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Original ledger classification: 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)

Recorded statement: 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 --…

Statement, assumptions and sources

Paper context: TFPT — Introduction and reading guide

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P2.PARTITION.01

Original ledger classification: 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)

Original claim text

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.

Original classification detail

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 as recorded

AX.P2.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; v23; v53; v115; v137; v216; v228

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Registered verification modules

  • v491_p2_partition_corollary.pyRegistered verification module · core
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    P2.PARTITION.01: g_car = 5 as a COROLLARY of the four marks -- the partition sharpening of v53 (sympy exact). The lemma: exactly 3 positive integers summing to 4 = uniquely {1,1,2} => e = (4,5,2) => g_car = e2 = 5, |Z2| = e3 = 2, N_fam = e2-e3 = 3 as corollaries; consistency fixed point (2^4-1)/5 = 3 = rank H^1. Negative controls (each assumption load-bearing): drop positivity -> 4 solutions, e2 in {0,3,4,5}; sum 3 -> e2 = 3; sum 5 -> not unique; 2/4 parts -> e2 {3,4}/{6}; Mehta-Seshadri rational weights -> 16/3 resp. 44/9 != 5 (integrality = Birkhoff-Grothendieck splitting-type typing). [E] v53 sharpening: v53 needs all three targets (4,5,2), the partition only e1 = 4. [E] sum rule 4 x (N-1)/2 = 4 iff N = 3; v115 route 4 tr(A0) = 4. Independence: #marks = 4 (v216, P1-side), rank H^1 = 3 (v137/v228), |a|_1 = 4 g_car-free (v115). [C] honest residual: P2 NOT axiom-free -- the weight-typing postulate remains; AX.P2.01 stays a declared axiom; N_fam direction reversal typed as over-determination. Exact (sympy); Wolfram-mirrored.

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