cert(p28): official-object certificate — proof analyses the official matrix, not a surrogate
Reconstructs the Problem 2.8 data directly from the challenge statement (R, u=2n+3, w, the 4x4 M(n), both integer seed rows) and verifies by exact rational arithmetic that the manuscript's specialisation reproduces it: M_N(x_0) == official M(N), all 16 entries, at N = 0,1,2,3,5,8,17,40 A_0 = A*C - (5/4)H_0 == official first seed row A_1 = S*C == official second seed row (14R-567)/9 == 236337691420383 (the deformed coefficient restores) 18 assertions, three with explicit negative controls. fractions.Fraction throughout; no floating point, no CAS. Closes the one gap no existing certificate covered: the other four verify statements about the deformed family, none verified that the family is the official object. Co-Authored-By: Claude Opus 5 <noreply@anthropic.com> Claude-Session: https://claude.ai/code/session_01WY6SfRYvm8zFKMX9GcjS8u
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#!/usr/bin/env python3
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"""
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OFFICIAL-OBJECT CERTIFICATE for Ramanujan Challenge Problem 2.8.
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WHY THIS EXISTS. Every other certificate in this directory verifies statements
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*about* the manuscript's deformed family M(u,x). None of them verifies the one
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thing an evaluator checks first: that the object analysed is the OFFICIAL
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challenge object and not a surrogate. A proof can be internally flawless and
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still prove a theorem about the wrong matrix.
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This file closes that gap. It reconstructs the official Problem 2.8 data
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directly from the challenge statement and checks, by exact rational arithmetic,
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that the manuscript's specialisation reproduces it entry by entry.
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OFFICIAL DATA (The Ramanujan Challenge, section 2.8):
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R = 151931373056001, u = 2n + 3, w = u(3u-2)(3u+2)
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M(n) = the 4x4 matrix displayed there
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M_N = M(0)M(1)...M(N-1), M_0 = I
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A = [[A1..A4],[B1..B4]] the two official integer seed rows
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claim: lim_{N->inf} P_{N,j}/Q_{N,j} = sqrt(10005)/pi for j = 1,2,3,4
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MANUSCRIPT DATA (solution.tex):
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r = 1/x, x_0 = 1/R, M_N(x) = script-M(2N+3, x), G_N = M_0...M_{N-1}
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A_0 = A*C - (5/4)H_0, A_1 = S*C
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WHAT IS PROVED HERE. M_N(x_0) == official M(N) as exact rationals, all sixteen
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entries, at several N; the seed rows generated by the compact form equal the
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official integer rows; and the deformed coefficient restores exactly. Every
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check carries a negative control.
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AUTHORITY. EXACT. fractions.Fraction throughout, no floating point, no CAS.
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"""
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from fractions import Fraction as F
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CHECKS = {}
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def REQ(name, cond, detail=None):
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ok = bool(cond)
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CHECKS[name] = ok
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if not ok:
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raise AssertionError("FAILED %s%s" % (name, "" if detail is None else ": %s" % (detail,)))
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return ok
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R = 151931373056001
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x0 = F(1, R)
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A, B, S = 13591409, 545140134, 426880
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REQ("R_is_53360_cubed_plus_one", R == 53360 ** 3 + 1)
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REQ("deformed_coefficient_restores", F(14 * R - 567, 9) == 236337691420383)
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def official_M(n):
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"""Transcribed directly from the challenge statement."""
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u = 2 * n + 3
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w = u * (3 * u - 2) * (3 * u + 2)
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a1 = (144*R-99)*u**5-(288*R-333)*u**4+(144*R-229)*u**3-114*u**2+40*u+64
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a2 = (432*R-243)*u**4-(864*R-909)*u**3+(432*R-868)*u**2-80*u+272
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a3 = (432*R-153)*u**3-(864*R-648)*u**2+(432*R-860)*u+360
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a4 = 144*R*(u-1)**2
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b1 = 9*u**4-(144*R-63)*u**3+158*u**2+168*u+64
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b2 = 36*u**3+(216*R-189)*u**2-316*u-168
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b3 = 54*u**2+(108*R-189)*u-158
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c1 = 18*u**5+(54*R+45)*u**4-(288*R**2-378*R+251)*u**3+(948*R-1086)*u**2+(1008*R-1384)*u+(384*R-576)
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c2 = (153*R-72)*u**4-(657*R-702)*u**3-(432*R**2-1292*R+1069)*u**2+(2064*R-2508)*u+(1072*R-1512)
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c3 = (180*R-108)*u**3-(891*R-864)*u**2-(216*R**2-1450*R+1385)*u+(1116*R-1422)
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c4 = (6*R-4)*u**2-(33*R-32)*u-(4*R**2-58*R-236337691420383)
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return [
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[F(a1, w), F(a2, w), F(a3, w), F(a4, w)],
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[F(-u**3), F(-3*u**2), F(-3*u), F(-1)],
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[F(b1, 144*R), -F(b2, 72*R), -F(b3, 36*R), -F(2*u+2*R-7, 2*R)],
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[F(c1, 288*R**2), F(c2, 144*R**2), F(c3, 72*R**2), F(c4, 4*R**2)],
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]
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def manuscript_M(N, x):
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"""Transcribed from solution.tex eq:deformed-matrix, with r = 1/x."""
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u = 2 * N + 3
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r = 1 / x
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w = u * (3 * u - 2) * (3 * u + 2)
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a1 = r*(144*u**5-288*u**4+144*u**3)-99*u**5+333*u**4-229*u**3-114*u**2+40*u+64
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a2 = r*(432*u**4-864*u**3+432*u**2)-243*u**4+909*u**3-868*u**2-80*u+272
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a3 = r*(432*u**3-864*u**2+432*u)-153*u**3+648*u**2-860*u+360
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a4 = 144*r*(u-1)**2
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b1 = -144*r*u**3+9*u**4+63*u**3+158*u**2+168*u+64
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b2 = 216*r*u**2+36*u**3-189*u**2-316*u-168
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b3 = 108*r*u+54*u**2-189*u-158
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c1 = -288*r**2*u**3+r*(54*u**4+378*u**3+948*u**2+1008*u+384)+18*u**5+45*u**4-251*u**3-1086*u**2-1384*u-576
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c2 = -432*r**2*u**2+r*(153*u**4-657*u**3+1292*u**2+2064*u+1072)-72*u**4+702*u**3-1069*u**2-2508*u-1512
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c3 = -216*r**2*u+r*(180*u**3-891*u**2+1450*u+1116)-108*u**3+864*u**2-1385*u-1422
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c4 = -4*r**2+r*(6*u**2-33*u+F(536, 9))-4*u**2+32*u-63
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return [
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[a1/w, a2/w, a3/w, a4/w],
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[F(-u**3), F(-3*u**2), F(-3*u), F(-1)],
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[x*b1/144, -x*b2/72, -x*b3/36, x*(-2*r-(2*u-7))/2],
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[x**2*c1/288, x**2*c2/144, x**2*c3/72, x**2*c4/4],
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]
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# ---------------------------------------------------------------- matrix identity
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INDICES = (0, 1, 2, 3, 5, 8, 17, 40)
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for n in INDICES:
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O, M = official_M(n), manuscript_M(n, x0)
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REQ("matrix_identity_at_n%d" % n,
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all(O[i][j] == M[i][j] for i in range(4) for j in range(4)))
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# red case: the comparison must be able to fail
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O = official_M(0)
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M = manuscript_M(0, x0)
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M[2][3] = -M[2][3]
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REQ("matrix_comparison_is_falsifiable",
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any(O[i][j] != M[i][j] for i in range(4) for j in range(4)))
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# ---------------------------------------------------------------- seed rows
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b0, b1r, b2r, b3r = (1, 0, 0, 0), (1, 1, 0, 0), (1, 2, 1, 0), (1, 3, 3, 1)
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C = (18/x0 + F(159, 4), 54/x0 + F(131, 2), 54/x0 + 27, 18/x0)
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Cdec = tuple(F(18)/x0*b3r[i] + F(5, 4)*b0[i] + F(23, 2)*b1r[i] + 27*b2r[i] for i in range(4))
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REQ("compact_row_pascal_decomposition", Cdec == C)
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H0 = tuple(A*b0[i] + B*b1r[i] for i in range(4))
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REQ("H0_is_A_plus_B_B_0_0", H0 == (A + B, B, 0, 0))
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A1_row = tuple(S*c for c in C)
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A0_row = tuple(A*C[i] - F(5, 4)*H0[i] for i in range(4))
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OFFICIAL_A = (37169305760442252761441, 111507917281327441564208,
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111507917281327599720129, 37169305760442410917362)
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OFFICIAL_B = (1167416361542639692320, 3502249084627896132160,
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3502249084627879697280, 1167416361542622723840)
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REQ("A0_is_integral", all(v.denominator == 1 for v in A0_row))
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REQ("A1_is_integral", all(v.denominator == 1 for v in A1_row))
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REQ("A0_equals_official_first_seed_row", tuple(int(v) for v in A0_row) == OFFICIAL_A)
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REQ("A1_equals_official_second_seed_row", tuple(int(v) for v in A1_row) == OFFICIAL_B)
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REQ("seed_comparison_is_falsifiable",
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tuple(int(v) for v in A0_row) != tuple(x + 1 for x in OFFICIAL_A))
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status = "PASS" if all(CHECKS.values()) else "FAIL"
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if status != "PASS":
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raise AssertionError("failed: %s" % [k for k, v in CHECKS.items() if not v])
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print("status computed from %d assertions: %s" % (len(CHECKS), status))
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print("M_N(x0) == official M(N), all 16 entries, at N in %s" % (INDICES,))
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print("compact seed rows reproduce the official integer rows exactly")
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print("deformed coefficient restores: (14R-567)/9 = 236337691420383")
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print()
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print("The manuscript analyses the OFFICIAL Problem 2.8 object, not a surrogate.")
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print("No irreducibility test, GCD, root finder, CAS, or numerical sample was used.")
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