"""v487 -- 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. The walk's one-step spectrum is the COMPLETE clock ladder below the wall; deck parity = rung index; the two search tones are one bend apart. Sharpens HYP.REWRITE.02; no gate closes. Background. v486 exhibited the unique lazy Z2-pair walk (stay, hop, leak) = (1/2, 1/6, 1/3) generating the full transfer spectrum {1, (2/3)^6, (1/3)^6}, but left the SPLIT SELECTION open: '(1/2, 1/6) vs uniform (1/3, 1/3) is selected by the physical spectrum, not yet derived from a first principle' (named candidate: sheet-symmetric choice -- vague). The v124 resummed clock reads the transfer eigenvalues as lambda_n = (1 - n/N_fam)^{p_2} with a POLE (wall) at n = N_fam. This module proves the split selection is clock-ladder faithfulness -- definitional for a full-spectrum generator: [E] 1. RUNG READING: the one-step spectrum of the lazy rule is EXACTLY the complete 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] 2. DECK PARITY = RUNG INDEX: the sigma-even pair mode (1,1) carries rung n = 1 (survival 2/3), the sigma-odd mode (1,-1) carries rung n = 2 (survival 1/3); each clock step n -> n+1 flips the deck parity. This is the structural home of the v486 parity assignment (omega_1 even-channel, omega_2 odd-channel) tested by the frb-parity-comb search. [E] 3. FORCING: demanding (i) the faithful ladder (no rung missing, no physical mode on the wall -- definitional for a full-spectrum generator), (ii) Z2 equivariance (symmetric pair block), (iii) non-negative rates, BOTH the split (s, h) = (1/2, 1/6) AND the assignment even -> rung 1, odd -> rung 2 are UNIQUE (the swapped assignment needs h = -1/6 < 0). The v486 'sheet-symmetric choice' candidate is superseded by this proof; the walk-level [O] closes. [E] 4. BEND COROLLARY: the two comb tones of the empirical search program are one established number apart, omega_1 / omega_2 = rate(2)/rate(1) = log_{3/2} 3 (the v107/v124 bend, 2.70951...) -- the tone ratio of the parity-comb search IS the bend of the pulsar-clock search. [O] 5. HONEST RESIDUAL: the channel count 3 = N_fam is forced by the clock wall (v124 pt 4); the ladder itself is an [I] identity on frozen spectra whose semiclassical derivation stays 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 (2026-07-15). Exact (sympy); mirrorable, deferred to the next verified Wolfram engine run. """ import sympy as sp from tfpt_constants import check, summary, reset, N_fam, g_car Z2 = g_car - N_fam # |Z2| = 2 P2 = 6 # |R+(A3)| hexagon size (v124) def rung(n): return sp.Rational(N_fam - n, N_fam) def rate(n): return -P2 * sp.log(rung(n)) def run(): reset() print("v487 HYP.REWRITE.03: the lazy split is forced by the resummed " "clock (ladder faithfulness); deck parity = rung index") s0, h0 = sp.Rational(1, 2), sp.Rational(1, 6) M = sp.Matrix([[1, 0, 0], [0, s0, h0], [0, h0, s0]]) Mu = sp.Matrix([[1, 0, 0], [0, sp.Rational(1, 3), sp.Rational(1, 3)], [0, sp.Rational(1, 3), sp.Rational(1, 3)]]) # 1. rung reading eigs = set(M.eigenvals().keys()) ladder = {rung(n) for n in range(N_fam)} wall = rung(N_fam) eigs_u = set(Mu.eigenvals().keys()) check("RUNG READING [E]: 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) -- the v486 " "degeneracy re-read as 'odd channel collapsed to the wall'", eigs == ladder and wall not in eigs and wall in eigs_u and rung(2) not in eigs_u) # 2. deck parity = rung index sigma = sp.Matrix([[0, 1], [1, 0]]) B = sp.Matrix([[s0, h0], [h0, s0]]) lam_even = (B * sp.Matrix([1, 1]))[0] lam_odd = (B * sp.Matrix([1, -1]))[0] check("DECK PARITY = RUNG INDEX [E]: [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; frb-parity-comb)", sp.simplify(B * sigma - sigma * B) == sp.zeros(2, 2) and lam_even == rung(1) and lam_odd == rung(2)) # 3. forcing (split + assignment unique) s, h = sp.symbols('s h', real=True) sol_A = sp.solve([sp.Eq(s + h, rung(1)), sp.Eq(s - h, rung(2))], [s, h], dict=True) sol_B = sp.solve([sp.Eq(s + h, rung(2)), sp.Eq(s - h, rung(1))], [s, h], dict=True) check("FORCING [E]: faithful ladder (no rung missing, no physical mode " "on the wall -- definitional for a full-spectrum generator) + Z2 " "equivariance + rates >= 0 force UNIQUELY both the split (s, h) " "= (1/2, 1/6) AND the assignment even -> rung 1, odd -> rung 2 " "(the swapped assignment needs h = -1/6 < 0, inadmissible); the " "v486 [O] 'split selection' closes at the walk level", len(sol_A) == 1 and sol_A[0][s] == sp.Rational(1, 2) and sol_A[0][h] == sp.Rational(1, 6) and sol_A[0][h] >= 0 and sol_B[0][h] < 0) # 4. bend corollary omega_1 = 2 * sp.pi / rate(1) omega_2 = 2 * sp.pi / rate(2) bend = sp.log(3) / sp.log(sp.Rational(3, 2)) check("BEND COROLLARY [E]: the two comb tones of the search program " "are one bend apart: omega_1/omega_2 = rate(2)/rate(1) = " "log_{3/2} 3 = 2.70951... (v107/v124) -- the parity-comb tone " "ratio IS the pulsar-clock bend", sp.simplify(omega_1 / omega_2 - bend) == 0) # 5. honest residual check("HONEST RESIDUAL [O]: the channel count 3 = N_fam is forced by " "the clock wall (v124 pt 4); the ladder is an [I] identity on " "frozen spectra with the semiclassical derivation open (R1/[P]); " "the arity {|Z2|, N_fam} = {2, 3} stays the anchor input; no " "independent choice remains at the walk level", {Z2, N_fam} == {2, 3} and sp.limit(-P2 * sp.log(1 - sp.Symbol('x') / N_fam), sp.Symbol('x'), N_fam, dir='-') == sp.oo) return summary("v487 HYP.REWRITE.03: the lazy split (1/2, 1/6) is " "FORCED -- the walk's spectrum is the complete v124 " "clock ladder below the wall (uniform rule = odd mode " "on the wall); deck parity = rung index (home of the " "parity assignment); omega_1/omega_2 = bend; the v486 " "walk-level [O] closes onto v124 structure (arity + " "clock derivation stay the open inputs)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)