TFPT scientific source excerpt
Original repository path: verification/v486_transfer_full_rule.py
Original file SHA-256: a8315978fb4ecbb4d178754483b03915c1016995667ac1ab7a0ea6e8dacb5825
Inclusive original lines: 56-131
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This bounded excerpt is source evidence; it is not a new proof/checker run.
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000056 | def run():
000057 |     reset()
000058 |     print("v486  HYP.REWRITE.02: the full transfer spectrum from the unique "
000059 |           "lazy Z2-pair walk")
000060 | 
000061 |     s, h = sp.symbols('s h', positive=True)
000062 | 
000063 |     # 1. uniqueness: spectrum {2/3, 1/3} forces (s, h, leak) = (1/2, 1/6, 1/3)
000064 |     sol = sp.solve([sp.Eq(s + h, sp.Rational(2, 3)),
000065 |                     sp.Eq(s - h, sp.Rational(1, 3))], [s, h], dict=True)
000066 |     s0, h0 = sol[0][s], sol[0][h]
000067 |     leak = 1 - s0 - h0
000068 |     check("UNIQUENESS [E]: the symmetric Z2-pair rule M(s,h) has survival "
000069 |           "spectrum {s+h, s-h}; demanding the PHYSICAL pair {2/3, 1/3} "
000070 |           "(v54/v221) forces UNIQUELY (stay, hop, leak) = (1/2, 1/6, 1/3) = "
000071 |           "(1/|Z2|, 1/(|Z2| N_fam), 1/N_fam) -- the lazy Z2-pair walk; atom "
000072 |           "form recorded (direction: spectrum -> rates)",
000073 |           len(sol) == 1 and s0 == sp.Rational(1, Z2)
000074 |           and h0 == sp.Rational(1, Z2 * N_fam)
000075 |           and leak == sp.Rational(1, N_fam))
000076 | 
000077 |     # 2. six-hand generation of the full physical spectrum
000078 |     B = sp.Matrix([[s0, h0], [h0, s0]])
000079 |     eigs6 = sorted((B ** HAND).eigenvals().keys(), key=lambda v: -sp.nsimplify(v))
000080 |     gap2 = sp.nsimplify(-sp.log(eigs6[0]))
000081 |     gap3 = sp.nsimplify(-sp.log(eigs6[1]))
000082 |     check("SIX-HAND GENERATION [E]: over the order-6 = 2 N_fam = |R+(A3)| "
000083 |           "hand the rule generates EXACTLY the physical spectrum eig(B^6) = "
000084 |           "{64/729, 1/729} = {(2/3)^6, (1/3)^6}; decay gaps 6 ln(3/2) (the "
000085 |           "recovery gap, v76) and 6 ln 3 (the subdominant gap, now "
000086 |           "generator-full)",
000087 |           eigs6 == [sp.Rational(64, 729), sp.Rational(1, 729)]
000088 |           and sp.simplify(gap2 - HAND * sp.log(sp.Rational(3, 2))) == 0
000089 |           and sp.simplify(gap3 - HAND * sp.log(3)) == 0)
000090 | 
000091 |     # 3. supersession of the uniform rule as full-spectrum generator
000092 |     uniform = sp.Matrix([[sp.Rational(1, 3), sp.Rational(1, 3)],
000093 |                          [sp.Rational(1, 3), sp.Rational(1, 3)]])
000094 |     eigs_u = set(uniform.eigenvals().keys())
000095 |     powers_dead = all(sp.Integer(0) in (uniform ** k).eigenvals()
000096 |                       for k in (1, 2, 3, 6))
000097 |     check("SUPERSESSION [E]: v327's uniform rule (1/3, 1/3) has spectrum "
000098 |           "{2/3, 0} -- the zero mode persists under every power, so NO "
000099 |           "iterate can produce the (1/3)^6 mode: the uniform rule stays the "
000100 |           "correct lambda_2 story (2/3 = |Z2|/N_fam from the arity) but is "
000101 |           "superseded as generator of the FULL spectrum",
000102 |           eigs_u == {sp.Rational(2, 3), sp.Integer(0)} and powers_dead)
000103 | 
000104 |     # 4. stochastic sanity
000105 |     M = sp.Matrix([[1, 0, 0], [0, s0, h0], [0, h0, s0]])
000106 |     rowsums = [sum(M.row(i)) for i in range(3)]
000107 |     check("STOCHASTIC SANITY [E]: the lazy rule is sub-stochastic (rates "
000108 |           ">= 0; row sums {1, 2/3, 2/3}), symmetric on the pair (detailed "
000109 |           "balance), and keeps the absorbing democratic family attractor of "
000110 |           "v327/v317 (eigenvalue 1)",
000111 |           all(v >= 0 for v in M) and rowsums == [1, sp.Rational(2, 3),
000112 |                                                  sp.Rational(2, 3)]
000113 |           and 1 in M.eigenvals())
000114 | 
000115 |     # 5. honest residual
000116 |     check("HONEST RESIDUAL [O]: the arity {|Z2|, N_fam} = {2, 3} stays the "
000117 |           "anchor input (as in v327); the lazy split (1/2, 1/6) vs uniform "
000118 |           "(1/3, 1/3) is SELECTED by the physical spectrum, not yet derived "
000119 |           "from a first principle -- named candidate: 'stay = 1/|Z2|' as the "
000120 |           "sheet-symmetric choice of the Z2 involution (recorded, not "
000121 |           "claimed)", {Z2, N_fam} == {2, 3})
000122 | 
000123 |     return summary("v486 HYP.REWRITE.02: the FULL transfer spectrum "
000124 |                    "{1,(2/3)^6,(1/3)^6} has one local generator -- the unique "
000125 |                    "lazy Z2-pair walk (stay, hop, leak) = (1/2, 1/6, 1/3) = "
000126 |                    "(1/|Z2|, 1/(|Z2| N_fam), 1/N_fam); over the order-6 hand "
000127 |                    "eig(B^6) = {64/729, 1/729} exactly; the previously "
000128 |                    "generator-less (1/3)^6 mode gets its origin (subdominant "
000129 |                    "gap 6 ln 3); v327's uniform rule = degenerate {2/3, 0} "
000130 |                    "case, superseded as full-spectrum generator; arity + "
000131 |                    "split selection stay [O]")