Original ledger status
AX.P2.01
g_car = 5 carrier interface (3+2) -- a DECLARED input but an OVER-DETERMINED bootstrap fixed point, not a free choice: forced three independent ways by the E8 closure (bootstrap mu^2-5mu+4=0, BOOT.01/v6), uniquely solved by the Pascal carrier condition 2^g=g^2+g+2 => g=5 (Lean-formalised, no sorry; FORM.LADDER.01), and the (g_car,N_fam)=(5,3) pair is the anchor moment data e2(a)=g_car (ARCH.CORE.01/v53; a=(1,1,2) the unique multiset with elem-symm (4,5,2)). Reduces with P1 to {a, pi}. SYNTHESIS NOTE (reduction state 2026-07-21; text consolidation, marker Axiom UNCHANGED): P1 + the four marks are [E]-anchored (v216); given the marks, g_car = 5 follows via the partition 4 = 2+1+1 plus the Deligne/Birkhoff-Grothendieck/stability typing (v491/v499, residual R1 [C]); the clock order, the Z8 tower, the deck class and the position half of the alignment bit are theorems (v506/v507/v510) -- the compiler's dimensionless inputs reduce to: four marks (derived from P1-side topology), ONE square-modulus datum (tau = i, equivalently: the clock / the CM-fixed half-period / the non-split extension class on the free seam circle / the flag-transitivity bit V4 -> D4, v512), and pi, plus the [C] identifications R1/R2 of v499. SYNTHESIS NOTE (2026-07-22, v512 -- marker UNCHANGED): the reduction state can equivalently be written 'four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi'. This describes the REDUCTION STATE; the axiom stays declared and QGEO.REALIZE.01 stays [C]/[O] [2026-08-02 XXXIX: the four-mark closure lands -- v624: under the D_{d+1}+A_{d-1} glue architecture three independent selectors (disc-group census, even-glue norm equation, Pythagorean (3,4,5)) force d = 4, with (1,1,2) the unique 3-part partition and c_3 = 1/(8 pi): the two-input presentation reduces to ONE boundary degree d = 4 plus pi, CONDITIONAL on the architecture; the axiom keeps its typing -- the residue moves to why the physical boundary realizes the minimal cyclically-glueable genus-0 divisor; see EXTREV.LATTICE.01.] SYNTHESIS NOTE (2026-08-28, v993 -- marker Axiom UNCHANGED): the architecture conditionality of v624 is lifted at the census level -- the FULL rank-8 ADE pair census with cyclic Z4 glue on BOTH factors selects (D5, A3) uniquely (D3 ≅ A3 aliases collapsed), so the D+A split is an OUTPUT of (rank 8 + cyclic Z4), not an INPUT. Cited-not-proved remain c_- = 8 and holomorphic uniqueness of (E8)_1. Axiom typing UNCHANGED. WAVE-6 (2026-08-30, v1005): relative pencil identities exact but CIRCULARITY-FLAGGED -- carrier 5 already sits in the column sums of Q_2 and C; NOT a P2 selector until Q, C, epsilon are derived from seam geometry independently. Axiom typing UNCHANGED. LATE EVENING (2026-08-30, v1014): W-bridge premise P-anch DERIVED (anchor chain: e1 = R^{-1}(1,1,2)^T uniquely via the winding/triple lock); P-dem narrowed to the character-blind determinant-response map -- same MMST/Quillen externalized leg as TEL-B/ALG-EXH/P1. Axiom typing UNCHANGED. MONDAY MORNING (2026-08-31, v1015): BOTH W-bridge premises now derived at finite level -- P-anch already (v1014 winding/triple lock); P-dem DERIVED (character-blind determinant response r=(1,1,1) exactly at collar sizes 12 and 16, democracy residual 0; mutants split (1,0,0) and (5/4,1,5/4)). The axiom-core remainder is ZERO modulo the externalized MMST identification. Axiom typing UNCHANGED (the finite derivations are shadows; the continuum identification is the same external leg).
Recorded status: Axiom (declared input; over-determined bootstrap fixed point, Lean-forced)
Dependencies, status detail and source
Canonical status: Axiom (declared) but bootstrap-forced (3 ways) + Pascal-unique (Lean): g=5 is not a free knob; reduction state 2026-07-22: four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi (v491/v499/v506/v507/v510/v512), marker unchanged | 2026-08-28 v993: v624 architecture conditionality lifted at the census level (D5, A3 unique in full rank-8 ADE); axiom typing UNCHANGED | WAVE-6 (2026-08-30 v1005): pencil circularity-flagged (5 already in Q_2/C column sums); NOT a P2 selector; axiom typing UNCHANGED | AFTERNOON (2026-08-30 v1010): pencil selector rehabilitated: column sums forced by prior constructions (ladder c=2 + n=3 force Q sums (9,5,1); C's 5 inherited from R col 3 -- non-circular); W-shift conditionally unique given democratic-image + carrier-anchor-kernel (1078 -> 1); remaining obligation = seam-geometric derivation of those premises; axiom typing UNCHANGED | LATE EVENING (2026-08-30 v1014): W-bridge premise P-anch derived (anchor chain); P-dem narrowed to the character-blind determinant-response map -- same MMST/Quillen externalized leg as TEL-B/ALG-EXH/P1. Axiom typing UNCHANGED. | MONDAY MORNING (2026-08-31 v1015): BOTH W-bridge premises derived at finite level (P-anch v1014; P-dem r=(1,1,1)); remainder 0 modulo MMST. Axiom typing UNCHANGED.
Dependencies: BOOT.01; ARCH.CORE.01; v491; v499; v510; v512; v993
Scripts: v1005_relative_pencil_bridges.py, v1010_simplicity_bridge_census.py, v1014_bridge_refinements.py, v1015_axiom_core_closure.py
Location: tfpt_1
Original CSV, line 3Original ledger status
P2.PARTITION.01
g_car = 5 AS A COROLLARY OF THE FOUR MARKS (2026-07-20; the partition sharpening of ARCH.CORE.01/v53): a multiset of exactly 3 POSITIVE INTEGERS summing to 4 is UNIQUE -- {1,1,2}, the unique partition of 4 into exactly 3 positive parts -- so the elementary symmetric functions e = (4,5,2) and hence g_car = e2 = 5, |Z2| = e3 = 2, N_fam = e2 - e3 = 3 are COROLLARIES, not inputs; the carrier formula closes as a fixed point ((2^(g_car-1)-1)/g_car = 3 = rank H^1 = #marks - 1). [E] NEGATIVE CONTROLS, each assumption load-bearing (sympy exact): drop positivity -> 4 solutions with e2 ambiguous {0,3,4,5}; sum 3 -> unique but e2 = 3; sum 5 -> not unique; 2 parts -> e2 in {3,4}; 4 parts -> e2 = 6; Mehta-Seshadri-typed RATIONAL weights (scaled by 4) -> e2 = 16/3 resp. 44/9 != 5 -- integrality comes from the SPLITTING-TYPE typing (Birkhoff-Grothendieck: bundles on P^1 split with integer degrees), NOT from MS weights (which are rational). [E] THE v53 SHARPENING: v53 pins (1,1,2) using ALL THREE targets (4,5,2); the sharper lemma needs only e1 = 4 (+ positivity + 3 parts) -- both searches return the same unique multiset. [E] THE SUM RULE: |a|_1 = 4 x (N-1)/2 = 4 iff N = 3 (equidistributed clock spectrum); the v115 mu4-average route |a|_1 = |mu4| tr(A0) = 4 agrees. INDEPENDENCE CHAIN (why not circular in g_car): #marks = 4 from v216 Gauss-Bonnet (P1-side premises only, checks 1/3/5 without g_car); the 3 = rank H^1(P^1 minus 4 marks) = #marks - 1 is pure topology (v137 explicit basis / v228 index gate); |a|_1 = 4 is g_car-free via par-deg-0/(U) (v115) or the mu4-average lemma. [C] HONEST RESIDUAL: P2 is NOT made axiom-free -- the residual is the weight-TYPING postulate (integrality = Birkhoff-Grothendieck splitting exponents; positivity a_i >= 1 = h^0(E) = 0, correlate: the trivial mu4 character absent from H^1, v137 check 1; sum rule = parabolic-degree-0 'given (U)', v115 [I]/[N]); AX.P2.01 stays a DECLARED AXIOM (over-determined, now by one more convergent route). DIRECTION-REVERSAL NOTE: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> partition -> g_car -- an over-determination check, not a circle, and it must never be double-counted in witness tallies. No gate closes, no marker moves.
Recorded status: Formal [E] (partition lemma + corollaries + negative controls + v53 sharpening, sympy exact, Wolfram-mirrored) + [C] (weight-typing postulate: splitting-type integrality, h^0 = 0 positivity, (U) sum rule; AX.P2.01 stays Axiom)
Dependencies, status detail and source
Canonical status: Partition corollary [E]/[C]: 4 = 1+1+2 unique in exactly 3 positive parts => g_car = 5, |Z2| = 2, N_fam = 3 as corollaries; all negative controls pass; sharpens v53 (only e1 = 4 needed); residual = the weight-typing postulate -- AX.P2.01 stays a declared axiom; the typing residual is itself sharpened by v499 (P2.TYPING.01, Deligne + BG + stability), v491 stays active
Dependencies: AX.P2.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; v23; v53; v115; v137; v216; v228
Scripts: v491_p2_partition_corollary.py
Location: introduction;tfpt_1;tfpt_2;origin_theory
Original CSV, line 553Original ledger status
P2.TYPING.01
THE P2 WEIGHT-TYPING POSTULATE HARDENED (2026-07-21; types the v491/P2.PARTITION.01 check-9 residual): the anchor weights a = (1,1,2) are the Birkhoff-Grothendieck SPLITTING EXPONENTS of the DELIGNE CANONICAL EXTENSION E of the flavor connection -- given the [E] anchors (4 marks, v216 Gauss-Bonnet; rank 3 = #marks - 1, v137/v228; cusp-class local monodromy char poly lam^3 - 1 + irreducibility |<U,M0>| = 24 = |S4| with sum |chi|^2 = 24 exact, v117; unitarity (U)), the three typing properties become THEOREMS of standard algebraic geometry: INTEGRALITY = Birkhoff-Grothendieck (every holomorphic bundle on P^1 splits with integer degrees, automatic once E is typed as a bundle); deg E = -4 = -#marks via the Deligne residue-trace formula (canonical residue exponents {0,1/3,2/3}, the UNIQUE lift of spec M0 to [0,1), trace 1 per mark; par-deg 0 = the v39 (U) consistency; infinity APPARENT, monodromy product = 1 exactly); POSITIVITY a_i >= 1 = h^0(E) = 0 via Mehta-Seshadri stability (irreducible + unitary => stable; every section-saturation has par-deg >= 0, all 324 cases; BG dictionary h^0 = 0 <=> a_i >= 1 exhaustive) -- so {a} = {1,1,2} is UNIQUE (v491 partition lemma) and g_car = e2 = 5 follows. [E] SELECTOR TYPED: the Schur-Horn permutohedron of 4 spec(A0) has integer points exactly {(2,2,0),(2,1,1)} (the two options of the tfpt_2 'Global computation' theorem); h^0 = 0 kills {0,2,2} (O(0) summand => h^0 = 1) -- the paper's 'balanced/minimal-variance' heuristic is now TYPED as stability/positivity. [E] THE h^1 DISCREPANCY, REAL + RESOLVED: h^1(O(-2)+O(-1)^2) = 1 != 3 (chi = deg + rk = -1) -- the naive typing 'generation space = H^1 of the rank-3 E' is FALSE (forcing it demands deg -6 and loses uniqueness: {1,1,4},{1,2,3},{2,2,2}, e2 in {9,11,12} never 5 -- negative control); the correct TWO-OBJECT typing: the LINE bundle L = O(-D), D = mu4, carries the generations (h^1(L) = 3 = N_fam = H^0(Omega^1(log D)) = the v137 basis; h^0(O(4)) = 5 = g_car, v228), while E carries them as its FIBER (rank 3) with the anchor as splitting type; three DIFFERENT H^1's coexist (3 generations L-side, 1 coherent h^1(E), 6 = |R+(A3)| local-system deformations). [E] BISWAS EXPLICIT (independent convergence witness): the weight denominator is 3 = N_fam (NOT 2 = |Z2|), so the Biswas cover is the Z3 cusp cover y^3 = (z-1)(z-i)(z+1)^2(z+i)^2 (genus 2 by RH, unramified over infinity; all-equal branch data impossible, 4c != 0 mod 3), NOT the pillowcase double w^2 = z^4-1 (genus 1, j = 1728, deck side); p_* O_Y = O + O(-2)^2 (chi cross-check -1 = -1); the equivariant regular-rep model W = (chi^0+chi^1+chi^2) x O_Y has parabolic partner with underlying bundle O + O(-2)^2: rank 3, deg -4, par-deg 0 all OK -- integrality AND the sum rule COME OUT of the correspondence -- but splitting {0,2,2} with h^0 = 1: exactly the UNSTABLE Schur-Horn companion; (1,1,2) is the Biswas image of the STABLE object, stability is the selector. [E] NEGATIVE CONTROLS: drop positivity -> 4 (31 with negatives) solutions; deg -3 -> e2 = 3, deg -5 -> not unique; n-mark scan: {1,..,1,2} unique for EVERY n >= 3 but e2 = (n-2)(n+1)/2 = 5 ONLY at n = 4 (the VALUE, not the uniqueness, is the n = 4 content); MS-rational exponents e2 = 16/3 resp. 44/9. [C]/[N] HONEST RESIDUALS (P2 stays an axiom, NO marker moves): R1 the module identification 'generation space = this H^1 with this connection' [C] (QGEO.SYM territory, QGEO.SYM.01 stays a corollary of the [O] keystone); CONVERGENCE THEOREM (2026-07-21, v503/QGEO.EMERGE.LIGHT.01): R1 is IDENTICAL with the QGEO modulus rest -- the order-4 clock on H^1 is the Coxeter element of W(A3) = the cusp-class carrier datum, and the QGEO emergence-light test shows marks + square are derived while the clock is the one input: QGEO-R1 == P2-R1, ONE common residual, not two; FERMIONIC HALVING (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01): the common R1 rest is mapped onto the SAME single alignment bit -- 'the collar deck is the central involution of the mark-D4' -- because the ORDER of the clock is now fermionically forced (U^2 = (-1)^F exact on the NS seam, nonsplit Z4; both Moebius square roots automatically order 4), leaving only the spatial alignment as the carrier input; ORIGIN CROSS-REFERENCE (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the common R1 bit is now characterised as a tetrad with a physical face (non-split NS Fock extension class) and its deck CLASS is derived (half-period translation of the seam double, v507); only the position stays the carrier input; FREEDOM HALVING (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the common R1 bit is HALVED -- its position half is topology (the collar deck is a covering deck, hence free on the seam circle; the edge class is excluded, v510), so R1 reduces to the square-modulus datum tau = i alone; the modulus stays the one [C] carrier input; R2 unitarity (U) [C] (v33/v40 premise); R3 the analytic-monodromy identification [N] (v117 check 4, ODE matching 1e-7). NON-CIRCULAR: neither v23's splitting-type input nor v115's diag(A0) = a/4 input is used; the chain is rank (topology) -> deg (residue trace) -> integrality (BG) -> positivity (stability) -> partition (v491); A0* appears only as the explicit consistency witness. The weight postulate shrinks from 'positive integers summing to 4 (why?)' to 'Deligne + BG + stability, given the cohomological typing'. DIRECTION-REVERSAL NOTE inherited from v491: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> bundle -> partition -> g_car -- an over-determination check, not a circle, never double-counted. AX.P2.01 stays a DECLARED AXIOM. SYNTHESIS NOTE (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED): the reduction state reads 'four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi' -- the R1 rest gains an equivalent discrete seam-intrinsic form (V4 -> D4 side-indistinguishability of the four equally-decorated marks), not a derivation.
Recorded status: Formal/Identity [E] (AG arithmetic: Deligne residue-trace degree, BG dictionary, stability skeleton, Schur-Horn lattice points, Biswas Z3-cover -- sympy exact, Wolfram-mirrored) + [C] (R1 module identification, QGEO.SYM territory; R2 unitarity (U)) + [N] reference (R3 analytic monodromy, v117 check 4); AX.P2.01 stays Axiom
Dependencies, status detail and source
Canonical status: Weight typing hardened [E]/[C]: rank 3 (topology) + deg E = -4 (Deligne) + integrality (BG) + positivity (MS stability, h^0 = 0) force E = O(-1)^2 + O(-2) => a = {1,1,2}, g_car = 5; balanced heuristic typed as stability; h^1 discrepancy resolved (two-object typing); Biswas Z3-cover explicit; residuals R1/R2 [C], R3 [N]; R1 mapped by v506 onto the single alignment bit (deck central in the mark-D4; the clock ORDER is fermionically forced); v510 halves R1: the position half is topology (edge class excluded), the rest = the square-modulus datum tau = i; v512 restates the rest as one discrete symmetry-lift bit (flag transitivity <=> tau = i) -- AX.P2.01 stays a declared axiom, QGEO.SYM stays [O]
Dependencies: AX.P2.01; P2.PARTITION.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; QGEO.SYM.01; v33; v40; v115; v117; v137; v216; v228; v491; QGEO.EMERGE.LIGHT.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512
Scripts: v499_p2_weights_deligne_bg.py
Location: introduction;tfpt_1;tfpt_2;origin_theory
Original CSV, line 554Original ledger status
QGEO.EMERGE.LIGHT.01
THE v215 DEFERRED EMERGENCE TEST (LEVER 7) EXECUTED LIGHT + RESIDUAL CONVERGENCE QGEO-R1 == P2-R1 (2026-07-21; analysis/kill-test class, v347/v264 style; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom): v215 lever 7 said the decisive (deferred) kill of the bedrock is to compute the raw P1 seam-collar DtN WITHOUT inputting the marks and check whether four SQUARE marks EMERGE; v216 already derives mark COUNT and EQUALITY from topology, so the genuinely open part is the MODULUS. The light version (17 checks) returns: RIGIDITY, NOT DYNAMICAL SELECTION. [N] NO DYNAMICAL SELECTION (zeta-log-det via the Dedekind-eta formula, mpmath 40 digits): logF = log(tau2 |eta|^4) is exactly S-invariant (dev < 1e-35), so t = 1 is AUTOMATICALLY critical on the rectangular family tau = it; there g'(1) = 0 and g''(1) = -1.2982113 < 0 (strict local maximum of det') BUT the tau1-direction has h''(0) = +0.29821132 > 0: tau = i is a SADDLE of the full moduli space; the critical points of logF are exactly the CM points (E2*(i) = 0, E2*(rho) = 4.7e-42, generic |E2*| = 0.163 != 0); the GLOBAL det' winner at fixed area is the HEXAGONAL torus: logF(rho) = -1.033519276 > logF(i) = -1.054688281 (diff 0.021169; Osgood-Phillips-Sarnak 1988) -- free-field dynamics alone does NOT select tau = i on the full (2,2,2,2) moduli space; criticality at the square is Palais symmetric criticality (the independent heat trace is t -> 1/t invariant and equally critical), not dynamics. [N] SHARP RIGIDITY (v280 lattice DtN machinery, N = 8): the mod-4 off-character indicator of the raw free-field DtN (the v215-K3 lever as a MODULUS FUNCTION) vanishes to 8.0e-15 EXACTLY at tau = i and increases strictly monotonically in both directions (min 8.4e-2 for every t != 1): an isolated zero, not a flat valley; R^2 = Z2 exactly as permutation matrices and R^4 = 1 (the v492 Z8 bridge on the lattice); j(it) >= 1728 with equality ONLY at t = 1, j(rho) = 0 -- only j = 1728 carries a Z4 automorphism (v214/v168). [E] THE Z2 CHAIN IS MODULUS-BLIND: the involution z -> -z has exactly 4 fixed points (2-torsion = the 4 marks) and [Z2, Delta_t] = 0 on EVERY torus -- v216 confirmed without inputting marks; the pillowcase det' is EXACTLY half the torus det' (+- pair structure, ratio 1.000000000000, dim(even) = (N^2+4)/2), so zeta_pillow = zeta_torus/2 and all tau conclusions transfer verbatim; [Z2, H_t] = 0 for all t (max 3.9e-15) -- marks, mark-locality and deck^2-invariance (v264/v280) exist on EVERY pillowcase; ONLY the order-4 clock distinguishes tau = i; Aut(E_rho) = Z6 (element orders 1,2,3,6) contains NO order-4 element: the dynamically selected hexagonal point CANNOT carry the mu4 clock -- dynamics and clock-symmetry are logically INDEPENDENT selectors; GIVEN the clock, P(iZ) = P(Z) forces a3 = a2 = a1 = 0 and the surviving family Z^4 + a0 has j = 1728 identically (v493 S1 mini): the modulus is NOT a second input. [C/O] RESIDUAL CONVERGENCE (the core finding): the order-4 element acting on H^1 is a COXETER element of W(A3) (char poly lam^3 + lam^2 + lam + 1, order h(A3) = 4 = |mu4| = N_fam + 1) -- exactly the P2 cusp-class/carrier datum of the P2 hardening (v499 R1: generation space = H^1(P^1 minus mu4) with the cusp connection): QGEO-R1 == P2-R1, ONE COMMON RESIDUAL -- 'why does the raw seam carry the order-4 clock' -- not two scattered ones. Chain: (i) mark count 4 derived (v216 [E]); (ii) square derived GIVEN the clock (v493 [E]); (iii) clock = carrier input [C]; (iv) rest = the dynamical realisation of the clock [O]. Route classification (v347 analogue): B-rigidity in the sharpened form 'RP seam data => order-4 clock' is the live route; free dynamics alone does NOT deliver the clock (the hexagonal finding). LEVER UPDATE (documented here; v215 itself unchanged): v215 lever 7 is lifted from 'deferred, not executable' to 'light version EXECUTED, result: rigidity instead of dynamical selection' -- marks emerge, the square does NOT emerge dynamically, it is pinned by the clock, and the clock is the input. No gate closes, no marker moves; QGEO.REALIZE.01 stays [C]/[O].
Recorded status: Formal [E] (exact structure: pillowcase = half torus pair structure, permutation identities R^2 = Z2, clock-forcing a3 = a2 = a1 = 0 with j = 1728 identically, Coxeter charpoly) + Numerical [N] (eta/E2*/j at 40 digits; v280-style lattice DtN curves) + [C] (clock = carrier input) + [O] (the dynamical realisation of the clock); NO marker moves -- QGEO.REALIZE.01 stays [C]/[O]
Dependencies, status detail and source
Canonical status: Emergence test light [E]/[N]/[C]/[O]: no dynamical selection of the square (global det' winner = hexagonal, tau = i a saddle, criticality = Palais symmetry); the square is pinned by rigidity alone (DtN mod-4 indicator = isolated zero at tau = i; j = 1728 iff t = 1; Z2 chain modulus-blind); residual convergence: QGEO-R1 == P2-R1 = ONE common rest, the order-4 clock (Coxeter of W(A3) = the v499 cusp-class carrier datum); v215 lever 7 lifted from deferred to light-executed -- marker unchanged, QGEO.REALIZE.01 stays [C]/[O]
Dependencies: QGEO.KILL.01; QGEO.REALIZE.01; QGEO.RIGID.01; P2.TYPING.01; CELEST.WP1.01; CELEST.WP2.01; v215; v216; v264; v280; v347; v492; v493; v499
Scripts: v503_qgeo_emergence_light.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 567Original ledger status
SEAM.CLOCK.RIGIDITY.01
SEAM CLOCK RIGIDITY -- ROUTE B OF THE v503 CLASSIFICATION EXECUTED (2026-07-21; two exact theorems in one module, Part A Moebius + Part B Fock, 34 checks; NO marker moves -- QGEO.REALIZE.01 stays [C]/[O], AX.P2.01 stays an axiom): v503 localised the one remaining identification input as 'why does the raw seam carry the order-4 clock' (QGEO-R1 == P2-R1). v506 reduces that input to ONE alignment bit. [E] THEOREM A (Moebius side, complete): the collar deck z -> -z has in PGL2(C) EXACTLY TWO square roots (z -> +iz and z -> -iz), both AUTOMATICALLY of order 4 (complete case analysis: tr g != 0 forces diagonal with a/d = +-i; tr g = 0 forces det = 0, impossible) -- 'order 4' is NO LONGER an input; a marks-preserving square root exists <=> the deck is the CENTRAL involution of the mark-stabiliser D4 (center = {id, deck}, exactly 2 order-4 elements = the two clocks, both squaring to the deck) => then the marks are one mu4 orbit, cross-ratio class {2, 1/2, -1}, j = 1728, tau = i, free 4-cycle, deck = clock^2, only the scale free; COUNTEREXAMPLE (mandatory honesty): {+-1, +-(3+2sqrt2)} is deck-free AND harmonic (j = 1728) with full D4 stabiliser, but there the deck is a NON-central (edge-type) involution => ZERO square roots -- harmonicity alone is NOT sufficient, the residual datum is deck-CENTRALITY, not the j-invariant; the RP filter selects nothing (both real structures pass both roots: the Aut(Z4) = {clock, clock^-1} ambiguity is irreducible, consistent with v492 NC3). [E] B3/B4 (n-arithmetic + controls): a free Z_n on 4 marks forces n | 4; marks-transitivity forces n = 4; faithfulness (a Moebius map fixing >= 3 points is the identity) kills every n >= 5; n = 3 fixes a mark (hexagonal shadow); n = 6 dies INDEPENDENTLY at the established cusp-class degree 3 (lam^5 - 1 vs lam^3 - 1, v117); the deck is modulus-blind (every {+-1, +-w} preserved -- v503 B5 made exact, ALL modulus-pinning power sits in the square-root bit); controls: no order-8 element preserves any 4-set (orbit sizes 8); hexagonal marks: stabiliser A4 (order 12), ZERO order-4 elements, squares only orders {1, 3}; generic marks: Klein V4, exponent 2 -- not even the deck has a root; stabiliser trichotomy V4/D4/A4 with order-4 counts (0, 2, 0): an order-4 element exists IFF the marks are harmonic, and then it is the clock pair. [E] THEOREM B (Fock side, the core result): the celestial sphere S^2 has b1 = 0, hence a UNIQUE spin structure whose restriction to every circle is the BOUNDING = NS/antiperiodic one (established P1 datum, v492/v490); ONE-PARTICLE: the NS half-rotation D has D^2 = -1 (charpoly (lam^2+1)^8) -- the 'Z2' deck is already order 4 before any Fock choice; the quarter-shift C has C^2 = D, C^8 = 1 (charpoly (lam^4+1)^4, zeta8 spectrum); FOCK (exact Cl(16) Fractions): the Bogoliubov implementer Utilde = prod_j (1 - gamma_j gamma_{j+8}) satisfies Utilde^2 = 256 gamma_1...gamma_16, i.e. U^2 = (-1)^F EXACTLY, and (cU)^2 = c^2 (-1)^F is NEVER 1 (center of Cl(16) = scalars, implementer unique up to phase) -- the Z4 lift is FORCED: 1 -> Z2^{(-1)^F} -> Z4<U> -> Z2^{deck} -> 1 is a NONSPLIT extension; Z8 TOWER: the quarter-shift lift V (unique even block implementer, rational nullspace) has V^2 prop Utilde, V^4 prop (-1)^F, projective order 8 = 2|mu4| -- the v492 Z8 spin bridge realised on the 16-Majorana Fock space, '8 = 2|mu4|' fermionically explained; R CONTROL: U_R^2 = +1 EXACTLY (split Z2 -- the entire forcing rests on the NS/bounding structure = established geometry, not a convention); 16-FOLD WAY: theta_v = 1 at nu = 16 (the gaugeable point, v490), the forced tower period 8 = the Fidkowski-Kitaev Z8; ROBUSTNESS: the forcing already holds on an 8-site NS circle -- 16 is the carrier count 2^(g_car-1), not a tuning. [C] THE ONE RESIDUAL (verbatim): 'the collar deck is the central involution of the mark-D4' (<=> square-root existence on the mark data) -- ONE alignment bit; the fermionic Z4 forces the ORDER, not the spatial alignment. NO OVERCLAIM: given the bit, order 4, uniqueness up to Aut(Z4), the free 4-cycle, tau = i (scale free) and deck = clock^2 are all THEOREMS; without it, nothing pins the modulus. Consequence for the residual map: QGEO-R1 == P2-R1 (v503) now reads as this single bit -- order/uniqueness/orbit/tau = i all follow (fermionic halving of the R1 rest). No gate closes, no marker moves. PRECISION (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the bit is now fully characterised as a tetrad (clock exists <=> tau = i AND deck = the CM-fixed half-period translation <=> Stab = D4 AND deck central <=> deck pairs harmonically separated <=> NS Fock lift nonsplit), and carries a physical face: the non-split NS Fock extension class (Fidkowski-Kitaev-type; the edge arrangement splits, U^2 = +1, zero roots) -- v507; the origin theorem shows every mark-free collar deck IS a half-period translation of the seam double, so the CLASS of the deck is derived and the bit is NO tautology -- only the position (which half-period) stays the carrier input. PRECISION (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the POSITION half of the alignment bit is now typed as topology -- the collar deck is a COVERING deck and covering decks act freely; the unique free involution of the seam circle is the antipode (Aut(C_16) = D_16, complete census) and NONSPLIT <=> FREE over all 17 involutions, so the edge class is topologically EXCLUDED (v510); the bit reduces to the square-modulus datum tau = i alone (harmonic + free => mu4 exactly) -- the modulus half stays the one [C] carrier input.
Recorded status: Formal/Identity [E] (Theorem A: complete PGL2(C) square-root classification, D4 centrality criterion, counterexample, n-arithmetic, stabiliser trichotomy -- sympy exact; Theorem B: U^2 = (-1)^F in exact Cl(16) Fractions, nonsplit Z4 extension, Z8 tower, R-sector split control, 16-fold-way arithmetic -- Wolfram-mirrored) + [C] (the one alignment bit: deck central in the mark-D4); QGEO.REALIZE.01 stays [C]/[O], P2.TYPING.01 R1 mapped onto the same bit
Dependencies, status detail and source
Canonical status: Clock rigidity [E]/[C]: the deck has exactly two Moebius square roots, both automatically order 4; marks-preserving root <=> deck = central involution of the mark-D4 (counterexample shows harmonicity alone insufficient); the NS Fock implementation is FORCED to order 4 (U^2 = (-1)^F exact in Cl(16), nonsplit; Z8 tower V^2 prop U, V^4 prop (-1)^F; R sector splits) -- the order-4 clock input reduces to ONE alignment bit ('deck central in the mark-D4'); the fermionic Z4 forces the order, not the alignment; QGEO-R1 == P2-R1 now = this single bit; no marker moves; v507 characterises the bit as a tetrad with a physical face (non-split NS Fock extension class) and derives the deck CLASS (half-period translation) -- only the position stays the input; v510 excludes the edge class topologically (covering-deck freeness = P1's closed seam circle): the bit reduces to the square-modulus datum tau = i alone
Dependencies: QGEO.REALIZE.01; QGEO.RIGID.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; SEAM.BIT.ORIGIN.01; v117; v168; v214; v215; v216; v490; v492; v493; v499; v503; v507; SEAM.BIT.FREEDOM.01; v510
Scripts: v506_seam_clock_rigidity.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 570Original ledger status
SEAM.TAU.FLAG.01
THE FLAG-TRANSITIVITY EQUIVALENCE WEB -- THE tau = i ATTACK EXECUTED ON THE FREE RP SEAM CIRCLE (2026-07-22; one module, 29 checks; NO marker moves -- the bit stays [C], QGEO.REALIZE.01 and P2.TYPING.01/AX.P2.01 text-precised only, SEAM.BIT.FREEDOM.01 cross-referenced): after v510 the last identification input is the square-modulus datum tau = i; which ESTABLISHED seam datum excludes the free-but-non-harmonic configurations? Test family: the deck-invariant 4-mark configurations on the free circle are, up to rotation, M(delta) = {+-1, +-e^(i delta)}, delta in (0, pi/2]; the v510 counterwitness {+-1, +-(3+4i)/5} is the member delta_cw = arctan(4/3). [E] FALSIFICATION OF BARE TRANSITIVITY: on the free RP circle the symmetry group is G(delta) = {z, -z, b/z, -b/z} ~= V4 for EVERY delta and the two inversions are pair-EXCHANGING -- V4 acts SIMPLY TRANSITIVELY on the four marks for every delta (complete census: any Moebius map commuting with the deck z -> -z is diagonal or antidiagonal); the expectation 'transitivity <= v216 local equality selects tau = i' is REFUTED; hardening: ALL undirected per-mark data (arc-pair multiset {delta, pi-delta}, chordal distance multiset, cone germ) are identical at the four marks for EVERY delta -- no local jet sees the bit. [E] THE SHARP SELECTOR: FLAG transitivity (marks AND their two sides indistinguishable) <=> delta = pi/2 <=> tau = i <=> the clock -- stabiliser jump 1 -> 2, flag/arc orbit collapse 2 -> 1, order-4 mark rotation (b^2 = -1 solveset {pi/2} exactly), mark-fixing mirror z -> 1/z <=> cos delta = 0. [E] 12-FOLD EQUIVALENCE WEB (every link exact): clock rotation, D4 (vs V4), mark mirror, flag transitivity, three spectral degeneracies (odd-doublet split of the seam covariance = (2/pi) ln cot(delta/2), zero only at pi/2 -- counterwitness exactly (2/pi) ln 2; arc-weighted seam-Laplacian degeneracy 4/pi; the v215-K3/v503 mod-4 indicator in CLOSED FORM (2 sqrt 2/pi)|ln tan(delta/2)| -- the v503 lattice curve analytically), rho-twist existence (mode identity (m^2 - n^2)(t - 1/t) = 0 <=> t = 1; Atiyah-Bott rationality is delta-BLIND -- Tr(sigma e^(t Delta)) = 1 for ALL delta, j_cw = 148176/25 rational non-CM: rationality does NOT select), cusp-4-cycle implementability (H4 rigidity: V4 with element orders {1,2,2,2} cannot induce the cusp 4-cycle; joint centraliser scalar, U^2 nonscalar => no projective involution -- the implementer has projective order exactly 4, the Z8-shadow consistent v510), harmonic, j = 1728. [E] NEGATIVE CONTROLS: hexagonal (marks not concyclic, fails the PRECONDITIONS of the battery -- consistent v503/v510); silver (harmonic but topologically dead -- position and modulus halves independent, consistent v510); 3 marks (impossible on the free circle: 0 invariant 3-subsets vs 28 invariant 4-subsets on Z16; on a plain circle bare transitivity ALREADY forces equal spacing at n = 3 -- '4 counts': the deck-pairing is exactly what makes bare transitivity free of charge at n = 4). [C] THE REST (verbatim, no overclaim): the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4: side-indistinguishability of the four equally-decorated marks); v216 local equality does NOT deliver it (no local jet can see the side bit); its negation is now concretely measurable (odd-split (2/pi) ln cot(delta/2) != 0). Scoreboard: the counterwitness passes all 7 established unsided tests and fails all 9 clock-equivalent ones; the free RP/Theta battery (v519/v521) is since documented as the EIGHTH side-blind test (7 -> 8), face #13 (the OS-reflection group closes to D4 exactly at the clock point) extends the equivalence web 12 -> 13 (v521), and the mark-decorated twist-state class (everything Wick-computable) is since documented as the NINTH side-blind test (8 -> 9, v525) -- the free-plus-twist class is exhausted. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (complete V4/D4 symmetry census, flag/arc orbit arithmetic, odd-split and K3-indicator closed forms, twist mode identity, H4 Schur rigidity, negative controls -- sympy exact + mpmath 30 digits, Wolfram-mirrored) + [C] (the symmetry-lift bit itself: flag transitivity stays a postulate; the gain is the FORM, not a derivation)
Dependencies, status detail and source
Canonical status: Flag transitivity [E]/[C]: bare mark-transitivity is FALSIFIED as a selector (pair-exchanging V4 simply transitive for every delta, no local jet sees the bit); the sharp selector is FLAG transitivity (marks AND sides) <=> delta = pi/2 <=> tau = i <=> the clock, welded into a 12-fold exact equivalence web (D4 enhancement, mark mirror, odd-split (2/pi) ln cot(delta/2), arc-Laplacian 4/pi, K3 indicator closed form, rho-twist existence, cusp-4-cycle implementability, harmonic, j = 1728) -- the last input reduces from a continuous modulus to ONE discrete symmetry-lift bit (V4 -> D4), concretely measurable in its negation; the bit stays the one [C] carrier input; the free RP/Theta battery (v521) is the eighth and the twist-state class (v525) the ninth side-blind test (the free-plus-twist class is exhausted); face #13 extends the web to 13; no marker moves
Dependencies: SEAM.BIT.FREEDOM.01; SEAM.BIT.ORIGIN.01; SEAM.CLOCK.RIGIDITY.01; QGEO.REALIZE.01; QGEO.EMERGE.LIGHT.01; P2.TYPING.01; AX.P2.01; v117; v215; v216; v483; v503; v506; v507; v510
Scripts: v512_seam_tau_flag.py
Location: introduction;tfpt_1;tfpt_3;tfpt_research_contracts
Original CSV, line 576Original ledger status
DIAMOND.PARA.SELFCODE.01
THE PARABOLIC ANCHOR SELF-CODE (2026-07-31; external structural review audited by discovery probe parabolic_selfcode_audit_probe.py 23/23, verdict SELFCODE-VERIFIED, and promoted; one module, 14 checks, ~1 s, exact integer/symbolic sympy, no floats). The v218 sheet-diamond direction operators U = Q diag(1,0,0) (rank-1 winding) and V = Q diag(0,1,1) (rank-2 sheet, Spec V = {0,1,2}) carry a previously unstated exact operator structure. (1) PARABOLIC THEOREM: A = Q<I,U,V> is the FULL line-stabiliser (maximal parabolic) algebra of block type 2|1 -- seven words independent (coordinate det -81), multiplicatively closed (49/49), all seven matrix units contained, dim 7 = 9-2; mutation of Q's third column breaks it. (2) DOUBLE SELF-CODE: Pi2 = V(V-I)/2 idempotent trace-1; H = I + Pi2 has spectrum EXACTLY {1,1,2} = a (ANCHOR.GEN.01/v23); chi_H = x^3 - 4x^2 + 5x - 2 with (e1,e2,e3) = (4,5,2) = (|mu4|, g_car, |Z2|) AND = (dim_R(GL3/P), dim(A/J), dim J) -- the anchor reproduced spectrally (dominant projector of V) and categorically (flag/Levi/radical of A) from ONE structure. (3) UNIQUENESS THEOREM: x^3 - 2m x^2 + (m^2+1) x - m = (x-m)(x^2-mx+1) identically; positive-integer spectra force m^2 - 4 = k^2, (m-k)(m+k) = 4, m = 2 UNIQUELY -- a = (1,1,2) is the only anchor this parabolic family can code. (4) THE 81 AS LATTICE INDEX: the SATURATED integral order (closure stable at word length 4) has elementary divisors (1,1,1,3,3,3,3): [P_Z : Z<U,V>] = 3^4 = 81, cokernel (Z/3)^4 -- equal to the v218 determinant-staircase sum 3+4+8+14+20+32 = N_fam^4 (cross-link typed; the mark-local mechanism, one Z/3 per mu4 mark, stays [O]: if no canonical local decomposition exists the 81 stays a fingerprint). (5) A2 BIT CARRIER: the basic algebra e A e is the A2 path algebra (dim Ext^1 = 1): dim-vector-(1,1) modules fall into EXACTLY two iso classes (split / one nonsplit) -- the alignment bit gains a canonical algebra carrier; uniqueness [E], SELECTION stays with RP/geometry (v528/v534) [C]. (6) TRACE CODE (re-encoding): p_n(a) = 2 + 2^n = 1 + tr(V^n); recurrence, Hankel 2^{n+1}; (1+trV)(1+trV^2)(1+trV^3) = 240 = |R(E8)|; tr(V^4-V^3) = 8 = rank E8 -- AnchorLadder read through V, one object fewer, NO new content. (7) CR TEST DESIGN (CONTRACT.F.01): CR(alpha_{-2},alpha_{-1};alpha_0,alpha_1) = 4/3 exactly on the v533 line; honesty wired (universal arithmetic-progression cross-ratio -- content = preregistered solver-state export); negative control 28/25; y_F forced fit-free. (8) LORENTZ FORM (reformulation): det X = t^2 - x^2 - y^2 under Phi; D(P,Q) = 2<Phi,Phi>_{1,2}; delta = sech^2(atanh r); pencil identities det(B-S) = det B det(I-Z), D(B,S) = det B tr Z, det S = det B det Z -- eigenvalue locking as the named prime-front instrument (no measurement, no RH statement). NAMED LIMITS AS CONTENT: NO P2 REDUCTION -- 'g_car = dim(A/J) derived' is conditional on the OPEN 'No Carrier Reconstruction' audit (rebuild Q/V from mu4, D4, Q_+ = A3 exponent grading, H^1, Q_- = sheet monodromy, WITHOUT g_car, a, rank E8 or carrier-aware budgets; Q's historical provenance runs through such budgets); AX.P2.01 and ANCHOR.GEN.01 markers UNTOUCHED; the 2..9 dimension table is COMPRESSION of one operator system, not nine independent hits (no-free-pattern rule); the bit selection, the mark-local 81 mechanism and the CR-test execution stay open and typed open; the word 'derived' appears nowhere for any physics statement. UPDATE (2026-07-31, v567): the review's Priority-1 audit is EXECUTED at Stage A -- verdict RECONSTRUCTS-TO-FAMILY: the anchor self-code is carrier-free; Q reconstructs up to one binary sheet-coupling choice; see QGEO.QRECON.01. FURTHER UPDATE (2026-07-31, v568): Priority 3 executed -- the A2 Ext arrow is NONZERO on the whole reconstruction class (the split member collapses to dim 5 and is excluded); Priority 2 sharpened -- the mu4-grading candidate fails, the F3[V]-action on the cokernel exists (see DIAMOND.BIT.SELECT.01). [2026-07-31: the S8 pencil identities are now executed on the REAL Paper-II surface by v569 (machine-tight at 2e-14 on 70 windows, verdict ONE-MODE); see PRIME.PENCIL.ONEMODE.01.] [2026-08-01 X: the per-mark Z/3 mechanism appears geometrically -- v597: the cusp-class group of the mu3-cover is (Z/3)^4 of order 81 with four free local classes; see QGEO.COVER.01.] [2026-08-01 XVI: the self-code algebra has a geometric realization -- v600: (U, V) as Gamma-even degree-2 cusp-Gram operators on the mu3-cover, GL3-conjugate to the compiler pair, with the SAME saturated Z-order structure (index 81 = 3^4, divisors (1,1,1,3,3,3,3)); existence [E], canonicity open; see QGEO.EMBED.01.]
Recorded status: Identity [E] (exact integer/symbolic algebra with mutation controls -- parabolic closure, double self-code, uniqueness enumeration + divisor argument, saturated Smith form, A2 quiver, trace code, cross-ratio identities, Lorentz/pencil identities) + [C] (the TFPT readings of the invariants; the bit-carrier and CR-test interpretations) + [O] (the No-Carrier-Reconstruction audit, the mark-local 81 mechanism, the CR-test execution); NO marker moves
Dependencies, status detail and source
Canonical status: [E] A = full 2|1 parabolic, dim 7, word det -81, closed 49/49, mutation-fragile; a = Spec(I + V(V-I)/2) = (1,1,2) with chi_H = x^3 - flag x^2 + Levi x - radical (4,5,2 double-coded); m = 2 unique (enumeration to 2000 + divisor argument); saturated Smith (1,1,1,3,3,3,3), index 81 = 3^4 = staircase sum; A2 Ext^1 = 1 (two module classes); p_n = 1 + tr V^n (240, rank 8 as traces); CR = 4/3 design with 28/25 control; Lorentz/pencil identities exact; 14/14 checks
Dependencies: DIAMOND.AXIS.01; ANCHOR.GEN.01; AX.P2.01; FLAV.QGEO.01; CONTRACT.F.01; SEAM.BIT.RPBLIND.01; FORM.LADDER.01; v23; v218; v224; v528; v533; v534
Scripts: v566_parabolic_anchor_selfcode.py
Location: tfpt_1;tfpt_2;tfpt_research_contracts
Original CSV, line 605Original ledger status
QGEO.QRECON.01
THE NO-CARRIER-RECONSTRUCTION AUDIT, STAGE A (2026-07-31; Priority 1 of the self-code review; one module, 9 checks, ~1 s, exact enumeration with integer prefilters + sympy verification; discovery probe carrier_free_q_reconstruction_probe.py 10/10, verdict RECONSTRUCTS-TO-FAMILY). THE QUESTION: can Q -- hence V and the v566 parabolic anchor self-code -- be reconstructed WITHOUT carrier/anchor inputs? THE CIRCULARITY LOCATED: v11's uniqueness certificate for Q enumerates over row sums (4,5,6), column sums (9,5,1) and chi_Q = (t-1)(t^2-5t+3) -- the 5 = g_car sits in the budgets and in chi_Q; all of it FORBIDDEN here. ALLOWED (each carrier-free, with corpus source): (S) Sigma = diag(1,-1,-1) [v10 sheet]; (W) Q(I+Sigma) = 2 N_fam W [v10 winding]; (E) Spec(Q_+) = {1,2,3} [FLAV.QGEO.01, A3 exponents]; (M) Q_- != 0, Q_-^2 = N_fam P, P^2 = P [FLAV.QGEO.01]; (Z) nonneg integers, bound 6 (robust at 9); (L) the seam-line demand [geometric, [C]]; (X) the saturation index N_fam^4 = 81 [v566 S4]. RESULTS [E]: (1) exactly 52 admissible matrices under (S)(W)(E)(M)(Z), corpus Q among them, first column forced to N_fam(1,1,1) by (W); (2) the seam-line demand cuts 52 -> 28 (real teeth); (3) THE ANCHOR IS CARRIER-FREE: all 28 parabolic members code the SAME anchor -- H' = I + Pi_dom(V') with char poly x^3 - 4x^2 + 5x - 2, spectrum (1,1,2) = a, ONE code class, no exceptions: the anchor self-code sits UPSTREAM of the carrier budgets, conditional on the declared geometric inputs; (4) THE RESIDUAL BIT: the 81-pin selects EXACTLY TWO members -- the corpus Q (q32 = 2 = |Z2|) and Q_alt (q32 = 0), difference matrix 2 E32 (the sheet-block coupling); every classical carrier-free certificate coincides on the pair (anticommutator spectrum {6,-4,-2} = (|R+(A3)|, -|mu4|, -|Z2|); commutator charpoly t(t^2+12); det = 3; saturation 81), while the ONE separating certificate is the carrier column budget (9,5,1) vs (9,3,1) -- exactly the forbidden input: the residual freedom is ONE binary coupled-vs-uncoupled sheet-block choice; (5) ablations: dropping (W) widens the bound-3 set 28 -> 1376 over 36 first columns (the winding input carries real load); bound robustness 6 -> 9 adds zero new (L)+(X) members; the forbidden inputs were evaluated nowhere. NAMED LIMITS AS LOAD-BEARING CONTENT: STAGE A ONLY -- whether (S)(W)(E)(M)(L) themselves derive from mu4/D4/H^1 parabolic geometry without family/sheet inputs is EXACTLY the open gate GATE.QGEO.01: the audit is RELOCATED there, not closed (N_fam = 3 is used throughout; family data was never forbidden -- the forbidden list is the CARRIER list); the carrier-free selector of the residual bit is OPEN, and its structural resonance with the split/nonsplit alignment-bit dichotomy (v528/v534, v566 S5) is a TYPED OBSERVATION, not a claim; NO P2 reduction -- AX.P2.01 and ANCHOR.GEN.01 markers untouched; the word 'derived' appears nowhere for any physics statement. UPDATE (2026-07-31, v568, DIAMOND.BIT.SELECT.01): the residual bit is located and given a canonical witness -- the twins generate different integral orders, and the ladder self-consistency (dominant mode = its own eigenvalue ladder (1,2,4)) selects the compiler Q uniquely on all 28 members; the split geometry (q32 = 1) is excluded by the demands already imposed; the physical selection stays open. Markers unchanged. [2026-07-31: Stage B narrowed by v574 -- the involution type is forced by a rank theorem and N_fam = 3 by the double self-consistency (ladder + sat = N^4); (E)(M)(L) stay the remaining gate; see QGEO.WINDING.SELF.01.]
Recorded status: Identity/enumeration [E] (exact admissible-set census 52/28/2 with integer prefilters verified by sympy on survivors; one anchor code class on all 28; separator dissection exact; ablations quantified) + [C] (the seam-line demand as declared geometric input; the bit-resonance observation) + [O] (Stage B = GATE.QGEO.01 relocation; the binary selector); verdict RECONSTRUCTS-TO-FAMILY; no marker moves
Dependencies, status detail and source
Canonical status: [E] 52 admissible under (S)(W)(E)(M)(Z) at B = 6; 28 under the seam-line demand; ONE anchor code x^3-4x^2+5x-2 = spectrum (1,1,2) on all 28 (carrier-free anchor); 81-pin -> exactly 2 (corpus Q vs q32 = 0 twin; all carrier-free certificates equal, separator = the forbidden column budget (9,5,1)); winding ablation 28 -> 1376 at bound 3; bound-stable at 9; 9/9 checks; verdict RECONSTRUCTS-TO-FAMILY
Dependencies: DIAMOND.PARA.SELFCODE.01; FLAV.QGEO.01; GATE.QGEO.01; FLAV.KQ.01; AX.P2.01; ANCHOR.GEN.01; v10; v11; v218; v566
Scripts: v567_carrier_free_q_reconstruction.py
Location: tfpt_1;tfpt_2
Original CSV, line 606Original 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
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
EXTREV.LATTICE.01
The external-review lattice audit (one module, 14 checks, <1 s; discovery probe external_lattice_audit_probe.py 14/14, verdict LATTICE-AUDIT-PASSED; external review 2026-08-02, three claimed derivations machine-checked, one follow-up retyped honestly). THE FOUR-MARK CLOSURE [E, conditional]: generalizing the glue to D_{d+1} + A_{d-1}, THREE independent selectors each force d = 4 -- the disc-group census (Z_d = Z_4 with cyclic D-side; negative controls d = 3, 5, 8), the even-glue norm equation (d+1)/4 + (d-1)/d = 2 (unique positive solution; no other even target admits an integer d), and the Pythagorean selector (d-1)^2 + d^2 = (d+1)^2 (the (3,4,5) = (cycles, marks, carrier) triple); (1,1,2) is the unique 3-part partition of 4 and c_3 = 1/(2 e_1 pi) = 1/(8 pi): under the named architecture the two-input presentation reduces to ONE boundary degree d = 4 plus pi; honest: conditional on the glue architecture, the residue moves to why the boundary realizes the minimal cyclically-glueable genus-0 divisor. THE ANCHOR BYTECODE [E]: chi_a = (t-1)^2(t-2) carries (4, 5, 2); p_n = 2 + 2^n with p_{n+2} = 3p_{n+1} - 2p_n; Newton recovers the e_k; p_1 p_2 p_3 = 240, p_4 - p_3 = 8, 240 + 8 = 248 (H-realization v602). THE LORENTZ CONGRUENCE [E + X]: P^T J_det P = J_fix EXACTLY with det P = -6 -- the prime-front determinant form and the v604 cover polarization lattice are the SAME rational quadratic form, the cover an index-6 sublattice (36 = 6^2, 6 = p_2(a)); the review's follow-up (integer R to B = M^dagger J) is ILL-POSED as stated (B COMPLEX symmetric, det 8 verified); partial positive: the third symmetric-Gauss ratio is an exact Q(omega) square; P-canonicity and the Hodge-chamber question are the named opens. CITATIONS VERIFIED: Suzuki arXiv 2606.09096 (screw-function Weil framework, Yoshida odd-criterion) and arXiv 2607.24830 (numerical realization) -- the basis of the PRIME.WEIL.OPERATOR contract. No marker moves; AX.P2.01 keeps its typing; no RH statement. [2026-08-02 XL: the prime-shadow audit lands -- v625: the 'primes appear AFTER E8' reading is exact at the theta level (Theta_E8 = E_4, shell counts 240 sigma_3(n) multiplicative, commuting Hecke channels with the theta as simultaneous eigenvector, L(E_4,s) = zeta(s) zeta(s-3)); the direction of explanation (geometry first, primes as readout) is fixed within the narrative; see PRIME.SHADOW.01.] [2026-08-02 XLI: the Lorentz congruence P^T J_det P = J_fix is additionally machine-proved in Lean (experiments/lean4-carrier-rigidity/TfptCarrier/LorentzCongruence.lean, kernel decide on the explicit integer matrices, lake build green, no sorry); P-canonicity and the operator construction of P executed by v635/v636 (see PRIME.PCANON.01 / PRIME.PCONSTRUCT.01).] [2026-08-02 XLV: look-elsewhere quantification -- v664: the observation 6 = Lorentz-congruence index = p_2(a) carries p_max = 0.169 under the most conservative null model: retyped as an observation WITHOUT statistical weight (the exact congruence P^T J_det P = J_fix itself is [E] and untouched).] [2026-08-28 CXL: v993 lifts the architecture conditionality at the census level -- full rank-8 ADE with cyclic Z4 both sides selects (D5, A3) uniquely; the D+A split is an OUTPUT. EXTREV.LATTICE.01's own v624 checks and typing UNCHANGED (they remain the architecture-conditional selectors).]
Recorded status: Exact [E] closure/bytecode/congruence identities + [X] the ill-posed retype + [C] conditional typing (sympy; citations web-verified)
Dependencies, status detail and source
Canonical status: LATTICE-AUDIT-PASSED: four-mark closure forces d = 4 three ways (conditional); bytecode exact (240 + 8 = 248); Lorentz congruence P^T J_det P = J_fix exact (index 6); B-test retyped ill-posed; Suzuki citations verified; 14/14 checks | 2026-08-28 v993: architecture conditionality lifted at census level (this row's v624 checks unchanged)
Dependencies: AX.P2.01; QGEO.EQORDER.01; QGEO.DUALFORMS.01; PRIME.UNIFC.01; v47; v602; v604; v618
Scripts: v624_external_lattice_audit.py
Location: tfpt_3;tfpt_research_contracts
Original CSV, line 640Original ledger status
TFPT.TOE.COMPLETE.01
THE MASTER TOE-COMPLETENESS CONTRACT [O] (registered 2026-08-29, review wave 4). Eight gates, all named; NOTHING claimed closed. T1 structure postulate -> P1/P2/compiler (AX.P1.01, AX.P2.01, TFPT.IRREDUCIBLE.01, DIMENSION.SELECTOR.4D.01). T2 seam = (E8)_1 proven (SEAM.EQUIV.01, SEAM.EQUIV.MMST.01, SEAM.EQUIV.TWISTOR.01, SEAM.SIMPLECURRENT.GENERATOR.01, SEAM.DETLINE.UNIFICATION.01, SEAM.STATE.RPMIXING.01). T3 quasilocal local unitary 3+1D Hamiltonian family (QFT4D.LATTICE.FUNDAMENTAL.01, TFPT4D.LATTICE.ACTION.01, DYN.UNITARY.DILATION.01, DYN.MARKOV.EMBED.01). T4 chiral SM with local measure + uniform mirror gap (CHIRAL4D.NOMIRROR.01). T5 IR Lorentz + confinement + clustering + nontrivial scattering (QFT4D.OS.RECON.01, SEAM.BULK4D.RECON.01). T6 all three gauge couplings + neutrino texture internally fixed (ALPHA.QUILLEN.EXACT.01, GAUGE.DETLINE.FIXPOINT.01, FLAV.NUSCALE.01-.06). T7 quantum massless spin-2 with universal coupling (GRAV.SPIN2.EMERGENCE.01, GRAV.NONCIRCULAR.01, GRAV.WEINBERG.WITTEN.01). T8 unique initial state + one SK functional generating all readouts (FTRANSFER.SK.RHO0.01, FTRANSFER.GENERATING.01, OBS.TRANSDUCTION.01, PRED.JOINTLIKELIHOOD.01). SEPARATION: TOE_Complete := AND(T1..T8); TOE_Validated := Complete AND independent holdouts passed. Dual rest: compiler Rest = v_geo oplus G_net oplus F_transfer sits beside Rest_TOE; this row is the named AND of the eight gates, not a closure. Children listed one-way here; children do not list this row back. EVENING (2026-08-30, v1012): gate-by-gate wave -- T5 four IR witnesses executed + the propagating-mode precondition (common-c needs dynamical gauge excitations); T7 interacting stress mechanism executed; T8 selection-principle vacuity typed (needs a non-entropic selection); T6/nu scaffold-level triple bound complete. Display stays [O]. LATE EVENING (2026-08-30, v1013/v1014): T3 mandatory-dynamics leg CLOSED at class level (LR + tau_t + A^G + state existence). Remaining axiom-core and continuum gaps now run through ONE externalized theorem (MMST/Quillen: TEL-B/ALG-EXH/P1 + character-blind det-response + Bismut-Freed/A0). Display stays [O]. MONDAY MORNING (2026-08-31, v1015): T1 structure-postulate route is now fully premise-supported at finite level (both W-bridge premises derived; axiom-core remainder 0 modulo externalized MMST). Display stays [O]. EXECUTED UPDATE (2026-09-05, v1022-v1025; no marker moves): T1 retains finite axiom-core/W-bridge support only; unconditional structure-postulate provenance and the dimension selector remain open. T2 closes C1, C2a and the interval-TV input and executes the analytic scalar C2b budget, while CF, DG and FE-GEN/ALG-EXH remain open and the old ALG2 Vandermonde implication is refuted. T3 remains Hamiltonian-class only. T4 gains a finite derivative-filtered block but no invariant mirror response gap, Weyl cone, DET realization, measure or continuum. T5 retains finite IR witnesses and the v1020 spectator obstruction but lacks prime completion, OS reconstruction, confinement and scattering. T6 gains a parent-internal intertwiner with projective full-Fock lift but no mass texture. T7 gains gap/TT no-gos but no graviton. T8 gains an arbitrary D4-breaking internal cap but no global rho_0. The historical one-external-theorem wording is restricted to the finite compiler/W-bridge lane. A single shared 3+1D parent satisfying T3-T8 is still missing. All T1-T8 gates and TOE remain Open [O]. ROUND-4 CURRENT (v1026-v1030): uniform relaxed TEL-B norm proved; canonical DET-CAR and signed z=1 wall only for fixed classical gauge fields with an explicitly changed additive parent; full Gauss site/cube dimensions3/0mod5; symmetric internal D4 cap noncanonical; local tensor target needs global zero-mode removal; ALG2 frame theorem conditional. All T1-T8 remain [O]; no shared complete3+1D parent.
Recorded status: Open (research contract; named AND of eight gates; no closure, no marker move) | Round-3 v1022-v1025 adds certified subgates and no-gos only; Round-4 v1026-v1030 adds narrow proved and conditional subresults only; all T1-T8 gates and TOE remain Open [O].
Dependencies, status detail and source
Canonical status: Open [O]: TOE_Complete := AND(T1..T8); TOE_Validated := Complete AND independent holdouts passed. Named AND of existing 4D/dilation/detline/SK/flavor/gravity/SK contracts; not a closure. Compiler Rest sits beside Rest_TOE. WAVE-5 (2026-08-29): T2 master scaling-limit theorem named SEAM.MMST.TYPEIII.CHARGED.01 [O] (children listed there, not back). Display unmoved. WAVE-6 (2026-08-30): T7 child GRAV.WEINBERG.WITTEN.01 [O] (Weinberg-Witten firewall before any spin-2 marker move). Display unmoved. | EVENING (2026-08-30 v1012): T5 four witnesses + propagating-mode precondition; T7 interacting stress mechanism; T8 selection-principle vacuity typed (needs a non-entropic selection); T6/nu scaffold-level triple bound complete. Display unmoved. | LATE EVENING (2026-08-30 v1013/v1014): T3 mandatory-dynamics leg closed at class level; remaining axiom-core and continuum gaps run through one externalized theorem (MMST/Quillen). Display unmoved. | MONDAY MORNING (2026-08-31 v1015): T1 structure-postulate route fully premise-supported at finite level. Display unmoved. | ROUND3 2026-09-05: current T1-T8 matrix recorded; no marker moves; shared 3+1D parent absent. The one-theorem statement is compiler/W-bridge scope only. | ROUND-4 CURRENT (v1026-v1030): uniform relaxed TEL-B norm proved; canonical DET-CAR and signed z=1 wall only for fixed classical gauge fields with an explicitly changed additive parent; full Gauss site/cube dimensions3/0mod5; symmetric internal D4 cap noncanonical; local tensor target needs global zero-mode removal; ALG2 frame theorem conditional. All T1-T8 remain [O]; no shared complete3+1D parent.
Dependencies: AX.P1.01; AX.P2.01; TFPT.IRREDUCIBLE.01; DIMENSION.SELECTOR.4D.01; SEAM.EQUIV.01; SEAM.EQUIV.MMST.01; SEAM.EQUIV.TWISTOR.01; SEAM.SIMPLECURRENT.GENERATOR.01; SEAM.DETLINE.UNIFICATION.01; SEAM.STATE.RPMIXING.01; QFT4D.LATTICE.FUNDAMENTAL.01; TFPT4D.LATTICE.ACTION.01; DYN.UNITARY.DILATION.01; DYN.MARKOV.EMBED.01; CHIRAL4D.NOMIRROR.01; QFT4D.OS.RECON.01; SEAM.BULK4D.RECON.01; ALPHA.QUILLEN.EXACT.01; GAUGE.DETLINE.FIXPOINT.01; FLAV.NUSCALE.01; FLAV.NUSCALE.04; FLAV.NUSCALE.03; FLAV.NUSCALE.02; FLAV.NUSCALE.05; FLAV.NUSCALE.06; GRAV.SPIN2.EMERGENCE.01; GRAV.NONCIRCULAR.01; GRAV.WEINBERG.WITTEN.01; FTRANSFER.SK.RHO0.01; FTRANSFER.GENERATING.01; OBS.TRANSDUCTION.01; PRED.JOINTLIKELIHOOD.01
Scripts: v1012_toe_gate_witnesses.py, v1013_thermodynamic_dynamics.py, v1014_bridge_refinements.py, v1015_axiom_core_closure.py, v1022_telb_round3_certificates.py, v1023_derivative_filtered_wall.py, v1024_parent_internal_boundaries.py, v1025_telb_c1_c2a_certificates.py, v1026_telb_round4_closure.py, v1027_signed_det_car_wall.py, v1028_gauss_d4_cap.py, v1029_local_tensor_constraints.py, v1030_alg2_sector_frame.py
Location: tfpt_research_contracts;tfpt_4_frontier;introduction
Original CSV, line 764Established within the stated premises
finding:pascal-carrier-rank-unique-unbounded-source-proof
CarrierRankUniqueness.carrier_rank_pascal_unique states and proves for every natural g that 2^g=g^2+g+2 iff g=5. Its proof combines the small cases with pascal_growth for all g>=6. This is unbounded; PascalLadder has a separate bounded K=2 statement. For g=5, N_fam=3 and g+N_fam=8 are computed.
Scope: Algebraic uniqueness under the stipulated Pascal/half-spinor exhaustion condition. Original source and proof text reviewed; no fresh Lean kernel run. AX.P2.01 remains a declared physical interface; the equation does not select itself from an original source.
Required premises and original source excerpts