"""v427 -- OVERDET.WITNESS.MAP.01: the honest over-determination accounting. A reviewer's sharpest methodological point: "every independent restriction multiplies the implausibility of coincidence" is only true when the witnesses come from GENUINELY DISJOINT grammars; witnesses that are projections of the SAME generator count as COMPRESSION, not Bayesian multiplication. This module makes that split machine-explicit -- the auditable map behind the over-determination claim. >>> REFINED (and partly walked back) BY v428 (OVERDET.WITNESS.RECLASS.01). An early version of this module classified the seven arithmetic witnesses below as DISJOINT grammars that MULTIPLY. Applying this module's own multiply-vs-compress test to itself (v428) shows that is wrong: by v236 (TOPO.BRIESKORN.01) the (2,3,5) Brieskorn singularity is the ONE generator behind the order-30 clock, so the seven are different number-theoretic READINGS of that one object -- they COMPRESS, they do not multiply. The check texts below therefore state the arithmetic only (each grammar's value is correct) and DEFER the multiply/compress verdict to v428. What honestly multiplies is the INPUT forced several independent ways (the "8" in c3 from rank(E8), h(D5), phi(30), Milnor) plus foreign witnesses (alpha^-1). See v428 for the corrected accounting. [E] 1. SEVEN GRAMMAR READOUTS. Seven syntactically distinct grammars each reproduce a carrier-skeleton integer, each computed FROM ITS OWN syntax: Gauss Z[i]: N(3+2i) = 3^2+2^2 = 13 (=Delta_Q) Eisenstein Z[w]: N(3+2w) = 3^2-3*2+2^2 = 7 (hex norm) Cyclotomy Q(z5): N(3+2 z5) = 2^4 Phi5(-3/2) = 55 (quark numerator) Galois (Z/5)^x: |Gal Q(z5)| = phi(5) = 4 (=|mu4|) Root lattice: |det Cartan(E8)| = 1 (holomorphic) Pascal/exterior: C(4,0)+C(4,1)+C(4,2) = 11; 2^4 = 16 = dim S+ Coxeter: h(E8) = 30 = 2*3*5; phi(30) = 8 = rank E8 (Whether these MULTIPLY or COMPRESS is settled in v428: compression.) [E] 2. SYNTACTIC DISTINCTNESS (NOT independence). The seven are pairwise different syntaxes (two quadratic rings, a degree-4 field, a finite group, a root lattice, a binomial/exterior algebra, a Coxeter invariant) -- but by v236 they are facets of ONE (2,3,5)/E8 object, so syntactic distinctness is NOT statistical independence (corrected in v428). [E] 3. COMPRESSION CLASS. The anchor a=(1,1,2) emits MANY readouts from ONE generator: elementary-symmetric (e1,e2,e3)=(4,5,2) (e2=g_car), power sums, and the E8 data (240 roots, 248 dim, rank 8) -- a compression GAIN, not independent witnesses. The flavor matrix R likewise (trace, det, minors from one R). v428 shows the seven grammar readouts above belong to THIS class too. [E] 4. THE HONEST SPLIT. The framework -- multiplicative witnesses vs. compression readouts -- is the right axis; the CLASSIFICATION of the seven is corrected by v428 (they compress). The genuinely multiplicative evidence is the input forced several independent ways (v428) + foreign witnesses, NOT one undifferentiated "everything multiplies". [C] 5. VERDICT. The multiply-vs-compress framework stands; this module's CLASSIFICATION of the seven arithmetic witnesses is refined by v428 (they are readings of one (2,3,5)/E8 object = compression). The auditable map still replaces the rhetorical claim and shows coincidence is an expensive explanation of the discrete core, but it does NOT by itself establish physics (the seam/anchor/transfer bridge problem) -- and the multiplicative strength is the multiply-forced inputs + foreign witnesses (v428), not the seven readings. Python-only (sympy/numpy exact arithmetic). """ import numpy as np import sympy as sp from tfpt_constants import check, summary, reset, g_car SKELETON = {1, 2, 3, 4, 5, 7, 8, 11, 13, 16, 30, 55, 240, 248} def cartan_E8(): return np.array([ [2, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0], [0, -1, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0], [0, 0, 0, 0, 0, -1, 2, -1], [0, 0, 0, 0, 0, 0, -1, 2]], dtype=float) def run(): reset() print("v427 OVERDET.WITNESS.MAP: disjoint-grammar witnesses multiply, " "same-generator projections compress") a, b = 3, 2 # ---- 1+2. seven disjoint-grammar witnesses, each from its own grammar ---- gauss = a**2 + b**2 # N(3+2i)=13 eisen = a**2 - a * b + b**2 # N(3+2w)=7 phi5 = lambda x: x**4 + x**3 + x**2 + x + 1 cyclo = int(b**4 * phi5(sp.Rational(-a, b))) # N(3+2 z5)=55 galois = int(sp.totient(5)) # |(Z/5)^x|=4 latt = int(round(np.linalg.det(cartan_E8()))) # |det Cartan E8|=1 pascal = sum(int(sp.binomial(4, kk)) for kk in range(3)) # 11 spinor = 2**4 # 16 = dim S+ coxeter = 30 # h(E8) cox_factor = sorted(sp.factorint(coxeter).keys()) # [2,3,5] cox_phi = int(sp.totient(coxeter)) # 8 = rank E8 witnesses = { "Gauss Z[i] N(3+2i)": gauss, "Eisenstein Z[w] N(3+2w)": eisen, "Cyclotomy Q(z5) N(3+2 z5)": cyclo, "Galois |(Z/5)^x|": galois, "Root lattice |det Cartan E8|": latt, "Pascal C(4,<=2)": pascal, "Coxeter phi(h(E8))": cox_phi, } landed = all(v in SKELETON for v in witnesses.values()) values_ok = (gauss == 13 and eisen == 7 and cyclo == 55 and galois == 4 and latt == 1 and pascal == 11 and spinor == 16 and cox_factor == [2, 3, 5] and cox_phi == 8) check("SEVEN GRAMMAR READOUTS [E]: seven distinct syntaxes each reproduce " "a carrier-skeleton integer FROM ITS OWN grammar -- N(3+2i)=13, N(3+2w)=7, " "N(3+2 z5)=55, |(Z/5)^x|=4, |det Cartan E8|=1, C(4,<=2)=11 (2^4=16), " "h(E8)=30=2*3*5 with phi(30)=8=rank E8 -- whether they MULTIPLY or COMPRESS " "is settled in v428 (compression: facets of one (2,3,5)/E8 object, v236)", landed and values_ok and len(witnesses) == 7) grammars = ["quadratic ring Z[i]", "quadratic ring Z[w]", "degree-4 field Q(z5)", "finite group (Z/5)^x", "root lattice E8", "exterior/binomial", "Coxeter invariant"] check("SYNTACTIC DISTINCTNESS [E]: the seven witnesses are pairwise distinct " "syntaxes (%d named grammars); but per v236 they are facets of ONE " "(2,3,5)/E8 object, so syntactic distinctness is NOT statistical " "independence -- the multiply/compress verdict is corrected in v428" % len(set(grammars)), len(set(grammars)) == 7) # ---- 3. compression class: one generator -> many readouts ---- anchor = [1, 1, 2] e1 = sum(anchor) e2 = anchor[0] * anchor[1] + anchor[0] * anchor[2] + anchor[1] * anchor[2] e3 = anchor[0] * anchor[1] * anchor[2] p1 = sum(x for x in anchor) p2 = sum(x**2 for x in anchor) compression = (e1 == 4 and e2 == 5 and e3 == 2 and p1 == 4 and p2 == 6 and e2 == g_car) check("COMPRESSION CLASS [E]: the anchor a=(1,1,2) emits MANY readouts from ONE " "generator -- elementary-symmetric (e1,e2,e3)=(4,5,2) (e2=g_car=5), power " "sums (p1,p2)=(4,6), and the E8 data (240,248,rank 8) -- a compression " "gain, NOT independent witnesses (they do not multiply)", compression) # ---- 4. the honest split ---- n_independent = len(witnesses) check("THE HONEST SPLIT [E]: the AXIS is right (multiplicative witnesses vs. " "compression readouts); the CLASSIFICATION of these %d grammar readouts is " "corrected by v428 -- they compress one (2,3,5)/E8 object. The genuinely " "multiplicative evidence is the input forced several independent ways + " "foreign witnesses (v428), not one undifferentiated 'everything multiplies'" % n_independent, n_independent >= 7 and compression and landed) # ---- 5. verdict (typed [C]) ---- check("VERDICT [C]: the multiply-vs-compress framework stands; this module's " "classification of the seven arithmetic witnesses is refined by v428 (they " "are readings of one (2,3,5)/E8 object = compression). The map still shows " "coincidence is an expensive explanation of the discrete core, but the " "multiplicative strength is the multiply-forced inputs + foreign witnesses " "(v428), and it does NOT by itself establish physics (the seam/anchor/" "transfer bridge problem)", values_ok and compression and n_independent >= 7) return summary("v427 OVERDET.WITNESS.MAP (seven grammar readouts; classification " "refined by v428 -- they compress one (2,3,5)/E8 object; the " "auditable over-determination map)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)