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Publication history: 2026-08-01

Original publication entries retain their wording and recorded status. Follow the current research history for later findings and revisions.

19 / 56 days

1 Aug 2026

22 updates
  1. Seam & QGEOXXXVI
    The covered seam lands (v623 promoted) — the lattice skeleton of the covering-level identification

    Summary

    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)

    Details

    • v623_covered_seam.py (QGEO.COVERSEAM.01, 9 checks; probe 9/9, COVERED-SEAM-SKELETON). Connectivity census (j=1,2: one circle; trivial and alternating: 3 components); the explicit 48-walk (three base turns, 12 crossings); L^4= deck and L^{12}= id on all 48 points with proper order 12; the N=48 mode-sum identity; three 16-mode deck classes with the offset-1/2 class = the base NS frequencies exactly; \mathrm{deck}^3=-1 on modes, bilinear eigenvalue =\omega; the 12 lifted marks in a single 12-cycle. Residue named: the interacting/CFT orbifold statement (twist-field OPE).
    • Sync. Ledger row + note on QGEO.SEAMID.01; tfpt_3 (audit block); diary; suite 616\to617 scripts; website/README/zenodo counters; mirrors regenerated.
  2. Seam & QGEOXXXV
    The seam identification lands (v622 promoted) — THE PHYSICAL SEAM IS THE CONFORMAL SEAM
    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

    Summary

    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

    Details

    • v622_seam_identification.py (QGEO.SEAMID.01, 10 checks; probe 10/10, SEAM-IDENTIFIED). Mode-sum identity for all d=1..15 with the closed form \sum_{j<8}\sin((2j{+}1)x)=\sin^2(8x)/\sin(x); Ramond must-fail; mark midpoints =\mu_4 exact; reflection axes (k{+}1)\pi/16 with cuts \mapsto\{\pi/4,\pi/2,3\pi/4,\pi\}; through-mark axes = the straddled cuts \{7,15\}; v617's bond reflection = cut k=11 with the v599 (k,5{-}k) pattern.
    • Sync. Ledger row + notes on QGEO.SEAMFORCE.01 and GATE.QGEO.01 (bedrock state after v617/v620/v622; no status move); tfpt_3 (audit block); diary; suite 615\to616 scripts; website/README/zenodo counters; mirrors regenerated.
  3. Celestial routeXXXII
    The interacting gamma slice lands at toy level (v615 promoted)
    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

    Summary

    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

    Key points

    1. 3
      Cannot fire on the surviving class — v608 upgraded from g=0 to the full grid); the mark transport exists interacting (kill test
    2. 7
      Does not fire); the odd transfer spectrum stays multiplicity-free; and the dead member has no RP home — the dynamical bit selection and the chirality data CO-LOCATE on the same member

    Details

    • v615_gamma_toy_interacting.py (WOIT.GAMMA.TOY.01, 11 checks; probe 11/11, GAMMA-TOY-LANDED). Chiral/mirror parents (I\pm iC, gap 1); v534 survivor anchor reproduced; sector-exact mirror flip (even (29,0,0) PD, odd (0,8,0) ND, max odd eigenvalue \le-1.5\cdot10^{-3}); \eta=-i convention control; mark transport \alpha_4 exact 16/16 with Gram and transport form Hermitian PD at every g; odd transfer spectrum simple at every g (honest finding: strong-coupling near-doubling at g=8 without closing; g=0 top 1.641438 = numerically the v524 clock top); dead-member control (m{=}2,s{=}+) indefinite. Kill tests (3)/(6)/(7) stay formally live on \mathcal A_{\mathrm{hol}}; kill test (6) untouched.
    • Sync. Ledger row + note on WOIT.GAMMA.FREE.01; tfpt_research_contracts (\gamma-stage note); tfpt_3 (audit block); diary; suite 608\to609 scripts; website/README/zenodo counters; mirrors regenerated.
  4. Seam & QGEOXXXI
    The Gauss–Manin transport round lands (v614 promoted) — THE DECK STEP IS THE BOUNDARY MONODROMY
    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

    Summary

    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

    Details

    • v614_transport_deck.py (QGEO.TRANSPORT.01, 12 checks; probe 12/12, TRANSPORT-CANONICAL). Exact rotation substitution identity; uniform deck \omega across all three omega-sheet rows (must-fail: t^3\ne\omega, t^5\ne\omega); \mathrm{char}(\mathrm{transport})=\mathrm{char}(r) exact; 25-digit branch-tracked certificates at \tau=1/3,1/2,2/3,1 (dps 80, max 5{\cdot}10^{-28}; \tau=1 at 2.7{\cdot}10^{-81}). The reading: the braid fixes the boundary, the rigid rotation does not — their homological difference is one boundary loop of local-system monodromy.
    • Sync. Ledger row + note on QGEO.PERNORM.01; tfpt_3 (audit block); diary; suite 607\to608 scripts; website/README/zenodo counters; mirrors regenerated.
  5. Seam & QGEOXXX
    The canonical period normalization lands (v613 promoted) — THE ANALYTIC BRIDGE CLOSES
    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

    Summary

    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

    Details

    • v613_canonical_periods.py (QGEO.PERNORM.01, 19 checks; probe 19/19, PERIOD-NORMALIZED). Deck-free rotation on segments (40-digit certificates) with the must-fail control (raw period rows are NOT r-eigenvectors, residual 0.42 — the naive ansatz falsified before the honest construction); \mathrm{char}(r)= \mathrm{char}(\omega M) exact with explicit conjugator G; character match antihol \leftrightarrow\zeta_{12}; omega-sheet signature (2,1) with the unique negative on the antiholomorphic line = J's unique positive line (v612): all three character-matched sign products negative. Residue named: the canonical fork-basis identification (basepoint bookkeeping for G).
    • Sync. Ledger row + notes on QGEO.POLFRAME.01 and QGEO.PERIODS.01; tfpt_3 (audit block); diary; suite 606\to607 scripts; website/README/zenodo counters; mirrors regenerated.
  6. Seam & QGEOXXIX
    The polarization frame lands (v612 promoted)
    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

    Summary

    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

    Details

    • v612_polarization_frame.py (QGEO.POLFRAME.01, 9 checks; probe 9/9, POLARIZATION-FRAMED). Simple spectrum, \det r=-1; exact diagonalization (honest typing: diagonality follows from J-unitarity, the content is the values and the signature distribution); product of diagonal values =\det J/|\det W|^2=1/2 exactly.
    • Sync. Ledger row + note on QGEO.PERIODS.01; tfpt_3 (audit block); diary; suite 605\to606 scripts; website/README/zenodo counters; mirrors regenerated.
  7. Seam & QGEOXXVIII
    The periods round lands (v611 promoted)
    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

    Summary

    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

    Details

    • v611_periods.py (QGEO.PERIODS.01, 6 checks; probe 6/6, PERIODS-CLASSICAL). Five symbolic Beta identities; three 40-digit rotation-character certificates; the classical \omega-CM period; the PSLQ-null cross-factor consistency; the period dictionary named as the Hodge/RP resource.
    • Sync. Ledger row + note on QGEO.LANDSCAPE.01; tfpt_3 (audit block); diary; suite 604\to605 scripts; website/README/zenodo counters; mirrors regenerated.
  8. Seam & QGEOXXVII
    The curve landscape lands (v610 promoted)
    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

    Summary

    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

    Details

    • v610_curve_landscape.py (QGEO.LANDSCAPE.01, 9 checks; probe 9/9, LANDSCAPE-MAPPED). Differential characters (12,12,6); splitting confirmed at 14 primes (ten exact zero-differences; surface traces (-8,4,-20,-32) at p\equiv1 mod 12); a_p=0 for p\equiv2 mod 3; census: anchor class 286= max nontrivial, \{1,1,3\} at 254; v598/v599 char hits retroactively confirmed as correctly typed candidates.
    • Sync. Ledger row + notes on QGEO.COVER.01/QGEO.DICT.01; tfpt_3 (audit block); diary; suite 603\to604 scripts; website/README/zenodo counters; mirrors regenerated.
  9. Seam & QGEOXXVI
    The translator factorization lands (v609 promoted)
    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)

    Summary

    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)

    Details

    • v609_translator_factorization.py (QGEO.CFACTOR.01, 8 checks; probe 8/8, C-FACTORIZATION-LANDED). Intertwiner space 1-dim; \det d=1; d\in M_3(\mathbb{Z}[\omega]); \Gamma(d)d=1; c=\lambda K_1d exact (\lambda=1/7-3\sqrt3 i/7); \mathrm{inner}(K_1d) reproduces \varphi on the pair; honest scope: F_2 carries the corpus datum by definition; the v606 arithmetic negative explains why the cyclotomic route had to fail.
    • Sync. Ledger row + notes on QGEO.TRANSLATOR.01/QGEO.MODESEP.01; tfpt_3 (audit block); diary; suite 602\to603 scripts; website/README/zenodo counters; mirrors regenerated.
  10. Celestial routeXXV
    The free gamma slice lands (v608 promoted) — the first executed step of the WOIT \gamma milestone
    Mirror modes are STRICTLY OS-negative sector-exactly (every mirror fermion vector has negative OS norm — no mirror survives reconstruction, kill test

    Summary

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

    Key points

    1. 3
      Cannot fire at the free level), the chiral generation is multiplicity-free (clock spectrum exactly \{1,\sqrt2-1\}), and the mark transport exists (\tau_2 is the mark-A\toB transport, PD — kill test
    2. 7
      Does not fire free); \gamma itself stays open on \mathcal A_{\mathrm{hol}}; side finding: 14c is unipotent mod (1-\omega)

    Details

    • v608_gamma_free_slice.py (WOIT.GAMMA.FREE.01, 9 checks; probe 9/9, GAMMA-FREE-SLICE-LANDED; the v524 machinery imported read-only). Chiral (16,0,0); anti-chiral (8,8,0) with even (8,0,0) PD and odd (0,8,0) strictly ND; clock spectrum \{1,\sqrt2-1\} multiplicity-free on the fermion domain; mark algebras PD (4,0,0) with \dim 4=|\mu_4|, products generate the full 16-dim basis, \alpha_2 transports mark A to mark B.
    • Sync. Ledger row + notes on WOIT.OS.TWISTOR.01/WOIT.BETA2.OS.01; tfpt_research_contracts (\gamma-stage free-slice note); tfpt_3 (audit block); diary (incl. the c-unipotency side finding); suite 601\to602 scripts; website/README/zenodo counters; mirrors regenerated.
  11. Seam & QGEOXXIV
    The selection rule lands (v607 promoted)
    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)

    Summary

    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)

    Details

    • v607_selection_rule.py (QGEO.SELRULE.01, 10 checks; probe 10/10, SELECTION-RULE-LANDED). \omega-multiplication acts as the identity on the torsion; the induced r-action is [[0,1,0],[1,0,2],[2,2,2]] mod 3 with only fixed class 0; x^2+1 irreducible over \mathbb{F}_3 gives structural uniqueness of the eigenline (\lambda=-1), census-confirmed; the top-mode class is that eigenline, the zero mode and the seam are not projectively fixed (nor under r^2).
    • Sync. Ledger row + notes on QGEO.MODESEP.01/QGEO.INVOL.01; tfpt_3 (audit block); diary; suite 600\to601 scripts; website/README/zenodo counters; mirrors regenerated.
  12. Seam & QGEOXXI
    The translator round lands (v605 promoted)
    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

    Summary

    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

    Details

    • v605_translator.py (QGEO.TRANSLATOR.01, 11 checks; probe 11/11, TRANSLATOR-LANDED). \varphi(A)=cAc^{-1} exact (1-dim intertwiner space, c invertible); 14c \mathbb{Z}[\omega]-integral (denominator 14=2\cdot7); seam lift zero in the free part, (2,2,0) in the torsion of \mathrm{coker}(1-T_\infty); order chain O_1\subset O_{\mathrm{joint}}\cap\mathrm{span}_1\subset P_1 with indices 243 and 81; C_V^TJ_{\mathrm{fix}}=J_{\mathrm{fix}}C_{V\mathrm{dual}} integer; c's independent geometric identity typed open.
    • Lean. CoverEmbedding.lean addendum: Jfix_symmetric, Jfix_det =72, CV_dual_certificate, CVdual_cubic, dual_joint_trace; lake build green (3375 jobs).
    • Sync. Ledger row + notes on QGEO.EQORDER.01/QGEO.SEAMMARKS.01/QGEO.DUALFORMS.01; tfpt_3 (audit block); diary; suite 598\to599 scripts; website/README/zenodo counters; mirrors regenerated.
  13. Seam & QGEOXX
    The equivariant-order round lands (v604 promoted)
    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)

    Summary

    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)

    Details

    • v604_equivariant_order.py (QGEO.EQORDER.01, 12 checks; probe 12/12, EQUIVARIANT-ORDER-LANDED). Anti-isometry exact; J_{\mathrm{fix}}=[[16,2,4],[2,-2,2],[4,2,-2]]; all 28 rotated words \mathbb{Z}[\omega]-integral, joint lattice full rank 18, Smith divisors (1^{15},3,3,3); \mathrm{coker}(1-T_\infty) divisors (1,1,1,3,3,3,0,0); each finite cusp twist torsion-free; three objects share (\mathbb{Z}/3)^3=27 (infinity monodromy, equivariant order, \det(1-\mathrm{deck})); F_2 vs J_{\mathrm{fix}}-menu and algebra-twist negatives quantified.
    • Sync. Ledger row + notes on QGEO.SEAMMARKS.01/QGEO.COVER.01; tfpt_3 (audit block); diary (incl. the exploration round XX: \infty-embedding, R-orbit classes, translator hunt); suite 597\to598 scripts; website/README/zenodo counters; mirrors regenerated.
  14. Seam & QGEOXIX
    The seam-and-marks round lands (v603 promoted)
    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

    Summary

    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

    Details

    • v603_seam_marks.py (QGEO.SEAMMARKS.01, 15 checks; probe 15/15, SEAM-REAL-LANDED). Seam: Uv=0, Vv=v unique, v=(0,(\sqrt3-i)/(\sqrt3+i),1), Rv\sim v, S_0v=-v; infinity: unreduced Burau braid relations, invariant column (1,1,1,1), full twist eigenvalues \{1,\omega^{\times3}\}; marks: E_1 integer restriction, in the word algebra with denominator lcm 3 (order-3 cokernel class); E_4 trivial, E_2/E_3 outside (naive global map fails, equivariant framing typed); translator: F_2=g^T\Sigma g matches no canonical menu form (all typed open).
    • Sync. Ledger row + notes on QGEO.DUALFORMS.01/QGEO.INVOL.01; tfpt_3 (audit block); diary; suite 596\to597 scripts; website/README/zenodo counters; mirrors regenerated.
  15. Seam & QGEOXVIII
    The duality-and-forms round lands (v602 promoted)
    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

    Summary

    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

    Details

    • v602_duality_forms.py (QGEO.DUALFORMS.01, 17 checks; probe 17/17, DUALITY-FORMS-LANDED). U_1 J-self-adjoint, V_1^J the exact product reversal, (U_1,V_1^J) conjugate to the compiler pair; H, Q, Q_+, ladder all exact in the J-projector language; Weil-twisted form positive definite (\{2\sqrt2-2,2,2+2\sqrt2\}); honest negatives: g^Tg\ne\lambda B, naive mark-local map fails, seam normal not a deck eigenvector (all typed open).
    • Lean (FORM.QGEO.04). TfptCarrier/CoverEmbedding.lean: the v600 integer core via determinant-free Smith certificates (W=TS, PT=\mathrm{diag}(1,1,1,3,3,3,3) with unimodular P, S-primitivity via unimodular minor); kernel decide; lake build green (3375 jobs).
    • Sync. Ledger row + notes on QGEO.CANON.01/QGEO.EMBED.01; tfpt_research_contracts (\mu_3-cover model paragraph after the bedrock chain), tfpt_3 (audit block); diary; suite 595\to596 scripts; website/README/zenodo counters; mirrors regenerated.
  16. Seam & QGEOXVII
    The canonicity round lands (v601 promoted)
    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

    Summary

    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

    Details

    • v601_equivariant_dual.py (QGEO.CANON.01, 14 checks; probe 14/14, EQUIVARIANT-DUAL-LANDED). GJ_k=(1-T_k)J^{-1}(1-T_k)^\dagger J idempotent rank-1 J-projectors rotating exactly under D_4; pair family U_k=3\,GJ_k, V_k=1+GJ_{k-1}-GJ_kGJ_{k+1}: \mathbb{Z}[\omega]-integral, compiler-exact, equivariant by construction; duality: conjugate to (U_c^T,V_c^T) exactly (must-fail to (U_c,V_c) documented); Klein group: all eight spectral-sign involutions integral, S_0 explicit, AB=S_0; honest: naive J-word-Gram inertia (3,2,2)\ne(4,0,3) (Weil twist named open).
    • Sync. Ledger row + note on QGEO.EMBED.01; tfpt_1/tfpt_2 (canonicity clauses), tfpt_3 (audit block); diary; suite 594\to595 scripts; website/README/zenodo counters; mirrors regenerated.
  17. Seam & QGEOXVI
    The joint embedding lands (v600 promoted)
    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

    Summary

    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

    Details

    • v600_joint_embedding.py (QGEO.EMBED.01, 15 checks; probe 15/15, JOINT-EMBEDDING-LANDED). The pair U^*=-2E_1+(G_1\Gamma(G_1)+\Gamma(G_1)G_1)+3, V^*=-E_3+\tfrac12(G_2\Gamma(G_2)+\Gamma(G_2)G_2)+1: chars and joint traces compiler-exact (\operatorname{tr}(UV)=3, \operatorname{tr}(UV^2)=6, \operatorname{char}(U{+}V)= \operatorname{char}Q); seven words independent; explicit invertible g with gU^*g^{-1}=U_c, gV^*g^{-1}=V_c; the real structure integer on \mathbb{Z}^6 with explicit saturated fixed-lattice basis (SNF certificate); integer restrictions; Smith divisors (1,1,1,3,3,3,3), index 81 — the v566 self-code order structure on the curve. Must-break control passes; canonicity open (existence claim only).
    • Sync. Ledger row + notes on QGEO.REAL.01, QGEO.DICT.01, QGEO.COVER.01 and DIAMOND.PARA.SELFCODE.01; tfpt_1/tfpt_2 (realization sentences), tfpt_3 (audit block); diary; suite 593\to594 scripts; website/README/zenodo counters; mirrors regenerated.
  18. Seam & QGEOXV
    The real structure lands (v599 promoted)
    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)

    Summary

    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)

    Details

    • v599_real_structure.py (QGEO.REAL.01, 18 checks; probe 18/18, REAL-STRUCTURE-LANDED). Existence and uniqueness of R(v)=M\,\overline{v} (1-dimensional intertwiner space, R^2=+1 exact, four twist intertwinings RT_kR^{-1}=T_{5-k}^{-1}); the grading functor \Gamma(A)=M\overline{A}M^{-1} locates V\sim G_2+G_3-\Gamma(G_3), H\sim 1+G_2-\Gamma(G_3) (the anchor spectrum) and Q_+\sim 1+G_2+G_3-\Gamma(G_3) exactly at char level; the \Gamma-even/odd cusp elements carry \{0,1,5\}, \{0,3,9\}, \pm\sqrt5, \pm\sqrt{27} and one nilpotent; the joint negative quantified (4\times6 candidates, zero pairs pass \operatorname{tr}(UV)=3 and \operatorname{char}(U+V)=\operatorname{char}Q; kill not triggered); the invariant hermitian form is unique with \det J=8 and signature (1,2) — Strategy II's Hodge prediction at form level, pinned.
    • Sync. Ledger row + notes on QGEO.DICT.01/QGEO.COVER.01; tfpt_3 (audit block); diary; suite 592\to593 scripts; website/README/zenodo counters; mirrors regenerated.
  19. Seam & QGEOXIII
    The cover dictionary reaches THREE exact entries (v598 promoted)
    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

    Summary

    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

    Details

    • v598_cover_dictionary.py (QGEO.DICT.01, 4 checks; probe 4/4, DICT-THREE-ENTRIES). All three entries exact (numeric prefilter, symbolic confirmation); supporting structure: opposite-twist products nilpotent, Q_-^2\sim x(x-3)^2; negatives quantified (7296 combinations, 0 matches for Q_+/V/H/Q); the Strategy-I kill criterion not triggered; GATE.QGEO does not move.
    • Sync. Ledger row + note on QGEO.COVER.01; tfpt_1/tfpt_2 (dictionary sentences updated), tfpt_3 (audit block); diary; suite 591\to592 scripts; website/README/zenodo counters; mirrors regenerated.
  20. Seam & QGEOXI
    The dictionary's first entry
    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

    Summary

    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

    Details

    • The hit. The cusp Gram (1-T_2)(1-T_2)^\dagger has characteristic polynomial x^2(x-3)=\operatorname{char}(U) exactly — the rank-one winding operator (trace 3=N_{\mathrm{fam}}) as a Gram of a D_4-conjugate cusp twist: the fifth fingerprint. Additional exact structure: opposite-twist products (1-T_1)(1-T_3) are nilpotent (x^3), G_1+G_3 has spectrum \{0,1,3\}. Honest negative: V/H/Q not located in the scanned canonical ranges (products/sums/\omega-twists of length \le2) — the V-dictionary likely requires the parabolic flag (the seam-cusp filtration); named next step; kill criterion unchanged.
    • Sync. Ledger note on QGEO.COVER.01; diary; mirrors regenerated.
  21. Seam & QGEOX
    Strategy I/II executed
    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

    Summary

    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

    Details

    • v597_mu4_cover_model.py (QGEO.COVER.01, 6 checks; probe 6/6, FINGERPRINTS-HIT). Genus 3, five cusps (\mu_4+\infty), Chevalley–Weil H^{1,0}=(1,2); Burau at t=\omega with verified braid relations, r^4=\omega\cdot1, r^{12}=1; cusp classes (\mathbb{Z}/3)^5/\text{one relation}=(\mathbb{Z}/3)^4, order 81=3^4 with the per-mark \mathbb{Z}/3 mechanism (\det_{\mathbb{Z}}(1-\text{deck})=27 complementarily); invariant form unique up to scale, eigenvalues \{-2, 2{-}2\sqrt2, 2{+}2\sqrt2\}, signature (1,2); single-cusp Gram spectrum \{0,0,2\} all-integer. OPEN and named: the Q/V operator dictionary, the real-structure/RP map to the v572 Gram data; kill criteria live, not triggered; GATE.QGEO does not move.
    • Sync. Ledger row + notes on QGEO.INVOL.01 and DIAMOND.PARA.SELFCODE.01; tfpt_1/tfpt_2 (candidate paragraphs), tfpt_3 (audit block); diary; suite 590\to591 scripts; website/README/zenodo counters; mirrors regenerated.
  22. Seam & QGEOIV
    The sheet involution exists unconditionally (v590 promoted)
    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

    Summary

    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

    Details

    • v590_involution_existence.py (QGEO.INVOL.01, 7 checks; probe 7/7, EXISTENCE-UNCONDITIONAL). All eight spectral sign choices on V's eigenspaces are integer, involutive and order-stabilizing on both twins; narrowing 6\to3\to2\to1: the trace +1 half dies by the v574 rank theorem, Se_3=-e_3 and +1 on \ker V single out S_0=[[1,-1,0],[0,-1,0],[0,0,-1]]; honest scope: the corpus RP reflection does not commute with V (different structure), the alt twin has the same classification (not a bit selector); the two demands' continuum derivation (\mu_4/D_4/H^1) stays the named open.
    • Sync. Ledger row + note on QGEO.WINDING.SELF.01; tfpt_1 (Stage-B passage), tfpt_2 (new paragraph), tfpt_3 (audit block); diary; suite 583\to584 scripts; website/README/zenodo counters; mirrors regenerated.