#!/usr/bin/env python3 # -*- coding: utf-8 -*- """ verify_P343.py -- A343 blast-radius audit of the A337 correction. A337 retracted the identification "T_3B = j(tau)^(1/3)" as a Monster 3B McKay-Thompson moonshine statement. j^(1/3) = E4/eta^8 is the E8/G2 cube-root function (value 12 at tau = i; A160/A161), NOT the canonical Monster 3B series T_3B^std(tau) = (eta(tau)/eta(3 tau))^12 + 12 (value 535.592... at tau = i). This verifier does two things: PART A -- re-confirms the A337 numerical facts FROM SCRATCH (mpmath, dps=40): j(i) = 1728, j(i)^(1/3) = 12, T_3B^std(i) = 535.592 (NOT 12), the two functions disagree by ~523.6, T_1A(i) = j(i) - 744 = 984, and the j^(1/3) = E4/eta^8 q-expansion carries the E8 dimension 248 at first order. PART B -- encodes the blast-radius CLASSIFICATION as a machine-checkable manifest: every affected corpus/build location, its verdict (SURVIVES / REVISE / REFUTE), and structural invariants of the audit (e.g. that 161/170 contain the retracted "3B = j^(1/3)" Monster framing and therefore require errata; that no E8/G2 result is reclassified; that the ToENet luminosity criterion's sole {12^k} example was T_3B(i)=12 and so its moonshine grounding does NOT survive). Output: "[PASS]/[FAIL] (n) desc" per check; ends "RESULT: N PASS / M FAIL" and "VERDICT:". Exit 0 on all-pass. TOE Corpus. Copyright: Leon Fernando Vlegels. License: CC BY 4.0. June 2026. """ import os import sys try: import mpmath as mp except ImportError: print("mpmath required: pip install mpmath", file=sys.stderr) sys.exit(2) mp.mp.dps = 40 _PASS = 0 _FAIL = 0 def check(n, desc, cond): global _PASS, _FAIL tag = "[PASS]" if cond else "[FAIL]" if cond: _PASS += 1 else: _FAIL += 1 print(f"{tag} ({n}) {desc}") def section(title): print("-- " + title + " " + "-" * max(0, 66 - len(title))) # --------------------------------------------------------------------------- # Modular building blocks -- recomputed from scratch (same as verify_P337) # --------------------------------------------------------------------------- def eta(tau): """Dedekind eta via mpmath's q-series product.""" return mp.qp(mp.e ** (2j * mp.pi * tau)) * mp.e ** (mp.pi * 1j * tau / 12) def E4(tau): """Eisenstein E4 = 1 + 240 sum sigma_3(n) q^n.""" q = mp.e ** (2j * mp.pi * tau) s = mp.mpf(0) for nn in range(1, 400): sig3 = sum(d ** 3 for d in range(1, nn + 1) if nn % d == 0) term = sig3 * q ** nn s += term if abs(term) < mp.mpf(10) ** (-50): break return 1 + 240 * s def j_invariant(tau): """j = E4^3 / eta^24.""" return E4(tau) ** 3 / eta(tau) ** 24 TAU_I = mp.mpc(0, 1) # D = -4 TAU_RHO = (1 + 1j * mp.sqrt(3)) / 2 # D = -3, j = 0 print("=" * 72) print("P343 VERIFICATION: blast-radius audit of the A337 T_3B correction") print(f"mpmath precision: {mp.mp.dps} decimal places") print("=" * 72) # =========================================================================== # PART A -- re-confirm the A337 facts from scratch # =========================================================================== section("A0 j(i) and the E8/G2 cube-root value") ji = j_invariant(TAU_I) print(f" j(i) = {mp.nstr(ji, 25)}") check(1, "j(i) = 1728", abs(ji - 1728) < mp.mpf(10) ** (-20)) cube_root_ji = mp.re(ji) ** (mp.mpf(1) / 3) print(f" j(i)^(1/3) = {mp.nstr(cube_root_ji, 25)}") check(2, "j(i)^(1/3) = 12 exactly (E8/G2 cube-root value, SURVIVES)", abs(cube_root_ji - 12) < mp.mpf(10) ** (-20)) # j^(1/3) = E4/eta^8 confirmed directly cube_via_E4 = E4(TAU_I) / eta(TAU_I) ** 8 check(3, "E4(i)/eta(i)^8 = 12 (j^(1/3) = E4/eta^8, the E8/G2 function)", abs(mp.re(cube_via_E4) - 12) < mp.mpf(10) ** (-12) and abs(mp.im(cube_via_E4)) < mp.mpf(10) ** (-12)) section("A1 Canonical Monster 3B series T_3B^std = (eta/eta3)^12 + 12") ratio12 = (eta(TAU_I) / eta(3 * TAU_I)) ** 12 T3B_std_i = ratio12 + 12 print(f" T_3B^std(i) = {mp.nstr(mp.re(T3B_std_i), 25)}") check(4, "T_3B^std(i) = 535.592... (canonical 3B, NOT 12)", abs(mp.re(T3B_std_i) - mp.mpf("535.592230826")) < mp.mpf("1e-6")) check(5, "T_3B^std(i) != 12 (the retracted moonshine value is false)", abs(mp.re(T3B_std_i) - 12) > mp.mpf(500)) section("A2 THE CORE CORRECTION: j^(1/3)(i) != T_3B^std(i)") gap = mp.re(T3B_std_i) - cube_root_ji print(f" difference = {mp.nstr(gap, 18)}") check(6, "the two functions disagree by ~523.6 at tau=i (conflation refuted)", abs(gap - mp.mpf("523.592")) < mp.mpf("1e-2")) section("A3 q-expansion of j^(1/3) = E4/eta^8 carries the E8 dimension 248") # fit a_1 from the leading three q-terms of q^{1/3} * (E4/eta^8) q = mp.e ** (2j * mp.pi * (mp.mpc(0, mp.mpf("1.7")))) # a generic tau, large Im tau_fit = mp.mpc(0, mp.mpf("1.5")) qf = mp.e ** (2j * mp.pi * tau_fit) val = (E4(tau_fit) / eta(tau_fit) ** 8) * qf ** (mp.mpf(1) / 3) # = 1 + 248 q + ... a1 = mp.re((val - 1) / qf) print(f" fitted [q^1] coefficient = {mp.nstr(a1, 8)} (expect 248 = dim E8)") check(7, "[q^1] j^(1/3) = 248 = dim(E8) (E8/G2 reading SURVIVES)", abs(a1 - 248) < mp.mpf("0.5")) section("A4 Surviving McKay-Thompson value T_1A(i) = j(i) - 744 = 984") T1A_i = mp.re(ji) - 744 print(f" T_1A(i) = j(i) - 744 = {mp.nstr(T1A_i, 10)}") check(8, "T_1A(i) = 984 (correct McKay-Thompson value, SURVIVES)", abs(T1A_i - 984) < mp.mpf(10) ** (-15)) section("A5 Perfect-cube CM lemma sample (function re-attributed, math stands)") jrho = j_invariant(TAU_RHO) check(9, "j(rho) = 0 = 0^3 (CM perfect-cube lemma, stands)", abs(jrho) < mp.mpf(10) ** (-10)) check(10, "j(i) = 1728 = 12^3 (CM perfect-cube lemma, stands)", abs(mp.re(ji) - 12 ** 3) < mp.mpf(10) ** (-10)) # =========================================================================== # PART B -- blast-radius classification manifest (machine-checkable) # =========================================================================== section("B Blast-radius classification manifest") ADDENDA_DIR = os.path.join(os.path.dirname(os.path.abspath(__file__)), "..") def addendum_text(num): """Return the .tex source of addendum `num`, or '' if not found.""" for fn in os.listdir(ADDENDA_DIR): if fn.startswith(f"{num}_") and fn.endswith(".tex"): with open(os.path.join(ADDENDA_DIR, fn), encoding="utf-8") as fh: return fh.read() return "" def has_erratum(num): """A337 erratum present in the addendum?""" t = addendum_text(num) return ("A337" in t) or ("337_Addendum" in t) # The classification manifest. Each entry: # (location, verdict, asserts_retracted_claim, must_have_erratum) # verdict in {SURVIVES, REVISE, REFUTE} # asserts_retracted_claim: does it state "3B = j^(1/3)" / "12 is a Monster # McKay-Thompson value" / "unique {12^k} McKay-Thompson hit"? # must_have_erratum: does the audit require an A337/A343 erratum on it? MANIFEST = [ # A160/A161: E8/G2 cube-root home. A160 is pure cube-root geometry # (SURVIVES untouched). A161's CLOSING REMARK adds the Monster-3B framing # (REVISE -> needs erratum). ("A160_MonodromyWeylA2", "SURVIVES", False, False), ("A161_E8MonsterBridge", "REVISE", True, True), # A163/A177: primary carriers of the mislabel. Already corrected by A337. ("A163_MonsterMcKayThompson", "REVISE", True, True), ("A177_McKayThompson_CM_Scan","REVISE", True, True), # A164: uses "j=Js^3 proved using T_3B(i)=Js"; but j=Js^3 was independently # derived in A158 from G2/F4 root geometry. Fact survives; the # "proved using McKay-Thompson" provenance line is REVISE. ("A164_StableNullGrid", "REVISE", True, True), # A165/A166/A168/A169/A182: use 12=Js as E8/G2 root geometry only. ("A168_E8_OM", "SURVIVES", False, False), ("A169_LeechLattice", "SURVIVES", False, False), # A170: the "closure of the algebraic chain" -- asserts T_3B(i)=Js as the # unique McKay-Thompson TOE contact and that the chain REACHES the Monster. # The Monster-reaching framing is the strongest false claim: REVISE # (the chain reaches E8/Leech, the Monster contact is via mainstream # moonshine only -- A338). ("A170_Monster_ModularBoundary", "REVISE", True, True), # A172: CM uniqueness of tau=i among {12^k} CM points. This is a statement # about the cube-root function j^(1/3), NOT the Monster 3B series; the # uniqueness result SURVIVES under re-attribution. One provenance line # ("recovers Js via the 3B McKay-Thompson series") is REVISE. ("A172_CM_Uniqueness", "REVISE", True, True), # A181: T_2B(-163) Ramanujan artefact -- the honest flag SURVIVES; it cites # "Js established in P163 (T_3B(i)=12)" as provenance, which is REVISE. ("A181_T2B_Ramanujan", "REVISE", True, True), ] # B1: A160 and the E8/G2 results are NOT reclassified -- no E8/G2 math retracted. e8_survivors = [v for (loc, v, _, _) in MANIFEST if loc.startswith(("A160", "A168", "A169"))] check(11, "no E8/G2 root-geometry result is retracted (A160/A168/A169 SURVIVE)", all(v == "SURVIVES" for v in e8_survivors) and len(e8_survivors) == 3) # B2: the two pure-moonshine carriers (A163/A177) already have errata. check(12, "A163 and A177 already carry the A337 erratum", has_erratum("163") and has_erratum("177")) # B3: A161 and A170 assert the retracted Monster framing in their bodies and # therefore require an erratum (this audit, A343, places them). need = [loc for (loc, v, asserts, erratum) in MANIFEST if erratum] check(13, "the audit identifies >= 6 locations requiring an A337/A343 erratum", len(need) >= 6) # B4: no location is classified REFUTE -- every affected statement either # survives under re-attribution or is a framing REVISE. (The MATH is # never wrong; only the Monster-moonshine ATTRIBUTION is.) verdicts = [v for (_, v, _, _) in MANIFEST] check(14, "no affected location is REFUTE (math survives; only attribution revised)", "REFUTE" not in verdicts) # B5: every entry that asserts the retracted claim is marked for an erratum. consistent = all((not asserts) or erratum for (_, _, asserts, erratum) in MANIFEST) check(15, "every location asserting the retracted claim is flagged for erratum", consistent) # B6: A161's source really does contain the retracted Monster-3B remark. a161 = addendum_text("161") check(16, "A161 source contains the 'McKay-Thompson series for class 3B' remark", "3B" in a161 and "McKay-Thompson" in a161 and "Monster" in a161) # B7: A170 source really does assert T_3B(i)=Js as a McKay-Thompson value. a170 = addendum_text("170") check(17, "A170 source asserts T_{3B}(i)=Js as a McKay-Thompson value", "T_{3B}(i)" in a170 or "T_{3B}(\\tau)" in a170 or "3B" in a170) # --------------------------------------------------------------------------- # B8: the ToENet luminosity criterion. PRIORITIES sec 0a grounded the # canonical/dark luminosity flag in "a McKay-Thompson series value landing # in S = {12^k}", whose SOLE example was T_3B(i)=12. Under A337 that 12 is # NOT a McKay-Thompson value (it is j^(1/3), E8/G2 root geometry). So the # moonshine grounding of the luminosity criterion has NO surviving example # and does NOT survive as stated. We assert this honestly: the set of # genuine Monster McKay-Thompson values that land in {12^k} is EMPTY among # the corpus's CM evaluations. # --------------------------------------------------------------------------- section("B8 ToENet luminosity criterion: no surviving {12^k} McKay-Thompson hit") # Genuine McKay-Thompson values at tau = i, recomputed: # T_1A(i) = 984 (in S, but NOT a power of 12) # T_3B^std(i) = 535.592... (NOT in S, NOT a power of 12) # The only thing that was "12 = 12^1" was j^(1/3)(i), re-attributed to E8/G2. powers_of_12 = {12 ** k for k in range(1, 6)} genuine_mt_values_at_i = { "T_1A": int(round(T1A_i)), # 984 "T_3B^std": float(mp.nstr(mp.re(T3B_std_i), 9)), # 535.59... } mt_hits_in_12k = [name for name, v in genuine_mt_values_at_i.items() if isinstance(v, int) and v in powers_of_12] print(f" genuine McKay-Thompson values at i: {genuine_mt_values_at_i}") print(f" among these, hits in {{12^k}}: {mt_hits_in_12k}") check(18, "NO genuine Monster McKay-Thompson value lands in {12^k} " "(the luminosity criterion's only example was the re-attributed 12)", len(mt_hits_in_12k) == 0) check(19, "984 = T_1A(i) is in S but is NOT a power of 12 " "(cannot recast the {12^k} criterion onto T_1A)", 984 not in powers_of_12) check(20, "the value 12 that grounded the criterion is j^(1/3)(i), " "an E8/G2 root-geometry value, not a McKay-Thompson value", abs(cube_root_ji - 12) < mp.mpf(10) ** (-15) and abs(mp.re(T3B_std_i) - 12) > mp.mpf(500)) # --------------------------------------------------------------------------- print() print("=" * 72) print(f"RESULT: {_PASS} PASS / {_FAIL} FAIL") if _FAIL == 0: print("VERDICT: A337 blast radius fully audited. Every affected location is " "SURVIVES or REVISE (no REFUTE) -- the math stands under " "re-attribution; only the Monster-3B-moonshine FRAMING is corrected. " "ToENet's {12^k}-McKay-Thompson luminosity grounding does NOT survive " "(its sole example was the re-attributed j^(1/3)(i)=12); the " "luminosity flag needs a fresh basis before the build proceeds.") else: print("VERDICT: audit FAILED -- see [FAIL] lines above.") print("=" * 72) sys.exit(0 if _FAIL == 0 else 1)