A registered result in its source context

HYP.REWRITE.02

Original ledger classification: Construction/uniqueness [E]/[O] (full spectrum from the unique lazy Z2-pair walk; split selection stays open)

Original claim text

THE FULL TRANSFER SPECTRUM FROM ONE FORCED LOCAL RULE (2026-07-15, generator-hunt round; sharpens HYP.REWRITE.01/v327): the verified seam transfer spectrum {1, (2/3)^6, (1/3)^6} (v54/v221) now has a single local recursive generator -- the LAZY Z2-PAIR WALK. [E] UNIQUENESS: for the 3-channel rule M(s,h) = [[1,0,0],[0,s,h],[0,h,s]] (one absorbing family channel + symmetric Z2 pair) the survival spectrum is {s+h, s-h}; demanding the PHYSICAL pair {2/3, 1/3} forces UNIQUELY (stay, hop, leak) = (1/2, 1/6, 1/3) = (1/|Z2|, 1/(|Z2| N_fam), 1/N_fam) -- all three rates exact atom expressions (recorded; honest direction: spectrum -> rates). [E] SIX-HAND GENERATION: over the order-6 = 2 N_fam = |R+(A3)| dynamic hand (v327's hand) the rule generates EXACTLY eig(B^6) = {64/729, 1/729} = {(2/3)^6, (1/3)^6}; the two decay gaps are 6 ln(3/2) (the recovery gap, v76) and 6 ln 3 (the subdominant gap -- previously WITHOUT any local-rule origin). [E] SUPERSESSION: v327's uniform rule (s = h = 1/3, spectrum {2/3, 0}) has a persistent zero mode under every power -- no iterate can produce the (1/3)^6 mode; its lambda_2 content (2/3 = |Z2|/N_fam from the arity) is unchanged; superseded as generator of the FULL spectrum. [E] sub-stochastic, symmetric (detailed balance on the pair), absorbing democratic family attractor unchanged (v317). [O] HONEST RESIDUAL: the arity {|Z2|, N_fam} = {2, 3} stays the anchor input (as in v327); ADDITIONALLY the lazy split (1/2, 1/6) vs the uniform split (1/3, 1/3) is SELECTED by the physical spectrum, not yet derived from a first principle -- named candidate: 'stay = 1/|Z2|' as the sheet-symmetric choice of the Z2 involution (recorded, not claimed). Discovery provenance: experiments/tfpt-discovery/generator_hunt_full_transfer_rule.py (with the anchor-bilinear coverage inventory and the lepton-exponent (5,3,2) = K-row = atom-triple observation, both typed, no unforced claims). No gate closes, no marker moves. UPDATE 2026-07-15 (v487/HYP.REWRITE.03): the split selection [O] CLOSES -- the lazy rule's one-step spectrum is the complete v124 clock ladder below the wall and ladder faithfulness + Z2 equivariance + rates >= 0 force (1/2, 1/6) uniquely (the uniform rule collapses the odd channel onto the wall); only the arity and the clock's semiclassical derivation remain open.

Original classification detail

Full-spectrum generator [E]: unique lazy Z2-pair walk (1/2, 1/6, 1/3); (1/3)^6 mode generator-full (gap 6 ln 3); v327 rule = degenerate case; arity [O]; split selection CLOSED by v487 (clock-ladder faithfulness)

Dependencies as recorded

HYP.REWRITE.01; HYP.FIBER.01; HYP.INJECT.01; DYN.SEMIGROUP.01; v54; v76; v221; v312; v317; v324; v327; v395