The weight typing premise of the carrier

P2.TYPING.01

Original ledger classification: 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

Original claim text

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.

Original classification detail

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 as recorded

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