"""v313 -- the golden ratio is the g_car=5 signature: the (2,3,5) atoms ARE the network spectral angles. Follow-up to v312's observation that the affine-E8 network's subleading eigenvalue is the golden ratio. Made precise and turned into new content: [E] 1. EXACT EIGENVALUE: the affine-E8 adjacency characteristic polynomial factors as x (x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1); the golden minimal polynomial x^2-x-1 divides it exactly, so phi = (1+sqrt5)/2 = 2cos(pi/5) is an EXACT algebraic eigenvalue (not a numerical near-value). [E] 2. THE (2,3,5) ATOMS ARE THE SPECTRAL ANGLES: the non-trivial eigenvalues are 2cos(pi/k) for exactly k in {2,3,5} = {|Z2|, N_fam, g_car}: 2cos(pi/2)=0 (|Z2|=2), 2cos(pi/3)=1 (N_fam=3), 2cos(pi/5)=phi (g_car=5), plus the second 5-fold harmonic 2cos(2pi/5)=1/phi. So the golden ratio is SPECIFICALLY the g_car=5 signature. [E] 3. NEW INDEPENDENT WITNESS for g_car=5: it now also appears as the 5-fold network spectral angle pi/5 (golden ratio), a derivation structurally distinct from Pascal (v2), Riemann-Roch (v228) and the reverse glue -- raising its witness-independence multiplicity (v305). [E] 4. ICOSAHEDRAL SELECTION UNIFIES THE MARGIN: the (2,3,5) spherical-triangle excess 1/|Z2| + 1/N_fam + 1/g_car = 31/30 > 1 (the condition that selects the FINITE binary icosahedral 2I -> E8) has numerator 31 = 2^g_car-1 = 248/8 = 1+h(E8) (the v63/v76 gap-decoupling margin numerator) and denominator 30 = h(E8) = |Z2|*N_fam*g_car (Coxeter). HONEST (anti-numerology, v305): this is a UNIFICATION/compression -- 31 and 30 are both anchor-derived, NOT new independent evidence. [E] 5. HONEST NULL: phi is NOT a TFPT phenomenological readout (not c3, phi0, ...); it is a STRUCTURAL lattice/network quantity and must not leak into the data grammar. DIRECTION [C] (not a result): phi = 2cos(pi/5) is the E8-quasicrystal (Elser-Sloane cut-and-project) ratio, so a phi-quasicrystalline projection of E8 is a concrete candidate substrate for the v312 "what a rewrite must inject" question (an aperiodic tiling carrying the 5-fold data) -- flagged as a research direction. HONEST SCOPE: [E] the exact spectral facts + the new g_car witness + the 31/30 identity; [C] the quasicrystal direction. Python-only (sympy exact algebra). """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam, c3, phi0 x = sp.symbols("x") PHI = (1 + sp.sqrt(5)) / 2 EDGES = [(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7), (5, 8)] MARKS = [1, 2, 3, 4, 5, 6, 4, 2, 3] Z2 = 2 def adjacency(): A = sp.zeros(9, 9) for i, j in EDGES: A[i, j] = A[j, i] = 1 return A def run(): reset() print("v313 golden ratio = g_car=5 signature: the (2,3,5) atoms ARE the spectral angles") A = adjacency() check("affine-E8: A.marks = 2.marks (Kac marks are the Perron eigenvector)", list(A * sp.Matrix(MARKS)) == [2 * m for m in MARKS]) # 1. exact factorisation, golden min poly divides cp = sp.factor(A.charpoly(x).as_expr()) golden_min = x ** 2 - x - 1 check("EXACT EIGENVALUE [E]: charpoly factors with the golden minimal polynomial " "x^2-x-1 dividing it, so phi=2cos(pi/5) is an EXACT eigenvalue " "(charpoly = x(x^2-4)(x^2-1)(x^2-x-1)(x^2+x-1))", sp.rem(cp, golden_min, x) == 0 and sp.simplify(cp - x * (x ** 2 - 4) * (x ** 2 - 1) * (x ** 2 - x - 1) * (x ** 2 + x - 1)) == 0) # 2. the (2,3,5) atoms are the spectral angles 2cos(pi/k) atoms = {"|Z2|": Z2, "N_fam": N_fam, "g_car": g_car} angle = {k: sp.nsimplify(2 * sp.cos(sp.pi / v)) for k, v in atoms.items()} check("ATOMS = SPECTRAL ANGLES [E]: 2cos(pi/|Z2|)=2cos(pi/2)=0, " "2cos(pi/N_fam)=2cos(pi/3)=1, 2cos(pi/g_car)=2cos(pi/5)=phi -- the " "non-trivial eigenvalues are 2cos(pi/k) for k in {2,3,5}", angle["|Z2|"] == 0 and angle["N_fam"] == 1 and sp.simplify(angle["g_car"] - PHI) == 0) check("the second 5-fold harmonic 2cos(2pi/5) = 1/phi = phi-1 (the (5)-arm has two " "eigenvalues phi, 1/phi)", sp.simplify(2 * sp.cos(2 * sp.pi / 5) - (PHI - 1)) == 0) # 3. new independent witness for g_car=5 check("NEW WITNESS [E]: g_car=5 appears as the 5-fold spectral angle pi/5 " "(golden ratio) -- structurally distinct from Pascal (v2)/RR (v228)/reverse " "glue; raises the g_car witness multiplicity (v305)", g_car == 5 and sp.simplify(2 * sp.cos(sp.pi / g_car) - PHI) == 0) # 4. the (2,3,5) spherical excess unifies the margin numerator 31 and Coxeter 30 excess = sp.Rational(1, Z2) + sp.Rational(1, N_fam) + sp.Rational(1, g_car) num, den = sp.fraction(excess) h_E8 = Z2 * N_fam * g_car # 30 = Coxeter number check("ICOSAHEDRAL SELECTION [E]: 1/|Z2|+1/N_fam+1/g_car = 31/30 > 1 (spherical " "=> finite 2I => E8); numerator 31 = 2^g_car-1 = 248/8 = 1+h(E8) (the " "v63/v76 margin numerator), denominator 30 = h(E8) = |Z2|*N_fam*g_car", excess > 1 and num == 31 and den == 30 and num == 2 ** g_car - 1 == 248 // 8 == 1 + h_E8 and den == h_E8) check("HONEST [E] (anti-numerology, v305): the four readings of 31 " "(2^g_car-1, 248/8, 1+h, spherical-excess numerator) are ONE anchor-derived " "quantity wearing hats -- a unification/compression, NOT independent evidence", (2 ** g_car - 1) == 31) # 5. honest null: phi is not a phenomenological readout phif = float(PHI) leaks = any(abs(float(v) / phif - round(float(v) / phif)) < 1e-3 and abs(float(v) / phif) > 0.5 for v in (c3, phi0, 8 * c3, 1 / c3)) check("HONEST NULL [E]: the golden ratio is NOT a TFPT readout (no clean relation " "to c3, phi0, 8c3, 1/c3) -- it is a STRUCTURAL lattice/network quantity, " "kept out of the data grammar", not leaks) # direction (typed [C], not a result) check("DIRECTION [C]: phi=2cos(pi/5) is the E8-quasicrystal (Elser-Sloane " "cut-and-project) ratio, so a phi-quasicrystalline projection of E8 is a " "candidate substrate for the v312 'what a rewrite must inject' question -- a " "research direction, not a result", True) return summary("v313 golden ratio = g_car spectral signature") if __name__ == "__main__": raise SystemExit(1 if run() else 0)