"""v486 -- HYP.REWRITE.02: the FULL physical transfer spectrum from ONE forced local rule -- the lazy Z2-pair walk. The (1/3)^6 mode, previously generator-less, gets its local-rule origin; v327's uniform rule is the degenerate boundary case, superseded as generator of the FULL spectrum. Sharpens HYP.REWRITE.01; no gate closes. Background. The verified seam transfer spectrum is {1, (2/3)^6, (1/3)^6} (v54/v221). v327 exhibited a minimal local rewrite for the DOMINANT decaying mode: its rule (uniform Z2-pair block, entries 1/3) has spectrum {1, 2/3, 0} -- it generates lambda_2 and KILLS lambda_3. The third mode had no local generator. Discovery result (generator hunt, 2026-07-15): [E] 1. 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 s = 1/2, hop h = 1/6, leak 1-s-h = 1/3, and the three rates are exactly the atom expressions s = 1/|Z2|, h = 1/(|Z2| N_fam), leak = 1/N_fam. (Honest direction: spectrum -> rates. That the unique rates land on atom expressions is recorded [E] as arithmetic; a first-principles forcing of the RULE is the [O].) [E] 2. SIX-HAND GENERATION: over the order-6 = 2 N_fam = |R+(A3)| dynamic hand (v327's hand) the rule generates EXACTLY the physical spectrum: 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 -- now with a generator). [E] 3. SUPERSESSION: v327's uniform rule (s = h = 1/3) is the DEGENERATE boundary case (spectrum {2/3, 0}); no power of it can produce the (1/3)^6 mode. Its lambda_2 content is unchanged (2/3 = |Z2|/N_fam emerges from the arity); the lazy rule extends it to the full spectrum with the SAME arity input. [E] 4. STOCHASTIC SANITY: the lazy rule is sub-stochastic (all rates >= 0, row sums 1 and 2/3), symmetric (detailed balance on the pair), with the absorbing family attractor of v327/v317 unchanged. [O] 5. HONEST RESIDUAL: as in v327 the arity {|Z2|, N_fam} = {2, 3} is the anchor input; ADDITIONALLY the lazy split (1/2, 1/6) vs the uniform split (1/3, 1/3) is here SELECTED by the physical spectrum, not yet derived from a first principle (candidate: the pair carries the Z2 sheet involution, so 'stay' = 1/|Z2| is the sheet-symmetric choice -- named, not claimed). NET EFFECT: the complete verified transfer spectrum {1, (2/3)^6, (1/3)^6} now has a single local recursive generator with atom rates; the family sector's dynamic content is one lazy walk iterated over the hexagon hand. 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 HAND = 2 * N_fam # 6 = |R+(A3)| def run(): reset() print("v486 HYP.REWRITE.02: the full transfer spectrum from the unique " "lazy Z2-pair walk") s, h = sp.symbols('s h', positive=True) # 1. uniqueness: spectrum {2/3, 1/3} forces (s, h, leak) = (1/2, 1/6, 1/3) sol = sp.solve([sp.Eq(s + h, sp.Rational(2, 3)), sp.Eq(s - h, sp.Rational(1, 3))], [s, h], dict=True) s0, h0 = sol[0][s], sol[0][h] leak = 1 - s0 - h0 check("UNIQUENESS [E]: the symmetric Z2-pair rule M(s,h) has survival " "spectrum {s+h, s-h}; demanding the PHYSICAL pair {2/3, 1/3} " "(v54/v221) forces UNIQUELY (stay, hop, leak) = (1/2, 1/6, 1/3) = " "(1/|Z2|, 1/(|Z2| N_fam), 1/N_fam) -- the lazy Z2-pair walk; atom " "form recorded (direction: spectrum -> rates)", len(sol) == 1 and s0 == sp.Rational(1, Z2) and h0 == sp.Rational(1, Z2 * N_fam) and leak == sp.Rational(1, N_fam)) # 2. six-hand generation of the full physical spectrum B = sp.Matrix([[s0, h0], [h0, s0]]) eigs6 = sorted((B ** HAND).eigenvals().keys(), key=lambda v: -sp.nsimplify(v)) gap2 = sp.nsimplify(-sp.log(eigs6[0])) gap3 = sp.nsimplify(-sp.log(eigs6[1])) check("SIX-HAND GENERATION [E]: over the order-6 = 2 N_fam = |R+(A3)| " "hand the rule generates EXACTLY the physical spectrum eig(B^6) = " "{64/729, 1/729} = {(2/3)^6, (1/3)^6}; decay gaps 6 ln(3/2) (the " "recovery gap, v76) and 6 ln 3 (the subdominant gap, now " "generator-full)", eigs6 == [sp.Rational(64, 729), sp.Rational(1, 729)] and sp.simplify(gap2 - HAND * sp.log(sp.Rational(3, 2))) == 0 and sp.simplify(gap3 - HAND * sp.log(3)) == 0) # 3. supersession of the uniform rule as full-spectrum generator uniform = sp.Matrix([[sp.Rational(1, 3), sp.Rational(1, 3)], [sp.Rational(1, 3), sp.Rational(1, 3)]]) eigs_u = set(uniform.eigenvals().keys()) powers_dead = all(sp.Integer(0) in (uniform ** k).eigenvals() for k in (1, 2, 3, 6)) check("SUPERSESSION [E]: v327's uniform rule (1/3, 1/3) has spectrum " "{2/3, 0} -- the zero mode persists under every power, so NO " "iterate can produce the (1/3)^6 mode: the uniform rule stays the " "correct lambda_2 story (2/3 = |Z2|/N_fam from the arity) but is " "superseded as generator of the FULL spectrum", eigs_u == {sp.Rational(2, 3), sp.Integer(0)} and powers_dead) # 4. stochastic sanity M = sp.Matrix([[1, 0, 0], [0, s0, h0], [0, h0, s0]]) rowsums = [sum(M.row(i)) for i in range(3)] check("STOCHASTIC SANITY [E]: the lazy rule is sub-stochastic (rates " ">= 0; row sums {1, 2/3, 2/3}), symmetric on the pair (detailed " "balance), and keeps the absorbing democratic family attractor of " "v327/v317 (eigenvalue 1)", all(v >= 0 for v in M) and rowsums == [1, sp.Rational(2, 3), sp.Rational(2, 3)] and 1 in M.eigenvals()) # 5. honest residual check("HONEST RESIDUAL [O]: the arity {|Z2|, N_fam} = {2, 3} stays the " "anchor input (as in v327); the lazy split (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: 'stay = 1/|Z2|' as the " "sheet-symmetric choice of the Z2 involution (recorded, not " "claimed)", {Z2, N_fam} == {2, 3}) return summary("v486 HYP.REWRITE.02: the FULL transfer spectrum " "{1,(2/3)^6,(1/3)^6} has one local generator -- the unique " "lazy Z2-pair walk (stay, hop, leak) = (1/2, 1/6, 1/3) = " "(1/|Z2|, 1/(|Z2| N_fam), 1/N_fam); over the order-6 hand " "eig(B^6) = {64/729, 1/729} exactly; the previously " "generator-less (1/3)^6 mode gets its origin (subdominant " "gap 6 ln 3); v327's uniform rule = degenerate {2/3, 0} " "case, superseded as full-spectrum generator; arity + " "split selection stay [O]") if __name__ == "__main__": raise SystemExit(1 if run() else 0)