Boundary normalization through index and KMS data

P1.INDEX.KMS.01

Original ledger classification: Exact [E] identity chain (formal Fraction x pi^k) + gated independence audit + measured finite evenness precursor; OPEN [O] contract with the one open bridge (even modular-response normalization) and the kill criterion named; c3 stays the P1 axiom

Original claim text

P1.INDEX.KMS.01: the closed normalization identity c3 = 1/(I_Jones * beta_angle) = 1/(4 * 2pi) = 1/(8pi), its convention audit, and the sharpened P1 contract, one probe (13/13 checks incl. the 2 must-fire controls, ~0.1 s; discovery probe p1_index_kms_probe.py, 2026-08-06, verdict P1-IDENTITY-CLOSED). THE IDENTITY [E]: I * beta * c3 = 4 * 2pi * 1/(8pi) = 1 EXACT in formal Fraction x pi^k arithmetic (no floats; S2.1-S2.3 are one identity in three rearrangements, counted as one content; the v208 area-entropy factor 1/4 = 1/I; the AX.P1.01 anchor route 1/(2 e1(a) pi) with e1 = 4 aligns anchor-4 with index-4). THE DEPENDENCY AUDIT (the load-bearing honesty step): the index I = 4 = |mu4| is an INDEPENDENT c3-free theorem (ZERO 'c3' occurrences in all four sources v154/v726/v746/v779, gated by file scan); beta_angle = 2pi is an INDEPENDENT measurement (v526: beta = N clock steps by detailed balance, wrong-beta {N/2, N/4, 2N} excluded; the v519 radian pin is the ONE shared convention and CANCELS in the unit-invariant statement 'response per full modular circle = 1/I'); c3 STAYS the P1 axiom -- the identity EXPLAINS its value, it does not derive it. THE FINITE EVENNESS PRECURSOR (measured, real): the tracial weight per mu4 deck sector on the l-windows satisfies w_1 = w_3 = 1/4 EXACTLY at every level l = 1..6, with the w_0/w_2 offset the symmetric zero-sum pair +-2^-(l+1) halving per level -- equal quarters asymptotically, NO anomaly offset (= the no-anomaly shape the contract's bridge needs); the index-4 conditional expectation is trace-preserving (dev < 1e-12); and the SEA STATE does NOT equidistribute (deviation 0.188 from 1/4 at the N = 1536 Haar pullback) -- the bridge must normalize the MODULAR RESPONSE, not vacuum weights. THE CONTRACT (registered [O]): an index-I simple-current extension carries per modular period exactly the normalized response 1/(I * beta_angle) = c3; the ONE open bridge is the even distribution of the modular-response (Quillen/determinant-line) normalization over the I deck sectors; KILL: any asymmetric sector offset (w_1 != w_3 at any level, or a non-vanishing offset as l -> inf). Controls fire: the wrong index 2 gives 1/(4pi), which the corpus books elsewhere (un-halved Gauss-Bonnet; SdS 1/(4pi) = 2 c3); wrong beta = pi breaks the chain at 1/2. c3 remains the P1 axiom; no marker move, NO RH claim. Python-only per GATE.WOLFRAM.02. NOTE (2026-08-28, v993): unique finite CE does NOT close the equidistribution bridge -- that identification stays the open [C] of this row.

Original classification detail

P1-IDENTITY-CLOSED: I*beta*c3 = 1 exact; the index c3-free (zero occurrences, four sources), beta independently measured (v526, wrong-beta excluded; the radian pin is the one shared convention and cancels); c3 stays the P1 axiom the identity EXPLAINS; finite evenness precursor measured: w1 = w3 = 1/4 exactly every level, symmetric zero-sum offset +-2^-(l+1) -- the no-anomaly shape; the sea state does NOT equidistribute (dev 0.188) -- the bridge must normalize the modular response; controls: index 2 -> 1/(4pi), booked elsewhere by the corpus; contract registered OPEN [O], kill = an asymmetric sector offset.

Dependencies as recorded

AX.P1.01; GATE.METRIC.11; GATE.METRIC.12; ANCHOR.GEN.01; v154; v726; v746; v779; v526; v239; v208; v8; v23