Every change and new result, by date.

The seam lifted to the \mu_3-cover is ONE 48-site NS circle for the uniform weights (the alternating pair DISCONNECTS it — a new independent kill), the lifted clock has EXACT ORDER 12 with L^4= deck (r^4=\omega as lattice combinatorics), the v622 kernel identity generalizes verbatim to N=48, and the deck-character mode grid is the 12-grid (\{1/6,1/2,5/6\}) with the untwisted sector equal to the base seam VERBATIM and the twisted pair carrying \omega on bilinears (the t=\omega/\bar\omega sheets); the fermionic deck lift satisfies \mathrm{deck}^3=-1 (the spin double)

The v519 chiral seam kernel is EXACTLY the discrete NS (antiperiodic) chiral mode sum on the 16-site circle, including the even-distance zeros (the flat band IS the NS spectrum; the Ramond sum fails as must-fail control); the four mark-bond midpoints sit EXACTLY at \mu_4 under the site map \theta_a=(2a+1)\pi/16 (no site touches a mark — the v519 “marks at bond midpoints” becomes literal geometry); the group dictionary is exact (clock =z\mapsto iz, RP reflections = diameter reflections); and the v534 STRADDLE LAW becomes geometry — straddled cuts = exactly the through-mark axes; the second v617 bedrock residue closes at the kernel + dictionary level, leaving the covering-level identification

The two C=1 violators are TRUNCATION ARTIFACTS of the prime-data cap (flip set = truncation set exactly, mechanism reproduced by injection at both scales, C=1 EXCEPTION-FREE on the complete-comb surface); the cyclic-N census makes the bedrock residue CONDITIONAL (seam-admissible primitive-Lorentz orders =\{3,5\} exactly, N=6 reduces to N=3, and the compiler's machine-checked 27/81 pin N=3 uniquely); and the interacting Klein–Landau semigroup CONTRACTS to the \mu_4-commensurate steps with the clock surviving as a positive transfer over the whole coupling grid

The internal SU

The gamma battery executed ON the RP-surviving interaction class (the alignment bit \delta=\pi/2, positive coupling, v534) — THE INTERACTING MIRROR FLIP: at the forced twist \eta=+i the mirror state's odd sector is STRICTLY NEGATIVE DEFINITE on every cut and every coupling g\in\{1/32..8\} (bounded away from zero: kill test

Transporting the segment cycles along the rigid quarter-rotation loop, the discriminant \lambda(\tau)=e^{2\pi i\tau} winds ONCE around 0 and the transported periods factorize EXACTLY, I_m(\tau)=e^{i\pi\tau(m+1)/2}e^{-2\pi i\tau j/3}I_m(0); at \tau=1 the deck factor is \omega=t^4 on EVERY sheet (the exponent 4=|\mu_4| forced by the puncture count); the canonical transport matrix in the segment basis is EXACTLY \omega M — the v613 dictionary r\sim\omega M is CANONICAL, no conjugator freedom left, and (\omega M)^4=\omega re-derives r^4=\omega; v613's residue F1 closed at the transport level

The quarter rotation acts DECK-FREE on the four branch segments (all 12 wrap factors exactly 1), and the exact dictionary is r=\mathrm{deck}\circ\mathrm{rotation} — \mathrm{char}(r)=\mathrm{char}(\omega M) EXACTLY, explaining r^4=\omega structurally, with the deck power unique; the period rows (hol dx/y^2, hol x\,dx/y^2, antihol \overline{dx/y}) realize the rotation eigenframe; the Hodge weights are explicit Gamma monomials (h_1=(3\sqrt3/16\pi)\Gamma(1/3)^2\Gamma(1/6)^2, complex Beta formula); and J=-h PER CHARACTER: the compiler polarization is MINUS the analytic Hodge form — Riemann–Hodge positivity realized, v612's named step B5 executed

The braid rotation has three simple eigenvalues \zeta_{12}\cdot\{1,-1,-i\}, the polarization J diagonalizes EXACTLY in the rotation eigenbasis (values (1+\sqrt3)/2, (1-\sqrt3)/2, -1), and THE UNIQUE POSITIVE (HODGE) DIRECTION IS THE PRIMITIVE \zeta_{12} EIGENSPACE — the analytic Hodge structure and the finite rotation agree on where positivity lives; the canonical period normalization (Beta weights) is the named next step

The analytic layer of the cover is CLASSICAL — all five basic period integrals are exact Beta values with Gamma denominators on the \zeta_{12} grid (\Gamma(11/12),\Gamma(7/12),\Gamma(5/12),\Gamma(1/12)); the \mu_4 rotation acts on periods with the exact characters i^{k+1}; \Omega_E=B(1/3,1/6)/2 exactly; and the cross-factor period ratio admits no low-degree integer relation (PSLQ null) — the two CM factors are period-independent, as the v610 splitting demands

The Jacobian SPLITS into a \mathbb{Q}(\zeta_{12})-CM abelian surface — the field of deck AND marks together — times the \mathbb{Q}(\omega)-CM elliptic quotient E: y^3=u^2-1, point-count-confirmed (a_p(C)=a_p(E) exactly for all ten tested p\not\equiv1 mod 12); the integer-spectrum census honestly relativizes single-char hits (7894 integer entries — single spectra are cheap; the anchor class is the most frequent nontrivial family at 286; the compiler's distinction is the joint structure); two new canonical period resources named

The geometric identity of c is CLOSED — the mirror intertwiner d (dA=\Gamma(A)d) is unique, unimodular (\det d=1), \mathbb{Z}[\omega]-integral and \Gamma-involutive (\Gamma(d)d=1 exactly), and the v605 element factorizes exactly as c=\lambda K_1d with K_1=J^{-1}\overline{F_2}^TM^{-1}: \theta_{\mathrm{corpus}}=J\mathrm{adj}\circ\mathrm{inner}(K_1)\circ\mathrm{inner}(d) — all three v605 residuals are now resolved (seam reversal v603, zero-mode rule v607, c-identity v609)

Mirror modes are STRICTLY OS-negative sector-exactly (every mirror fermion vector has negative OS norm — no mirror survives reconstruction, kill test

The infinity torsion is a genuine \mathbb{F}_3 space, the deck rotation acts with char (x+1)(x^2+1) mod 3 — EXACTLY ONE projective eigenline, and it IS the top mode; the three modes carry mutually exclusive canonical labels (seam = R-real, top = deck-eigen, zero = the only unlabeled) and S_0 flips exactly the labeled pair: BOTH v590 demands expressed through cover structure (typed reading; the gate residual is the continuum derivation)

The infinity torsion SEPARATES the three V-eigenlines (all pure 3-torsion at infinity, classes pairwise distinct — the selection carrier the real structure could not provide; the canonical “+1” rule stays open); the arithmetic-identity route for c is closed (cubic field disc -783, Galois S_3, not abelian); the equidistribution conjecture is posed with the explicit constant C=1 in the prime-front paper; and the contract survey executes its bookkeeping (the disc-7 preregistration's discrete half marked executed via v571/v575/v578; the WOIT \gamma milestone documented as the designed next step)

The gate audit documents the cover program as MODEL-COMPLETE on the operator half (ledger note on GATE.QGEO.01, no status move — the residual is the conformal seam identification, the zero-mode selection, and the identity of c); honest negatives quantified (c without independent geometric identity; the zero-mode line is not R-real); tfpt_1 gains the consolidated “cover program at a glance” paragraph; and the prime-front uniformity sharpens: \varepsilon\cdot h\le0.98 on all 68 floor-passed windows — |q_{\mathrm{real}}/q_{\mathrm{model}}|\le h^{-1} uniformly on the declared surface (note on PRIME.LOCKPROJ.01)

The corpus RP anti-automorphism IS the J-adjoint twisted by an explicit inner element c (\varphi=J\mathrm{adj}\circ\theta_{\mathrm{pull}} is inner, c unique up to scale, 14c integral over \mathbb{Z}[\omega] — the single-sheet negatives resolved, the \bar\omega detour unnecessary); the seam is a NONZERO pure 3-torsion class at infinity (torsion coordinates (2,2,0)); the 81 survives frame-joining as the saturation defect (chain indices 3^5, 3^4); the J_{\mathrm{fix}} duality certified with a Lean addendum

R IS an anti-isometry of the polarization (M^\dagger JM=J^T exact — correcting a diary side remark), J restricts to the integer form J_{\mathrm{fix}} with \det=72=8\cdot9; the four mark frames generate a FULL order of index 27=3^3 in M_3(\mathbb{Z}[\omega]); and the 3-torsion localizes at infinity (\mathrm{coker}(1-T_\infty)=\mathbb{Z}^2\oplus(\mathbb{Z}/3)^3, finite cusps torsion-free): 81 at the marks, 27 at infinity — the integral infinity/order-torsion bridge; translator negatives extended (the \bar\omega-sheet named)

The seam line is a REAL line of the cover (unique common eigenline, R-fixed) and S_0 reverses it — the v590 seam-reversal demand holds geometrically; the infinity cusp separates in the unreduced Burau module; the first per-mark \mathbb{Z}/3 witness lands (E_1, class order 3 in the word-order cokernel); the corpus-transpose translator negatives are quantified

The J-adjoint exchanges the two parabolic types exactly ((U_1,V_1^J)\sim compiler pair); the dictionary completes in the J-projector language (the corpus anchor formula H=1+V(V-1)/2 holds verbatim on the curve); Weil positivity holds exactly (the Riemann bilinear relations on the \omega-sheet); the canonical bilinear form B=M^\dagger J is symmetric with \det B=8; the v600 integer core is Lean-certified (FORM.QGEO.04); the contracts paper gains the \mu_3-cover model paragraph

The J-adapted cusp projectors give a manifestly D_4-equivariant presentation of the pair (mark choice = winding choice), which realizes the DUAL parabolic exactly (conjugate to the transposed compiler pair; the direct intertwiner is singular); the v590 Klein four-group lives integrally on the curve; the Weil-twisted RP comparison and the seam identification stay open

The compiler pair (U,V) is REALIZED on the \mu_3-cover — a \Gamma-even degree-2 pair, \mathbb{Z}[\omega]-integral, seven-word rank 7, simultaneous conjugation to the compiler pair solved exactly, integer restrictions on the saturated \mathrm{Fix}(R) lattice, and saturation index 81=3^4 with the exact v566 divisor pattern; canonicity typed open, the gate does not move

The anti-holomorphic D_4 reflection on the \mu_3-cover exists and is unique up to scale with R^2=+1; its grading functor closes all three dictionary gaps at char level (V, H with the anchor spectrum (1,1,2), Q_+); the \Gamma-decomposition carries 5, 9, 27; the joint algebra embedding stays open (quantified negative); and the invariant form has \det J=8 — a sheet-diamond determinant-list member — with Lorentz signature (1,2)

U\sim G_2, Q_-\sim G_1-G_3 — the \mu_4 sheet coupling is the diamond-diagonal Gram difference — and the binary AnchorLadder (1,2,4)\sim1+G_1+G_3; the grading half is absent from 7296 canonical combinations, with the real structure as the named suspect

No Q/V hit; kill not triggered, the flag-adapted search is the designed next session); the real-structure map documented as blocked on the dictionary; and the self-contained equidistribution problem statement added to tfpt_prime_front (sec:theory-open) — the v596 identity family posed TFPT-free for external consumption

U IS LOCATED — the winding operator appears as a cusp Gram on the \mu_3-cover; exploration round, documented as a note on QGEO.COVER.01

The \mu_3-cover y^3=x^4-1 hits FOUR compiler fingerprints (v597 promoted) — rank-3 Eisenstein H_1, projective D_4 via the deck, cusp-class group (\mathbb{Z}/3)^4 of order 81, and the unique invariant hermitian form of LORENTZ signature (1,2) matching Paper II's polarization: the first geometric candidate that could close GATE.QGEO and the RP slot with one construction

The 1D lock projection (v596 promoted, T199 — the near-identity along the closed direction, median 7.7\times10^{-4}, coupling spectrum collective); the v590 involution group formalized in Lean with its Klein-four closure A\cdot B=S_0; and the big-picture strategy survey (the open core funnels into GATE.QGEO + the RP slot; two new geometric strategies preregistered)

Repository pushed (v576–v594, 111 files); the residual bookkeeping executed (v595 promoted, T198: the v593 residual was the \sigma MAPPING — identity-level closed forms 1.00000–1.00017, honest amendment of the v593 attribution); derivation sketches added to the theorem section; the determinant law formalized in Lean (det_S_closed, build green)

The cutoff completes the density layer END-TO-END (v593, T196), the FIRST UNCONDITIONAL STATEMENT via Büthe 2018 (v594, T197), the consolidated theorem section in tfpt_prime_front, and the Lean-4 formalization of the round's algebraic cores (ParityWeightLaws.lean, builds green against Mathlib)

\det S(a) closed with the e^{2a} coefficient identically zero and exact leading law 640\pi^2e^a/a^3; the real dominance tracks the closed formula at correlation 0.9985 (the v570 growth derived); entry-level arithmetic certificate with margin 47.6\times; PLUS the Paper-II consolidation section (the deterministic–arithmetic dissection as one narrative)

S^{\mathrm{pole}}=-32\pi^2ae^a\,gg^T with \det\equiv0 symbolically; v_2/v_1=-(a^2+16\pi^2)/(2(a^2+4\pi^2)) with limit exactly -1/2 — the v577 null-ray conjecture DERIVED, the v586 drift explained; plus the honest B/C rounds: deeper zero sums plateau (structural, note on v589) and the grading shortcut for the involution demands is falsified (note on v590)

The integral V-commuting involutions are exactly (\mathbb{Z}/2)^3 (rigidity, both twins); the v574 type theorem + seam reversal + zero-mode fixing single out the explicit S_0 with exact \Sigma-signature uniquely — GATE.QGEO.01's involution-existence input relocates to DERIVED

The arithmetic depth layer is IDENTIFIED as the explicit-formula zero comb — correlation 0.82 with the first 200 damped zeros, a factor 5.7 of the lock depth recovered at the smallest window; the zero-oscillation bound becomes a quantified target

G_{ij}(\beta) carries the comb side at the pole exponent and the archimedean side as the TRIVIAL-ZERO LADDER; closed B to 2\times10^{-5}, the closed defect to 0.2\% on all 70 windows, census decay h^{-1.40} — the explicit formula's skeleton, pole vs \Gamma-factor, in closed form

The diagonal lag weight is the single closed expression W_{kk}(d)=(2/N)[(N{-}1{-}d)\cos(\omega_k d)+\sin(\omega_k(d{+}1))/\sin\omega_k], machine-exact for all lags; the exact cell-sum reproduces the prime-free model's entries and determinant to 0.03\%