Original ledger status
DIAMOND.BIT.SELECT.01
THE RESIDUAL RECONSTRUCTION BIT: LOCATED, SEPARATED, AND GIVEN A CANONICAL EXACT WITNESS (2026-07-31; Priorities 2+3 of the self-code review fused with the v567 residual bit; one module, 11 checks, ~1 s; discovery probe bit_selector_probe.py 10/10, verdict SELECTOR-FOUND). (1) THE BIT IS INTEGRALLY VISIBLE [E]: the two 81-twins of the v567 audit (q32 = 2 = |Z2| vs q32 = 0) generate DIFFERENT integral orders -- Hermite bases differ at EQUAL index 81 and equal Smith divisors (1,1,1,3,3,3,3): real integral structure where every classical carrier-free invariant coincides. (2) THE SELECTOR [E]: the dominant mode of the sheet direction satisfies V_c (1,2,2c)^T = 2 (1,2,2c)^T identically in the coupling c; the LADDER SELF-CONSISTENCY -- the dominant mode reproduces its own eigenvalue as the geometric ladder (2^0,2^1,2^2) = (1,2,4) -- forces c = 2: on ALL 28 members of the v567 parabolic carrier-free class EXACTLY ONE satisfies it, the compiler Q, with no 81-pin and no column budget. (3) THE WITNESS IS PRIOR STRUCTURE [C] (no-free-pattern): (1,2,4) is the binary AnchorLadder step sequence p_{n+1} - p_n = 2^n (Lean-formalised), 2 = |Z2| the sheet datum, and the v566 trace code p_n(a) = 1 + tr(V^n) = 2 + 2^n rides the same ladder (n = 1..6 re-verified exactly). (4) THE SPLIT GEOMETRY LOCATED AND EXCLUDED [E]: an invariant module complement of the seam line exists IFF q32 = 1 (exact solve: a = 0, b = 1, q32 = 1; the plane span((1,0,0),(0,1,1)) exactly invariant there), and that member's direction algebra COLLAPSES to dimension 5 (Smith (1,1,1,3,3), no full parabolic, no 81): the seam-line + saturation demands admit ONLY nonsplit geometry -- the v566 A2 Ext arrow is NONZERO on the whole reconstruction class; neither twin splits (the complement system is inconsistent for q32 in {0,2}): the bit does NOT toggle split/nonsplit -- it toggles WHICH nonsplit integral order is realised and whether the dominant mode carries the full binary ladder. (5) PRIORITY-2 CANDIDATE TESTS [E], honest both ways: the mu4-grading candidate deg(E_jk) = g_k - g_j mod 4 FAILS (the order is not a graded sublattice: 5 of 7 Hermite vectors homogeneous) -- the review's kill condition does NOT fire and the 81 stays a FINGERPRINT pending a mechanism; left multiplication by V PRESERVES the order lattice and induces a canonical F3-linear action on the cokernel (Z/3)^4 -- the F3[V]-character computation is EXECUTED: the induced action has characteristic polynomial x(x-1)^2(x+1) mod 3 -- eigenvalues {0,1,1,-1}, NOT the regular mu4 pattern x^4 - 1 that 'one Z/3 per mark' would require (not even invertible), U acts as ZERO, and both twins give the identical structure: after two candidates (grading, V-action) the 81 stays a FINGERPRINT; the kill condition (no canonical local decomposition at all) is closer but not proven. NAMED LIMITS AS LOAD-BEARING CONTENT: the PHYSICAL selection is NOT derived -- why the dominant mode should satisfy the ladder self-consistency is the relocated open question, the same structural slot as the RP-selection of the alignment bit (v528/v534); the resonance between the reconstruction bit and the alignment bit is a TYPED OBSERVATION, not an identification; Stage B of v567 (GATE.QGEO.01) untouched; NO P2 reduction; AX.P2.01 and ANCHOR.GEN.01 markers untouched; the word 'derived' appears nowhere for any physics statement. [2026-07-31: the RP form is spectrally blind but integrally ANCHOR-VISIBLE on the twins (v572, Smith slot 18 vs 2 = 9 = N_fam^2); the dynamical RP-slot stays open; see DIAMOND.BIT.RPFORM.01.] [2026-07-31 II: the review's Ext-sensitive marked resolvent chi_Ext(z) = tr(P_seam R N_Ext R) vanishes IDENTICALLY for both twins with the natural markings (P_seam = E33, N_Ext = the radical lattice generators -- which are IDENTICAL between the twins: span{3E31, 3E32}); the review's own kill criteria fire -- the Gibbs pencil (v572) remains the one working marked object; sandbox record, no module.]
Recorded status: Identity [E] (exact integer/symbolic algebra -- Hermite/Smith lattice comparison, the symbolic dominant-mode identity, the uniqueness census on the 28-member class, the split-complement solve, the candidate tests) + [C] (the anchored ladder selector as candidate canonical selection; its physical justification open) + [O] (the selection mechanism; the F3[V]-character question; Stage B); verdict SELECTOR-FOUND; no marker moves
Dependencies, status detail and source
Canonical status: [E] twins integrally distinct (H0 != H2 at equal index 81); v(c) = (1,2,2c) identically; ladder self-consistency <=> c = 2, unique on 28/28; witness = AnchorLadder 2^n / |Z2| / trace code; split member q32 = 1 (complement solve a=0,b=1) collapses to dim 5 -- excluded; mu4-grading candidate fails (5/7 homogeneous); F3[V]-action computed: charpoly x(x-1)^2(x+1) mod 3, eigenvalues {0,1,1,-1} != regular mu4, U-action zero, twins identical; 11/11 checks; verdict SELECTOR-FOUND
Dependencies: QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; DIAMOND.AXIS.01; SEAM.BIT.RPBLIND.01; FORM.LADDER.01; v10; v218; v528; v534; v566; v567
Scripts: v568_bit_selector_ladder.py
Location: tfpt_1;tfpt_2
Original CSV, line 607Original ledger status
FTR.CRDISC.01
The preregistered cross-ratio check (self-code review Priority 4 / v566 S9), executed on its DISCRETE half (one module, 9 checks, <1 s; discovery probe cr_discrete_corners_probe.py 9/9, verdict CR-PASSES): the four solver corners are the in-repo Sheet-Diamond kernels J,K,C,F = M(1,t) = R + Q diag(1,t,t), t = -2..1 (FTR.PATH.01/v224, dets (2,4,14,32)); the corpus's one canonical declared solver reading, the Koide anchor response ell(M) = a^T M 1 (v183), reads (26, 35, 44, 53): CR = 4/3 EXACTLY, and the fit-free reconstruction forces y_F = 53 from (26, 35, 44) alone -- the Koide numerator is Moebius-determined by the three upstream corners (a NEW exact cross-corner identity); honesty theorem: ell is affine on the path (step 9 = N_fam^2) and EVERY linear functional passes -- the check separates linear from nonlinear exports; negative controls with teeth: the determinant path fails at 28/25 exactly as preregistered, the dominant-eigenvalue export fails at 1.322162. THE HONEST BOUNDARY: the DISCRETE half only -- the external continuous solvers (pole transfer, Boltzmann, relic ODE, lattice) export no states here; the continuous half of the check stays open exactly as CONTRACT.F.01 fences it; contract markers unchanged. [2026-07-31: the CONTINUOUS half is since executed by v575 -- UNIFLOW-DEGENERATE, closing onto the v425 single-flow theorem; see FTR.CRCONT.01.]
Recorded status: Identity [E] + typed boundary [C] (exact symbolic identities on compiler kernels; the continuous-solver half open)
Dependencies, status detail and source
Canonical status: CR-PASSES (discrete half): declared export (26,35,44,53), CR = 4/3 exact, y_F = 53 Moebius-forced; controls 28/25 and 1.3222 fail as they must; continuous half open; 9/9 checks
Dependencies: CONTRACT.F.01; FTR.PATH.01; DIAMOND.PARA.SELFCODE.01; v183; v224; v533; v566
Scripts: v571_cr_discrete_corners.py
Location: tfpt_research_contracts
Original CSV, line 608Original ledger status
DIAMOND.BIT.RPFORM.01
The RP form on the 81-twins -- spectrally BLIND, integrally ANCHOR-VISIBLE (one module, 11 checks, ~1 s; discovery probe rp_bit_form_probe.py 11/11, verdict ANCHOR-VISIBLE): with the corpus reflection theta(X) = Sigma X^T Sigma (v10) and the declared states (trace; anchor (1,1,2); ladder (1,2,4); unit), the RP kernel K(x,y) = omega(theta(x) y) on the Hermite basis of each twin order has IDENTICAL inertia on both twins for every state (trace (4,0,3); vector states (3,1,3)) and is never PSD -- no kinematic v534-style positivity selection exists (honest negative, consistent with v567's certificate coincidence and v534's nonperturbative lesson); but the INTEGRAL Smith towers of the anchor-state Gram SEPARATE the twins -- q32 = 2: (2,2,18,18,18,18,0) vs q32 = 0: (2,2,2,18,18,18,0), one slot 18 vs 2 = a factor 9 = N_fam^2 (the compiler twin carries an EXTRA 3^2 of divisibility); the ladder state separates identically while trace/unit are blind: the bit is visible to the RP form ONLY through anchor-coupled states; trace discriminant -6561 = -81^2 on both twins; q32 = 1 collapses to a 5-dim order (v568 reproduced); and the blindness is ROBUST under the compiler's own flow -- the V-semigroup-deformed kernels K_n = omega(theta(x) V^n y), n = 1..3, keep identical inertia on both twins for every state (the deformation moves the inertia, never differentially): a dynamical selection needs genuine interaction, not a semigroup weighting. THE GIBBS PENCIL SEPARATES (the first one-parameter dynamical distinction): along the anchor-Hamiltonian Gibbs family omega_g ~ tr(e^{-gH} .), H = I + Pi_dom (v566), det(K_0 + s K_Pi) factors (6561/4096)(s^2+16s+16)^2(15s^2-16s-16) on the compiler twin vs -(6561/4096)(s^2+16s+16)^3 on the alt (6561 = 81^2) -- the compiler's extra factor has its root s* = (8 - 4 sqrt 19)/15 INSIDE the physical Gibbs range: the compiler twin's kernel degenerates at a finite critical temperature, the alt twin's never does (a rank-drop marker, honest: not a positivity selection) -- and THE CRITICAL MODE IS IDENTIFIED: the null space at s* is one-dimensional, rank 1, supported entirely on the seam row (the radical / A2-Ext-carrier row of v566 S5), fixes the seam line (X e3 = e3), is a fixed point of the sheet flow (V X = X) and is annihilated by the dominant projector: the Gibbs criticality lives exactly in the Ext-carrier slot -- the typed resonance with the alignment bit deepens to a SHARED CARRIER; and the half-sided restrictions stay blind (Levi (3,2) / radical (1,1) inertia identical on both twins across the Gibbs range): the RP-positivity selection resists every kinematic and half-sided design tried -- what remains is the genuinely INTERACTING construction (the v534 FK-quartet analogue), typed open. NAMED LIMITS: no physical selection claimed; the dynamical/interacting RP question stays the open RP-slot; the alignment-bit resonance stays typed; no marker moves. [2026-07-31 V: the interacting follow-up is executed -- v584: rank-one Ext-source coupling separates the twins directionally in the unit box (exact thresholds, pencil-factor numerators, positive-coupling healing of exactly the compiler twin); see DIAMOND.BIT.EXTSOURCE.01.]
Recorded status: Identity [E] (exact integral/spectral computations; honest negative on the kinematic selection; RP-slot open)
Dependencies, status detail and source
Canonical status: ANCHOR-VISIBLE: spectrally blind (same inertia, never PSD), integrally separated exactly through anchor/ladder states (Smith slot 18 vs 2, factor 9 = N_fam^2); trace discriminant -81^2; 11/11 checks (half-sided restrictions blind too; the interacting construction is the one remaining RP route) (V-flow robustness; Gibbs pencil: compiler Gibbs-critical at s* = (8-4sqrt19)/15, alt never; critical mode = seam-row flow-fixed direction, the Ext-carrier slot); dynamical RP-slot open
Dependencies: DIAMOND.BIT.SELECT.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v10; v23; v534; v568
Scripts: v572_rp_bit_form.py
Location: tfpt_1;tfpt_2
Original CSV, line 609Original ledger status
QGEO.WINDING.SELF.01
Stage B of the No-Carrier-Reconstruction audit, first narrowing (one module, 11 checks, ~6 s; discovery probe stageb_winding_probe.py 12/12, verdict FORCED): the involution TYPE is killed by a rank theorem -- any involution with a two-dimensional positive eigenspace gives V' = Q P_- of rank <= 1 (char poly x^2 (x - t) identically in Q), so the binary trace code Spec V = {0,1,2} (v566) is unreachable: only the corpus type carries the self-code; (E) BOUNDS the winding normalisation (the winding demand forces the first column to (N,N,N), so q11 = N must itself be an A3 exponent -- N in {4,5,6} give EMPTY classes at the declared B = 6); the ladder self-consistency (v568) is ROBUST -- for every N in {1,2,3} it selects EXACTLY ONE parabolic member (28/16/14 -> 1/1/1); and the saturation self-consistency sat = N^4 (v567's (X) read as a self-consistency) holds ONLY at N = 3 (N=1: 32 != 1; N=2: 512 != 16; N=3: 81 = 3^4): the DOUBLE self-consistency pins (N, Q) = (3, compiler Q) uniquely across the whole N-ablated family -- N_fam = 3 relocates from declared input to consistency-forced within the reconstruction; equivalently the V-spectra {0,2,3}/{0,1,3}/{0,1,2}: only N = 3 carries the binary trace code p_n = 2 + 2^n. THE HONEST DECOMPOSITION (spectrum ablation): freeing the exponent spectrum itself, the surviving double-selfconsistent family is EXACTLY spectrum = {1,2,N} with shape Q(N) = [[N,1,0],[N,2,0],[N,2,1]] (the full 35-spectrum scan in the probe finds no other) -- the self-consistency forces the SHEET spectrum {1,2} (binary trace code) and the compiler SHAPE, and the A3 exponent demand (E) contributes exactly ONE datum, N = 3. REDUNDANCY -- (M) AND (L) FALL TOO: relative to the self-code demands (parabolic + ladder + sat = 81) the monodromy budget and the seam-line demand are REDUNDANT at the declared bound (dropping both widens the class to 1274 members and the self-code still picks exactly the compiler Q): the declared geometric input list shrinks to (S)-existence, (W)-form and (E) -- and THE MAXIMAL ABLATION drops (E) too: (W: column 3(1,1,1)) + integrality + the self-code package alone pin the compiler Q uniquely (29 ladder candidates, one hit): what stays declared is the involution EXISTENCE, the winding FORM and N_fam = 3 (independently anchored in the E8 glue, v1: 240 = 16 x 5 x 3); and the winding FORM's content is located exactly: dropping it entirely leaves a FOUR-element column family {(0,1,1),(0,3,3),(3,3,3),(5,1,5)} of which exactly ONE is uniform -- the compiler: the winding demand is precisely the UNIFORMITY of the column (1 of 4), a genuine declared input with sharply located content. NAMED LIMITS: (E) stays DECLARED ((M)/(L) relocate to redundant-given-the-self-code at B = 6) geometric inputs -- deriving them from mu4/D4/H^1 geometry is exactly the remaining GATE.QGEO.01 (narrowed by two items, NOT closed); NO P2 reduction, AX.P2.01/ANCHOR.GEN.01 markers untouched. [2026-08-01: the involution-existence input is DERIVED -- v590: rigidity (Z/2)^3, narrowing to a unique explicit S0 via the v574 type theorem plus two named demands; GATE.QGEO.01 narrows further; see QGEO.INVOL.01.]
Recorded status: Identity [E] + typed gate [C] (rank theorem symbolic; enumerations at declared bound; the remaining gate named)
Dependencies, status detail and source
Canonical status: FORCED: type killed by rank theorem; N <= 3 by (E); double self-consistency (ladder + sat = N^4) pins (N, Q) = (3, compiler Q) uniquely; 11/11 checks; (E)/(M)/(L) ALL redundant, winding form = the uniformity choice (1 of 4) at B = 6 -- the reconstruction needs only involution existence, winding form with N_fam = 3 (E8-glue) and the self-code; gate narrowed by FIVE items; GATE.QGEO.01 narrowed by two items, not closed
Dependencies: QGEO.QRECON.01; DIAMOND.BIT.SELECT.01; DIAMOND.PARA.SELFCODE.01; FLAV.QGEO.01; v567; v568
Scripts: v574_stageb_winding.py
Location: tfpt_1;tfpt_2
Original CSV, line 610Original ledger status
FTR.CRCONT.01
The CONTINUOUS half of the preregistered cross-ratio check, executed against the in-repo solver dynamics (one module, 7 checks, <1 s; discovery probe cr_continuous_uniflow_probe.py 7/7, verdict UNIFLOW-DEGENERATE): the frozen protocol exports -- the fixed-point multipliers of the native contractions, recomputed from primitives (F_pole = e^{-Delta} = (2/3)^6 via the v99 generator; F_Boltzmann = the SAME contraction by the v425 single-flow theorem; F_QCD b3 = -7; F_relic adiabatic 1) -- form the quadruple {(2/3)^6, (2/3)^6, 1, -7} with a FORCED repetition, and a repeated quadruple gives CR in {0, 1, oo} in every corner assignment (theorem + all 24 permutations): the continuous half can never pass natively. THE CLOSURE: it closes onto v425 -- four independent projective solver states do not exist natively; the only four independent external data are the dimensionful anchors (v_geo, M_R, C_p, theta_i), [O] walls, not canonical projective states; the meaningful CR content was and remains the discrete half (FTR.CRDISC.01, y_F = 53 forced). A future continuous test would need anchor-level exports from genuinely external runs -- outside the native scope, stated honestly; contract markers unchanged. [2026-07-31 II: the jet-level successor is executed natively by v578 (MOEBIUS-IN-OWN-CLOCK: no shared wall-time Schwarzian, each instance projective in its own clock); see FTR.JETS.01.]
Recorded status: Identity [E] + typed closure [C] (exports recomputed; degeneracy theorem exact; the closure onto v425 typed)
Dependencies, status detail and source
Canonical status: UNIFLOW-DEGENERATE: exports {(2/3)^6,(2/3)^6,1,-7} forced degenerate by the single-flow theorem; CR in {0,1,oo} in every assignment; the continuous half closes onto v425; 7/7 checks; anchors stay [O]
Dependencies: FTR.CRDISC.01; CONTRACT.F.01; DYN.TRANSFER.UNIVERSAL.01; v99; v159; v320; v425; v571
Scripts: v575_cr_continuous_uniflow.py
Location: tfpt_research_contracts
Original CSV, line 611Original ledger status
FTR.JETS.01
The jet/Schwarzian contract, executed natively (one module, 6 checks, <1 s; discovery probe native_jets_probe.py 6/6, verdict MOEBIUS-IN-OWN-CLOCK; the second review's Priority 6): the exact v99 Koide trajectory solves the generator identically and has wall-time Schwarzian {q,t} = -Delta^2/2 = -18 log(3/2)^2 exactly (constant, trajectory-independent; F_Boltzmann carries the SAME value by the v425 single-flow theorem); in its own exponential clock u = e^{Delta t} the flow is Moebius (Schwarzian 0); the 1-loop QCD flow is Moebius in its OWN RG time already (Schwarzian 0; b0 = 7 = -b3); the relic instance is adiabatic-constant (jets degenerate). The native wall-time quadruple {-Delta^2/2, -Delta^2/2, 0, degenerate} shares NO Schwarzian, and the discrepancy is exactly one exponential clock change: each instance is PROJECTIVE IN ITS OWN CLOCK -- the shared structure is Moebius-in-own-clock, the obstruction is the clock, and the clocks are anchored by the external data (v_geo, M_R, alpha_s(M_Z), theta_i). A future joint jet test needs external anchor-level jets, exactly as v575 fenced; contract markers unchanged.
Recorded status: Identity [E] + typed closure [C] (exact trajectories and Schwarzians; the joint test external)
Dependencies, status detail and source
Canonical status: MOEBIUS-IN-OWN-CLOCK: seam {q,t} = -Delta^2/2 (own clock 0), QCD Schwarzian 0 in own time, relic degenerate; no shared wall-time jet invariant; clock = the obstruction; 6/6 checks
Dependencies: FTR.CRCONT.01; FTR.CRDISC.01; DYN.TRANSFER.UNIVERSAL.01; v99; v159; v425; v575
Scripts: v578_native_jets.py
Location: tfpt_research_contracts
Original CSV, line 612Original ledger status
DIAMOND.BIT.EXTSOURCE.01
The interacting follow-up of v572, executed -- RANK-ONE EXT-SOURCE COUPLING ON THE GIBBS PENCIL SEPARATES THE 81-TWINS DIRECTIONALLY INSIDE THE UNIT COUPLING BOX (one module, 8 checks, ~1 s; discovery probe ext_source_response_probe.py 8/8, verdict UNIT-BOX-SEPARATED): the interacting kernel K(s,J) = K0 + s K_Pi + J v v^T (v from the shared Ext arrows E31/E32 of the radical row) has EXACT rational flip thresholds J_c(s) = -1/(v^T K(s)^{-1} v) whose numerators are exactly the v572 pencil factors (compiler: 15s^2-16s-16; alt: s^2+16s+16); numerically verified (16 flip events, one eigenvalue negative -> positive each). THE SEPARATION: in the crossed phase s in (-1, s*] the compiler twin is healed by positive coupling J_c <= 15/16 < 1 through either arrow; the alt twin cannot be flipped by any positive coupling in the open unit box (E31 threshold >= 1 with equality exactly at s = 0, E32 threshold negative throughout): positive Ext coupling inside |J| < 1 selects exactly the compiler twin. Exact anchors: J_c(-1) = 15/16; alt J_c(0) = 1 and -1; both compiler thresholds vanish exactly at s*. The direction is strictly positive (negative coupling never heals) -- consistent with the v534 toy RP selection of positive seam coupling. Observed structure: the compiler E31 response pole sits on the golden polynomial s^2 - s - 1 (s = -1/phi = -0.618, 0.011 above s* = -0.629); typed as observed, no claim built on it. Mechanism: the v572 critical mode is Ext-visible (pairings 1/4 and 1/2 + sqrt(19)/4), hence J_c(s*) = 0 exactly. HONEST TYPING: a RESPONSE-level selection (inertia restoration in a declared coupling box), NOT an OS/RP positivity theorem -- no kernel is ever PSD; the alignment-bit contract stays open.
Recorded status: Exact [E] + typed consistency [C] (thresholds closed rational; verified numerically; response level only)
Dependencies, status detail and source
Canonical status: UNIT-BOX-SEPARATED: positive Ext coupling inside |J| < 1 heals exactly the compiler twin (J_c <= 15/16, thresholds exact with v572 pencil-factor numerators); alt twin rigid; golden-polynomial response pole observed; response-level, contract stays open; 8/8 checks
Dependencies: DIAMOND.BIT.RPFORM.01; DIAMOND.BIT.SELECT.01; QGEO.QRECON.01; v534; v568; v572
Scripts: v584_ext_source_response.py
Location: tfpt_1;tfpt_2
Original CSV, line 613Original ledger status
QGEO.INVOL.01
The sheet-involution existence question of GATE.QGEO.01, narrowed to a two-demand selection with UNCONDITIONAL existence (one module, 7 checks, <1 s; discovery probe involution_existence_probe.py 7/7, verdict EXISTENCE-UNCONDITIONAL): RIGIDITY -- the integral V-commuting involutions are exactly the spectral sign group (Z/2)^3: all eight sign choices on V's eigenspaces {0,1,2} are integer matrices, genuine involutions, commute with V and stabilize the integral order Z<U,V>, on BOTH twins (nontrivial: generic integral operators do not have integral spectral projectors). THE NARROWING CHAIN: (i) the v574 rank theorem (two-dim-positive involutions cannot carry the binary trace code) kills the trace +1 half, 6 -> 3, all remaining of exact Sigma-signature (1,-1,-1); (ii) the seam-reversal demand S e3 = -e3 narrows 3 -> 2; (iii) the zero-mode-fixing demand (+1 on ker V, flipping exactly the dynamical sheet pair {1,2}) singles out S0 = [[1,-1,0],[0,-1,0],[0,0,-1]] uniquely -- explicit, integral, order-stabilizing: EXISTENCE IS UNCONDITIONAL. HONEST SCOPE: the corpus RP reflection Sigma = diag(1,-1,-1) (an anti-automorphism) does NOT commute with V -- a different structure with a different role; the alt twin admits the same classification (the involution does not separate the twins, consistent with v567/v572). RELOCATION: GATE.QGEO.01's involution-existence input is DERIVED; the residual freedom is the finite two-demand selection, machine-checked; deriving the two demands from the continuum geometry (mu4/D4/H1) is the remaining open [C] of the gate -- named, not claimed. [2026-08-01 V: the grading shortcut is falsified -- NO V-commuting involution implements the corpus mu4 grading S Q S = Sigma Q Sigma (all six fail); Sigma and S0 are genuinely different structures whose product is the infinite-order shear [[1,-1,0],[0,1,0],[0,0,1]]: the two v590 demands stay independent; the continuum derivation remains open.] [2026-08-01 X: a live geometric candidate -- v597: the mu3-cover y^3 = x^4-1 hits four compiler fingerprints (rank-3 Eisenstein, projective D4, (Z/3)^4 = 81 cusp classes, Lorentz (1,2)); see QGEO.COVER.01.] [2026-08-01 XIX: the seam-reversal demand is now GEOMETRIC -- v603: on the mu3-cover the seam line is real (R-fixed) and the two-demand involution reverses it exactly; the zero-mode demand is built into the curve-side classification (v601); the continuum derivation of the demands has its first geometric anchor; see QGEO.SEAMMARKS.01.] [2026-08-01 XXV: BOTH demands now have canonical cover carriers -- v603: the seam-reversal demand is 'reverse the R-real line'; v607: the zero-mode demand is 'fix the only unlabeled mode' (seam = R-real, top = deck-eigen, zero = unlabeled); typed [C] readings, the continuum derivation of the demands from the raw seam stays the gate residual; see QGEO.SELRULE.01.]
Recorded status: Exact [E] + typed relocation [C] (exact sympy; demands declared; continuum derivation open)
Dependencies, status detail and source
Canonical status: EXISTENCE-UNCONDITIONAL: rigidity (Z/2)^3 on both twins; narrowing 6 -> 3 -> 2 -> 1 via v574 type theorem + seam reversal + zero-mode fixing; S0 explicit with Sigma-signature; the two demands' continuum derivation stays open; 7/7 checks
Dependencies: QGEO.WINDING.SELF.01; QGEO.QRECON.01; DIAMOND.PARA.SELFCODE.01; v566; v567; v574
Scripts: v590_involution_existence.py
Location: tfpt_1;tfpt_2
Original CSV, line 614Original ledger status
QGEO.COVER.01
Strategy I/II, first executed step -- the mu3-cover of P^1 branched at mu4 (the curve y^3 = x^4 - 1) carries FOUR independent compiler fingerprints (one module, 6 checks, <1 s; discovery probe mu4_cover_model_probe.py 6/6, verdict FINGERPRINTS-HIT): (1) genus 3 with five cusps (mu4 + infinity; monodromies (w,w,w,w,w^2) multiplying to 1) and Chevalley-Weil H^{1,0} = (1,2) per character -- H_1 is a RANK-3 module over the Eisenstein integers, the compiler's 3-dimensional integral frame; (2) the D4 rotation closes PROJECTIVELY VIA THE DECK: in the reduced Burau model at t = omega (braid relations machine-verified) the rotation braid satisfies r^4 = omega x identity and r^12 = 1; (3) THE TORSION: the cusp-class group is (Z/3)^5 modulo one global relation = (Z/3)^4 of order 81 = 3^4 -- the v566 saturation index with exactly the per-mark Z/3 mechanism the self-code program asked for (four free local classes); det_Z(1 - deck) = 27 complementarily; (4) THE SIGNATURE: the invariant hermitian form of the Burau module is unique up to scale with eigenvalues {-2, 2-2sqrt2, 2+2sqrt2} -- LORENTZ SIGNATURE (1,2), matching the Chevalley-Weil Hodge numbers AND the rank-3 polarization signature of Paper II: Strategy II's first prediction (RP positivity as Hodge/Riemann positivity) confirmed at the form level. First operator pass: the single-cusp Gram has all-integer spectrum {0,0,2}; the compiler V-spectrum {0,1,2} is not yet located -- the operator dictionary (which canonical homology operator is Q/V) and the real-structure/RP map to the v572 Gram data are the NAMED NEXT STEPS. Kill criteria live (Smith/spectral mismatch for I; signature mismatch for II -- not triggered). GATE.QGEO does NOT move; no marker changes. [2026-08-01 XI, the dictionary's first entry: U IS LOCATED -- the cusp Gram (1-T2)(1-T2)^dagger has characteristic polynomial x^2(x-3) = char(U) exactly (the winding operator as a cusp Gram; fifth fingerprint); additional exact structure: opposite-twist products (1-T1)(1-T3) are NILPOTENT (char x^3), and G1+G3 has spectrum {0,1,3}; V/H/Q not located in the scanned canonical ranges (products/sums/omega-twists of length <= 2) -- the V-dictionary likely requires the parabolic flag (the seam-cusp filtration), named as the next step.] [2026-08-01 XIII: dictionary round 2 -- v598: THREE entries now exact (U ~ G2; Q_- ~ G1-G3, the sheet coupling as the diamond-diagonal Gram difference; the binary ladder ~ 1+G1+G3); the grading half absent from 7296 canonical combinations; see QGEO.DICT.01.] [2026-08-01 XV: the real structure is constructed and the invariant form is pinned -- v599: R unique with R^2 = +1, det J = 8 (a sheet-diamond determinant list member), Lorentz (1,2) confirmed exactly; see QGEO.REAL.01.] [2026-08-01 XVI: the operator dictionary lands at ALGEBRA level -- v600: the compiler pair (U, V) realized on the cover with the exact v566 saturation structure (index 81, divisors (1,1,1,3,3,3,3)); the [P]-model of GATE.QGEO has an explicit construction; canonicity and the two-demand derivation stay open; see QGEO.EMBED.01.] [2026-08-01 XXI: the torsion picture completes -- v604: 81 = 3^4 localizes at the finite marks (divisor classes; per-frame word order) and 27 = 3^3 at infinity (monodromy-defect cokernel = equivariant-order cokernel = det(1-deck)); see QGEO.EQORDER.01.] [2026-08-01 XXVIII: the curve's wider landscape is mapped -- v610: the Jacobian splits into a Q(zeta_12)-CM surface (the field of deck AND marks) times the Q(omega)-CM elliptic quotient, point-count-confirmed; the integer-spectrum census honestly relativizes single-char hits (the anchor class is the most frequent nontrivial family; the joint structure is the distinction); see QGEO.LANDSCAPE.01.] [2026-08-01 XXXV: the bedrock direction reverses -- v617: the conformal seam axioms FORCE the mu3-cover (Z4-Mobius rigidity, uniform weights j in {1,2}, genus 3, seam reflections = the v599 pattern); the QGEO.SYM.01 residue narrows to the cyclic-3 choice and the physical<->conformal seam identification; see QGEO.SEAMFORCE.01.]
Recorded status: Exact [E] fingerprints + typed candidate [C] (Burau convention validated; Q-identification open)
Dependencies, status detail and source
Canonical status: FINGERPRINTS-HIT: rank-3 Eisenstein module; projective D4 via deck (r^4 = omega); cusp-class group (Z/3)^4 = 81 with the per-mark mechanism; Lorentz (1,2) polarization (Chevalley-Weil + Paper II simultaneously); single-cusp Gram {0,0,2}; Q/V dictionary and RP map named open; 6/6 checks
Dependencies: QGEO.INVOL.01; QGEO.WINDING.SELF.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v574; v590
Scripts: v597_mu4_cover_model.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 615Original ledger status
QGEO.DICT.01
The cover dictionary, round 2 -- THREE compiler objects located as canonical cusp-Gram combinations on the mu3-cover (one module, 4 checks, <1 s; discovery probe cover_dictionary_probe.py 4/4, verdict DICT-THREE-ENTRIES): with T_1..T_4 the D4-conjugate cusp twists and G_k their Grams -- U ~ G_2 (char x^2(x-3), the winding operator as a single-cusp Gram; v597 entry); Q_- ~ G_1 - G_3 (char x(x^2-3)): THE MU4 SHEET COUPLING IS THE DIFFERENCE OF THE OPPOSITE CUSP GRAMS -- the diamond diagonal; ladder ~ 1 + G_1 + G_3 (char (x-1)(x-2)(x-4)): the binary AnchorLadder (1,2,4) = (2^0,2^1,2^2) of v568 is identity plus the diagonal sum. Supporting exact structure: opposite-twist products (1-T_1)(1-T_3) are nilpotent (cube zero), Q_-^2 has char x(x-3)^2. THE NEGATIVES QUANTIFIED: Q_+ ({1,2,3}), V ({0,1,2}), H ({1,1,2}) and Q itself are NOT contained in 7296 scanned canonical combinations (the integer Gram lattice a + bG_1 + cG_2 + dG_3 + eG_4 and its non-hermitian (1-T_i)(1-T_j) extension; numeric prefilter, exact confirmation; 0 matches) -- the grading half needs a generator outside the finite-cusp twist algebra, with the anti-holomorphic D4 reflections (the real structure) as the named suspect. The Strategy-I kill criterion is NOT triggered (the canonical space is not exhausted). GATE.QGEO does not move; no marker changes. [2026-08-01 XV: the named suspect is CONSTRUCTED -- v599: the real structure exists and is unique up to scale (R^2 = +1); its grading functor closes all three gaps at char level (V, H with the anchor spectrum, Q_+); the joint algebra embedding stays open (quantified negative); see QGEO.REAL.01.] [2026-08-01 XVI: the dictionary completes at algebra level -- v600: a Gamma-even degree-2 pair realizes (U, V) jointly (conjugation exact, index 81); canonicity open; see QGEO.EMBED.01.] [2026-08-01 XXVIII: retroactive typing confirmed -- v610's census shows single integer spectra are common on the cover; this module's char-level entries were correctly typed candidates, upgraded to the joint embedding only by v600; see QGEO.LANDSCAPE.01.]
Recorded status: Exact [E] entries + quantified negatives (numeric prefilter, symbolic confirmation)
Dependencies, status detail and source
Canonical status: DICT-THREE-ENTRIES: U ~ G2, Q_- ~ G1-G3 (diamond diagonal), ladder ~ 1+G1+G3 -- all exact; grading half absent from 7296 canonical combinations (kill not triggered; real structure the named suspect); 4/4 checks
Dependencies: QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v566; v568; v597
Scripts: v598_cover_dictionary.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 616Original ledger status
QGEO.REAL.01
The real structure on the mu3-cover -- the v598 suspect constructed (one module, 18 checks, ~3 s; discovery probe real_structure_probe.py 18/18, verdict REAL-STRUCTURE-LANDED). EXISTENCE AND UNIQUENESS [E]: complex conjugation on the base fixes the punctures {1,-1} and swaps {i,-i} (an orientation-reversing D4 reflection); it must act semi-linearly, R(v) = M conj(v), sending each cusp twist to the INVERSE of its reflected partner, R T_k R^-1 = T_(5-k)^-1. The intertwining system has a 1-DIMENSIONAL solution space: R exists and is unique up to scale; the canonical normalization satisfies R^2 = +1 exactly (a genuine real structure, not quaternionic); all four twist intertwinings hold exactly. THE GRADING FUNCTOR [E]: Gamma(A) = M conj(A) M^-1 (the curve analogue of the compiler grading A -> Sigma A Sigma) closes the three v598 dictionary gaps at char level: V ~ G2 + G3 - Gamma(G3) with char x(x-1)(x-2); H ~ 1 + G2 - Gamma(G3) with char (x-1)^2(x-2) -- THE ANCHOR SPECTRUM (1,1,2); Q_+ ~ 1 + G2 + G3 - Gamma(G3) with char (x-1)(x-2)(x-3). THE CONSTANTS [E]: the Gamma-even/odd cusp elements E_k = G_k + Gamma(G_k), O_k = G_k - Gamma(G_k) carry the compiler constants: E2 spectrum {0,1,5} (the carrier g_car = 5), E4 {0,3,9} = {0, N_fam, N_fam^2}, O3 nilpotent (char x^3), O2 char x(x^2-5), O4 char x(x^2-27). THE JOINT NEGATIVE [X, quantified]: char-matching is NOT yet an algebra embedding -- in the +-1 Gamma-extended integer lattice (8 generators) there are exactly 4 U-candidates and 6 V-candidates, and ZERO pairs satisfy the joint compiler relations tr(UV) = 3 and char(U+V) = (x-1)(x^2-5x+3); the natural pair (G2, G2+G3-Gamma(G3)) gives char x(x^2-6x+4), an exact documented mismatch. The flag-adapted simultaneous-conjugation search remains the named next step; the Strategy-I kill criterion is NOT triggered. THE LORENTZ FORM [E, Strategy II]: the invariant hermitian form J (T_k^dagger J T_k = J for all four twists) is unique up to real scale, integral over Z[omega], with det J = 8 -- A MEMBER OF THE SHEET-DIAMOND DETERMINANT LIST (3,4,8,14,20,32) -- and char (x+2)(x^2-4x-4): LORENTZ SIGNATURE (1,2), matching the Paper-II rank-3 polarization signature AND the Chevalley-Weil Hodge prediction (h^{1,0}, h^{0,1}) of Strategy II simultaneously. GATE.QGEO does not move; no marker changes. [2026-08-01 XVI: the degree-2 Gamma-even sector closes the joint gap -- v600: the compiler pair (U, V) is realized (GL3-conjugation solved exactly, saturation index 81 on the R-fixed lattice); the v599 joint negative is thereby located precisely at degree 1; see QGEO.EMBED.01.]
Recorded status: Exact [E] existence/uniqueness/char identities + [X] quantified joint negative + [C] the TFPT readings (exact sympy; numeric prefilter with symbolic confirmation for the scan)
Dependencies, status detail and source
Canonical status: REAL-STRUCTURE-LANDED: R unique (R^2 = +1), four intertwinings exact; Gamma closes V/H/Q_+ at char level (H carries the anchor spectrum); constants 5/9/27 in the Gamma-decomposition; joint embedding open (4x6 candidates, 0 joint pairs); det J = 8, Lorentz (1,2); 18/18 checks
Dependencies: QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v566; v597; v598
Scripts: v599_real_structure.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 617Original ledger status
QGEO.EMBED.01
The joint embedding -- the compiler pair (U, V) is realized on the mu3-cover (one module, 15 checks, ~4 s; discovery probe joint_embedding_probe.py 15/15, verdict JOINT-EMBEDDING-LANDED). THE PAIR [E]: in the Gamma-even DEGREE-2 sector (the v599 grading functor applied to cusp-Gram products), U* = -2E1 + (G1 Gamma(G1) + Gamma(G1) G1) + 3 and V* = -E3 + (1/2)(G2 Gamma(G2) + Gamma(G2) G2) + 1 are both integral over Z[omega], Gamma-even, with char U* = x^2(x-3), char V* = x(x-1)(x-2), char(U*+V*) = (x-1)(x^2-5x+3) = char Q, and the compiler joint traces tr(U*V*) = 3, tr(U*V*^2) = 6. THE ALGEBRA EMBEDDING [E]: the seven words {1, U*, V*, U*V*, V*U*, V*^2, V*U*V*} are independent (rank 7 = the full 2|1 parabolic) and the simultaneous-conjugation system is EXACTLY solved: an explicit invertible g with g U* g^-1 = U_c and g V* g^-1 = V_c -- the v566 parabolic self-code algebra lives on the cover. THE INTEGRAL FORM [E]: the real structure R as an integer matrix on Z^6 has R^2 = 1; its fixed lattice has the explicit saturated basis (1,2,2,0,0,0), (0,1,1,-1,0,0), (0,0,1,1,-1,1) (SNF certificate (1,1,1)); U*, V* restrict to INTEGER 3x3 matrices with the same chars. THE 81 [E, the hard test]: the Z-order of the seven restricted words has Smith elementary divisors (1,1,1,3,3,3,3) -- saturation index 81 = 3^4, EXACTLY the v566 compiler self-code index with the same divisor pattern. HONEST LIMITS: must-break control passes (the v599 degree-1 pair fails the joint criterion); CANONICITY IS OPEN and typed open -- the degree-2 scan admits many joint pairs, so this is an EXISTENCE claim, not uniqueness; GATE.QGEO does not move until the canonicity round and the two-demand continuum derivation land; no marker changes. [2026-08-01 XVII: the canonicity question is answered -- v601: the J-adapted projector presentation is D4-equivariant by construction (mark choice = winding choice) and realizes the DUAL parabolic exactly; the Klein group lives integrally on the curve; see QGEO.CANON.01.] [2026-08-01 XVIII: the integer core is Lean-certified -- FORM.QGEO.04 in TfptCarrier/CoverEmbedding.lean: W = T*S, Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P, S-primitivity via unimodular minor (determinant-free, kernel decide; lake build green).]
Recorded status: Exact [E] existence realization (integrality, conjugation, saturation index all exact sympy) + typed open canonicity [O]; NO marker moves
Dependencies, status detail and source
Canonical status: JOINT-EMBEDDING-LANDED: (U*, V*) Gamma-even degree-2, Z[omega]-integral, chars + joint traces compiler-exact, word rank 7, conjugation solved, integer on saturated Fix(R), Smith (1,1,1,3,3,3,3) index 81 = v566; canonicity open; 15/15 checks
Dependencies: QGEO.REAL.01; QGEO.DICT.01; QGEO.COVER.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; GATE.QGEO.01; v566; v597; v598; v599
Scripts: v600_joint_embedding.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 618Original ledger status
QGEO.CANON.01
The canonical equivariant realization (one module, 14 checks, ~12 s; discovery probe equivariant_dual_probe.py 14/14, verdict EQUIVARIANT-DUAL-LANDED). THE REPAIR: the naive cusp-Gram presentation of v600 is not D4-equivariant because the standard hermitian dagger is basis-dependent -- the Burau image preserves J; the canonical building blocks are the J-ADAPTED cusp projectors GJ_k = (1-T_k) J^-1 (1-T_k)^dagger J. THE PROJECTORS [E]: all four are idempotent rank-1 J-self-adjoint projectors (char x^2(x-1)) rotating exactly under D4 (r GJ_k r^-1 = GJ_(k+1)); opposite cusps J-orthogonal (tr(GJ1 GJ3) = 0), neighbors overlap with trace 1. THE EQUIVARIANT PAIR [E]: U_k = 3 GJ_k, V_k = 1 + GJ_(k-1) - GJ_k GJ_(k+1) -- Z[omega]-integral, compiler spectra, compiler joint data (char(U+V) = charQ, tr(UV) = 3, tr(UV^2) = 6), seven-word rank 7, and D4-EQUIVARIANT BY CONSTRUCTION (r^k-conjugation = mark shift, verified for all marks): the mark choice IS the compiler's winding choice. THE DUALITY [E]: the equivariant pair is NOT conjugate to (U_c, V_c) (the intertwiner space is 1-dimensional with identically singular solutions -- must-fail documented) but EXACTLY conjugate to the transposed pair (U_c^T, V_c^T): the equivariant presentation realizes the DUAL (plane-stabilizer) parabolic, the v600 Gamma-even pair the line-stabilizer -- the cover carries BOTH parabolic types, exchanged by duality. THE KLEIN GROUP ON THE CURVE [E]: all eight spectral-sign involutions of V_1 are Z[omega]-integral (v590 rigidity reproduced); the two v590 demands select the explicit integral S_0 (trace -1, commuting with V), and A B = S_0 closes the Klein four-group on the cover. HONEST OPENS [C]: the naive J-word Gram has inertia (3,2,2), not the corpus v572 RP pattern (4,0,3) -- the Hodge/RP comparison needs the Weil-twisted polarization (named); the seam-line geometric identification stays open. GATE.QGEO does not move; no marker changes. [2026-08-01 XVIII: the duality mechanism is identified -- v602: the J-adjoint exchanges the two parabolic types exactly ((U1, V1^J) ~ compiler pair); the dictionary completes in the J-projector language; Weil positivity holds exactly; see QGEO.DUALFORMS.01.]
Recorded status: Exact [E] equivariance/duality/Klein identities + [C] typed opens (exact sympy; one numeric inertia count of an exact hermitian matrix)
Dependencies, status detail and source
Canonical status: EQUIVARIANT-DUAL-LANDED: GJ_k idempotent equivariant projectors; pair family compiler-exact and equivariant by construction; conjugate to the TRANSPOSED compiler pair (dual parabolic); Klein group integral on the curve (8/8, S_0 explicit, A B = S_0); J-word-Gram inertia (3,2,2) typed open vs v572; 14/14 checks
Dependencies: QGEO.EMBED.01; QGEO.REAL.01; QGEO.INVOL.01; DIAMOND.PARA.SELFCODE.01; v590; v597; v599; v600
Scripts: v601_equivariant_dual.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 619Original ledger status
QGEO.DUALFORMS.01
The duality-and-forms round (one module, 17 checks, ~8 s; discovery probe duality_forms_probe.py 17/17, verdict DUALITY-FORMS-LANDED). THE DUALITY MECHANISM [E]: U_1 = 3 GJ_1 is J-self-adjoint; the J-adjoint of V_1 = 1 + GJ_4 - GJ_1 GJ_2 is the exact product reversal 1 + GJ_4 - GJ_2 GJ_1; and (U_1, V_1^J) is EXACTLY conjugate to the compiler line-stabilizer pair while (U_1, V_1) is conjugate to the transposed pair (v601) -- the J-adjoint exchanges the two parabolic types inside ONE equivariant structure: the cover's realization is J-self-dual as a pair of pairs. DICTIONARY COMPLETION [E]: H = 1 + V(V-1)/2 has char (x-1)^2(x-2) -- the corpus anchor formula holds VERBATIM on the curve; Q = U + V has char (x-1)(x^2-5x+3); Q_+ ~ 2 - GJ_1 + GJ_2 GJ_3 has char (x-1)(x-2)(x-3); the binary AnchorLadder ~ 2 - GJ_1 + 2 GJ_2 GJ_3 has char (x-1)(x-2)(x-4); the S_0-grading halves of Q are computed and documented. WEIL POSITIVITY [E, Strategy II at module level]: the Weil operator C from J's spectral split (Hodge numbers (1,2)) gives a POSITIVE DEFINITE twisted form JC with exact eigenvalues {2sqrt2-2, 2, 2+2sqrt2} = |Spec J| -- the Riemann bilinear positivity statement holds on the omega-sheet. THE CANONICAL BILINEAR FORM [E]: B(v,w) = (Rv)^dagger J w has matrix B = M^dagger J, SYMMETRIC and nondegenerate with det B = 8 -- the curve's complex orthogonal structure combining the real structure with the polarization. HONEST NEGATIVES [X/O, quantified]: (i) g^T g is NOT proportional to B -- the corpus transpose structure is not the (R,J)-bilinear form (corpus RP word-Gram inertia (4,3,0) documented); (ii) the naive mark-local cokernel map fails (U_2, U_3 lie outside the seven-word algebra, U_4 inside with integer coefficients) -- the carrier of the mark-local Z/3 mechanism stays open; (iii) the seam-plane normal is explicit but NOT a deck eigenvector -- the seam identification stays open. COMPANION LEAN CERTIFICATE (FORM.QGEO.04): the v600 integer core is certified in TfptCarrier/CoverEmbedding.lean -- compiler relations of (C_U, C_V), the word factorization W = T*S, the Smith certificate P*T = diag(1,1,1,3,3,3,3) with unimodular P (P*Pinv = 1), and S-primitivity via a unimodular minor: determinant-free, kernel decide, lake build green (3375 jobs). GATE.QGEO does not move; no marker changes. [2026-08-01 XIX: the seam question advances -- v603: the seam line is a REAL line of the cover (R-fixed), S_0 reverses it geometrically; the infinity cusp separates in the unreduced module; the first per-mark Z/3 witness lands (E_1, order 3); see QGEO.SEAMMARKS.01.] [2026-08-01 XXII: the corpus-transpose translator question is RESOLVED -- v605: theta_corpus = Jadj o inner(c) exactly, c unique up to scale, 14c integral; c's independent geometric identity is the remaining open; see QGEO.TRANSLATOR.01.]
Recorded status: Exact [E] duality/dictionary/positivity identities + [X/O] quantified negatives and typed opens (exact sympy; two numeric inertia counts of exact matrices) + Lean certificate FORM.QGEO.04
Dependencies, status detail and source
Canonical status: DUALITY-FORMS-LANDED: J-adjoint exchanges the parabolic types (exact conjugations both ways); dictionary complete in J-projector language (anchor formula verbatim); Weil-twisted form positive definite (exact); B = M^dag J symmetric, det 8; corpus-transpose / mark-local / seam questions typed open; Lean FORM.QGEO.04 green; 17/17 checks
Dependencies: QGEO.CANON.01; QGEO.EMBED.01; QGEO.REAL.01; DIAMOND.PARA.SELFCODE.01; v572; v590; v600; v601
Scripts: v602_duality_forms.py
Location: tfpt_1;tfpt_2;tfpt_3;tfpt_research_contracts
Original CSV, line 620Original ledger status
QGEO.SEAMMARKS.01
The seam-and-marks round (one module, 15 checks, ~7 s; discovery probe seam_marks_probe.py 15/15, verdict SEAM-REAL-LANDED). THE SEAM LINE IS REAL [E]: the line-stabilizer pair (U_1, V_1^J) has a UNIQUE common eigenline v with U v = 0 and V v = v -- the compiler seam span(e3), transported; explicitly v = (0, (sqrt3-i)/(sqrt3+i), 1). The real structure FIXES this line (R v ~ v): the seam is a real line of the cover. The two-demand involution REVERSES it (S_0 v = -v): the v590 seam-reversal demand holds GEOMETRICALLY on the curve. Honest scope: S_0 does not commute with R (the sheet involution is not Gamma-even), and the seam's identification with the infinity cusp stays open. THE INFINITY CUSP SEPARATED [E]: the unreduced Burau of B_4 at t = omega (braid relations verified) has invariant column (1,1,1,1) and invariant row (1, omega, conj omega, 1); the full twist r^4 acts with eigenvalues {1 (x1), omega (x3)} -- the infinity direction canonically separated from the omega-sheet. THE FIRST PER-MARK Z/3 WITNESS [E]: E_1 = G_1 + Gamma(G_1) restricts to an INTEGER matrix on the saturated fixed lattice and lies in the seven-word algebra with coefficient-denominator lcm EXACTLY 3 -- an ORDER-THREE class in the word-order cokernel P/O: the v566/v597 per-mark mechanism gets its first exact witness. Honest structure: E_4's class is trivial (integer coefficients), E_2/E_3 lie OUTSIDE the mark-1-anchored algebra -- the naive global map fails; the equivariant per-mark framing (each mark's class in its own rotated frame) is the typed reading. TRANSLATOR NEGATIVES [X, quantified]: the pullback form F_2 = g^T Sigma g is symmetric but not proportional to any of 7 canonical menu forms, and neither B^-1 F_2 nor F_2 B^-1 lies in the word algebra -- the corpus-transpose translator needs a new canonical object, named open. GATE.QGEO does not move; no marker changes. [2026-08-01 XXI: the integral infinity bridge lands -- v604: the 3-torsion localizes at infinity (coker(1-T_inf) = Z^2 + (Z/3)^3, finite cusps torsion-free), the four frames generate a FULL order of index 27 = 3^3, and R is an anti-isometry (J_fix integer, det 72); see QGEO.EQORDER.01.] [2026-08-01 XXII: the seam-infinity identification lands INTEGRALLY -- v605: the seam lift is a NONZERO pure 3-torsion class in coker(1-T_inf) (free coords zero, torsion (2,2,0)); see QGEO.TRANSLATOR.01.] [2026-08-01 XXIV: the zero-mode selection gains its carrier -- v606: the infinity torsion separates all three modes (pure torsion, distinct classes); the canonical '+1' rule is the remaining step; see QGEO.MODESEP.01.]
Recorded status: Exact [E] seam/infinity/witness identities + [X] quantified translator negatives + [C] typed per-mark framing (exact sympy)
Dependencies, status detail and source
Canonical status: SEAM-REAL-LANDED: seam line unique, real (R-fixed), S_0-reversed (v590 demand geometric); infinity cusp separated (unreduced Burau, full twist {1, omega x3}); E_1 order-3 cokernel class (first per-mark witness); E_4 trivial, E_2/E_3 outside; translator negatives quantified; 15/15 checks
Dependencies: QGEO.DUALFORMS.01; QGEO.CANON.01; QGEO.EMBED.01; QGEO.INVOL.01; v590; v600; v601; v602
Scripts: v603_seam_marks.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 621Original ledger status
QGEO.EQORDER.01
The equivariant order and the infinity bridge (one module, 12 checks, ~4 s; discovery probe equivariant_order_probe.py 12/12, verdict EQUIVARIANT-ORDER-LANDED). CORRECTION + THE INTEGRAL FORM [E]: R IS an anti-isometry of the polarization -- the correct compatibility equation M^dagger J M = J^T holds EXACTLY (a v599-diary side remark had tested the wrong equation M conj(J) M^dagger vs J); consequently J restricts to a REAL SYMMETRIC INTEGER form on the saturated fixed lattice: J_fix = [[16,2,4],[2,-2,2],[4,2,-2]], det 72 = 8 x 9 (the polarization determinant times the (1-omega)-covolume factor). THE EQUIVARIANT ORDER IS FULL [E]: the Z-lattice generated by the seven words of ALL FOUR mark frames (the r^k-conjugates of the v601/v602 equivariant pair; all 28 words Z[omega]-integral) has FULL rank 18 in M_3(Z[omega]) with Smith divisors (1^15, 3, 3, 3) -- saturation index 27 = 3^3: one frame gives the 7-dim parabolic of index 81 = 3^4 (v600), all four give the full matrix algebra of index 27. THE 3-TORSION LOCALIZES AT INFINITY [E]: on the unreduced integral module Z[omega]^4, coker(1 - T_inf) = Z^2 + (Z/3)^3 (Smith divisors (1,1,1,3,3,3,0,0), torsion 27) while each FINITE cusp twist is torsion-free (divisors (1,1,0^6)) -- three objects share (Z/3)^3 = 27: the infinity-monodromy cokernel, the equivariant-order cokernel, and det_Z(1-deck) (v597). THE READING [C]: the two compiler constants localize on the cover -- 81 = 3^4 at the FINITE marks (cusp divisor classes, v597; per-frame word order, v600) and 27 = 3^3 at INFINITY (monodromy defect; full equivariant order): the integral infinity <-> order-torsion bridge. TRANSLATOR NEGATIVES EXTENDED [X, quantified]: the corpus RP metric pulled back to the integral fixed lattice is NOT proportional to J_fix or its Klein twists (6 tested), and neither J_fix^-1 F2 nor F2 J_fix^-1 lies in the word algebra -- after the (B, Gamma, Klein, J_fix) sweep the corpus transpose structure resists all single-sheet constructions; the conjugate character sheet (omega-bar) is the named next candidate. GATE.QGEO does not move; no marker changes. [2026-08-01 XXII: the translator is FOUND -- v605: the corpus RP anti-automorphism is the J-adjoint twisted by an explicit inner element c (14c integral; the omega-bar detour was unnecessary); the seam is 3-torsion at infinity; the 81 survives frame-joining; see QGEO.TRANSLATOR.01.] [2026-08-02 XXXIX: the fixed-lattice form J_fix gains an arithmetic identity -- v624: P^T J_det P = J_fix EXACTLY with det P = -6 (J_det the prime-front determinant form): the two Lorentz lattices are the same rational quadratic form, the cover an index-6 sublattice (72/2 = 6^2, 6 = p_2(a)); P-canonicity and the Hodge-chamber question preregistered; see EXTREV.LATTICE.01.]
Recorded status: Exact [E] anti-isometry/order/localization identities + [C] typed localization reading + [X] quantified translator negatives (exact sympy; Smith normal forms over Z)
Dependencies, status detail and source
Canonical status: EQUIVARIANT-ORDER-LANDED: M^dag J M = J^T exact (R anti-isometry); J_fix integer, det 72; four-frame order FULL rank 18, divisors (1^15,3,3,3), index 27 = 3^3; coker(1-T_inf) = Z^2+(Z/3)^3, finite cusps torsion-free; 81 at marks / 27 at infinity; translator negatives extended (omega-bar named); 12/12 checks
Dependencies: QGEO.SEAMMARKS.01; QGEO.DUALFORMS.01; QGEO.EMBED.01; QGEO.COVER.01; v597; v600; v601; v602; v603
Scripts: v604_equivariant_order.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 622Original ledger status
QGEO.TRANSLATOR.01
The translator round (one module, 11 checks, ~11 s; discovery probe translator_probe.py 11/11, verdict TRANSLATOR-LANDED). THE TRANSLATOR EXISTS [E, centerpiece]: pulling the corpus RP anti-automorphism theta(X) = Sigma X^T Sigma back along the compiler conjugation and composing with the J-adjoint gives an algebra automorphism phi = Jadj o theta_pull -- and phi is INNER: the intertwiner space is 1-DIMENSIONAL, its generator c is invertible, and phi(A) = c A c^-1 holds EXACTLY on the pair. Hence THE CORPUS RP STRUCTURE IS THE J-ADJOINT COMPOSED WITH THE INNER TWIST BY c -- the RP anti-automorphism is fully expressed in curve language (polarization + one explicit element); the normalization 14c is Z[omega]-INTEGRAL (denominator 14 = 2 x 7, the parabolic dimension). Honest scope: c is unique up to scale by construction but not Gamma-even and not in the word algebra -- its independent geometric identity is the named open (the earlier single-sheet negatives v603/v604 are thereby RESOLVED: the missing object was inner, not a new form; the omega-bar detour was unnecessary). THE SEAM-INFINITY BRIDGE [E]: in the Smith coordinates of coker(1 - T_inf) = Z^2 + (Z/3)^3, the seam lift (-conj omega, -conj omega, 0, -omega) has ZERO free coordinates and torsion coordinates (2,2,0) mod 3: the seam line is a NONZERO pure 3-torsion class of the infinity monodromy cokernel -- the integral seam <-> infinity identification. THE 81 SURVIVES [E]: inside the frame-1 span the order chain O_1 subset (O_joint cap span_1) subset P_1 has indices 3^5 = 243 and 3^4 = 81: joining all four frames refines each frame's order by exactly 3^5 and the residual saturation defect is EXACTLY the v566 index 81. THE J_FIX DUALITY CERTIFIED [E]: C_V^T J_fix = J_fix C_Vdual with the explicit integer C_Vdual = [[1,0,0],[-2,0,0],[2,1,2]] (char x(x-1)(x-2), dual joint trace 3, char(C_U + C_Vdual) = x(x-2)(x-4)); companion Lean addendum in TfptCarrier/CoverEmbedding.lean (Jfix_symmetric, Jfix_det = 72, CV_dual_certificate, CVdual_cubic, dual_joint_trace; lake build green, 3375 jobs). GATE.QGEO does not move; no marker changes. [2026-08-01 XXIV: the mode separation lands -- v606: all three V-eigenlines are pure 3-torsion at infinity with pairwise distinct classes (the selection carrier R could not provide); c's arithmetic-identity route is closed (cubic field S_3, not abelian); see QGEO.MODESEP.01.] [2026-08-01 XXVII: the geometric identity of c is CLOSED -- v609: c = lambda K_1 d with K_1 = J^-1 conj(F_2)^T M^-1 and d the UNIQUE unimodular Z[omega]-integral Gamma-involutive mirror intertwiner of the realized pair; see QGEO.CFACTOR.01.]
Recorded status: Exact [E] translator/bridge/chain/duality identities + [C] typed opens (exact sympy; Smith transforms over Z; Lean addendum green)
Dependencies, status detail and source
Canonical status: TRANSLATOR-LANDED: theta_corpus = Jadj o inner(c) exact (c unique, 14c integral); seam = nonzero 3-torsion at infinity ((2,2,0) pattern); order chain indices 3^5/3^4 (the 81 survives); J_fix duality certified + Lean; c's geometric identity open; 11/11 checks
Dependencies: QGEO.EQORDER.01; QGEO.SEAMMARKS.01; QGEO.DUALFORMS.01; QGEO.EMBED.01; v572; v600; v601; v602; v603; v604
Scripts: v605_translator.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 623Original ledger status
QGEO.MODESEP.01
The mode-separation round (one module, 7 checks, ~2 s; discovery probe mode_separation_probe.py 7/7, verdict MODES-SEPARATED). ALL THREE MODES ARE PURE TORSION AT INFINITY [E]: the three eigenlines of the sheet operator V (eigenvalues 0, 1, 2), lifted to the unreduced integral module along the unique intertwiner (1-parameter, rank 3), all have ZERO free part in coker(1 - T_inf) = Z^2 + (Z/3)^3 -- like the seam (v605), the zero mode and the top mode are pure 3-torsion classes at infinity. THE SEPARATION [E]: the three classes are PAIRWISE DISTINCT elements of (Z/3)^3 (fixed Smith frame: (2,0,2), (2,2,0), (1,2,0) for V = 0, 1, 2; distinctness is frame-invariant) and none is trivial -- the infinity-cusp torsion is a SELECTION CARRIER distinguishing the zero mode from the top mode: exactly the separation the real structure could NOT provide (only the seam is R-fixed; the 0/2 eigenlines are neither R-fixed nor R-swapped, v605-followup). HONEST SCOPE [C]: the specific coordinate triples are Smith-frame-dependent; the canonical rule singling out the zero mode's class (the '+1' of the v590 demand) from the structure alone is the named remaining step. THE ARITHMETIC ROUTE FOR c IS CLOSED [X, honest negative]: the canonical cubic invariant char(7 Gamma(c) c) = y^3 + 3y^2 + 6y - 1 (integer monic, irreducible) has discriminant -783 = -3^3 x 29, NOT a square -- Galois group S_3, not abelian, so no cyclotomic-subfield identity for c exists (Kronecker-Weber); c's geometric identity stays the named open. GATE.QGEO does not move; no marker changes. [2026-08-01 XXV: the canonical rule LANDS -- v607: the deck rotation has exactly ONE projective eigenline on the torsion (char (x+1)(x^2+1) mod 3) and it is the top mode; the zero mode is the only unlabeled mode; see QGEO.SELRULE.01.] [2026-08-01 XXVII: the c-identity question is resolved geometrically -- v609 (factorization through the unique mirror intertwiner); the arithmetic negative of this module stands as the reason the cyclotomic route had to fail; see QGEO.CFACTOR.01.]
Recorded status: Exact [E] torsion/separation identities + [C] typed frame caveat + [X] honest arithmetic negative (exact sympy; Smith transforms over Z)
Dependencies, status detail and source
Canonical status: MODES-SEPARATED: all three V-eigenlines pure 3-torsion at infinity, classes pairwise distinct ((2,0,2)/(2,2,0)/(1,2,0)); canonical '+1' rule open; cubic field of c disc -783, S_3, not abelian (arithmetic route closed); 7/7 checks
Dependencies: QGEO.TRANSLATOR.01; QGEO.SEAMMARKS.01; QGEO.EQORDER.01; v590; v603; v604; v605
Scripts: v606_mode_separation.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 624Original ledger status
QGEO.SELRULE.01
The selection rule round (one module, 10 checks, ~2 s; discovery probe selection_rule_probe.py 10/10, verdict SELECTION-RULE-LANDED). THE F_3 STRUCTURE [E]: the torsion of coker(1 - T_inf) is a genuine F_3 vector space -- omega-multiplication acts as the IDENTITY ((1-omega) acts as zero) -- and the deck rotation r (commuting with T_inf = r^4) acts F_3-linearly on it. THE RIGIDITY [E]: the induced action is the explicit matrix [[0,1,0],[1,0,2],[2,2,2]] mod 3; its only fixed class is 0; char mod 3 = (x+1)(x^2+1) with x^2+1 IRREDUCIBLE over F_3: the action has EXACTLY ONE projective eigenline (eigenvalue -1) -- structural uniqueness, census-confirmed. THE LABELING [E]: the unique deck eigenline IS the top-mode class (V = 2); the zero mode and the seam are NOT projectively fixed (neither under r nor r^2). With v603/v605 the three modes carry MUTUALLY EXCLUSIVE canonical labels: seam (V = 1) = the R-REAL line; top (V = 2) = the unique DECK-EIGEN torsion class; zero (V = 0) = the only UNLABELED mode. THE RULE [C, typed reading]: the two-demand involution S_0 acts as -1 exactly on the two labeled modes and +1 on the unlabeled one -- the v590 zero-mode demand ('+1 on ker V') is the statement 'fix the unlabeled mode', the seam-reversal demand is 'reverse the R-real line' (v603): BOTH demands are now expressed through canonical cover structure. Honest scope: a typed reading of the selection, not a continuum derivation; eigen-ness, uniqueness and distinctness are Smith-frame-invariant, the coordinate triples are not. GATE.QGEO does not move; no marker changes.
Recorded status: Exact [E] F_3/rigidity/labeling identities + [C] typed selection reading (exact sympy / F_3 arithmetic; Smith transforms over Z)
Dependencies, status detail and source
Canonical status: SELECTION-RULE-LANDED: torsion = F_3 space (omega trivial); r-action explicit, char (x+1)(x^2+1) mod 3, unique projective eigenline = top mode; seam/zero not fixed; labels mutually exclusive (R-real / deck-eigen / unlabeled); S_0 flips the labeled pair; 10/10 checks
Dependencies: QGEO.MODESEP.01; QGEO.TRANSLATOR.01; QGEO.INVOL.01; v590; v603; v605; v606
Scripts: v607_selection_rule.py
Location: tfpt_1;tfpt_2;tfpt_3
Original CSV, line 625Original ledger status
WOIT.GAMMA.FREE.01
The free gamma slice of the WOIT roadmap (one module, 9 checks, ~1 s; discovery probe gamma_free_slice_probe.py 9/9, verdict GAMMA-FREE-SLICE-LANDED; the v524 machinery imported read-only). THE BASELINE [E]: the chiral N = 8 OS Gram is PD (16,0,0) -- H_phys exists (v524 Q1.3 reproduced). THE FLIP LOCALIZES [E]: the anti-chiral (mirror) state's Gram has inertia (8,8,0) and the split is SECTOR-EXACT: even (8,0,0) PD, odd (0,8,0) STRICTLY NEGATIVE DEFINITE -- the v519 odd-sector definiteness flip stated sector-exactly on the full N = 8 half algebra. MIRROR EXCLUSION [E, the gamma-A statement at the free level]: every nonzero mirror fermion vector has strictly negative OS norm; hence in the doubled (vector-like) system any OS-positive subspace meets the mirror odd sector in {0}: NO mirror fermion mode survives OS reconstruction -- kill test (3) ('the reconstruction produces a vector-like mirror generation') CANNOT fire at the free level, upgraded from the v519 state-level shadow to the SECTOR level. ONE GENERATION, MULTIPLICITY-FREE [E]: the compressed clock spectrum on the odd (fermion) transfer domain is EXACTLY {1, sqrt(2)-1} -- two distinct eigenvalues, no degeneracy. MARK INCIDENCE AT THE FREE LEVEL [E]: the two positive-half mark algebras (monomials on {4,5} and {6,7}) have PD OS Grams (4,0,0) each (dim 4 = |mu4|); their monomial products generate the FULL 16-dim H_phys basis (the marks jointly generate the net, meeting in the vacuum line); and the quarter-turn transfer tau_2 IS the mark-A -> mark-B transport (alpha_2: {4,5} -> {6,7}), Hermitian PD on the mark algebra -- the incidence-compatible extension exists at the free level, kill test (7) does not fire there. FIREWALL: the FREE slice only -- the gamma milestone itself (A_hol, the interacting net, the spacetime extension) stays OPEN; all seven contract kill tests stay formally live on A_hol; WOIT.OS.TWISTOR.01 does not move; no marker changes. [2026-08-01 XXXIV: the interacting gamma slice lands at toy level -- v615: on the RP-surviving interaction class (the alignment bit, v534) the mirror odd sector is strictly ND on every cut and every coupling (kill test (3) cannot fire on the surviving class), the mark transport is Hermitian PD at every g (kill test (7) does not fire), and the odd transfer spectrum stays multiplicity-free; the dead member has no RP home -- bit selection and chirality data co-locate; see WOIT.GAMMA.TOY.01.]
Recorded status: Exact [E] sector/spectrum/incidence statements on the explicit v524 net + [C] the gamma reading (free slice; A_hol untouched)
Dependencies, status detail and source
Canonical status: GAMMA-FREE-SLICE-LANDED: chiral (16,0,0); anti-chiral (8,8,0) sector-exact (odd strictly ND -- mirror exclusion); clock spectrum {1, sqrt2-1} multiplicity-free; mark algebras PD dim 4, products generate, tau_2 = mark transport PD; gamma open on A_hol; 9/9 checks
Dependencies: WOIT.OS.TWISTOR.01; WOIT.BETA2.OS.01; WOIT.THETA.FREE.01; v519; v524; v565
Scripts: v608_gamma_free_slice.py
Location: tfpt_research_contracts;tfpt_3
Original CSV, line 626