#!/usr/bin/env python3 """ verify_P016.py -- Paper 16: shell completion and Mersenne-prime prediction. This verifier checks the explicit shell arithmetic, prime-exponent tests, and the statistical-validation table against the displayed model definition. """ from __future__ import annotations import math import sys from pathlib import Path PASS = FAIL = 0 _N = 0 def check(desc, cond): global PASS, FAIL, _N _N += 1 ok = bool(cond) PASS += ok FAIL += not ok print(f" [{'PASS' if ok else 'FAIL'}] {_N:>2}. {desc}") return ok class Verifier: """Same check semantics as verify_common.Verifier; modern output style.""" def __init__(self, name): print(name) def check(self, label, computed, claimed, *, rel=1e-3, abs_tol=None, detail=""): if abs_tol is not None: ok = abs(computed - claimed) <= abs_tol err = abs(computed - claimed) err_detail = f"abs err={err:.6g}, tol={abs_tol:.6g}" else: if claimed == 0: ok = abs(computed) <= (rel or 1e-12) err_detail = f"abs value={abs(computed):.6g}, tol={rel:.6g}" else: err = (computed - claimed) / abs(claimed) ok = abs(err) <= (rel or 0) err_detail = f"rel err={100 * err:+.6g}%, tol={100 * (rel or 0):.6g}%" return self.record(label, ok, computed, claimed, err_detail + (f"; {detail}" if detail else "")) def record(self, label, ok, computed="", claimed="", detail=""): check(label + (f" -- {detail}" if detail else ""), ok) if computed != "" or claimed != "": print(f" computed: {computed}") print(f" claimed : {claimed}") return ok def summary(self): print(f"\n{'='*60}\nRESULT: {PASS} PASS / {FAIL} FAIL") return 1 if FAIL else 0 v = Verifier("P016 -- Shell Completion") ROOT = Path(__file__).resolve().parents[2] TEX = (ROOT / "toe" / "16_Paper_ShellCompletion.tex").read_text() PI = math.pi ALPHA_INV = 4.0 * PI**3 + PI**2 + PI ALPHA = 1.0 / ALPHA_INV KAPPA = ALPHA ** 1.25 KAPPA_PRINTED = 0.00213306 # Known Mersenne-prime exponents through the pre-2024 M51. The entry # 42,643,801 is easy to miss because it was discovered after larger ones. M51_EXPONENTS = [ 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933, ] PREDICTED_TRIPLET = [160_964_569, 260_446_093, 421_410_673] ACTUAL_2024_M52 = 136_279_841 PHI = (1.0 + math.sqrt(5.0)) / 2.0 def shell_distance(value: float) -> float: frac = value - math.floor(value) return min(frac, 1.0 - frac) def is_probable_prime(n: int) -> bool: if n < 2: return False small_primes = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37] for p in small_primes: if n % p == 0: return n == p d = n - 1 s = 0 while d % 2 == 0: s += 1 d //= 2 # Deterministic for these < 2^32 exponents; deliberately stronger than # needed so the witness set is obvious in the audit output. for a in small_primes: if a >= n: continue x = pow(a, d, n) if x == 1 or x == n - 1: continue for _ in range(s - 1): x = pow(x, 2, n) if x == n - 1: break else: return False return True def binomial_tail(n: int, observed: int, p: float) -> float: return sum( math.comb(n, k) * (p**k) * ((1.0 - p) ** (n - k)) for k in range(observed, n + 1) ) s51 = sum(M51_EXPONENTS) * KAPPA triplet_sum = sum(PREDICTED_TRIPLET) s54 = s51 + triplet_sum * KAPPA rounded_proof_s54 = 1_237_518.625 + triplet_sum * KAPPA_PRINTED distances = [] running = 0.0 for exponent in M51_EXPONENTS: running += exponent * KAPPA distances.append(shell_distance(running)) v.check("alpha inverse", ALPHA_INV, 137.0363037769, rel=8e-12) v.check("printed kappa", KAPPA, KAPPA_PRINTED, rel=2e-4) v.record( "M51 exponent list has 51 entries", len(M51_EXPONENTS) == 51 and M51_EXPONENTS[-1] == 82_589_933, computed=f"count={len(M51_EXPONENTS)}, last={M51_EXPONENTS[-1]}", claimed="51 known exponents through p51=82,589,933", ) v.check("S51 from exact kappa", s51, 1_237_518.625, abs_tol=2.0e-5) v.check("triplet sum", triplet_sum, 842_821_335, rel=0) v.check("Z3 coherence", triplet_sum % 3, 0, rel=0) v.check("target shell from exact kappa", round(s54), 3_035_110, rel=0) v.check("S54 exact shell distance", shell_distance(s54), 4.8e-6, rel=1e-3) v.check("first phi ratio", PREDICTED_TRIPLET[1] / PREDICTED_TRIPLET[0], PHI, rel=3e-7) v.check("second phi ratio", PREDICTED_TRIPLET[2] / PREDICTED_TRIPLET[1], PHI, rel=3e-7) v.record( "predicted exponents pass Miller-Rabin witness set", all(is_probable_prime(p) for p in PREDICTED_TRIPLET), computed=[(p, is_probable_prime(p)) for p in PREDICTED_TRIPLET], claimed="all three exponents prime", ) v.check("p54 decimal digits", math.floor(PREDICTED_TRIPLET[2] * math.log10(2.0)) + 1, 126_857_254, rel=0) v.record( "prize threshold", PREDICTED_TRIPLET[2] * math.log10(2.0) > 1.0e8, computed=PREDICTED_TRIPLET[2] * math.log10(2.0), claimed="p54 log10(2) > 1e8", ) v.check("target shell mod 5", 3_035_110 % 5, 0, rel=0) v.check("target shell mod 13", 3_035_110 % 13, 0, rel=0) v.record( "M52 discrepancy is acknowledged in the TeX", "differs from the announced" in TEX and "136{,}279{,}841" in TEX, computed="note-on-M52 paragraph present", claimed="paper flags predicted p52 differs from discovered M52", ) v.check( "proof line works with printed kappa 0.00213306", rounded_proof_s54, 3_035_109.99999, abs_tol=1.0e-4, detail="Expected fail: the headline shell closure uses the exact kappa, not the rounded number shown in the proof line.", ) v.record( "theorem's p52 ordinal agrees with discovered M52", PREDICTED_TRIPLET[0] == ACTUAL_2024_M52, computed=f"predicted first triplet member={PREDICTED_TRIPLET[0]}, discovered M52={ACTUAL_2024_M52}", claimed="p52 = 160,964,569", detail="Expected fail: the paper acknowledges this later, but the theorem still labels the value p52.", ) v.check( "observed d_n < 0.01 count", sum(d < 0.01 for d in distances), 14, rel=0, detail="Expected fail under the cumulative S_n definition and the 51 listed exponents.", ) v.check( "observed d_n < 0.05 count", sum(d < 0.05 for d in distances), 19, rel=0, detail="Expected fail under the cumulative S_n definition and the 51 listed exponents.", ) v.check( "mean shell distance", sum(distances) / len(distances), 0.187, rel=2e-3, detail="Expected fail: the cumulative exact-kappa mean is about 0.221.", ) v.check( "random expected count for d_n < 0.01 on [0,0.5]", 51 * (0.01 / 0.5), 10.2, rel=1e-12, detail="Expected fail: a uniform shell distance on [0,0.5] gives expectation 1.02, not 10.2.", ) v.check( "random expected count for d_n < 0.05 on [0,0.5]", 51 * (0.05 / 0.5), 10.2, rel=1e-12, detail="Expected fail: this expectation is 5.1, not 10.2.", ) v.check( "one-sided p-value for 14 distances below 0.01", binomial_tail(51, 14, 0.01 / 0.5), 0.03, rel=0.1, detail="Expected fail: under the stated null, this tail is about 1e-12 and also does not match the observed count.", ) v.record( "unique exhaustive search is reproducible from supplied data", False, computed="the TeX gives aggregate candidate counts but no search grid, tolerances beyond six bullet constraints, or candidate list", claimed="exhaustive search yields one unique triplet", detail="Expected reproducibility fail.", ) v.record( "Mersenne primality of 2^p-1 is established by exponent primality", False, computed="Miller-Rabin verifies only that p is prime; it does not test whether 2^p-1 is a Mersenne prime", claimed="prediction is p54 = 421,410,673", detail="Expected logical-scope fail.", ) sys.exit(v.summary())