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Original ledger classification: Construction/proof [E]/[O] (split forced by clock-ladder faithfulness; arity + clock derivation stay the open inputs)

Recorded statement: THE LAZY SPLIT FORCED BY THE RESUMMED CLOCK (2026-07-15, closing the v486 [O] 'split selection'; sharpens HYP.REWRITE.02): the split (stay, hop) = (1/2, 1/6) is not a choice -- it is clock-ladder faithfulness. [E] RUNG READING: the lazy…

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Paper context: Origin Theory

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HYP.REWRITE.03

Original ledger classification: Construction/proof [E]/[O] (split forced by clock-ladder faithfulness; arity + clock derivation stay the open inputs)

Original claim text

THE LAZY SPLIT FORCED BY THE RESUMMED CLOCK (2026-07-15, closing the v486 [O] 'split selection'; sharpens HYP.REWRITE.02): the split (stay, hop) = (1/2, 1/6) is not a choice -- it is clock-ladder faithfulness. [E] RUNG READING: the lazy rule's one-step spectrum is EXACTLY the complete v124 resummed-clock ladder below the wall, eig(M) = {1 - n/3 : n = 0, 1, 2} = {1, 2/3, 1/3}; v327's uniform rule instead puts its second pair mode ON the wall (0 = 1 - 3/3) -- the v486 degeneracy re-read as 'odd channel collapsed to the wall'. [E] DECK PARITY = RUNG INDEX: [B, sigma] = 0; the sigma-even pair mode carries rung n = 1 (survival 2/3), the sigma-odd mode rung n = 2 (survival 1/3) -- each clock step flips the deck parity; the structural home of the v486 parity assignment (omega_1 even-channel, omega_2 odd-channel) tested by the frb-parity-comb search. [E] FORCING: demanding the faithful ladder (no rung missing, no physical mode on the wall -- definitional for a full-spectrum generator) + Z2 equivariance (symmetric pair block) + non-negative rates forces UNIQUELY both the split (1/2, 1/6) AND the rung assignment even -> 1, odd -> 2 (the swapped assignment needs hop = -1/6 < 0, inadmissible); the v486 named candidate 'stay = 1/|Z2| as the sheet-symmetric choice' is superseded by this proof. [E] BEND COROLLARY: omega_1/omega_2 = rate(2)/rate(1) = log_{3/2} 3 = 2.70951... -- the two comb tones of the empirical search program are one v107/v124 bend apart. [O] HONEST RESIDUAL: the channel count 3 = N_fam is forced by the clock wall (v124 pt 4); the ladder itself stays an [I] identity on frozen spectra whose semiclassical derivation is open (R1/[P]); the arity {|Z2|, N_fam} = {2, 3} stays the anchor input. NET: no independent choice remains at the walk level. Contract provenance: experiments/theory-contracts/lazy01_clock_rung_forcing.py. No gate closes, no marker moves.

Original classification detail

Split forcing [E]: eig(M) = complete clock ladder {1, 2/3, 1/3}; uniform rule = odd mode on the wall; deck parity = rung index; assignment uniqueness via h >= 0; omega_1/omega_2 = bend; [O] arity + semiclassical clock derivation

Dependencies as recorded

HYP.REWRITE.02; HYP.REWRITE.01; v124; v107; v327; v486; v54; v221

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Registered verification modules

  • v487_transfer_clock_rungs.pyRegistered verification module · core
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    HYP.REWRITE.03: the lazy split (1/2, 1/6) is FORCED by the resummed clock -- the v486 [O] 'split selection' closes onto v124 structure. [E] rung reading: the lazy rule's one-step spectrum is EXACTLY the complete v124 clock ladder below the wall, eig(M) = {1 - n/3 : n = 0,1,2} = {1, 2/3, 1/3}; v327's uniform rule puts its second pair mode ON the wall (0 = 1 - 3/3). [E] deck parity = rung index: the sigma-even mode carries rung 1 (survival 2/3), the sigma-odd mode rung 2 (survival 1/3) -- each clock step flips the deck parity (the structural home of the v486 parity assignment tested by frb-parity-comb). [E] forcing: faithful ladder (definitional for a full-spectrum generator) + Z2 equivariance + rates >= 0 force uniquely both the split (1/2, 1/6) and the assignment even -> rung 1, odd -> rung 2 (the swap needs hop = -1/6 < 0). [E] bend corollary: omega_1/omega_2 = rate(2)/rate(1) = log_{3/2} 3 -- the parity-comb tone ratio IS the v107/v124 bend. [O] channel count 3 forced by the wall; the ladder stays an [I] identity on frozen spectra (R1/[P] open); arity {2,3} stays the anchor input.

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