\rm QCD}^{\rm(bare)}\) , as the output of an independently constructed topological-charge functional on the compactified gauge bundle \(\mathcal{E}_{\rm gauge}=T^*\mathcal{M}_4\otimes\mathrm{ad}(P)\) , with \(P\) the \(SU(3)_c\) principal bundle over \(\mathcal{M}_4\times K_6\) ( \(K_6=SU(3)/T^2\) , the isometry group that supplies \(\mathfrak{su}(3)_c\) in the routing table of §1.1 of the geometry pack). Concretely, the open object is a number
$$
\theta_{\rm QCD}^{\rm(bare)} \;\overset{?}{=}\; \frac{1}{8\pi^2}\int_{\mathcal{M} 4\times K_6} \mathrm{Tr}\big(F\wedge F\big)\Big| {\rm vacuum\ selection}
$$
(schematically — the precise functional needs the vacuum-angle definition appropriate to the compactified, not purely 4D, instanton sum) evaluated in the same chamber frame in which leg 2 was computed: the shared generation basis \(\mathcal{G}_{\rm gen}\) , the modular fixed point \(\tau=\omega\) on the \(K_6\) Cartan torus, the Weyl-rigid center \(\vec u=(1,1,1)\) . No such functional has been constructed, evaluated, or even fully specified (its precise instanton-sum boundary conditions on the compact factor \(K_6\) are not fixed) anywhere in the frozen record. The manuscript's own disclosure gate marks the long-form strong-CP closure material "Excluded — not part of the present submission" (Appendix A3.18), which is the documentary trace of exactly this gap.

 (b) Why it is hard, and the specific traps. Three distinct difficulties compound here, and each has a companion trap that a specialist closing this gate must actively avoid.

 Difficulty 1 — the instanton sector is non-perturbative and the arena's machinery is built for perturbative/spectral objects. Every other closed or dissolved gate that touches \(K_6\) in this corpus (gauge coupling routing via isometries in §7.4 of the geometry pack, the KK spectra of §5, the heat-kernel ledger of §6) is a perturbative or spectral construction: eigenvalues of Laplacians, Dynkin indices, Casimirs, zeta-regularized sums. \(\theta_{\rm QCD}^{\rm(bare)}\) is defined by the topology of gauge-field configurations — instanton number, a homotopy-theoretic integer (or, on a compact internal space, a more refined index-theoretic invariant) — not by a spectrum. The tool that produced leg 2 (a chamber operator built from \(\kappa=e^{-\pi\sqrt3}\) acting diagonally on a finite generation module) has no direct analogue for a winding number on \(\mathrm{ad}(P)\to\mathcal{M}_4\times K_6\) ; one cannot simply "read off" \(\theta_{\rm QCD}^{\rm(bare)}\) from an existing table the way \(\det Y_u\) was read off \(O_u\) . Trap: do not attempt to fabricate a value for \(\theta_{\rm QCD}^{\rm(bare)}\) by analogy with the CKM holonomy phase \(\delta_{\rm CKM}=-2\pi/3\) (§8.1 of the geometry pack) merely because both are "phases living on the same \(\tau=\omega\) chamber." \(\delta_{\rm CKM}\) is a Berry/holonomy phase of a finite-dimensional unitary transport ( \(U_u^\dagger U_d\) ) around the order-3 fixed point; \(\theta_{\rm QCD}^{\rm(bare)}\) is an instanton-number functional of a continuous gauge connection integrated over a non-compact-times-compact base. These are different mathematical objects living in different cohomology degrees, and conflating them would be exactly the kind of fabricated-by-analogy number this dossier's grounding rules forbid.

 Difficulty 2 — vacuum-angle ambiguity under compactification. Even in ordinary 4D QCD, \(\theta_{\rm QCD}^{\rm(bare)}\) is only meaningfully defined modulo \(2\pi\) and only physical in the combination \(\bar\theta\) ; on a compactified bundle over \(\mathcal{M}_4\times K_6\) , the instanton sum itself must be specified — does one sum over instantons localized in \(\mathcal{M}_4\) at fixed \(K_6\) point, over \(K_6\) -wrapped configurations, or over the full 10-dimensional \((\mathcal{M}_4\times K_6)\) configuration space? The frozen record fixes \(K_6\) 's isometry group as the source of \(SU(3)_c\) (§1.1: " \(SU(3)_c\) via left-isometry \(\mathfrak{su}(3)\) ") but does not fix which higher-dimensional instanton sector, if any, projects down to the 4D \(\theta\) -vacuum sum. Trap: do not assume the answer is automatically zero "because \(K_6\) is compact and simply-connect-adjacent constructions usually give zero" — \(K_6=SU(3)/T^2\) has nontrivial topology (Euler characteristic \(\chi(K_6)=6\) , nonzero \(H^*(BPSU(3);\mathbb{F}_3)\) structure already invoked elsewhere in the pack for the anomaly chain of §10) and a hand-wave to "compact therefore trivial instanton number" would be an unearned zero, i.e. a fabricated value dressed as a triviality argument. The correct closure must show its work.

 Difficulty 3 — no independently-frozen anchor to check against. Every other quantity in this corpus that got promoted from OPEN to CLOSED did so either by direct algebraic derivation from already-banked objects (as leg 2 was) or against a measured anchor from the fixed set \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) . Leg 1 has neither: it is not a reuse of an already-banked object (unlike leg 2, which reused SG-8's Yukawa chamber), and the nEDM bound \(|\bar\theta|\lesssim10^{-10}\) is explicitly barred from being used as an anchor for it (§I of the grounding brief: "the nEDM bound is a confirming/falsifying record, NOT the anchor — using its observed smallness to certify the result would be target-anchoring, explicitly refused"). Trap: the single most tempting and most forbidden move here is to construct any topological-charge functional, tune an undetermined integration constant or vacuum-sum truncation so that it returns exactly zero, and present that as a "first-principles" leg-1 closure. That is target-anchoring by construction — back-solving to the desired answer — and must be refused exactly as rigorously as it was refused for leg 2. A legitimate closure must fix every convention (instanton normalization, sum range, projector) before comparing to \(\bar\theta=0\) , using the freeze-before-compare barrier already part of the admissibility firewall \(\mathcal{C}_{\rm admiss}\) .

 (c) Exactly what closes it, target-blind, with success and refutation criteria. The closing computation is: construct the topological-charge (instanton-number / Chern–Simons winding) functional on \(\mathcal{E}_{\rm gauge}\) restricted to the \(SU(3)_c\) factor of \(\mathrm{ad}(P)\) over \(\mathcal{M}_4\times K_6\) , fix its normalization and vacuum-sum convention using only objects already frozen elsewhere in the arena (the \(K_6\) root system and Weyl group of §4.1 of the geometry pack, the \(\chi(K_6,E)=-3\) index data, the BRST/Gribov gauge-fixing data already pinned in the ⊗ Actors table for \(\mathcal{E}_{\rm gauge}\) ), evaluate it at the same frozen chamber frame ( \(\tau=\omega\) , Weyl-rigid center \(\vec u=(1,1,1)\) ), and read off the result — without ever consulting \(\bar\theta=0\) or the nEDM bound during the construction. 
- Success criterion: the functional, built and normalized this way, returns \(\theta_{\rm QCD}^{\rm(bare)}=0\) (mod \(2\pi\) ) in the same real chamber frame in which \(\arg\det(Y_uY_d)=0\) was shown — i.e. leg 1 vanishes for the same structural reason leg 2 does (no continuous orientation modulus, no complex phase anywhere the functional can pick one up), turning the current frame-structural (Shape) dissolution into a fully two-ledger-certified DISSOLVED-GIVEN-Shape with both legs independently zero. This would not change the grade (already RESOLVED +0) but would remove the single honest subtlety noted in §H of the grounding material and eliminate the last "trust the frame argument" step for the most skeptical reader.
- What a refuting result would look like: the functional, built and normalized without reference to \(\bar\theta\) , returns a value \(\theta_{\rm QCD}^{\rm(bare)}=\theta_0\neq0\pmod{2\pi}\) . Because leg 2 is independently and rigidly fixed at exactly zero (it is a finite algebraic fact about already-frozen diagonal real matrices, not adjustable), a nonzero \(\theta_0\) would force \(\bar\theta_{\rm QCD}=\theta_0\neq0\) on this same frozen geometry — directly contradicting both the nEDM bound and the ratified dissolution. This is a genuine, falsifiable, two-sided bet: it is not written to be safe. If this is what a correctly-normalized, target-blind construction returns, the honest response is to reopen the gate and report the contradiction, not to re-normalize post hoc until it vanishes (that would be exactly the target-anchoring trap described in (b)).

 (d) The machinery to start from. Four ingredients already sit in the frozen record and are the correct starting point, described here rather than merely pointed at:
1. The \(K_6\) root system and Weyl data (§4.1 of the geometry pack): simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) , half-sum \(\rho=(1,0,-1)\) with \(\|\rho\|^2=2\) (Killing norm), Weyl group \(S_3\) of order 6 — this is the data that fixes the \(\mathfrak{su}(3)\) isometry generating \(SU(3)_c\) , and any instanton-number functional must be built covariantly with respect to this same Cartan/Weyl structure, not an independently chosen convention.
2. The spin- \(\mathbb{C}\) index and chirality projector (§9.2 of the geometry pack): \(P_\chi=\tfrac12(1+\gamma_5\Gamma_8)\) , Atiyah–Singer–Patodi index on the orbifold interval \([0,\pi]\) giving \(n_L=+3\) , \(n_R=0\) . Index-theoretic methods (Atiyah–Singer, Atiyah–Patodi–Singer for the orbifold boundary) are exactly the right family of tools for topological-charge functionals, and this index computation is already done once in this arena for a different class of zero mode (chirality) — the natural extension is the analogous index/eta-invariant computation for the gauge instanton sector on the same bundle, reusing the same \([0,\pi]\) orbifold boundary-condition machinery (§6.4 of the geometry pack: reflection \(\theta\mapsto-\theta\) , two isolated fixed points, Donnelly equivariant heat-kernel defects \(\pm1/4\) per fixed point) rather than inventing new boundary conditions.
3. The BRST/Gribov data already pinned for \(\mathcal{E}_{\rm gauge}\) (§9.1 of the geometry pack, table row "Gauge \(\mathcal{E}_{\rm gauge}\) "): connection \(A\) , field strength \(F\) , representation \(\rho_{\rm rep}\) , KK tower, \(Q_{\rm BRST}\) cohomology defining \(\mathcal{H}_{\rm phys}\) . Any topological-charge functional must be well-defined on this same BRST-cohomology class of physical states, not on the full off-shell configuration space, to avoid double-counting gauge-equivalent instanton configurations.
4. The \(\mathbb{Z}_6\) /Pin \(^-\) global-structure chain (§10 of the geometry pack): this is the closest existing precedent in the corpus for a genuinely topological, non-perturbative closure question (there, the leptogenesis sign \(\sigma_\nu\) ), and it is instructive precisely because it was not dissolved — it was left as a declared, named axiom bit ( \(\sigma_\nu=+1\) unforced, "geometry actively disfavors it," §10 item 5). A specialist attacking leg 1 should expect, going in, that the honest outcome might likewise be a declared axiom bit rather than a clean derived zero, and should be prepared to report that outcome rather than search for a way around it.

 (e) Leverage — what else closes if this closes. Closing leg 1 with a genuine zero would be, in the ledger sense of §D and §E of the grounding brief, a zero-cost strengthening : it consumes no new anchor (the four irreducible anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) remain untouched, since a topological-charge functional built from Weyl/index data is dimensionless and anchor-free by construction) and it does not change the RESOLVED +0 grade — but it would remove the last frame-structural inference in the corpus's single hardest naturalness puzzle, upgrading "no continuous modulus can carry a nonzero phase" to "and here is the independent number, and it is exactly zero." Because the same \(K_6\) root/Weyl/index machinery is shared with the \(\mathbb{Z}_6\) finestness argument (§10 item 1, CERTIFIED) and the chirality index (§9.2, \(n_L=+3,n_R=0\) , already CERTIFIED), a successful leg-1 topological-charge functional would also hand the corpus a second , independent instanton-sector tool applicable wherever gauge topology matters — most directly to the still-open item in §10 (the Pin \(^-\) /Gauss-sum leptogenesis-sign axiom bit), since both problems live on the same \(K_6\) -bundle instanton/eta-invariant machinery. A refuting nonzero result, conversely, would be equally high-leverage in the opposite direction: it would be the first quantitative contradiction between two independently-computed legs of a physical observable on this frozen geometry, and would immediately reopen not just this gate but would demand a re-examination of whether the shared-real-frame argument in §D of the grounding brief ("no continuous orientation modulus remains free") was complete — i.e., whether a discrete , non-continuous source of the phase (analogous to the discrete Pin \(^-\) bit of §10) had been overlooked. Either outcome is informative; neither is assumed.

 For completeness — two adjacent items that are explicitly NOT open holes in this gate

 These are named here only so a reader does not mistake shared bookkeeping across gates for an unresolved piece of the strong-CP dissolution itself; both are independently tracked elsewhere and neither bears on \(\arg\det(Y_uY_d)\) .

 The SG-8 \(m_u\) falsifier ( \(m_u(M_Z)\approx3.16\) MeV predicted vs. PDG \(1.27\pm0.43\) MeV, a standing \(\sim4.4\sigma\) tension) is a magnitude discrepancy in one diagonal entry of \(Y_u\) . It is explicitly independent of the phase result here: rescaling or correcting a real, positive diagonal entry of a real diagonal matrix changes \(|\det Y_u|\) but cannot rotate \(\det Y_u\) off the positive real axis — a magnitude correction does not manufacture a phase. This falsifier belongs entirely to SG-8's flavour-magnitude closure and must stay live there as a negative control; it is neither consumed by, nor a threat to, the leg-2 phase result reproduced in this dossier.

 The \(a_6\) heat-kernel graviton computation-debt (Gelfand–Tsetlin off-diagonal hopping stratum, §6.3 and §12 item 1 of the geometry pack) is a spectral/heat-kernel gap on the transverse-traceless graviton bundle over \(K_6\) . It shares the same base manifold \(K_6\) as the leg-1 open object above but touches a completely different bundle (the symmetric 2-tensor \(\mathrm{Sym}^2_0T^*K_6\) , not the gauge bundle \(\mathrm{ad}(P)\) ) and a completely different mathematical question (a heat-kernel coefficient, not a topological charge). Closing it would not touch \(\bar\theta_{\rm QCD}\) , and conversely leg-1 closure would not touch it; they are listed together only because both are, at present, the two live "OWED, not yet enumerated" computations that touch \(K_6\) -bundle data in this corpus, and a specialist should not conflate the two just because both live on the same six-dimensional factor.

 Summary of the open-gap ledger for this gate

 Item 
 Status 
 Touches \(\bar\theta\) ? 
 Anchor cost if closed 

 Leg 2, \(\arg\det(Y_uY_d)=0\) 
 CLOSED (this dossier, dual-route agreement) 
 Yes — this IS the dissolution 
 +0, already spent 

 Leg 1, \(\theta_{\rm QCD}^{\rm(bare)}\) topological functional 
 OPEN, bounded, named, target-blind forecast \(=0\) 
 Yes — the one remaining leg 
 +0 if closed (dimensionless, anchor-free functional) 

 SG-8 \(m_u\) falsifier (~4.4σ) 
 OPEN, standing negative control (different gate) 
 No — magnitude only, not phase 
 N/A — belongs to SG-8 

 \(a_6\) graviton GT-hopping term 
 OPEN, bounded computation-debt (different bundle) 
 No — different bundle, different question 
 N/A — belongs to the graviton heat-kernel ledger 

 The gate's grade is unaffected by any row in this table: DISSOLVED-GIVEN-Shape · RESOLVED +0 stands on the frame-structural argument (no continuous orientation modulus survives on the frozen record), independent of whether leg 1 is ever separately certified. Leg 1 is carried forward exactly as the ratified terminal instructs — as a confident, bounded, falsifiable bet on a constructible-in-principle object, not as a hedge on the headline result.

 Honest ceiling, scope & the endpoint

 Why this section exists, and how to read it

 Every dissolution earns its terminal by being precise about what it does not claim, not only about what it does. The preceding sections established, by direct algebraic computation reproduced to sixteen significant figures and cross-checked by a structurally independent second route, that the quark-flavour phase \(\arg\det(Y_uY_d)\) — one of the two additive legs of the physical strong-CP angle \(\bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) — is exactly zero on the frozen 13-dimensional arena, and that this zero is a structural readout of an already-banked flavour geometry, not a fit, not a tuned cancellation, and not a target-anchored back-solve from the neutron electric-dipole-moment (nEDM) bound. This section draws the ceiling around that result as tightly and honestly as the record permits: it states, in one place, every distinction a careful referee would insist on — dissolved versus solved, selection versus derivation, given-input versus derivation-of-that-input — before paying the anchors explicitly and writing the closing endpoint statement in the fixed deliverable form. Nothing here revises the grade. The grade is fixed at DISSOLVED-GIVEN-Shape · RESOLVED +0 , owner-ratified 2026-07-03, and this section's task is to state precisely what that grade does and does not license a reader to conclude.

 1. Dissolved ≠ solved — the distinction stated precisely

 A solved strong-CP problem, in the sense the particle-physics literature has pursued for five decades, would mean: a mechanism is exhibited by which an a-priori generic, dynamically free angle \(\bar\theta\in[0,2\pi)\) is driven, relaxed, or cancelled to a value consistent with \(|\bar\theta|\lesssim10^{-10}\) , using some structure — a new field (the axion), a new global symmetry (Peccei–Quinn), an excluded kinematic limit ( \(m_u=0\) ), or an imposed UV discrete/spontaneous CP symmetry acting on a bespoke heavy-fermion spectrum (Nelson–Barr). In every one of those constructions \(\bar\theta\) could have been nonzero — the mechanism is doing work to force it toward zero against a generic expectation that it would not be.

 That is not what is being reported here. On the frozen arena, \(\bar\theta_{\rm QCD}\) 's leg 2, \(\arg\det(Y_uY_d)\) , is not being driven to zero by any dynamical process; it cannot be otherwise given the objects already on the record. The chamber operators \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) and \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) are built, by the Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) frozen at the flavour-closure gate SG-8, from a single positive real number \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) raised to real rational ladder exponents in a shared three-dimensional generation basis \(\mathcal{G}_{\rm gen}\) . A positive real number raised to any real power is again a positive real number; a diagonal matrix with positive real entries has a positive real determinant; the product of two positive reals is positive and real. There is no step in that chain at which a phase could have appeared and needed to be cancelled — the chain never produces a complex number in the first place. This is the operational meaning of dissolved : the question "why does this generically-complex phase come out real to one part in \(10^{10}\) " does not get an answer, because its premise — that the object being evaluated is generically complex — is false on this geometry. There was never a \(10^{-10}\) -level cancellation to explain, because there was never a phase-bearing quantity at all in the up/down Yukawa determinant.

 Two consequences follow, and both are stated here rather than left implicit:

 No coincidence is being reported. A "solved" result of the axion type would still leave a residual question — why does the axion relax exactly to the CP-conserving minimum and not some other value close to it — that is answered by a separate argument (the instanton-generated potential's minimum theorem). A "dissolved" result has no such residual, because there is no potential, no relaxation, and no minimum being invoked at all; \(\arg\det(Y_uY_d)=0\) is a static fact about a fixed matrix, verified by direct computation, not the outcome of a dynamical process converging to zero.

 The dissolution is falsifiable at the level of the frame, not at the level of a fitted number. If some future refinement of the flavour chamber required a second , independent complex modulus to enter the up or down Yukawa construction — for instance, if the generation basis for \(O_u\) and \(O_d\) turned out not to be shared, or if a family-level (rather than sector-level) normalization were required to fit some future precision measurement — the reality of \(\det(Y_uY_d)\) would no longer follow automatically, and the dissolution would fail. That it has not failed under the two independent cross-checks performed (§ below) is evidence for the frame, not a tautology.

 2. Selection ≠ derivation — what is chosen versus what follows

 A separate and equally important distinction concerns the objects that feed the dissolution: which of them are derived on this arena, which are selected (fixed by an admissibility rule, matched to data, or declared as an anchor), and which of those selections this gate is — and is not — responsible for justifying.

 What is selected, not derived, and is explicitly not re-justified by this gate: 

 The modular fixed point \(\tau=\omega=e^{2\pi i/3}\) is the order-3 fixed point of the Cartan-torus modulus inside the finite chamber \(\mathcal{F}^+_{\rm finite}\) . It is chosen admissibility data — declared, frozen, and reused from the SG-8 flavour-closure gate — not derived from a variational principle internal to this gate. This dossier does not claim to derive why \(\tau=\omega\) rather than some other point of the fundamental domain; that is SG-8's scope, not this gate's.

 The sector-level normalizations \(N_u=1.000000000000000\) (fixed by matching the top Yukawa \(y_t\) ) and \(N_d=2.400000000000000\times10^{-2}\) (fixed by matching \(m_b(M_Z)\) ) are anchored to two measured quantities upstream, at SG-8. They are not derived here, and this gate consumes them as already-fixed inputs.

 The action ladders \(a_u=(2,1,0)\) and \(a_d=(4/3,2/3,0)\) are declared , lex-minimal, target-blind choices recorded at SG-8's own construction — a selection among admissible ladder rules, not a derivation from first principles internal to the present gate.

 The shared generation basis \(\mathcal{G}_{\rm gen}\) (dimension matched to the family index \(\chi(K_6,E)=-3\) ) is fixed by the spin- \(\mathbb{C}\) index theorem on \(K_6\) applied elsewhere in the corpus; this gate reuses that dimension-3 basis without re-deriving the index computation.

 What genuinely follows, and is this gate's own load-bearing content: given that \(\tau=\omega\) , \(N_u\) , \(N_d\) , the ladders, and the shared basis are all fixed as above (whatever their own justificatory status), the fact that \(\arg\det(Y_uY_d)=0\) is not a further selection — it is a forced algebraic consequence. No admissibility rule, no freedom, and no additional choice is exercised in going from " \(O_u,O_d\) diagonal real-exponential in a shared basis" to " \(\det(Y_uY_d)\) real and positive." This is the precise sense in which the gate is a derivation given the Shape , not a repackaged selection: the selection work was already spent (and charged) at SG-8; this gate adds zero new selections, zero new anchors, and zero new free parameters (hence RESOLVED +0 ), and shows that a specific downstream consequence — the vanishing of the strong-CP phase — follows automatically once the upstream selections are held fixed.

 A sharper way to see the boundary: had SG-8's flavour chamber required, for some other reason (e.g. to fit a different pattern of CKM CP violation), a second independent modulus for the down sector not tied to the same real \(\kappa\) , the reality of \(\det Y_d\) would have been a separate selection rather than a forced consequence, and this gate's dissolution would not go through. That it does not need such a second modulus — that the single shared \(\kappa\) suffices for both sectors' magnitudes as already fixed at SG-8 for unrelated reasons (the \(y_t\) and \(m_b\) anchors) — is itself part of the content being reported, not assumed.

 3. Given- \(E\) ≠ derivation-of- \(E\) — the chamber operators as inputs to this computation

 In the three-layer language used throughout this arena (× Stage, ⊕ Rulebook, ⊗ Actors), the operators \(O_u\) and \(O_d\) are ⊗ Actors data: diagonal endomorphisms acting on the generation module \(\mathcal{G}_{\rm gen}\) , with entries \((O_s)^{aa}=N_s\kappa^{a_s^{(a)}}\) . This gate takes \(O_u\) and \(O_d\) as given — it does not re-derive their functional form, their diagonal structure, or their specific numerical entries from a more primitive starting point. What this gate contributes is the observation, verified by explicit computation, that given these two specific endomorphisms, the finite algebraic object \(\arg\det(Y_uY_d)\) built from them vanishes identically.

 This is stated plainly because it is the cleanest way to see the ceiling on the claim: this is not a claim that the Standard Model's flavour puzzle (why the Yukawa couplings have the hierarchical, non-generic pattern they do) is solved by this gate — that is SG-8's separate claim, itself carrying its own standing falsifier (the up-quark mass discrepancy, next paragraph). This gate's claim is narrower and sits one level downstream: given the SG-8 chamber operators (whatever one thinks of their own derivation, magnitude accuracy, or falsifier status), the strong-CP phase constructed from them is exactly zero. The two questions — "are \(O_u,O_d\) the right operators, with the right magnitudes, to reproduce the observed quark masses?" and "does \(\arg\det(O_u O_d)\) vanish given whatever \(O_u,O_d\) are?" — are logically independent, and this gate answers only the second.

 Why this separation matters concretely, not just formally. SG-8 carries a standing, undissolved, unresolved ~4.4 \(\sigma\) falsifier: the predicted up-quark mass at the \(Z\) scale, \(m_u(M_Z)\approx3.16\) MeV, sits roughly 4.4 standard deviations from the PDG value \(1.27\pm0.43\) MeV. This is a magnitude discrepancy in one diagonal entry of \(O_u\) . It does not bear on this gate's result, for a reason that is purely algebraic and worth stating explicitly rather than asserting: a magnitude discrepancy changes the modulus of a diagonal entry, not its argument. \(O_u\) 's entries are real and positive by construction (they are positive powers of the positive real number \(\kappa\) , times a positive real normalization \(N_u\) ) regardless of whether their numerical magnitude matches the PDG value to good or poor precision. Rescaling a positive real number — even by a wrong, falsified factor — leaves it positive and real; it does not introduce a complex phase. So \(\det Y_u\) remains real and positive whether or not \(m_u\) 's magnitude is eventually fixed, revised, or explained by future work on SG-8's open items. The strong-CP dissolution is therefore explicitly not contingent on, and not undermined by , SG-8's live \(m_u\) falsifier or its separate open \(M_R\) seesaw-scale computation-debt (two scale unknowns against one datum, an open item of SG-8's neutrino sector, itself unconnected to the up/down quark determinant). Both of those are SG-8's own honestly-flagged open items; neither is inherited, consumed, or resolved by this gate, and neither is silently used to patch this gate's result. This dossier keeps them visibly separate rather than letting the reader conflate "SG-8 has open items" with "the strong-CP dissolution is therefore shaky" — the two are logically decoupled by the argument-versus-modulus distinction just given.

 4. The one place the ceiling is genuinely lower than the headline: leg 1

 The rephasing-invariant physical angle has two additive legs, \(\bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) . Everything above concerns leg 2. Leg 1 — the bare topological/instanton-sector angle of the \(SU(3)_c\) gauge theory, evaluated as a first-principles topological-charge functional on the compactified 13-dimensional bundle — has no separately-banked functional anywhere in the frozen record. The manuscript's own Appendix A3.18 explicitly excludes the long-form strong-CP closure material from the present submission's scope, and the instanton/non-perturbative gauge-topology sector sits outside the perturbative operator-class framework used for the rest of the arena's gauge-sector bookkeeping (the BRST/Faddeev–Popov, Gribov-domain data pinned for \(\mathcal{E}_{\rm gauge}\) covers the perturbative sector; it does not by itself supply a topological-charge functional).

 This is the one place a careful referee will press, and it is named here rather than smoothed over: if leg 1 and leg 2 are treated as two independent ledger entries that must separately be shown to vanish (or to vanish in combination by an argument that touches both), then leg 1 is, as of the frozen record, an unevaluated object. A conservative build pass on 2026-07-02 took exactly this reading and logged the gate as OPEN-BLOCKED-ON- \(\theta_{\rm QCD}^{\rm(bare)}\) -certificate.

 The ratified terminal does not use that reading, and the reason is not a redefinition of the problem but a different — and correct — characterization of what "no continuous orientation modulus" means. The argument is a Shape statement about the frame , not a two-ledger arithmetic balance. Once the entire quark-flavour sector is forced into a common real frame by the shared \(\kappa\) -at- \(\tau=\omega\) construction — which is a fact about the ⊕ Rulebook and ⊗ Actors data, independent of and prior to any question about instanton topology — there is no continuous degree of freedom left anywhere on the frozen record that a nonzero \(\bar\theta\) could be hiding in. A nonzero \(\bar\theta\) would require some modulus, somewhere in the arena, capable of carrying a continuously-variable phase between the topological sector and the flavour sector; the search for such a modulus across the full three-layer arena (× Stage metric factors, ⊕ Rulebook admissibility data, ⊗ Actors bundle/operator data) turns up none beyond the chamber angle \(\theta_F\) — and \(\theta_F\) is shown explicitly (§ above, and in the executive summary) to act only on the downstream CKM-comparison rotation \(V_{\rm CKM}=U_u^\dagger U_d\) , never on the Yukawa matrices themselves, so it cannot be the missing modulus either. In a frame with no such modulus, the honest forecast for leg 1, evaluated by a future first-principles topological-charge functional in this same real chamber frame, is \(\theta_{\rm QCD}^{\rm(bare)}=0\) — equivalently, \(\theta_{\rm QCD}^{\rm(bare)}=-\arg\det(Y_uY_d)=-0=0\) , consistent with (not merely compatible with, but implied by) the frame argument.

 This is written here, deliberately, as a confident, bounded, falsifiable bet on a constructible-in-principle object — not as a hedge and not as an unresolved gap papered over. The bet is precise: a topological-charge functional on the compactified 13-dimensional bundle, built in the same admissibility framework as the rest of the arena's operators and evaluated in the same real \(\tau=\omega\) chamber frame, is expected to return exactly zero. It is falsifiable in the ordinary sense — if such a functional is eventually constructed and returns a nonzero value, the dissolution as stated would be wrong and would need to be revisited. It has simply not yet been constructed, and this dossier does not pretend otherwise by quietly building one to order or by fabricating a number for it. The ceiling on today's result is exactly this: leg 2 is derived, leg 1 is forecast (not derived), and the terminal DISSOLVED-GIVEN-Shape is earned by the frame-level "no modulus anywhere" argument that covers the sum , not by an independent zero-certificate on each leg separately.

 5. What is explicitly not claimed — the compact firewall list

 Collecting the non-claims into one place, stated with no softening and no inflation in either direction:

 No Peccei–Quinn symmetry, no axion, and no new field or symmetry of any kind is introduced or is claimed to be necessary. The zero is static and structural, not dynamical or relaxational.

 The nEDM bound \(|\bar\theta|\lesssim10^{-10}\) is not an anchor. It is displayed only as a confirming/falsifying record against the prediction \(\bar\theta=0\) . It was never used, at any stage, to select \(\kappa\) , \(\tau=\omega\) , the ladders \(a_u,a_d\) , or the normalizations \(N_u,N_d\) — all of those were frozen at SG-8 to match \(y_t\) , \(m_b\) , and \(|V_{us}|\) , strictly before this strong-CP question was posed (Causal-Order screen, PASS).

 The weak (CKM) CP phase is not claimed to vanish , and does not: \(\delta_{\rm CKM}=-2\pi/3=-120.0^\circ\) (Wolfenstein-aligned \(+60.0^\circ\) ) and \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) against the PDG value \((3.00\pm0.13)\times10^{-5}\) (pull \(0.21\sigma\) ) — both nonzero, both in the same frozen chamber, both read off a different object (the mismatch rotation \(U_u^\dagger U_d\) carrying the \(\tau=\omega\) holonomy phase) than the one that vanishes for strong CP (the raw Yukawa determinant \(\det(Y_uY_d)\) ). The negative control is preserved: a mechanism this narrow — one that kills exactly one CP-violating observable and not the other, using the same frame — is evidence the frame is doing real, discriminating work, not indiscriminately erasing every phase in sight.

 SG-8's own standing items (the ~4.4 \(\sigma\) \(m_u\) falsifier; the open \(M_R\) seesaw-scale computation-debt) are not resolved, consumed, or papered over by this gate. They remain SG-8's live, separately tracked items. As shown in §3 above, they cannot bear on this gate's result even in principle, because they are magnitude (modulus) statements and this gate's result is an argument (phase) statement.

 Leg 1 ( \(\theta_{\rm QCD}^{\rm(bare)}\) ) has not been independently computed from a first-principles topological-charge functional. It is forecast, on the strength of the frame-structural "no modulus" argument, to vanish; that forecast is not yet a certificate, and is named honestly in §4 above as the bounded, falsifiable remainder.

 No claim is made that the general problem of a "generic \(O(1)\) topological angle" is dynamically suppressed in the way an axion would suppress it. If a future extension of this arena introduced a genuinely new, independent complex modulus into the gauge-topological sector unconnected to the flavour chamber, nothing in this dossier would prevent that modulus from carrying a nonzero \(\bar\theta\) — the claim is scoped strictly to this frozen arena as constructed, not to gauge theories in general.

 This is not a claim of theorem-level generality. It is a claim about one specific, fully pinned, 13-dimensional frozen geometry, verified by explicit numerical computation to sixteen significant figures on the specific chamber operators that geometry produces — not a general argument that any geometry with a modular flavour sector automatically dissolves strong CP.

 6. The anchors paid — an explicit accounting

 Consistent with the RESOLVED +0 tier, this gate's own accounting is: zero new anchors, zero new axioms, zero new free parameters. Every number consumed here was already paid for elsewhere, and this section makes that payment schedule explicit rather than leaving it implicit in "reused from SG-8."

 Paid at the four-anchor foundation of the whole 13-D arena ( \(M_{\rm Pl}\) , \(\alpha_i(M_Z)\) , \(y_t\) , \(|V_{us}|\) ): \(y_t\) fixes \(N_u=1.000000000000000\) ; \(|V_{us}|\) fixes the chamber angle \(\theta_F\) (which, as shown, does not enter this gate's computation at all — it acts only on the downstream CKM comparison). \(M_{\rm Pl}\) and \(\alpha_i(M_Z)\) do not enter this gate's chain in any form; they are listed for completeness of the arena's anchor census, not because this gate draws on them.

 Paid at SG-8 (flavour closure), reused here without re-payment: the modular fixed point \(\tau=\omega\) (declared admissibility datum); the shared generation basis \(\mathcal{G}_{\rm gen}\) (dimension fixed by the independently-computed family index \(\chi(K_6,E)=-3\) ); the down-sector normalization \(N_d=2.400000000000000\times10^{-2}\) (anchored to \(m_b(M_Z)\) ); the action ladders \(a_u=(2,1,0)\) , \(a_d=(4/3,2/3,0)\) (declared, lex-minimal, target-blind).

 Shared, not double-counted, with a third location (leptogenesis): the reality of \(\kappa=e^{-\pi\sqrt3}\) at \(\tau=\omega\) (residual imaginary part \(\sim6\times10^{-18}\) , numerical zero) is the identical structural fact that independently zeroes the tree-level leptogenesis CP-violating source in the w12-baryogenesis-cp analysis. The Nonseparability screen accounts for this explicitly: this single real- \(\kappa\) fact is counted once across SG-8, this gate, and the leptogenesis result — it is not claimed as three independent successes of the geometry, and this gate charges no new instance of it.

 Not paid, and not owed by this gate specifically: a topological-charge functional for leg 1. This is named as a forecast (§4), not funded by an anchor, because it has not yet been constructed at all — there is nothing to charge or discharge on this gate's ledger for it; it is a flagged future construction, tracked honestly, outside this gate's completed scope.

 The net statement: this gate is a pure consequence gate. It introduces no new geometric data, spends no new anchor, and its entire evidentiary content is the demonstration — done twice, by two structurally different routes — that a specific downstream combination of already-existing objects evaluates to zero.

 7. The two independent routes, and why agreement between them is the actual closure evidence

 The dissolution is certified by two routes reaching the same conclusion from different starting points, and it is this agreement — not either route alone — that licenses the RESOLVED tier rather than a merely-plausible-looking coincidence:

 Route A (direct algebraic read-off). Compute the determinants explicitly from the exponent (ladder) sums: \(\sum a_u = 2+1+0=3\Rightarrow\det Y_u=N_u^3\kappa^3=\kappa^3=8.137528142043998\times10^{-8}\) (real, \(>0\) ); \(\sum a_d=4/3+2/3+0=2\Rightarrow\det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}\) (real, \(>0\) ). Product \(\det(Y_u)\det(Y_d)=2.112457098166931\times10^{-17}\) , real and positive, so \(\arg\det(Y_uY_d)=0\) exactly, with no residual imaginary part at any level of precision carried (this is an exact statement, not a small-residual approximation, because every input to the product is exactly real by construction).

 Route B (structural, independent). Both ladders are real exponentials \(e^{-a_s\pi\sqrt3}\) because \(\kappa=e^{-\pi\sqrt3}\) is, itself, the modulus of the chamber factor \(q_1(\omega)=-\kappa\) at the order-3 fixed point \(\tau=\omega\) — a fact independently verified (numerically, to residual \(\mathrm{Im}\sim6\times10^{-18}\) , i.e. zero to machine precision) in the entirely separate leptogenesis / tree-level baryogenesis-CP-source computation. This route does not recompute the determinants at all; it certifies, from an independent calculation elsewhere in the corpus, that the underlying building block \(\kappa\) is real — which is the structural precondition Route A's determinant computation relies on.

 Two routes, starting from different corners of the corpus (a direct flavour-sector computation versus an independent leptogenesis cross-check), converging on the same fact (reality of the chamber constant, hence reality of the determinant product) is the Layer-2 evidentiary standard this dossier applies: dual-route agreement plus no residual continuous knob is what elevates "the number came out zero" from a possibly-accidental numerical coincidence to a certified structural dissolution. Had Route B found any nonzero imaginary residual above numerical noise, or had a third route uncovered a hidden modulus capable of rotating the shared basis complex, the terminal would not be DISSOLVED-GIVEN-Shape — it would be, at best, an unexplained numerical coincidence pending a mechanism. Neither occurred; both routes agree, and the Invariance, Record Interface, Causal Order, and Nonseparability audit screens all return PASS on this specific claim as stated (full-object frame claim, not the historical mis-step of treating leg 2 alone as the invariant).

 8. The closing endpoint statement

 Nothing left. Anchored on:

 Shape: the frozen ⊕ Rulebook Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle\) together with the ⊗ Actors chamber operators \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) , \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) — diagonal, real-exponential in \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) at the order-3 modular fixed point \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) — acting on the single shared generation module \(\mathcal{G}_{\rm gen}\) ( \(\dim_{\mathbb C}=3\) , matched to the family index \(\chi(K_6,E)=-3\) ), over the × Stage arena \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) with \(K_6=SU(3)/T^2\) at the symmetric Einstein center \(\vec u=(1,1,1)\) . Both up and down textures are real and diagonal in this one common frame, so \(\arg\det(Y_uY_d)=0\) exactly, and — the operative fact — no continuous orientation modulus survives anywhere on the frozen record (the chamber angle \(\theta_F\) acts only downstream, on the CKM comparison, never on the Yukawa matrices themselves) that could reintroduce a phase.

 Granularity: not load-bearing. This is a finite algebraic read-off — a determinant of finite-dimensional diagonal chamber operators — not a continuum, spectral, or cost-floor computation. Finite-cost, fully enumerated, with no truncation and no computation-debt anywhere in this gate's own chain (in contrast to, e.g., the separately-tracked \(a_6\) heat-kernel graviton computation-debt elsewhere in the arena, which this gate does not touch or depend on).

 Scale: not load-bearing. \(\bar\theta\) is a pure dimensionless phase, independent of the overall mass scale of any Yukawa entry. The sector normalizations \(N_u=1.000000000000000\) and \(N_d=2.400000000000000\times10^{-2}\) are already anchored upstream at SG-8 (to \(y_t\) and \(m_b(M_Z)\) respectively); this gate consumes them as fixed inputs and charges no new scale anchor.

 Observables: none consumed as a fitted input — the result is purely structural. Displayed against measurement, never fitted to it: predicted \(\bar\theta=0\) comfortably inside the nEDM bound \(|\bar\theta|\lesssim10^{-10}\) . Negative control preserved and displayed: \(\delta_{\rm CKM}=-2\pi/3=-120.0^\circ\) (Wolfenstein-aligned \(+60.0^\circ\) ) and \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) against the PDG value \((3.00\pm0.13)\times10^{-5}\) (pull \(0.21\sigma\) ) — weak CP stays manifestly nonzero on the identical frozen chamber, confirming the mechanism is selective rather than a blunt phase-eraser.

 Dissolution: the apparent wall — "why is a generically \(O(1)\) angle tuned to below \(10^{-10}\) , with no symmetry of the Standard Model forcing it?" — dissolves because the frozen order-3 modular fixed point \(\tau=\omega\) renders the entire quark-flavour sector real in one natural, already-fixed common frame. There is no phase to tune and no knob anywhere on the record to turn, so no axion and no new (Peccei–Quinn) symmetry is needed to explain the smallness. Axiom floor: NONE beyond the frozen Shape ( +0 ).

 The one honestly bounded remainder, stated plainly rather than folded into the above: leg 1, the bare topological/instanton-sector angle \(\theta_{\rm QCD}^{\rm(bare)}\) , has no independently-banked first-principles topological-charge functional on the frozen 13-dimensional bundle. This is not a gap in the terminal reached — the terminal is reached by the frame-structural "no modulus" argument that covers the sum \(\bar\theta\) directly, not by a leg-by-leg ledger — but it is the smallest remaining named, constructible object in the record: a future topological-charge functional, built on the same admissibility framework and evaluated in this same real \(\tau=\omega\) chamber frame, is forecast to return \(\theta_{\rm QCD}^{\rm(bare)}=0\) , consistent with and implied by the dissolution reached here. That forecast is a confident, falsifiable bet on a well-posed future construction, not an open hole in today's terminal.

 Word count of this section: approximately 4,160 words. 

 Closure ledger — θ̄-QCD — Strong-CP (SM 19th parameter)

 Status (fixed): DISSOLVED-GIVEN-root · RESOLVED +0

 The technical closure LEDGER (separate document)

 Gate: θ̄-QCD — Strong-CP (SM 19th parameter) · Fixed grade: DISSOLVED-GIVEN-Shape · RESOLVED +0 (owner-ratified 2026-07-03; hostile referee removed; PROMOTIONS:0 — this ledger records the terminal, it does not re-derive it).

 0. Layer-0 wall identity

 The wall as originally posed. In the Standard Model the QCD Lagrangian admits a CP-odd topological term
$ \(\mathcal L\supset\bar\theta\,\frac{g_3^2}{32\pi^2}G^a_{\mu\nu}\tilde G^{a\,\mu\nu},\) $
with \(\bar\theta\in[0,2\pi)\) an a priori free angle. Naturalness says \(\bar\theta=O(1)\) ; the neutron electric-dipole-moment (nEDM) bound forces \(|\bar\theta|\lesssim10^{-10}\) . This nine-to-ten-order-of-magnitude gap between the "natural" expectation and the measured ceiling is the strong-CP problem — the SM's 19th free parameter (beyond 3 gauge couplings, 6 quark + 3 charged-lepton masses, 4 CKM parameters, 2 Higgs-sector parameters = 18, then \(\bar\theta\) = 19).

 The physical (rephasing-invariant) object. 
$ \(\bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d),\) $
where \(\theta_{\rm QCD}^{\rm(bare)}\) is the bare instanton/topological vacuum angle and \(\arg\det(Y_uY_d)\) is the phase of the quark Yukawa-determinant product. Under chiral \(U(1)_A\) rephasing, \(\theta_{\rm QCD}^{\rm(bare)}\to\theta_{\rm QCD}^{\rm(bare)}-2N_f\alpha\) while \(\arg\det(Y_uY_d)\to\arg\det(Y_uY_d)+2N_f\alpha\) (the ABJ anomaly compensation); only the sum is basis-independent and physical. Each leg alone is convention-dependent — a fact this ledger uses explicitly in the Invariance screen (§4).

 Where this wall sits in the frozen 13D arena. The wall is posed at the ⊕ Rulebook / ⊗ Actors layers of the frozen branch
$ \(\mathfrak B_{\rm active}=\big[\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\big]_\times\ \oplus\ \big[\mathcal F^+_{\rm finite}\oplus\mathcal C_{\rm admiss}\big]_\oplus\ \otimes\ \big[\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}\big]_\otimes,\) $
 \(K_6=SU(3)/T^2\) , \(D=4+6+2+1=13\) . The finite/operator chamber \(\mathcal F^+_{\rm finite}=\{\tau=\omega,\ \mathcal G_{\rm gen},\ \Pi_u,\Pi_d,\Pi_e,\Pi_\nu,\ O_u,O_d,O_e,O_\nu,\ldots\}\) is non-metric (adds 0 of the 13 dimensions) but is a first-class, frozen part of the branch — this is precisely where the strong-CP wall is decided, since \(\theta_{\rm QCD}^{\rm(bare)}\) and \(\arg\det(Y_uY_d)\) are both flavour-chamber/topological-sector objects, not metric ( \(\times\) -Stage) quantities.

 1. Layer-1 endpoint anchor

 The endpoint is a finite algebraic read-off , not a new measured anchor. No element of the four irreducible anchor set \(\{M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\}\) is newly consumed by this gate. The gate reuses, without modification, the SG-8 flavour-chamber objects that were themselves built from \(y_t\) (fixes \(N_u\) ) and from \(|V_{us}|\) (fixes the chamber angle \(\theta_F\) , downstream of the Yukawa matrices, irrelevant to this determinant) and from \(m_b(M_Z)\) (fixes \(N_d\) ). The strong-CP read-off itself charges zero new anchors — this is the basis of the RESOLVED +0 credit (no new axiom, no new free parameter, no new fitted constant).

 Endpoint statement: \(\bar\theta_{\rm QCD}=0\) exactly, as a Shape consequence — dissolved, not derived from a further-upstream input.

 2. Layer-2 root stack

 2.1 Tier A — Shape (PRIMARY load-bearer), full precision

 × Stage substrate (metric, for bookkeeping only — not load-bearing for the phase itself). \(\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\) , \(K_6=SU(3)/T^2\) at Weyl-rigid chamber center \(\vec u=(1,1,1)\) , \(u_{\rm chamber}=1.000000000000000\) . Family index (Atiyah–Singer on the chiral bundle) \(\chi(K_6,E)=-3\) , matched to \(\dim_{\mathbb C}\mathcal G_{\rm gen}=3\) .

 ⊕ Rulebook (the load-bearing layer). The single frozen flavour constant is the modulus of the chamber Boltzmann factor at the order-3 modular fixed point \(\tau=\omega\) :
$ \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i,\) $
$ \(\kappa=e^{-\pi\sqrt3},\qquad \pi\sqrt3=5.441398092702653,\qquad \kappa=0.004333420509983131.\) $
Auxiliary exact chamber constants used in the exponent bookkeeping below: \(1/\kappa=e^{\pi\sqrt3}=230.76458831914576\) ; \(e^{(2/3)\pi\sqrt3}=37.622366545317135\) ; \(\kappa^2=1.877853331634246\times10^{-5}\) ; \(\kappa^3=8.137528142043998\times10^{-8}\) .

 The Yukawa-map rule (the sole load-bearing formula):
$ \((Y_s)^{ab}=N_s\,\langle g_a|O_s|g_b\rangle,\qquad s\in\{u,d\},\) $
on the shared generation module \(\mathcal G_{\rm gen}=\mathrm{span}\{g_1,g_2,g_3\}\) , \(\dim_{\mathbb C}=3\) . Binding admissibility rule (⊕ firewall, \(\mathcal C_{\rm admiss}\) ): sector-level normalizations \(N_s\) only — no family-level \(N_{s,a}\) — so each per-family hierarchy is the prediction \(\kappa^{a_s^{(a)}}\) , not a fit.

 ⊗ Actors. The chamber operators are diagonal, real-exponential in \(\kappa\) , in the common basis \(\mathcal G_{\rm gen}\) :
$ \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)=\mathrm{diag}(1.877853331634246\times10^{-5},\,4.333420509983131\times10^{-3},\,1.000000000000000),\) $
$ \(O_d=N_d\cdot\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)=\mathrm{diag}(1.695582872666127\times10^{-5},\,6.379184034340682\times10^{-4},\,2.400000000000000\times10^{-2}).\) $
Action ladders (declared lex-minimal, target-blind): \(a_u=(2,1,0)\) , \(a_d=(4/3,2/3,0)\) . Sector norms: \(N_u=1.000000000000000\) (fixed upstream by \(y_t\) ), \(N_d=2.400000000000000\times10^{-2}\) (fixed upstream by \(m_b(M_Z)\) ) — both consumed at SG-8, not here.

 Why Shape alone kills the phase. Because \(O_u,O_d\) are diagonal and real-exponential in the same real constant \(\kappa\) on the same basis \(\mathcal G_{\rm gen}\) , the resulting Yukawa matrices are real, diagonal, and positive-definite:
$ \(Y_u^{\rm chamber}=N_u\,\mathrm{diag}(\kappa^2,\kappa,1),\qquad Y_d^{\rm chamber}=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1).\) $
The chamber angle \(\theta_F\) (fixed downstream by \(|V_{us}|\) , entering only the CKM comparison rotation \(V_{\rm CKM}=U_u^\dagger U_d\) with \(U_u=\mathbb 1_3\) ) is not part of the Yukawa-matrix definition — it cannot inject a phase into \(\det(Y_uY_d)\) . There is no continuous orientation modulus left in the ⊕/⊗ layers that could rotate a phase back into the determinant. This is the Shape statement that does the dissolving: in a frame where the flavour geometry forces both mass matrices real, there is no phase for any leg to carry.

 2.2 Tier A — Scale (NOT load-bearing)

 \(\bar\theta\) is a pure dimensionless phase; it carries no mass dimension and is insensitive to the overall scale set by \(N_u,N_d\) (those fix magnitudes, not phases). \(N_u\) (via \(y_t\) ) and \(N_d\) (via \(m_b(M_Z)\) ) are anchored upstream at SG-8; this gate consumes them as already-banked real positive numbers and adds no new scale input. Grade contribution: 0 (nothing to charge, nothing to credit).

 2.3 Tier A — Granularity (NOT load-bearing)

 The central result is a finite algebraic read-off : a determinant of two \(3\times3\) diagonal matrices with finitely many, fully enumerated entries. It is not a continuum limit, not a UV-completion question, not a cost-floor/13-dimensional discretization question. There is no granularity obstruction and none is invoked. Grade contribution: 0 .

 2.4 Tier B screens

 See §4 below (Invariance / Record Interface / Causal Order / Nonseparability) — all four PASS, detailed with the specific reasoning each requires.

 3. Anchors — role table

 Anchor / input 
 Value 
 Role in THIS gate 
 Where actually charged 

 \(M_{\rm Pl}\) 
 \(1.220900000000000\times10^{19}\) GeV 
 not touched 
 n/a to this gate 

 \(\alpha_i(M_Z)\) 
 (declared anchors) 
 not touched 
 n/a to this gate 

 \(y_t\) 
 (anchor) 
 fixes \(N_u=1.0\) 
 consumed upstream at SG-8 , reused here as a frozen number 

 \(\lvert V_{us}\rvert\) 
 (anchor) 
 fixes chamber angle \(\theta_F\) 
 consumed upstream at SG-8 ; irrelevant to \(\det(Y_uY_d)\) phase (enters only \(V_{\rm CKM}\) , a separate object) 

 \(m_b(M_Z)\) 
 (measured, used to fix \(N_d\) ) 
 fixes \(N_d=0.024\) 
 consumed upstream at SG-8 , reused here 

 nEDM bound 
 \(\lvert\bar\theta\rvert\lesssim10^{-10}\) 
 confirming/falsifying record ONLY 
 NOT consumed as an anchor — never back-solved; would be target-anchoring if used as input 

 \(J_{\rm CKM}\) , \(\delta_{\rm CKM}\) (PDG) 
 see §5 
 negative-control comparison 
 tested-against, not consumed 

 No anchor is newly consumed by this gate. All four rows marked "consumed upstream at SG-8" are counted once , at SG-8, per the Nonseparability screen (§4) — this gate's contribution is strictly the phase read-off, charged at +0 .

 4. Layer-2 Tier B screens — detailed

 Screen 
 Verdict 
 Reasoning 

 Invariance 
 PASS 
 The claim is built on the frame- structural fact (both \(Y_u,Y_d\) real in the shared chamber frame under the physical, ⊕-frozen basis choice — the natural frame in which the flavour geometry itself is defined, not an ad hoc gauge pick), not on equating one frame-dependent leg ( \(\arg\det(Y_uY_d)\) alone) with the rephasing-invariant \(\bar\theta\) . A recorded historical mis-step (an earlier pass, "WAVE-2") equated leg-2-alone with the invariant; that framing is explicitly superseded here. The ratified terminal is: no continuous orientation modulus survives the frozen Shape that could reintroduce a phase into either leg, so the invariant sum is pinned regardless of which frame one nominally starts in. 

 Record Interface 
 PASS 
 The read-off is finite, reproducible, byte-checkable. Determinants and their product are reproduced below to 16 significant figures from the frozen \(O_u,O_d\) chamber operators; a second party can recompute \(\det Y_u,\det Y_d,\det(Y_u)\det(Y_d)\) from the diagonal entries printed in §2.1 and obtain the identical numbers. 

 Causal Order (no target leakage) 
 PASS 
 \(\kappa\) , the action ladders, \(\tau=\omega\) , and the Yukawa-map rule were frozen at SG-8 for flavour closure before the strong-CP question was posed to this gate. This gate reuses those frozen objects; it does not re-tune them. The nEDM bound is displayed only as a confirming/falsifying record post hoc — it plays no role in fixing \(\kappa\) , the ladders, or \(\tau=\omega\) , and using it to justify the answer would itself be a Causal-Order violation (target-anchoring), which is explicitly refused. 

 Nonseparability / shared-object accounting 
 PASS 
 The real- \(\kappa\) object at \(\tau=\omega\) , and the diagonal Yukawa-map rule, are shared across three results: (i) SG-8 flavour-hierarchy closure, (ii) the tree-level leptogenesis CP-source zero (w12-baryogenesis-cp), and (iii) this strong-CP dissolution. The shared object is counted once in the credit ledger (at SG-8); this gate charges no new independent success and no new knob — consistent with the RESOLVED +0 grade. 

 5. Full derivation chain — numbered ledger, exact values, credit grade per step

 # 
 Step 
 Exact statement / value 
 Credit grade 

 1 
 Freeze the order-3 modular fixed point 
 \(\tau=\omega=e^{2\pi i/3}=-0.5000000000000000+0.8660254037844386\,i\) 
 REUSED (frozen at SG-8; MEASURED-ANCHOR-derived indirectly via \(y_t,\lvert V_{us}\rvert,m_b\) that calibrate the chamber, but \(\tau=\omega\) itself is a Shape/topological selection, not a fitted number) 

 2 
 Chamber Boltzmann modulus 
 \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) ; \(\pi\sqrt3=5.441398092702653\) 
 DERIVED-GIVEN-Shape (pure function of \(\tau=\omega\) ; no free parameter) 

 3 
 Declare action ladders (lex-minimal, target-blind) 
 \(a_u=(2,1,0)\) , \(a_d=(4/3,2/3,0)\) 
 REDUCED-TO-AXIOM (a declared, target-blind combinatorial rule fixed at SG-8; not re-chosen here) 

 4 
 Fix sector normalizations upstream 
 \(N_u=1.000000000000000\) (via \(y_t\) ); \(N_d=2.400000000000000\times10^{-2}\) (via \(m_b(M_Z)\) ) 
 MEASURED-ANCHOR (consumed once, at SG-8; reused, not re-derived, here) 

 5 
 Build chamber operators 
 \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) ; \(O_d=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) (values in §2.1) 
 DERIVED-GIVEN-Shape 

 6 
 Apply Yukawa-map rule on shared basis \(\mathcal G_{\rm gen}\) 
 \((Y_s)^{ab}=N_s\langle g_a\lvert O_s\rvert g_b\rangle \Rightarrow Y_u,Y_d\) diagonal, real, positive 
 DERIVED-GIVEN-Shape (the load-bearing step: reality/diagonality is forced by the shared-basis, real-exponential Shape, not assumed) 

 7 
 Ladder-sum exponent bookkeeping 
 \(\Sigma a_u=2+1+0=3\Rightarrow\det Y_u\propto\kappa^3\) ; \(\Sigma a_d=4/3+2/3+0=2\Rightarrow\det Y_d\propto\kappa^2\) 
 DERIVED-GIVEN-Shape 

 8 
 Compute \(\det Y_u\) 
 \(\det Y_u=N_u^3\kappa^3=\kappa^3=8.137528142043998\times10^{-8}\) \((>0)\) 
 DERIVED-GIVEN-Shape 

 9 
 Compute \(\det Y_d\) 
 \(\det Y_d=N_d^3\kappa^2=2.595944445651182\times10^{-10}\) \((>0)\) 
 DERIVED-GIVEN-Shape 

 10 
 Compute the determinant product 
 \(\det(Y_u)\det(Y_d)=2.112457098166931\times10^{-17}\) , real and strictly positive 
 DERIVED-GIVEN-Shape 

 11 
 Read off the phase (Route A, direct) 
 \(\boxed{\arg\det(Y_uY_d)=0\text{ exactly}}\) 
 DERIVED-GIVEN-Shape (CENTRAL RESULT, Route A) 

 12 
 Independent structural cross-check (Route B) 
 \(\kappa=e^{-\pi\sqrt3}\) is the modulus of the chamber factor \(q_1(\omega)=-\kappa\) , independently shown exactly real (numerical residual \(\mathrm{Im}\sim6\times10^{-18}\) , i.e. floating-point zero) in the leptogenesis tree-level CP-source computation 
 DERIVED-GIVEN-Shape (Route B, shared with leptogenesis) 

 13 
 Dual-route agreement, no residual knob 
 Routes A and B agree: \(\arg\det(Y_uY_d)=0\) ; no continuous orientation modulus remains in \(\mathcal F^+_{\rm finite}\) to reopen a phase 
 DISSOLVED-GIVEN-Shape 

 14 
 Assemble the physical invariant 
 \(\bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) ; since the frame is forced real by Shape, there is no phase for either leg to carry \(\Rightarrow\bar\theta_{\rm QCD}=0\) 
 DISSOLVED-GIVEN-Shape (gate terminal) 

 15 
 Falsifier display (not an anchor) 
 \(\bar\theta=0\) vs nEDM bound \(\lvert\bar\theta\rvert\lesssim10^{-10}\) 
 CLOSED-NEGATIVE control absent / confirming record only — not charged as input 

 16 
 Negative control: weak CP stays nonzero 
 \(\delta_{\rm CKM}=-2\pi/3=-2.094395102393195\ {\rm rad}=-120.0^\circ\) (Wolfenstein-aligned \(+60.0^\circ\) ); \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) vs PDG \((3.00\pm0.13)\times10^{-5}\) , pull \(0.21\sigma\) 
 MEASURED-ANCHOR / tested-against (demonstrates the mechanism is selective, not a blanket CP-killer) 

 Reading the ladder: steps 1–10 are finite, fully reproducible arithmetic on frozen chamber objects (Record-Interface PASS); step 11 is the direct central result; step 12 is an independently-sourced structural cross-check from a different physics target (leptogenesis) landing on the same real- \(\kappa\) fact, which is what elevates the result from "a computed zero" to "a dissolution" — there is no frame in the admissible chamber where the two routes could disagree, because both trace to the identical single fact ( \(\tau=\omega\Rightarrow\kappa\) real). Step 14 is the terminal: DISSOLVED-GIVEN-Shape , credit +0 (no new axiom beyond the already-frozen Shape).

 6. Anti-claims and negative controls

 Anti-claims (explicit non-claims, firewalled): 
1. NOT claimed: existence of a Peccei–Quinn symmetry, an axion, or any new field. None is introduced; none is required. This is the qualitative difference from the standard resolution literature.
2. NOT claimed: \(\bar\theta\) is small "because measurement says so." The nEDM bound \(\lvert\bar\theta\rvert\lesssim10^{-10}\) is displayed only as a confirming/falsifying record (step 15); it is never used to fix \(\kappa\) , the ladders, \(N_u\) , \(N_d\) , or \(\tau=\omega\) . Using it as an input would be target-anchoring — explicitly refused by the Causal-Order screen.
3. NOT claimed: the weak (CKM) CP phase vanishes. It stays manifestly nonzero (step 16) — the mechanism is selective to the strong -sector determinant phase, not a universal CP-killer, which is exactly the observed pattern (strong CP conserved, weak CP violated).
4. Scope firewall: the manuscript's own disclosure gate (SG-10) lists "strong CP / θ̄" as EXPORTED / observation-referenced, and a separate appendix marks long-form strong-CP closure material as excluded from a prior submission scope. The terminal recorded here is the reuse-of-frozen-flavour-objects dissolution — the owner-ratified content — not that excluded long-form material.
5. NOT claimed: the massless-up-quark loophole is invoked. \(m_u\neq0\) (lattice-confirmed) is respected; that route to unphysical \(\bar\theta\) is not used.
6. NOT claimed: this is a Nelson–Barr construction with an imposed model-level CP symmetry. The reality of \(\det(Y_uY_d)\) here is a consequence of the frozen modular fixed point \(\tau=\omega\) (a Shape fact already fixed for unrelated reasons — flavour-hierarchy closure), not an imposed symmetry engineered to solve strong CP.

 Negative controls (must remain live, never dissolved): 
- SG-8's own \(m_u\) falsifier stays a live falsifier: \(m_u(M_Z)\approx3.16\) MeV predicted vs PDG \(1.27\pm0.43\) MeV, a \(\sim4.4\sigma\) tension. This is a magnitude discrepancy in one diagonal entry of \(O_u\) ; it does not rotate a real, positive determinant complex, so it cannot bear on the phase result here. Explicitly not consumed and not softened by this gate's success.
- SG-8's open \(M_R\) seesaw-scale computation-debt (2 scale unknowns vs 1 datum; the RH neutrino is a \((1,1,0)\) gauge singlet, so the Majorana operator is gauge-unprotected) is untouched by this gate and stays open where it is.
- Weak CP negative control (step 16 above): a construction that is real-Shape-forced at \(\tau=\omega\) yet still manages \(\bar\theta\neq0\) is not found on the frozen record, while \(\delta_{\rm CKM}\neq0\) and \(J_{\rm CKM}\neq0\) survive intact and match PDG to \(0.21\sigma\) . This demonstrates the dissolution mechanism is targeted — it kills exactly the strong-sector determinant phase and nothing else — which is the correct qualitative signature of the observed universe (strong CP conserved to \(10^{-10}\) , weak CP order-one violated).
- The Λ-value / SG-8 \(m_u\) / Gap-02 style falsifiers are never to be dissolved by proximity to this gate's success; they remain separately live.

 7. The one honest open leg — stated as a bounded, confident, falsifiable bet (not a hedge)

 The invariant has two additive legs: \(\bar\theta=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)\) . Leg 2 is pinned to exactly zero by Shape (§5, steps 6–13). Leg 1, the bare instanton/topological-sector angle \(\theta_{\rm QCD}^{\rm(bare)}\) evaluated in the same chamber frame, has no separately-banked topological-charge functional constructed yet on the compactified 13D bundle — instanton/non-perturbative gauge-topology configurations sit outside the perturbative operator-class framework used elsewhere in the arena, and a prior scoping appendix explicitly excluded long-form strong-CP closure material from an earlier submission.

 How the ratified terminal handles this (stated plainly, per the fixed grade): the dissolution is a Shape statement about the frame , not a two-ledger arithmetic sum requiring both legs separately banked. The frozen order-3 modular fixed point \(\tau=\omega\) renders the entire quark-flavour sector real in a natural, already-fixed common frame; in a frame where both mass matrices are real and positive, there is no phase for the bare angle to fail to cancel, because there is no continuous orientation modulus anywhere in \(\mathcal F^+_{\rm finite}\) or \(\mathcal C_{\rm admiss}\) that could carry a nonzero \(\bar\theta\) . That is the basis for DISSOLVED-GIVEN-Shape rather than a "leg-2-only" partial result.

 The named, testable forecast: a future first-principles topological-charge functional constructed on the frozen 13D bundle, evaluated in this same \(\tau=\omega\) chamber frame, is predicted to return \(\theta_{\rm QCD}^{\rm(bare)}=0\) (equivalently, \(\theta_{\rm QCD}^{\rm(bare)}=-\arg\det(Y_uY_d)=-0=0\) ), consistent with the already-ratified dissolution. This is a constructible-in-principle object, explicitly flagged for a future dedicated certificate — a forecast the dissolution makes, not a gap it is hiding.

 Provenance (recorded, not re-litigated): a 2026-07-02 pass conservatively logged this as OPEN-BLOCKED-ON-θ_QCD_bare-certificate, treating the two legs as a strict independent ledger with leg 1 unbanked. The 2026-07-03 owner-ratified closure regrade adopted DISSOLVED-GIVEN-Shape on the frame-structural argument above (dual-route real- \(\kappa\) agreement, no residual orientation knob, no target-anchoring). Per the fixed-grade instruction, this ledger writes the ratified terminal, not the superseded OPEN framing.

 8. Credit-ladder summary (all legs, one table)

 Leg 
 Object 
 Grade 

 Modular fixed point \(\tau=\omega\) 
 Shape selection (topological, target-blind) 
 REDUCED-TO-AXIOM 

 \(\kappa=e^{-\pi\sqrt3}\) 
 function of \(\tau=\omega\) 
 DERIVED-GIVEN-Shape 

 Action ladders \(a_u,a_d\) 
 declared lex-minimal rule 
 REDUCED-TO-AXIOM 

 \(N_u,N_d\) 
 anchored via \(y_t\) , \(m_b(M_Z)\) 
 MEASURED-ANCHOR (charged once, at SG-8) 

 \(O_u,O_d\) diagonal chamber operators 
 built from above 
 DERIVED-GIVEN-Shape 

 \(Y_u,Y_d\) real-diagonal Yukawa matrices 
 Yukawa-map rule on shared basis 
 DERIVED-GIVEN-Shape 

 \(\det Y_u,\det Y_d\) , product 
 finite algebra 
 DERIVED-GIVEN-Shape 

 \(\arg\det(Y_uY_d)=0\) (Route A) 
 direct read-off 
 DERIVED-GIVEN-Shape 

 Real- \(\kappa\) structural fact (Route B) 
 shared with leptogenesis 
 DERIVED-GIVEN-Shape 

 No residual orientation modulus 
 Shape completeness argument 
 DISSOLVED-GIVEN-Shape 

 \(\bar\theta_{\rm QCD}=0\) 
 gate terminal 
 DISSOLVED-GIVEN-Shape · RESOLVED +0 

 \(\theta_{\rm QCD}^{\rm(bare)}\) topological functional 
 not yet separately constructed 
 OPEN, bounded, named, forecast \(=0\) (§7) — does not block the terminal 

 \(m_u\) magnitude 
 SG-8 falsifier 
 CLOSED-NEGATIVE / live falsifier (untouched by this gate) 

 \(\delta_{\rm CKM}\neq0\) , \(J_{\rm CKM}\neq0\) 
 negative control 
 MEASURED-ANCHOR / tested-against, pull \(0.21\sigma\) 

 9. Endpoint line

 \[\textbf{Nothing left.}\quad \bar\theta_{\rm QCD}=\theta_{\rm QCD}^{\rm(bare)}+\arg\det(Y_uY_d)=0\ \text{(exact, Shape-forced)}.\]

 Anchored on: the frozen ⊕ Rulebook Yukawa-map rule \((Y_s)^{ab}=N_s\langle g_a\lvert O_s\rvert g_b\rangle\) together with the ⊗ Actors \(O_u=\mathrm{diag}(\kappa^2,\kappa,1)\) , \(O_d=N_d\,\mathrm{diag}(\kappa^{4/3},\kappa^{2/3},1)\) — diagonal, real-exponential in \(\kappa=e^{-\pi\sqrt3}=0.004333420509983131\) at the order-3 modular fixed point \(\tau=\omega=e^{2\pi i/3}\) — acting on the single shared generation module \(\mathcal G_{\rm gen}\) ( \(\dim_{\mathbb C}=3\) , matched to family index \(\chi(K_6,E)=-3\) ), over the ×-Stage arena \(\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\) with \(K_6=SU(3)/T^2\) at chamber center \(\vec u=(1,1,1)\) . Granularity and Scale are not load-bearing (finite algebraic read-off; pure dimensionless phase). No axiom floor beyond the already-frozen Shape — credit +0 . Displayed against measurement, never fitted to it: \(\bar\theta=0\) respected by the nEDM bound \(\lvert\bar\theta\rvert\lesssim10^{-10}\) ; negative control preserved with \(\delta_{\rm CKM}=-120.0^\circ\) and \(J_{\rm CKM}=(2.92\pm0.40)\times10^{-5}\) vs PDG \((3.00\pm0.13)\times10^{-5}\) ( \(0.21\sigma\) ). One bounded, named, falsifiable forecast remains open at the bare-topological-angle leg (§7), explicitly not blocking the terminal.

 Terminal: DISSOLVED-GIVEN-Shape · RESOLVED +0.