#!/usr/bin/env python3 """ verify_P183.py — Addendum 183: G1_P171 Near-Miss PSLQ Investigation ======================================================================= Investigates G1_P171 = j(i)/(4·BREATH_PERIOD) - 1 ≈ 0.3456% and the near-miss δ = j(i) - 4·BREATH_PERIOD ≈ 5.951 ≈ h(G₂) = 6. TOE constants (exact): ALPHA_INV = 4π³ + π² + π BREATH_PERIOD = π · ALPHA_INV = 4π⁴ + π³ + π² 4·BREATH_PERIOD = 16π⁴ + 4π³ + 4π² j(i) = 1728 (exact integer) h(G₂) = 6 (Coxeter number of G₂) Copyright: Léon Fernando Vlegels. License: MIT. May 2026. """ import sys from mpmath import mp, pi, mpf, gamma, sqrt, log, zeta, exp, identify, pslq, nstr, sin mp.dps = 55 # 55 decimal places PASS = FAIL = 0 def check(n, desc, cond): global PASS, FAIL ok = bool(cond); PASS += ok; FAIL += (not ok) print(f" [{'PASS' if ok else 'FAIL'}] {n:>2}. {desc}") # --------------------------------------------------------------------------- # 1. Core constants # --------------------------------------------------------------------------- print("S1 Core TOE constants at mp.dps=55") ALPHA_INV = 4*pi**3 + pi**2 + pi BREATH_PERIOD = pi * ALPHA_INV # = 4π⁴ + π³ + π² FOUR_BP = 4 * BREATH_PERIOD # = 16π⁴ + 4π³ + 4π² j_i = mpf(1728) # j(τ=i) = 12³ exact J_short = mpf(12) h_G2 = mpf(6) # Coxeter number of G₂ = J_short/2 h_F4 = mpf(12) # Coxeter number of F₄ = J_short G1 = j_i / FOUR_BP - 1 # G1_P171 delta = j_i - FOUR_BP # the near-miss gap residual = delta - h_G2 # how far δ is from 6 frac_res = abs(residual) / abs(delta) # fraction of gap that is residual print(f"ALPHA_INV = {nstr(ALPHA_INV, 50)}") print(f"BREATH_PERIOD = {nstr(BREATH_PERIOD, 50)}") print(f"4·BREATH_PERIOD = {nstr(FOUR_BP, 50)}") print(f"j(i) = {j_i} (exact)") print(f"G1_P171 = {nstr(G1, 50)}") print(f"G1_P171 (percent) = {nstr(G1 * 100, 50)} %") print(f"δ = j(i) - 4·BP = {nstr(delta, 50)}") print(f"h(G₂) = {h_G2} (exact)") print(f"δ - h(G₂) = {nstr(residual, 50)}") print(f"|δ - h(G₂)| / |δ| = {nstr(frac_res, 50)}") print(f"4·BREATH_PERIOD - 1722 = {nstr(FOUR_BP - 1722, 50)}") print() # --------------------------------------------------------------------------- # 2. Auxiliary transcendentals # --------------------------------------------------------------------------- print("S2 Auxiliary transcendentals for PSLQ bases") Gamma14 = gamma(mpf('1/4')) Gamma34 = gamma(mpf('3/4')) log2 = log(mpf(2)) logpi = log(pi) zeta3 = zeta(3) e2pi = exp(2*pi) # δ_QLC from P175: sin⁴(π/14) basis elements sin_pi14 = sin(pi/14) sin4_pi14 = sin_pi14**4 print(f"Γ(1/4) = {nstr(Gamma14, 50)}") print(f"Γ(3/4) = {nstr(Gamma34, 50)}") print(f"log(2) = {nstr(log2, 50)}") print(f"log(π) = {nstr(logpi, 50)}") print(f"ζ(3) = {nstr(zeta3, 50)}") print(f"e^(2π) = {nstr(e2pi, 50)}") print(f"sin(π/14) = {nstr(sin_pi14, 50)}") print(f"sin⁴(π/14) = {nstr(sin4_pi14, 50)}") print() # --------------------------------------------------------------------------- # 3. PSLQ — Basis A: trivial polynomial check # --------------------------------------------------------------------------- print("S3 PSLQ basis A: {1, δ, π, π², π³, π⁴}") print(" Expected: PSLQ recovers 1728·1 - 1·δ - 16·π⁴ - 4·π³ - 4·π² = 0") basis_A = [mpf(1), delta, pi, pi**2, pi**3, pi**4] labels_A = ['1', 'δ', 'π', 'π²', 'π³', 'π⁴'] rel_A = pslq(basis_A, maxcoeff=200, maxsteps=1000) if rel_A is not None: expr_A = " + ".join(f"({rel_A[k]})*{labels_A[k]}" for k in range(len(rel_A)) if rel_A[k] != 0) residual_A = sum(rel_A[k] * basis_A[k] for k in range(len(rel_A))) print(f" Relation found: {rel_A}") print(f" Symbolic: {expr_A} = 0") print(f" Verification residual: {nstr(residual_A, 10)}") print(" → VERDICT A: Trivial polynomial identity recovered ✓") else: print(f" No relation found (maxcoeff=200)") print(" → VERDICT A: Unexpected — PSLQ failed to recover trivial identity") check(1, "basis A recovers the trivial polynomial identity at maxcoeff=200 " "(script-designated expectation; see VERDICT A above)", rel_A is not None) print() # --------------------------------------------------------------------------- # 4. PSLQ — Basis B: residual (δ−6) over transcendental basis # --------------------------------------------------------------------------- print("S4 PSLQ basis B: {δ−6, π, π², π³, π⁴, Γ(1/4), log(2), ζ(3)}") print(" Question: Is the residual (δ−6) expressible over CM/Gamma basis?") res6 = delta - h_G2 # = δ − 6 basis_B = [res6, pi, pi**2, pi**3, pi**4, Gamma14, log2, zeta3] labels_B = ['δ−6', 'π', 'π²', 'π³', 'π⁴', 'Γ(1/4)', 'log(2)', 'ζ(3)'] rel_B = pslq(basis_B, maxcoeff=500, maxsteps=1000) if rel_B is not None: expr_B = " + ".join(f"({rel_B[k]})*{labels_B[k]}" for k in range(len(rel_B)) if rel_B[k] != 0) residual_B = sum(rel_B[k] * basis_B[k] for k in range(len(rel_B))) print(f" Relation found: {rel_B}") print(f" Symbolic: {expr_B} = 0") print(f" Verification residual: {nstr(residual_B, 10)}") print(" → VERDICT B: STRUCTURED — δ−6 has expression over this basis") else: print(f" No relation found (maxcoeff=500)") print(" → VERDICT B: No relation over Gamma/CM basis at coeff ≤ 500") print() # --------------------------------------------------------------------------- # 5. PSLQ — Basis C: cross-check with δ_QLC from P175 # --------------------------------------------------------------------------- print("S5 PSLQ basis C: {G1, ALPHA_INV, π, π², π³, Γ(1/4), sin(π/14), sin⁴(π/14)}") print(" Question: Is G1_P171 related to P175's δ_QLC basis?") basis_C = [G1, ALPHA_INV, pi, pi**2, pi**3, Gamma14, sin_pi14, sin4_pi14] labels_C = ['G1', 'α⁻¹', 'π', 'π²', 'π³', 'Γ(1/4)', 'sin(π/14)', 'sin⁴(π/14)'] rel_C = pslq(basis_C, maxcoeff=500, maxsteps=1000) if rel_C is not None: expr_C = " + ".join(f"({rel_C[k]})*{labels_C[k]}" for k in range(len(rel_C)) if rel_C[k] != 0) residual_C = sum(rel_C[k] * basis_C[k] for k in range(len(rel_C))) print(f" Relation found: {rel_C}") print(f" Symbolic: {expr_C} = 0") print(f" Verification residual: {nstr(residual_C, 10)}") print(" → VERDICT C: STRUCTURED — G1_P171 lives in this basis") else: print(f" No relation found (maxcoeff=500)") print(" → VERDICT C: No relation over P175 δ_QLC basis at coeff ≤ 500") print() # --------------------------------------------------------------------------- # 6. PSLQ — Basis D: Is δ = 6·(something)? # --------------------------------------------------------------------------- print("S6 PSLQ basis D: {δ/6, π, π², ALPHA_INV/12, h_G2=6, h_F4=12}") print(" Question: Is δ a rational multiple of 6 (Coxeter number)?") basis_D = [delta/6, pi, pi**2, ALPHA_INV/12, h_G2, h_F4] labels_D = ['δ/6', 'π', 'π²', 'α⁻¹/12', '6', '12'] rel_D = pslq(basis_D, maxcoeff=500, maxsteps=1000) if rel_D is not None: expr_D = " + ".join(f"({rel_D[k]})*{labels_D[k]}" for k in range(len(rel_D)) if rel_D[k] != 0) residual_D = sum(rel_D[k] * basis_D[k] for k in range(len(rel_D))) print(f" Relation found: {rel_D}") print(f" Symbolic: {expr_D} = 0") print(f" Verification residual: {nstr(residual_D, 10)}") print(" → VERDICT D: STRUCTURED — δ is rational multiple of 6 / combination") else: print(f" No relation found (maxcoeff=500)") print(" → VERDICT D: δ is NOT a simple rational multiple of 6 at coeff ≤ 500") print() # --------------------------------------------------------------------------- # 7. PSLQ — Basis E: extended, with e^{-π√163} # --------------------------------------------------------------------------- print("S7 PSLQ basis E: {δ−6, π, π², π³, π⁴, ALPHA_INV, Γ(1/4), log(2), e^{-π√163}}") print(" Question: Is the residual (δ−6) expressible over extended basis?") e_heegner = exp(-pi * sqrt(mpf(163))) basis_E = [res6, pi, pi**2, pi**3, pi**4, ALPHA_INV, Gamma14, log2, e_heegner] labels_E = ['δ−6', 'π', 'π²', 'π³', 'π⁴', 'α⁻¹', 'Γ(1/4)', 'log(2)', 'e^{-π√163}'] rel_E = pslq(basis_E, maxcoeff=1000, maxsteps=2000) if rel_E is not None: expr_E = " + ".join(f"({rel_E[k]})*{labels_E[k]}" for k in range(len(rel_E)) if rel_E[k] != 0) residual_E = sum(rel_E[k] * basis_E[k] for k in range(len(rel_E))) print(f" Relation found: {rel_E}") print(f" Symbolic: {expr_E} = 0") print(f" Verification residual: {nstr(residual_E, 10)}") print(" → VERDICT E: STRUCTURED — δ−6 has expression over extended basis") else: print(f" No relation found (maxcoeff=1000)") print(" → VERDICT E: No relation at coeff ≤ 1000 over extended basis") print() # --------------------------------------------------------------------------- # 8. mpmath.identify probes # --------------------------------------------------------------------------- print("S8 mpmath.identify probes") probes = { 'G1_P171' : G1, 'δ' : delta, 'δ−6' : residual, 'G1·π' : G1 * pi, 'G1·π²' : G1 * pi**2, 'δ/π' : delta / pi, '(δ−6)/π²' : residual / pi**2, '(δ−6)/Γ(1/4)' : residual / Gamma14, '(δ−6)·α⁻¹' : residual * ALPHA_INV, '6·G1' : 6 * G1, } for name, val in probes.items(): result = identify(val, tol=1e-14) print(f" identify({name}) = {result if result else 'no match'}") print() # --------------------------------------------------------------------------- # 9. Structural check: j(i) = 4π²·(4π²+π+1) + 6 ? # --------------------------------------------------------------------------- print("S9 Structural near-identity check") print(" Is j(i) = 4π·ALPHA_INV + h(G₂)? i.e., 1728 = 4πα⁻¹ + 6 ?") lhs = j_i rhs_check = FOUR_BP + h_G2 diff_check = lhs - rhs_check print(f" j(i) = {j_i}") print(f" 4π·ALPHA_INV + 6 = {nstr(rhs_check, 50)}") print(f" Difference (j(i) - rhs) = {nstr(diff_check, 50)}") print(f" Is j(i) = 4·BP + 6 exact? {'YES' if diff_check == 0 else 'NO — diff = ' + nstr(diff_check, 20)}") print() # Factor: 4·BREATH_PERIOD = 4π²(4π²+π+1) factor_check = 4*pi**2 * (4*pi**2 + pi + 1) print(f" 4π²(4π²+π+1) = {nstr(factor_check, 50)}") print(f" 4·BREATH_PERIOD = {nstr(FOUR_BP, 50)}") print(f" Difference = {nstr(FOUR_BP - factor_check, 15)}") print() # --------------------------------------------------------------------------- # 10. Final summary # --------------------------------------------------------------------------- print("S10 Final summary") print(f" G1_P171 = j(i)/(4·BREATH_PERIOD) - 1") print(f" = {nstr(G1, 50)}") print(f" ≈ {float(G1*100):.6f} %") print() print(f" δ = j(i) - 4·BREATH_PERIOD = {nstr(delta, 50)}") print(f" h(G₂) = 6 (exact Coxeter number)") print(f" δ - h(G₂) = {nstr(residual, 50)}") print(f" |δ - h(G₂)| / |δ| = {nstr(frac_res, 15)}") print() print(" Basis A (trivial polynomial): relation recovered ✓ (as expected)") print(" Basis B (δ−6 over CM/Gamma):", "STRUCTURED" if rel_B else "NO RELATION — COINCIDENCE evidence") print(" Basis C (G1 over P175 basis):", "STRUCTURED" if rel_C else "NO RELATION") print(" Basis D (δ/6 Coxeter probe) :", "STRUCTURED" if rel_D else "NO RELATION") print(" Basis E (δ−6 extended) :", "STRUCTURED" if rel_E else "NO RELATION — extended") print() any_structured = any([rel_B, rel_C, rel_D, rel_E]) if any_structured: print(" OVERALL VERDICT: STRUCTURED") print(" The near-miss δ ≈ 6 has additional integer-relation structure") print(" beyond the trivial polynomial identity.") else: print(" OVERALL VERDICT: COINCIDENCE (no additional structure found)") print(" δ = 1728 - 16π⁴ - 4π³ - 4π² is a polynomial in π with no") print(" additional ℚ-linear structure over the tested bases.") print(f" The residual δ − 6 ≈ {float(residual):.6f} is itself a transcendental") print(" polynomial in π with no CM/Gamma decomposition at coeff ≤ 1000.") print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") # Investigation script: PSLQ verdicts are findings, not gates. Baseline exits 0. sys.exit(0)