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v781_one_object_clock_census.py
verification/v781_one_object_clock_census.py
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#!/usr/bin/env python3
"""v781 -- E8.ONEOBJECT.01: the d = 4 forcing census + the 60-line clock synchronization kill, ONE module from two probes (21/21 + 23/23 checks, ~9 s; discovery probes one_object_reduction_probe.py ONE-OBJECT-PARTIAL and clock_fiber_product_probe.py CLOCK-FIBER-DEAD, both 2026-08-05). THE FORCING CENSUS (part 1, exact Fraction arithmetic, no floats): for d = 2..12 the discriminant forms are disc(A_{d-1}) = Z_d and disc(D_{d+1}) = Z4 (d even) / Z2 x Z2 (d odd), and the TFPT glue pattern (isotropic glue group, evenness q = 0 mod 2Z, unimodularity |H|^2 = 4d) admits an even unimodular DIAGONAL glue of A_{d-1} (+) D_{d+1} iff d = 4 -- even stronger, ANY even unimodular glue exists only at d = 4; the coarse square layer (|H|^2 = 4d a perfect square) passes only d in {4, 9}, and d = 9 dies at isotropy (disc(D_10) q-values {0, 1, 1/2, 1/2}: the forced order-2 part of an |H| = 6 subgroup cannot be even-isotropic). At d = 4 the census reproduces v92 EXACTLY: precisely TWO even Lagrangian glues, both diagonal cyclic Z4, the graphs of the two anti-isometries k -> k and k -> 3k (q(A3) = 3/4, q(D5) = 5/4, 3/4 + 5/4 = 2 = the E8 root norm), intersecting in the order-2 halfway glue {0, (2,2)} (v92's SO(16)_1 rung); the explicit v1 lattice certificate closes: glued basis (drop root #2, add g = (w_A3, spinor_D5)) with integer Gram, even diagonal, det = 1, norm-2 count per glue coset [52, 64, 60, 64], total 240, ALL 240 integral over the glued basis -- THE even unimodular E8. Controls fire: dropping evenness opens d = 9 (an odd unimodular glue exists there), dropping unimodularity opens d in {7, 8, 9, 12}. Corollaries at the forced d, typed honestly: g_car = d+1 = 5, N_fam = d-1 = 3, rank = 2d = 8, glue index = sqrt(4d) = 4 = d = |mu4| are verified arithmetic; c_3 = 1/(2 pi d)|_{d=4} = 1/(8 pi) is a corpus-legitimate REWRITING (c_3 = 1/(2 e_1(a) pi) with e_1(a) = 4 = |mu4| = d, tfpt_1 ~L189-194 / v23 anchor-first refinement), NOT a new derivation of P1 (the corpus Gauss-Bonnet chain reads the 4 pi as total sphere curvature, not d; ONE-OBJECT-FORCED predeclared unreachable). THE CLOCK KILL (part 2, three levels, each independently fatal to L_60 = Z12 x_{Z6} Z30 on the 60 Gaussian lines): (1) the seam clock c = J o sigma (order 12; c^4 = sigma, c^9 = J, c^6 = -1 exact, v634 S7) acts on the 60 lines only through Z3 = <sigma> (line-trivial powers {0, 3, 6, 9}; sigma line census {3: 19, 1: 3}, v633 Q4); (2) the deployed Coxeter clock w = s_1 ... s_8 (order 30 on the 240 roots, Kostant census {30: 8}, w^15 = -1) does NOT descend to the lines (line-preserving powers {0, 15} only; w J w^-1 not in {J, J^-1}); (3) the existence sweep decides EVERY realization: the FULL line-compatible group (the 60 unitary reflections + the conj coset of N(<J>)) has order 23040 = 2 x 11520 (the antiunitary coset is genuinely new) with max element order 12 (spectrum {1: 1, 2: 751, 3: 800, 4: 4560, 5: 2304, 6: 7520, 8: 2880, 10: 2304, 12: 1920}) -- orders 15, 30, 60 ALL absent, while a simply-transitive Z60 needs line order 60 and any root-level order-30 clock would induce line order 30 or 15: no realization can exist. The abstract layer stays exact: the fiber product Z12 x_{Z6} Z30 has order 60 and IS cyclic Z60 (phi(60) = 16 generators), while the wrong pairing Z12 x Z30 has order 360 with invariant factors (6, 60) (CC1 fires); CC2 fires 5/5 (frozen-seed random order-12 unitary words break r^4 = sigma, r^9 = J); CC3 fires (the wrong complex structure J' preserves the roots and the free-60 census but breaks [J', sigma] = 0); the mu4 layer 240 = 4 x 60 is exact; the joint <c, w> orbit census on the 240 roots is {240: 1}. No marker move, NO RH claim. Python-only per GATE.WOLFRAM.02.
PROVENANCE: discovery probes one_object_reduction_probe.py (2026-08-05, 21/21, 4.3 s, ONE-OBJECT-PARTIAL; no disclosed corrections) + clock_fiber_product_probe.py (2026-08-05, 23/23, 4.7 s, CLOCK-FIBER-DEAD; the CC2 SAMPLER CORRECTION disclosed in the probe docstring carried verbatim: "the first run's word length <= 5 letters found NO order-12 words in 4000 frozen-seed tries; the sampler was widened to words of length <= 12 with 20000 tries. The control's SEMANTICS (random order-12 unitary must break r^4 = sigma, r^9 = J) is unchanged; the frozen spec text and SHA are untouched."); both re-run identically at promotion. Promoted verbatim (part 2 wrapped in a function scope). Numbers unchanged. run() encodes both patterns (v757 precedent).
Original one_object_reduction_probe docstring (verbatim):
one_object_reduction_probe -- THE d = 4 FORCING CENSUS: are P1 and P2
two functors of ONE four-point boundary object?
CLAIM CHAIN UNDER TEST (frozen; see FROZEN_SPEC, SHA-256 printed BEFORE
any lattice data is computed): for a boundary divisor with d marked
points the cycle side gives A_{d-1} (N_fam = d-1 = rank H_1 of the
d-punctured P^1) and the function/carrier side gives D_{d+1}
(g_car = d+1). The corpus already reads 3 and 5 this way at d = 4:
* X_f deg 4 boundary: X_f^o ~= P^1 \\ mu_4, H_1 = Z^3, "hence A_3 and
N_fam = 3" (tfpt_1_architecture_e8.tex ~L894, ~L4234);
* g_car = rank D_5 = 5, q(D5) = g_car/|mu4|, q(A3) = N_fam/|mu4| with
n+1 = 4 = |mu4| on the A-side (tfpt_1 ~L4201-4205);
* |mu4| = (g_car + N_fam)/2 = 4 (tfpt_1 ~L4235, v53_compiler_core).
THE NEW FORCING ARGUMENT to verify exactly: disc(A_{d-1}) ~= Z_d, while
disc(D_{d+1}) is Z_4 for d+1 odd (d even) and Z_2 x Z_2 for d+1 even
(d odd). The TFPT glue pattern (v1_e8_glue + v92_glue_uniqueness:
cyclic isotropic glue group, evenness q = 0 mod 2Z, unimodularity
|H|^2 = |disc A| * |disc D|) admits a DIAGONAL glue of
A_{d-1} (+) D_{d+1} iff Z_d ~= Z_4 with anti-isometric quadratic forms,
i.e. iff d = 4 -- yielding E8 = the D5 (+) A3 glue. Any other d
admitting an even unimodular diagonal glue KILLS the forcing claim.
CENSUS: for d = 2..12 compute both discriminant groups and their
quadratic/linking forms EXACTLY from the Cartan matrices (Fraction
arithmetic, no floats), enumerate ALL subgroups of the direct-sum
discriminant form, and classify:
* even unimodular glues (Lagrangian, q = 0 mod 2Z)
* ... of which DIAGONAL (trivial intersection with both factors)
* odd unimodular glues (Lagrangian, q = 0 mod Z, b = 0 mod Z,
NOT even) [control: evenness]
* nontrivial even glues of ANY index [control: unimod]
VERDICT ENUMS (frozen):
ONE-OBJECT-FORCED = glue census forces d = 4 AND the c_3 = 1/(2 pi d)
link is corpus-DERIVED from the boundary object.
Predeclared UNREACHABLE for this corpus snapshot:
P1 is a primitive postulate (tfpt_1 ~L169) whose
Gauss-Bonnet hardening 1/(|Z2| * 4 pi) reads the
"4 pi" as total sphere curvature, NOT as d
(tfpt_1 ~L670-689).
ONE-OBJECT-PARTIAL = the glue census forces d = 4 exactly (controls
fire), corollary g_car = d+1 = 5 is verified
arithmetic, but the c_3 link stays INTERPRETIVE:
c_3 = 1/(2 e_1(a) pi) with e_1(a) = 4 = |mu4| IS
corpus-legitimate (tfpt_1 ~L189-194, v23
anchor-first refinement), so 1/(2 pi d)|_{d=4}
= 1/(8 pi) is a corpus-legitimate REWRITING --
but "the glue-forced d derives P1" would be a NEW
derivation, which this probe must NOT smuggle.
ONE-OBJECT-DEAD = some d != 4 admits an even unimodular diagonal
glue (the forcing claim dies), or d = 4 fails.
TEST-VOID = a corpus-reproduction gate fails (G2/G3) or a
must-fire control does not fire (G4/G5).
GATES (frozen before running):
G1 d-table: {d in 2..12 admitting an even unimodular DIAGONAL glue}
must be decided exactly; forcing holds iff it equals {4}.
G2 at d = 4 the census must reproduce v92 EXACTLY: precisely TWO
even Lagrangian glues, both diagonal cyclic Z4 -- the graphs of
the two anti-isometries k -> k and k -> 3k.
G3 explicit lattice certificate at d = 4 (v1 reproduction): gluing
A3 (+) D5 by one such glue gives an even lattice with Gram det 1
and EXACTLY 240 norm-2 vectors, all integral over the glued
basis -- i.e. E8.
G4 CONTROL (must fire): dropping evenness (isotropy mod Z only)
opens at least one d != 4 (an odd unimodular glue exists there).
G5 CONTROL (must fire): dropping unimodularity (any nontrivial even
isotropic glue subgroup) opens at least one d != 4.
G6 square obstruction layer: |H|^2 = 4d needs 4d to be a perfect
square, i.e. d itself a square: only d in {4, 9} survive the
coarse layer in 2..12; the census must show HOW d = 9 dies
(no isotropic-even Lagrangian in Z9 (+) Z2 x Z2).
G7 corollaries stated as verified arithmetic at the forced d:
g_car = d+1 = 5, N_fam = d-1 = 3, rank = 2d = 8,
glue index = sqrt(4d) = 4 = d = |mu4|; c_3 typing per the
PARTIAL/FORCED rule above (honesty gate: interpretation label).
FIREWALL: experiments/tfpt-discovery probe; EXPLORATION ONLY; writes
nothing; touches no verification/, paper, ledger, changelog or website
surface; no marker moves; no physics claims. ROOTCLASS-MIXED fence
(v775_gaussian_class_d5_purity): no code->matter semantics is used or
implied anywhere -- this census is pure lattice/discriminant-form
arithmetic. A d != 4 hit is a valid, well-powered KILL and will be
reported honestly.
Sources (read-only): verification/v1_e8_glue.py (glue conditions +
lattice certificate pattern), v92_glue_uniqueness.py (Lagrangian
classification of Z4 x Z4, q = (5x^2+3y^2)/8 mod 1 in conformal-weight
normalization = half of our q mod 2Z), v53_compiler_core.py (atoms),
tfpt_1_architecture_e8.tex ~L160-194 (P1/P2, anchor-first refinement),
~L670-689 (Gauss-Bonnet hardening), ~L4190-4245 (glue norms as
compiler atoms), ~L894/4234 (H_1 = Z^3 reading).
Run:
experiments/tfpt-discovery/.venv/bin/python \\
experiments/tfpt-discovery/one_object_reduction_probe.py
Original clock_fiber_product_probe docstring (verbatim):
clock_fiber_product_probe -- THE 60-LINE CLOCK SYNCHRONIZATION CENSUS:
is L_60 ~= Z_12 x_{Z_6} Z_30 realized on the 60 Gaussian E8 lines?
HYPOTHESIS UNDER TEST (frozen; FROZEN_SPEC SHA-256 printed BEFORE any
root data is computed): the 60 Gaussian E8 lines (the free mu4-quotient
of the 240 roots, v629 R3 / v633 Q1) carry a canonical simply-transitive
(or at least orbit-defining) action of the fiber product
L_60 ~= Z_12 x_{Z_6} Z_30
of the SEAM CLOCK (order 12; corpus: v623 V3 "L^4 = deck and L^12 = id
... proper order 12" on the covered seam; E8 avatar: the compiler clock
c = J o sigma with sigma = c^4, J = c^9, <c> = Z12, v629 R1 / v634 S7)
and the COXETER CLOCK (order 30; the E8 Coxeter element, constructed
exactly as the product of the 8 Bourbaki simple reflections deployed in
v1_e8_glue), over their shared Z_6 (sheet x family phase).
DEPLOYED CONVENTIONS (frozen, all corpus-verbatim):
* Root model: D8+spinor, exact Fractions (v633 verbatim): 112 integer
roots +-e_i +- e_j and 128 half-integer roots (+-1/2)^8 with an
even number of minus signs / even coordinate sum.
* J_all = i-rotation on the four coordinate pairs ((a,b) -> (-b,a));
sigma = 3-cycle of pairs 1 -> 2 -> 3 -> 1 fixing pair 4; lines =
the 60 free J-orbits, canonical rep = lexicographic minimum over
the J-orbit (v633 canon).
* Seam clock avatar: c = J o sigma (order 12). Deployment gates:
c^4 = sigma, c^9 = J, c^6 = -1 (v634 S7 identities).
* Coxeter clock: w = s_1 s_2 ... s_8, the simple reflections of the
Bourbaki E8 basis of v1_e8_glue.e8_simple_roots(), applied in
Bourbaki index order. Deployment gates: order 30 on the 240
roots, Kostant orbit census {30: 8}, w^15 = -1 (E8 exponents all
odd).
* Z6 target: <c>/<c^6 = -1> and <w>/<w^6>; "sheet x family" = the
-1 (sheet) and sigma (family) content.
* Everything exact (int/Fraction permutations); the only RNG is the
frozen-seed control CC2.
TESTS:
(i) FIBER CONDITION: both clocks must descend to the 60 lines and
induce the SAME Z6 quotient action (compatibility square
Z12 -> Z6 <- Z30 verified on the actual line permutations).
(ii) SIMPLE TRANSITIVITY: the joint line action of the two clock
images must be isomorphic to the fiber product Z12 x_{Z6} Z30
(order 60, computed exactly: it is CYCLIC Z60) acting on
itself -- or the actual orbit type is named honestly.
(iii) MU4 LAYER: 240 = 4 x 60 via the mu4 phases (J free with 60
orbits of size 4) as the final layer.
EXISTENCE SWEEP (decisive for any realization, not just the deployed
one): the full line-compatible world is N(<J>) inside the root-set
automorphisms; its line image is generated by the 60 unitary
reflections (v633 Q7: order 11520) together with the complex
conjugation representative conj: (a,b) -> (a,-b) per pair (which obeys
conj o J o conj^-1 = J^-1). Any order-30 root-level clock acting on
the 60 lines would have line order 30 or 15 (its cyclic group meets
the line-kernel <J> in a subgroup of order gcd-limited to <= 2), and a
simply-transitive Z60 needs line order 60. The exact element-order
spectrum of the full line-compatible group therefore DECIDES the
hypothesis for EVERY realization of the clocks.
CENSUS CONTROLS (must fire):
CC1 wrong pairing: the FULL product Z12 x Z30 has order 360 != 60
and invariant factors (6, 60) != (60): it cannot act simply
transitively on 60 lines; the fiber product Z12 x_{Z6} Z30 is
cyclic of order 60 (exact SNF-level arithmetic).
CC2 a random order-12 element of the unitary line world in place of
the seam clock must BREAK the deployment identities
(r^4 != sigma or r^9 != J) -- the compatibility square is not
generic (frozen-seed word sampling in the 60 reflections).
SAMPLER CORRECTION (disclosed): the first run's word length
<= 5 letters found NO order-12 words in 4000 frozen-seed tries;
the sampler was widened to words of length <= 12 with 20000
tries. The control's SEMANTICS (random order-12 unitary must
break r^4 = sigma, r^9 = J) is unchanged; the frozen spec text
and SHA are untouched.
CC3 a wrong complex structure J' (i-rotation on a scrambled
coordinate pairing) must break the corpus deployment (fail to
preserve the root set, or fail the free-60-orbit census, or fail
[J', sigma] = 0).
VERDICT ENUMS (frozen):
CLOCK-FIBER-EXACT = both clocks act on the 60 lines, induce the
same Z6, and the joint image is simply
transitive ~= Z60 = Z12 x_{Z6} Z30.
CLOCK-FIBER-PARTIAL = both act on the lines and the Z6 squares agree,
but the joint action is NOT simply transitive
-- the true orbit type is named.
CLOCK-FIBER-DEAD = a clock fails to act on the 60 lines at all
(descent failure), or the Z6 compatibility
fails, or the required element orders
(15/30/60) are absent from the FULL
line-compatible group (no realization can
work).
TEST-VOID = a must-fire control does not fire or a corpus
deployment gate fails.
FIREWALL: experiments/tfpt-discovery probe; EXPLORATION ONLY; writes
nothing; touches no verification/, paper, ledger, changelog or website
surface; no marker moves; NO physics claims. ROOTCLASS-MIXED fence
(v775_gaussian_class_d5_purity, binding): no code->matter or
line->matter semantics is used or implied -- the 60 lines are treated
as pure lattice/group-theory objects; any clock structure found here
carries NO matter reading. A DEAD verdict is a valid, well-powered
outcome and is reported honestly.
Sources (read-only): verification/v633_orbit60_quotient.py (lines,
canon, hermitian form, reflections, 11520 line group), v629 (compiler
clock c = J o sigma, free mu4 quotient), v634_st31_structure.py
(sigma = c^4, J = c^9, G31 order 46080, max element order 24),
v647_st31_degree24.py (line 6-clock), v623_covered_seam.py (seam clock
L^4 = deck, L^12 = id), v1_e8_glue.py (Bourbaki simple roots).
Run:
experiments/tfpt-discovery/.venv/bin/python \\
experiments/tfpt-discovery/clock_fiber_product_probe.py
"""
import hashlib
import itertools
import time
from fractions import Fraction as Fr
CHECKS = []
def check(name, ok, detail=""):
CHECKS.append((name, bool(ok)))
print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
(" -- " + detail) if detail else ""), flush=True)
return bool(ok)
def section(title):
print("=" * 78)
print(title)
print("=" * 78, flush=True)
# ======================================================================
# S0: FROZEN SPEC -- hashed BEFORE any lattice data is computed
# ======================================================================
FROZEN_SPEC = """\
ONE-OBJECT REDUCTION FROZEN SPEC (one_object_reduction_probe)
frozen 2026-08-05, BEFORE computing any lattice data in this run.
CLAIM: the TFPT glue pattern (cyclic isotropic glue group, evenness
q = 0 mod 2Z, unimodularity |H|^2 = |disc|) admits an even unimodular
DIAGONAL glue of A_{d-1} (+) D_{d+1} iff d = 4; the boundary object
with d marks then forces g_car = d+1 = 5 and N_fam = d-1 = 3 as the
two functor readouts of ONE four-point object.
CENSUS RANGE: d = 2..12 inclusive. All arithmetic exact (Fraction).
DIAGONAL glue := Lagrangian H with H int (A(+)0) = 0 = H int (0(+)D).
EVEN glue := q(h) = 0 mod 2Z for all h in H.
UNIMODULAR := |H|^2 = |disc(A_{d-1})| * |disc(D_{d+1})| = 4d.
ODD-relaxed := q(h) = 0 mod Z and b(h,h') = 0 mod Z, not even.
VERDICT RULE (frozen):
TEST-VOID if G2 or G3 fails, or G4 or G5 does not fire;
ONE-OBJECT-DEAD if {d with even unimodular diagonal glue} != {4};
ONE-OBJECT-PARTIAL otherwise. ONE-OBJECT-FORCED is predeclared
unreachable: the c_3 = 1/(2 pi d) link is a corpus-legitimate
REWRITING (c_3 = 1/(2 e_1(a) pi), e_1(a) = 4 = |mu4| = d; tfpt_1
~L189-194, v23) but NOT a corpus derivation of P1 from the boundary
object; this probe must not smuggle a new derivation of P1.
"""
SPEC_SHA = hashlib.sha256(FROZEN_SPEC.encode()).hexdigest()
# ======================================================================
# exact linear algebra helpers (Fractions, no floats anywhere)
# ======================================================================
def cartan_A(n):
M = [[0] * n for _ in range(n)]
for i in range(n):
M[i][i] = 2
for i in range(n - 1):
M[i][i + 1] = M[i + 1][i] = -1
return M
def cartan_D(n):
"""v1_e8_glue convention: A_n chain, detach last node, re-attach at
the fork (n >= 3; D_3 ~= A_3)."""
M = cartan_A(n)
M[n - 2][n - 1] = M[n - 1][n - 2] = 0
M[n - 3][n - 1] = M[n - 1][n - 3] = -1
return M
def frac_inv(G):
"""exact matrix inverse via Gauss-Jordan."""
n = len(G)
A = [[Fr(G[i][j]) for j in range(n)] + [Fr(int(i == k)) for k in range(n)]
for i in range(n)]
for c in range(n):
p = next(r for r in range(c, n) if A[r][c] != 0)
A[c], A[p] = A[p], A[c]
pv = A[c][c]
A[c] = [x / pv for x in A[c]]
for r in range(n):
if r != c and A[r][c] != 0:
f = A[r][c]
A[r] = [A[r][j] - f * A[c][j] for j in range(2 * n)]
return [row[n:] for row in A]
def det_int(G):
"""exact integer determinant (fraction-free would be nicer; Fraction
Gaussian elimination is exact and fine at n <= 11)."""
n = len(G)
A = [[Fr(G[i][j]) for j in range(n)] for i in range(n)]
det = Fr(1)
for c in range(n):
p = next((r for r in range(c, n) if A[r][c] != 0), None)
if p is None:
return 0
if p != c:
A[c], A[p] = A[p], A[c]
det = -det
det *= A[c][c]
for r in range(c + 1, n):
if A[r][c] != 0:
f = A[r][c] / A[c][c]
A[r] = [A[r][j] - f * A[c][j] for j in range(n)]
assert det.denominator == 1
return int(det)
def mod1(x):
return x - (x.numerator // x.denominator)
def mod2(x):
m = x % 2
return m
class DiscForm(object):
"""discriminant group L*/L of an even lattice with Gram G, with the
quadratic form q: A -> Q/2Z and linking form b: A x A -> Q/Z,
computed exactly from the Cartan/Gram matrix."""
def __init__(self, G):
self.G = G
self.n = len(G)
Ginv = frac_inv(G)
gens = [tuple(Ginv[i][j] for i in range(self.n))
for j in range(self.n)] # dual basis vectors, root coords
# BFS closure of the generators mod Z^n
elems = {tuple(Fr(0) for _ in range(self.n))}
frontier = list(elems)
while frontier:
nxt = []
for e in frontier:
for g in gens:
s = tuple(mod1(a + b) for a, b in zip(e, g))
if s not in elems:
elems.add(s)
nxt.append(s)
frontier = nxt
self.elems = sorted(elems)
self.index = {e: i for i, e in enumerate(self.elems)}
self.zero = self.index[tuple(Fr(0) for _ in range(self.n))]
m = len(self.elems)
self.add = [[0] * m for _ in range(m)]
for i, u in enumerate(self.elems):
for j, v in enumerate(self.elems):
self.add[i][j] = self.index[
tuple(mod1(a + b) for a, b in zip(u, v))]
def q(self, i):
v = self.elems[i]
s = Fr(0)
for a in range(self.n):
for b in range(self.n):
s += v[a] * self.G[a][b] * v[b]
return mod2(s)
def b(self, i, j):
u, v = self.elems[i], self.elems[j]
s = Fr(0)
for a in range(self.n):
for b in range(self.n):
s += u[a] * self.G[a][b] * v[b]
return mod1(s)
def order_of(self, i):
k, cur = 1, i
while cur != self.zero:
cur = self.add[cur][i]
k += 1
return k
def structure(self):
from collections import Counter
return dict(Counter(self.order_of(i) for i in range(len(self.elems))))
class SumForm(object):
"""orthogonal direct sum of two DiscForms; elements are index pairs."""
def __init__(self, A, D):
self.A, self.D = A, D
self.elems = [(i, j) for i in range(len(A.elems))
for j in range(len(D.elems))]
self.index = {e: k for k, e in enumerate(self.elems)}
self.zero = self.index[(A.zero, D.zero)]
def add(self, x, y):
(a1, d1), (a2, d2) = self.elems[x], self.elems[y]
return self.index[(self.A.add[a1][a2], self.D.add[d1][d2])]
def q(self, x):
a, d = self.elems[x]
return mod2(self.A.q(a) + self.D.q(d))
def b(self, x, y):
(a1, d1), (a2, d2) = self.elems[x], self.elems[y]
return mod1(self.A.b(a1, a2) + self.D.b(d1, d2))
def closure(self, gens):
H = {self.zero}
frontier = list(H)
while frontier:
nxt = []
for h in frontier:
for g in gens:
s = self.add(h, g)
if s not in H:
H.add(s)
nxt.append(s)
frontier = nxt
return frozenset(H)
def all_subgroups(self):
subs = {frozenset({self.zero})}
work = [frozenset({self.zero})]
while work:
H = work.pop()
for e in range(len(self.elems)):
if e in H:
continue
H2 = self.closure(list(H) + [e])
if H2 not in subs:
subs.add(H2)
work.append(H2)
return subs
def classify(d):
"""full glue census for A_{d-1} (+) D_{d+1}."""
A = DiscForm(cartan_A(d - 1))
D = DiscForm(cartan_D(d + 1))
S = SumForm(A, D)
total = len(S.elems)
lag_order_sq = total # |H|^2 == |A|*|D| condition on the order
res = {
"d": d,
"discA": A.structure(),
"discD": D.structure(),
"orderA": len(A.elems),
"orderD": len(D.elems),
"square": int(round(total ** 0.5)) ** 2 == total,
"even_unimod": [],
"even_unimod_diagonal": [],
"odd_unimod": [],
"even_nontrivial_any": [],
}
for H in S.all_subgroups():
hl = sorted(H)
even = all(S.q(h) == 0 for h in hl)
modZ = (all(S.q(h).denominator == 1 for h in hl)
and all(S.b(x, y) == 0 for x in hl for y in hl))
lag = len(H) * len(H) == lag_order_sq
intA = sum(1 for h in hl if S.elems[h][1] == D.zero)
intD = sum(1 for h in hl if S.elems[h][0] == A.zero)
diagonal = (intA == 1 and intD == 1) # only 0 in each factor
if even and len(H) > 1:
res["even_nontrivial_any"].append(H)
if lag and even:
res["even_unimod"].append(H)
if diagonal:
res["even_unimod_diagonal"].append(H)
if lag and modZ and not even:
res["odd_unimod"].append(H)
return A, D, S, res
# =====================================================================
def main():
T0 = time.time()
section("S0: FROZEN SPEC (SHA-256 %s)" % SPEC_SHA)
print(FROZEN_SPEC)
# ==================================================================
# S1: THE d-CENSUS 2..12
# ==================================================================
section("S1: the d-census (exact discriminant-form glue theory, d = 2..12)")
TABLE = {}
FORMS = {}
for d in range(2, 13):
A, D, S, res = classify(d)
FORMS[d] = (A, D, S)
TABLE[d] = res
print(" d=%2d disc(A_%d)=%s (Z_%d) disc(D_%d)=%s 4d=%2d square=%s "
"evenUnimod=%d (diag %d) oddUnimod=%d evenAnyIndex(nontriv)=%d"
% (d, d - 1, res["discA"], res["orderA"], d + 1, res["discD"],
4 * d, res["square"], len(res["even_unimod"]),
len(res["even_unimod_diagonal"]), len(res["odd_unimod"]),
len(res["even_nontrivial_any"])), flush=True)
# structural sanity: disc(A_{d-1}) = Z_d; disc(D_{d+1}) = Z4 (d even) /
# Z2xZ2 (d odd)
ok_A = all(TABLE[d]["orderA"] == d and max(TABLE[d]["discA"]) == d
for d in range(2, 13))
ok_D = all(TABLE[d]["orderD"] == 4 and
(max(TABLE[d]["discD"]) == (4 if d % 2 == 0 else 2))
for d in range(2, 13))
check("S1.1 disc(A_{d-1}) ~= Z_d for all d (order d, cyclic)", ok_A)
check("S1.2 disc(D_{d+1}) ~= Z4 for d even, Z2 x Z2 for d odd (order 4)",
ok_D)
forced_set = sorted(d for d in TABLE
if len(TABLE[d]["even_unimod_diagonal"]) > 0)
any_glue_set = sorted(d for d in TABLE if len(TABLE[d]["even_unimod"]) > 0)
g1 = check("S1.3 [G1] {d : even unimodular DIAGONAL glue exists} = %s "
"(forcing claim needs exactly {4})" % forced_set,
forced_set == [4])
check("S1.4 [G1+] even stronger: {d : ANY even unimodular glue exists} "
"= %s (all glues at the forced d are diagonal)" % any_glue_set,
any_glue_set == [4])
# G6: the coarse square layer
square_set = sorted(d for d in TABLE if TABLE[d]["square"])
check("S1.5 [G6] coarse layer: |H|^2 = 4d is a perfect square only for "
"d in %s (d must itself be a square)" % square_set,
square_set == [4, 9])
d9 = TABLE[9]
# how d = 9 dies: no even Lagrangian; exact reason: the order-2 part of
# any |H| = 6 subgroup must sit in disc(D_10) (Z9 has no 2-torsion) and
# no element of disc(D_10) has q = 0 mod 2Z.
D10 = FORMS[9][1]
d10_qvals = sorted(str(D10.q(i)) for i in range(len(D10.elems)))
check("S1.6 [G6] d = 9 dies at isotropy: 0 even Lagrangians; "
"disc(D_10) q-values %s contain no 0 besides the origin -> the "
"forced order-2 part of |H| = 6 cannot be even-isotropic"
% d10_qvals,
len(d9["even_unimod"]) == 0
and all(D10.q(i) != 0 for i in range(len(D10.elems))
if i != D10.zero))
# ==================================================================
# S2: G2 -- d = 4 must reproduce v92 EXACTLY
# ==================================================================
section("S2: [G2] d = 4 reproduces v92 (two diagonal Lagrangians, "
"graphs of k -> k and k -> 3k)")
A4, D4, S4 = FORMS[4]
res4 = TABLE[4]
# corpus q-values: q(A3) = 3/4, q(D5) = 5/4 on primitive generators
qA_prim = sorted({str(A4.q(i)) for i in range(len(A4.elems))
if A4.order_of(i) == 4})
qD_prim = sorted({str(D4.q(i)) for i in range(len(D4.elems))
if D4.order_of(i) == 4})
check("S2.1 q on primitive generators: q(A3) = {3/4} mod 2Z, "
"q(D5) = {5/4} mod 2Z (v1: 3/4 + 5/4 = 2 = E8 root norm)",
qA_prim == ["3/4"] and qD_prim == ["5/4"],
"qA %s qD %s" % (qA_prim, qD_prim))
lags = res4["even_unimod"]
check("S2.2 EXACTLY two even Lagrangian glues at d = 4 (v92: <(1,1)> "
"and <(1,3)>), both diagonal, both cyclic Z4",
len(lags) == 2 and len(res4["even_unimod_diagonal"]) == 2
and all(max(len({h for h in H}), 0) == 4 for H in lags))
# graph check: each Lagrangian is the graph of an anti-isometry
# Z4 -> Z4; the two anti-isometries are k -> k and k -> 3k on suitable
# generators (equivalently: their intersection is {0, (2,2)})
a_gen = next(i for i in range(len(A4.elems)) if A4.order_of(i) == 4)
d_gen = next(i for i in range(len(D4.elems)) if D4.order_of(i) == 4)
d_gen3 = D4.add[D4.add[d_gen][d_gen]][d_gen]
exp1 = S4.closure([S4.index[(a_gen, d_gen)]])
exp2 = S4.closure([S4.index[(a_gen, d_gen3)]])
check("S2.3 the two Lagrangians ARE the graphs <(a, s)> and <(a, 3s)> "
"for a fixed primitive pair (a, s) (v92's <(1,1)>, <(1,3)>)",
{exp1, exp2} == set(lags))
inter = set(lags[0]) & set(lags[1])
check("S2.4 their intersection is the order-2 halfway glue "
"{0, (2,2)} (v92's unique isotropic Z2 = the SO(16)_1 rung)",
len(inter) == 2 and all(S4.q(h) == 0 for h in inter))
# ==================================================================
# S3: G3 -- explicit lattice certificate at d = 4 (v1 reproduction)
# ==================================================================
section("S3: [G3] explicit glue A3 (+) D5 + <(w1, s)> = E8 "
"(240 roots, even, det 1)")
# orthonormal ambient R^4 (+) R^5; A3 = sum-zero integer vectors of Z^4,
# D5 = even-sum integer vectors of Z^5.
A3_basis = [(1, -1, 0, 0), (0, 1, -1, 0), (0, 0, 1, -1)]
D5_basis = [(1, -1, 0, 0, 0), (0, 1, -1, 0, 0), (0, 0, 1, -1, 0),
(0, 0, 0, 1, -1), (0, 0, 0, 1, 1)]
wA = (Fr(3, 4), Fr(-1, 4), Fr(-1, 4), Fr(-1, 4)) # A3 weight, q = 3/4
wD = (Fr(1, 2),) * 5 # D5 spinor, q = 5/4
glue = tuple(list(wA) + list(wD)) # norm 2 = E8 root
def dot(u, v):
return sum(Fr(a) * Fr(b) for a, b in zip(u, v))
check("S3.1 glue vector g = (w_A3, spinor_D5) has norm q(A3) + q(D5) "
"= 3/4 + 5/4 = 2 (an E8 root already)",
dot(wA, wA) == Fr(3, 4) and dot(wD, wD) == Fr(5, 4)
and dot(glue, glue) == 2)
full9 = [tuple(list(b) + [0] * 5) for b in A3_basis] + \
[tuple([0] * 4 + list(b)) for b in D5_basis]
# glued basis: replace one root vector by g so that the 8-set has det 1
best = None
for drop in range(8):
basis = [full9[i] for i in range(8) if i != drop] + [glue]
Gm = [[dot(u, v) for v in basis] for u in basis]
dt = det_int([[int(x) if x.denominator == 1 else x for x in row]
for row in Gm]) if all(
x.denominator == 1 for row in Gm for x in row) else None
if dt is None:
# Gram has half-integers off-diagonal? (g vs roots gives
# integers here, but keep the exact fallback)
n = len(basis)
Af = [[Fr(Gm[i][j]) for j in range(n)] for i in range(n)]
dtf = Fr(1)
for c in range(n):
p = next((r for r in range(c, n) if Af[r][c] != 0), None)
if p is None:
dtf = Fr(0)
break
if p != c:
Af[c], Af[p] = Af[p], Af[c]
dtf = -dtf
dtf *= Af[c][c]
for r in range(c + 1, n):
if Af[r][c] != 0:
f = Af[r][c] / Af[c][c]
Af[r] = [Af[r][j] - f * Af[c][j] for j in range(n)]
dt = dtf
if dt == 1:
even_diag = all(Gm[i][i] % 2 == 0 for i in range(8))
integer = all(x.denominator == 1 for row in Gm for x in row)
if even_diag and integer:
best = (drop, basis, Gm)
break
check("S3.2 glued basis found (drop root #%s, add g): Gram is integer, "
"even diagonal, det = 1 -> even UNIMODULAR rank 8"
% (best[0] if best else "-"), best is not None)
# count norm-2 vectors coset by coset: L' = L + Z g, L'/L = Z4
def coset_norm_counts(k, side):
"""multiset {norm: count} of vectors in k*w + lattice, norm <= 2."""
if side == "A":
w, dim = wA, 4
else:
w, dim = wD, 5
counts = {}
for m in itertools.product(range(-3, 4), repeat=dim):
if side == "A" and sum(m) != 0:
continue
if side == "D" and sum(m) % 2 != 0:
continue
v = tuple(Fr(k) * w[i] + m[i] for i in range(dim))
nrm = dot(v, v)
if nrm <= 2:
counts[nrm] = counts.get(nrm, 0) + 1
return counts
root_total = 0
per_coset = []
for k in range(4):
cA = coset_norm_counts(k, "A")
cD = coset_norm_counts(k, "D")
n_k = sum(cA[na] * cD[nd] for na in cA for nd in cD if na + nd == 2)
per_coset.append(n_k)
root_total += n_k
check("S3.3 norm-2 count per glue coset k = 0..3: %s, total = %d "
"(v1: A3+D5 roots 12+40 = 52 in k = 0; glued total must be 240)"
% (per_coset, root_total),
per_coset[0] == 52 and root_total == 240)
# integrality of all 240 roots over the glued basis
if best:
_, basis, Gm = best
Ginv8 = frac_inv([[Fr(x) for x in row] for row in Gm])
roots_all = []
for k in range(4):
for mA in itertools.product(range(-3, 4), repeat=4):
if sum(mA) != 0:
continue
vA = tuple(Fr(k) * wA[i] + mA[i] for i in range(4))
nA = dot(vA, vA)
if nA > 2:
continue
for mD in itertools.product(range(-3, 4), repeat=5):
if sum(mD) % 2 != 0:
continue
vD = tuple(Fr(k) * wD[i] + mD[i] for i in range(5))
if nA + dot(vD, vD) == 2:
roots_all.append(tuple(list(vA) + list(vD)))
integral = True
for r in roots_all:
proj = [dot(r, bv) for bv in basis]
coef = [sum(Ginv8[i][j] * proj[j] for j in range(8))
for i in range(8)]
if not all(c.denominator == 1 for c in coef):
integral = False
break
check("S3.4 all %d enumerated norm-2 vectors are INTEGRAL over the "
"glued basis: the glue closes to THE even unimodular E8 "
"lattice (v1 certificate reproduced)" % len(roots_all),
integral and len(roots_all) == 240)
# ==================================================================
# S4: CONTROLS G4 / G5 -- the constraints must BIND
# ==================================================================
section("S4: controls -- relaxing evenness/unimodularity must open "
"other d (must fire)")
odd_set = sorted(d for d in TABLE if d != 4 and len(TABLE[d]["odd_unimod"]))
g4 = check("S4.1 [G4, must fire] dropping EVENNESS opens d in %s "
"(odd unimodular glues exist there; e.g. d = 9: "
"A8 (+) D10 + <(3,0),(0,v)> is an ODD unimodular rank-18 "
"glue)" % odd_set, len(odd_set) > 0)
sub_set = sorted(d for d in TABLE
if d != 4 and len(TABLE[d]["even_nontrivial_any"]))
g5 = check("S4.2 [G5, must fire] dropping UNIMODULARITY opens d in %s "
"(nontrivial even isotropic glue subgroups of smaller index "
"exist there)" % sub_set, len(sub_set) > 0)
check("S4.3 the intermediate at d = 4 itself: exactly one nontrivial "
"even isotropic subgroup BELOW the Lagrangians (v92's halfway "
"SO(16)_1 rung, |H| = 2)",
sum(1 for H in TABLE[4]["even_nontrivial_any"] if len(H) == 2) == 1)
# ==================================================================
# S5: [G7] corollaries at the forced d -- typed honestly
# ==================================================================
section("S5: [G7] corollaries at the forced d = 4 (typed)")
d_star = forced_set[0] if len(forced_set) == 1 else None
if d_star is not None:
check("S5.1 [verified arithmetic] g_car = d+1 = %d, N_fam = d-1 = %d, "
"rank = 2d = %d, glue index = sqrt(4d) = %d = d = |mu4|"
% (d_star + 1, d_star - 1, 2 * d_star,
int(round((4 * d_star) ** 0.5))),
d_star + 1 == 5 and d_star - 1 == 3 and 2 * d_star == 8
and int(round((4 * d_star) ** 0.5)) == 4)
check("S5.2 [corpus-anchored] N_fam = d-1 = rank H_1(P^1 minus d "
"points) = 3 (tfpt_1 ~L894/4234: H_1 = Z^3 hence A_3); "
"q(A3) = N_fam/|mu4|, q(D5) = g_car/|mu4| with n+1 = 4 = "
"|mu4| = d (tfpt_1 ~L4201-4205)", d_star == 4)
# the c3 honesty gate: arithmetic identity is [E]-grade, the LINK is
# an interpretation -- frozen in the spec, restated here.
c3_arith = Fr(1, 2 * d_star) # times 1/pi: 1/(2 pi d) = 1/(8 pi)
check("S5.3 [verified arithmetic] 1/(2 pi d)|_{d=4} = 1/(8 pi): "
"coefficient 1/(2d) = %s = 1/8" % c3_arith,
c3_arith == Fr(1, 8))
check("S5.4 [HONESTY GATE -- interpretation, NOT derivation] the "
"1/(2 pi d) READING of P1 is corpus-legitimate only as a "
"rewriting: c_3 = 1/(2 e_1(a) pi) with e_1(a) = 4 = |mu4| "
"(tfpt_1 ~L189-194, v23 anchor-first refinement) and "
"|mu4| = #marks = d; but the corpus derivation chain of P1 "
"is Gauss-Bonnet 1/(|Z2| * oint K dA) = 1/(2 * 4 pi) where "
"4 pi is TOTAL SPHERE CURVATURE, not d (tfpt_1 ~L670-689); "
"P1 status: primitive postulate (tfpt_1 ~L169). The glue "
"census therefore does NOT derive P1; the c_3 link is typed "
"INTERPRETATION", True)
# ==================================================================
# S6: VERDICT (frozen rule)
# ==================================================================
section("S6: VERDICT (frozen enum: ONE-OBJECT-FORCED / ONE-OBJECT-"
"PARTIAL / ONE-OBJECT-DEAD / TEST-VOID)")
g2_ok = all(ok for name, ok in CHECKS if name.startswith("S2."))
g3_ok = all(ok for name, ok in CHECKS if name.startswith("S3."))
controls_fire = g4 and g5
if not (g2_ok and g3_ok and controls_fire):
VERDICT = "TEST-VOID"
elif forced_set != [4]:
VERDICT = "ONE-OBJECT-DEAD"
else:
VERDICT = "ONE-OBJECT-PARTIAL"
print("VERDICT: %s" % VERDICT)
print(" glue-forced d set: %s (needed {4})" % forced_set)
print(" c_3 link: corpus-legitimate rewriting via e_1(a) = |mu4| = d "
"(tfpt_1 ~L192), typed INTERPRETATION -- FORCED predeclared "
"unreachable")
print(" controls: evenness relaxation opens %s, unimodularity "
"relaxation opens %s" % (odd_set, sub_set))
n_pass = sum(1 for _, ok in CHECKS if ok)
print("%d/%d checks passed" % (n_pass, len(CHECKS)))
print("FROZEN_SPEC SHA-256: %s" % SPEC_SHA)
print("elapsed: %.1f s" % (time.time() - T0))
print("ALL CHECKS PASSED" if n_pass == len(CHECKS) else
"SOME CHECKS FAILED")
return 0 if n_pass == len(CHECKS) else 1
_run_part1 = main
def _run_part2():
# PART 2 -- clock_fiber_product_probe.py (verbatim; module-level names
# local to this function scope)
import hashlib
import itertools
import time
from collections import Counter, deque
from fractions import Fraction as Fr
from math import gcd
T0 = time.time()
CHECKS = []
def check(name, ok, detail=""):
CHECKS.append((name, bool(ok)))
print("[%s] %s%s" % ("PASS" if ok else "FAIL", name,
(" -- " + detail) if detail else ""), flush=True)
return bool(ok)
def section(title):
print("=" * 78)
print(title)
print("=" * 78, flush=True)
# ==================================================================
# S0: FROZEN SPEC -- hashed BEFORE any root data is computed
# ==================================================================
FROZEN_SPEC = """\
CLOCK FIBER-PRODUCT FROZEN SPEC (clock_fiber_product_probe)
frozen 2026-08-05, BEFORE computing any root data in this run.
HYPOTHESIS: L_60 ~= Z12 x_{Z6} Z30 acts canonically (simply
transitively, or at least orbit-definingly) on the 60 Gaussian E8
lines; the seam clock (corpus avatar c = J o sigma, order 12, c^4 =
sigma, c^9 = J) and the Coxeter clock (w = product of the 8 Bourbaki
simple reflections, order 30, w^15 = -1) are its two legs over the
shared Z6 (sheet x family phase); 240 = 4 x 60 via mu4 is the final
layer.
DECISION RULE (frozen):
TEST-VOID if a deployment gate (root census, clock identities,
Coxeter order/census) fails or a control CC1-CC3 does
not fire;
CLOCK-FIBER-EXACT if both clocks descend to the 60 lines, the Z6
squares agree, and the joint line image is simply
transitive of order 60 (necessarily Z60);
CLOCK-FIBER-PARTIAL if both descend and the Z6 squares agree but
the joint action is not simply transitive (name the
orbit type);
CLOCK-FIBER-DEAD otherwise (descent failure / Z6 mismatch /
required orders 15, 30, 60 absent from the FULL
line-compatible group -- the existence sweep kills every
realization, not only the deployed one).
MEASURED DRIVERS (not gates; reported exactly): the line order of c,
the descent subgroup {k : w^k preserves the line partition}, the full
line-compatible group order and its element-order spectrum, the joint
<c, w> orbit census on the 240 roots.
"""
SPEC_SHA = hashlib.sha256(FROZEN_SPEC.encode()).hexdigest()
section("S0: FROZEN SPEC (SHA-256 %s)" % SPEC_SHA)
print(FROZEN_SPEC)
# ==================================================================
# S1: deployment -- roots, lines, seam clock (corpus reproduction gates)
# ==================================================================
section("S1: deployment gates (corpus reproduction, v629/v633/v634)")
roots = []
for v in itertools.product(range(-1, 2), repeat=8):
if sum(a * a for a in v) == 2 and sum(v) % 2 == 0:
roots.append(tuple(Fr(a) for a in v))
half = Fr(1, 2)
for y in itertools.product((0, -1), repeat=8):
v = tuple(Fr(a) + half for a in y)
if sum(a * a for a in v) == 2 and sum(v) % 2 == 0:
roots.append(v)
roots = sorted(roots)
RS = set(roots)
root_idx = {r: i for i, r in enumerate(roots)}
N = len(roots)
def J_all(v):
out = []
for i in range(0, 8, 2):
a, b = v[i], v[i + 1]
out += [-b, a]
return tuple(out)
def sigma(v):
return (v[4], v[5], v[0], v[1], v[2], v[3], v[6], v[7])
def canon(v):
o = v
best = v
for _ in range(3):
o = J_all(o)
if o < best:
best = o
return best
check("S1.1 240 roots in the D8+spinor model (v633 verbatim)", N == 240)
reps = sorted({canon(v) for v in roots})
line_of = {}
for i, rep in enumerate(reps):
o = rep
for _ in range(4):
line_of[o] = i
o = J_all(o)
orb_sizes = Counter()
for rep in reps:
orb = {rep, J_all(rep), J_all(J_all(rep)), J_all(J_all(J_all(rep)))}
orb_sizes[len(orb)] += 1
check("S1.2 [mu4 layer, test (iii)] J free with EXACTLY 60 orbits of "
"size 4: 240 = 4 x 60 (v629 R3 / v633 Q1)",
len(reps) == 60 and orb_sizes == Counter({4: 60}))
def perm_of(f):
return tuple(root_idx[f(roots[i])] for i in range(N))
def compose(p, q):
"""(p o q)(x) = p[q[x]]."""
return tuple(p[q[i]] for i in range(N))
def perm_order(p):
n = len(p)
seen = [False] * n
o = 1
for s in range(n):
if seen[s]:
continue
ln, cur = 0, s
while not seen[cur]:
seen[cur] = True
cur = p[cur]
ln += 1
o = o * ln // gcd(o, ln)
return o
def orbit_census(perms, n):
seen = [False] * n
cens = Counter()
for s in range(n):
if seen[s]:
continue
comp = {s}
dq = deque([s])
seen[s] = True
while dq:
x = dq.popleft()
for p in perms:
for y in (p[x],):
if not seen[y]:
seen[y] = True
comp.add(y)
dq.append(y)
cens[len(comp)] += 1
return cens
P_J = perm_of(J_all)
P_sig = perm_of(sigma)
P_c = compose(P_J, P_sig) # c = J o sigma
P_id = tuple(range(N))
P_minus = perm_of(lambda v: tuple(-a for a in v))
def perm_pow(p, k):
r = P_id
for _ in range(k):
r = compose(r, p)
return r
check("S1.3 seam clock avatar c = J o sigma has order 12 on the roots "
"(v629 R1)", perm_order(P_c) == 12)
check("S1.4 clock identities c^4 = sigma, c^9 = J, c^6 = -1 "
"(v634 S7; L^4 = deck matches: c^4 IS the order-3 family deck)",
perm_pow(P_c, 4) == P_sig and perm_pow(P_c, 9) == P_J
and perm_pow(P_c, 6) == P_minus)
# ==================================================================
# S2: the seam clock on the 60 lines
# ==================================================================
section("S2: the seam clock's line action (measured driver)")
LINES = [frozenset({rep, J_all(rep), J_all(J_all(rep)),
J_all(J_all(J_all(rep)))}) for rep in reps]
LINESET = set(LINES)
def line_perm_of_rootperm(p):
"""line permutation induced by a root permutation IF it preserves
the line partition, else None."""
out = []
for L in LINES:
img = frozenset(roots[p[root_idx[r]]] for r in L)
if img not in LINESET:
return None
out.append(LINES.index(img))
return tuple(out)
lc = line_perm_of_rootperm(P_c)
lc_ok = lc is not None
lc_ord = perm_order(lc) if lc_ok else 0
lc_kernel = sorted(k for k in range(12)
if line_perm_of_rootperm(perm_pow(P_c, k))
== tuple(range(60))) if lc_ok else []
check("S2.1 c descends to the 60 lines (J = c^9 is line-trivial by "
"construction)", lc_ok)
lsig = line_perm_of_rootperm(P_sig)
check("S2.2 MEASURED: line order of the seam clock = %d; line-trivial "
"powers %s; line image = <sigma's line action> (census on lines "
"%s)" % (lc_ord, lc_kernel,
dict(orbit_census([lc], 60)) if lc_ok else "-"),
lc_ok, "c^3 = J^3 sigma^3 = J^3 is line-trivial, so the Z12 seam "
"clock acts on the lines through Z12/<c^3> = Z3 only")
same_as_sigma = lc_ok and orbit_census([lc], 60) == orbit_census([lsig], 60)
check("S2.3 the seam clock's line action has the sigma line census "
"{3: 19, 1: 3} (v633 Q4)",
same_as_sigma and orbit_census([lsig], 60) == Counter({3: 19, 1: 3}))
# ==================================================================
# S3: the Coxeter clock (deployment gates)
# ==================================================================
section("S3: the Coxeter clock w = s_1 ... s_8 (Bourbaki, v1 basis)")
def bourbaki_simple_roots():
e = [[Fr(int(i == j)) for j in range(8)] for i in range(8)]
a1 = [Fr(1, 2) * (e[0][j] - e[1][j] - e[2][j] - e[3][j] - e[4][j]
- e[5][j] - e[6][j] + e[7][j]) for j in range(8)]
a2 = [e[0][j] + e[1][j] for j in range(8)]
a3 = [e[1][j] - e[0][j] for j in range(8)]
a4 = [e[2][j] - e[1][j] for j in range(8)]
a5 = [e[3][j] - e[2][j] for j in range(8)]
a6 = [e[4][j] - e[3][j] for j in range(8)]
a7 = [e[5][j] - e[4][j] for j in range(8)]
a8 = [e[6][j] - e[5][j] for j in range(8)]
return [tuple(a) for a in (a1, a2, a3, a4, a5, a6, a7, a8)]
def refl(alpha):
def f(v, alpha=alpha):
s = sum(a * b for a, b in zip(v, alpha)) # norm 2 simple roots
return tuple(v[j] - s * alpha[j] for j in range(8))
return f
simples = bourbaki_simple_roots()
check("S3.1 the 8 Bourbaki simple roots are norm-2 roots of the model "
"(v1 basis deployed verbatim)",
all(sum(a * a for a in s) == 2 and s in RS for s in simples))
P_w = P_id
for s in simples:
P_w = compose(P_w, perm_of(refl(s))) # w = s1 o s2 o ... o s8
w_ord = perm_order(P_w)
w_census = orbit_census([P_w], N)
check("S3.2 Coxeter clock order 30 on the 240 roots (h(E8) = 30)",
w_ord == 30)
check("S3.3 Kostant orbit census {30: 8} (8 = rank E8 free 30-orbits)",
w_census == Counter({30: 8}), str(dict(w_census)))
check("S3.4 w^15 = -1 (all E8 exponents odd)",
perm_pow(P_w, 15) == P_minus)
# ==================================================================
# S4: descent of the Coxeter clock to the 60 lines (measured driver)
# ==================================================================
section("S4: does the Coxeter clock act on the 60 lines? (test (i))")
descent_powers = sorted(k for k in range(30)
if line_perm_of_rootperm(perm_pow(P_w, k))
is not None)
w_descends = 1 in descent_powers
# the descent powers form a subgroup <w^m> of <w>
m = descent_powers[1] if len(descent_powers) > 1 else 0
line_shadow = {}
for k in descent_powers:
lp = line_perm_of_rootperm(perm_pow(P_w, k))
line_shadow[k] = perm_order(lp)
check("S4.1 MEASURED: powers of w preserving the line partition: %s "
"(a subgroup <w^%s> of <w>); their induced line orders: %s"
% (descent_powers, m if m else "-", line_shadow), True)
check("S4.2 the Coxeter clock itself descends to the 60 lines: %s "
"(descent FAILURE means the strict fiber-product hypothesis "
"cannot be built from the deployed Coxeter clock)"
% w_descends, True, "measured driver, feeds the frozen verdict "
"rule, not a gate")
wJw = compose(P_w, compose(P_J, perm_pow(P_w, 29)))
normalizes = (wJw == P_J) or (wJw == perm_pow(P_J, 3))
check("S4.3 MEASURED: w J w^-1 in {J, J^-1}: %s (normalizing <J> is "
"exactly the line-partition compatibility)" % normalizes, True)
# ==================================================================
# S5: existence sweep -- the FULL line-compatible group and its
# element-order spectrum (decides every realization)
# ==================================================================
section("S5: existence sweep over N(<J>): line group + order spectrum")
def herm(u, v):
re = Fr(0)
im = Fr(0)
for i in range(4):
a, b = u[2 * i], u[2 * i + 1]
c, d = v[2 * i], v[2 * i + 1]
re += a * c + b * d
im += b * c - a * d
return (re, im)
def refl2(x, v):
"""order-2 unitary reflection R_v(x) = x - H(x,v) v (v633 Q6)."""
re, im = herm(x, v)
out = []
for i in range(4):
c, d = v[2 * i], v[2 * i + 1]
out += [x[2 * i] - (re * c - im * d),
x[2 * i + 1] - (re * d + im * c)]
return tuple(out)
def conj_map(v):
"""complex conjugation per pair: (a, b) -> (a, -b);
conj o J o conj^-1 = J^-1."""
out = []
for i in range(0, 8, 2):
out += [v[i], -v[i + 1]]
return tuple(out)
P_conj = perm_of(conj_map)
cJc = compose(P_conj, compose(P_J, P_conj)) # conj is an involution
check("S5.1 conj preserves the roots and conjugates J to J^-1 "
"(the antiunitary leg of N(<J>))",
cJc == perm_pow(P_J, 3) and compose(P_conj, P_conj) == P_id)
line_gens = set()
for v in reps:
lp = line_perm_of_rootperm(perm_of(lambda x, v=v: refl2(x, v)))
line_gens.add(lp)
lp_conj = line_perm_of_rootperm(P_conj)
line_gens.add(lp_conj)
line_gens = sorted(line_gens)
ident60 = tuple(range(60))
seen_g = {ident60}
dq = deque([ident60])
CAP = 60000
while dq and len(seen_g) <= CAP:
p = dq.popleft()
for g in line_gens:
q = tuple(p[g[i]] for i in range(60))
if q not in seen_g:
seen_g.add(q)
dq.append(q)
full_order = len(seen_g)
check("S5.2 the FULL line-compatible group (60 unitary reflections + "
"conj) has order %d (v633 Q7 unitary part: 11520; N(<J>) adds "
"the conj coset)" % full_order, full_order in (11520, 23040),
"11520 means conj's line image is already unitary-realized; "
"23040 means the antiunitary coset is genuinely new")
spec = Counter(perm_order(p) for p in seen_g)
has15 = spec.get(15, 0) > 0
has30 = spec.get(30, 0) > 0
has60 = spec.get(60, 0) > 0
maxord = max(spec)
check("S5.3 MEASURED element-order spectrum of the full line world: "
"%s" % dict(sorted(spec.items())), True)
kill_orders = not (has15 or has30 or has60)
check("S5.4 DECISIVE: orders {15, 30, 60} present? 15:%s 30:%s 60:%s "
"(max order %d). Any root-level order-30 clock acting on the "
"lines would induce line order 30 or 15 (its cyclic group meets "
"the line kernel <J> in order <= 2 since 4 does not divide 30); "
"a simply-transitive Z60 = Z12 x_{Z6} Z30 needs line order 60. "
"Absence of ALL THREE kills EVERY realization"
% (has15, has30, has60, maxord), True)
# joint <c, w> orbit census on the 240 roots (honest orbit type)
joint_census = orbit_census([P_c, P_w], N)
check("S5.5 MEASURED: joint <seam clock, Coxeter clock> orbit census "
"on the 240 roots: %s" % dict(joint_census), True)
# ==================================================================
# S6: the abstract fiber product + control CC1 (wrong pairing)
# ==================================================================
section("S6: abstract layer -- Z12 x_{Z6} Z30 vs Z12 x Z30 (control CC1)")
fib = [(a, b) for a in range(12) for b in range(30) if a % 6 == b % 6]
fib_orders = Counter()
for (a, b) in fib:
k = 1
x, y = a, b
while (x, y) != (0, 0):
x, y = (x + a) % 12, (y + b) % 30
k += 1
fib_orders[k] += 1
fib_cyclic = fib_orders.get(60, 0) > 0
check("S6.1 the fiber product Z12 x_{Z6} Z30 has order %d and IS "
"cyclic Z60 (max element order 60, phi(60) = %d generators)"
% (len(fib), fib_orders.get(60, 0)),
len(fib) == 60 and fib_cyclic and fib_orders.get(60, 0) == 16)
cc1 = check("S6.2 [CC1, must fire] wrong pairing: |Z12 x Z30| = 360 "
"!= 60 and its invariant factors are (6, 60) != (60): the "
"full product can NEVER act simply transitively on the 60 "
"lines; only the fiber product has the right size",
12 * 30 == 360 and gcd(12, 30) == 6
and (12 * 30 // 6 == 60))
# ==================================================================
# S7: controls CC2 / CC3 (must fire)
# ==================================================================
section("S7: controls CC2 (random order-12 unitary) and CC3 (wrong J)")
class LCG(object):
def __init__(self, seed):
self.s = seed
def next(self, n):
self.s = (1103515245 * self.s + 12345) % (1 << 31)
return self.s % n
refl_perms = sorted({perm_of(lambda x, v=v: refl2(x, v)) for v in reps})
rng = LCG(20260805)
fired_cc2 = 0
tried = 0
found12 = 0
while found12 < 5 and tried < 20000:
tried += 1
p = refl_perms[rng.next(len(refl_perms))]
for _ in range(rng.next(11) + 1):
p = compose(p, refl_perms[rng.next(len(refl_perms))])
if perm_order(p) == 12:
found12 += 1
breaks = (perm_pow(p, 4) != P_sig) or (perm_pow(p, 9) != P_J)
if breaks:
fired_cc2 += 1
cc2 = check("S7.1 [CC2, must fire] %d/%d random order-12 unitary "
"elements (frozen-seed words in the 60 reflections) BREAK "
"the seam-clock deployment identities (r^4 = sigma, "
"r^9 = J): the compatibility square is NOT generic"
% (fired_cc2, found12), found12 >= 3 and fired_cc2 == found12)
def J_wrong(v):
"""i-rotation on the scrambled pairing (0,2),(1,3),(4,6),(5,7)."""
out = list(v)
for (i, j) in ((0, 2), (1, 3), (4, 6), (5, 7)):
out[i], out[j] = -v[j], v[i]
return tuple(out)
jw_preserves = all(J_wrong(r) in RS for r in roots)
if jw_preserves:
P_jw = perm_of(J_wrong)
jw_free60 = (perm_order(P_jw) == 4 and
orbit_census([P_jw], N) == Counter({4: 60}))
jw_comm = compose(P_jw, P_sig) == compose(P_sig, P_jw)
jw_broken = not (jw_free60 and jw_comm)
else:
jw_broken = True
cc3 = check("S7.2 [CC3, must fire] the wrong complex structure J' "
"(pairing (0,2),(1,3),(4,6),(5,7)) breaks the deployment: "
"root-preservation %s%s"
% (jw_preserves,
"" if not jw_preserves else
", free-60 census %s, [J',sigma] = 0 %s"
% (jw_free60, jw_comm)), jw_broken)
# ==================================================================
# S8: VERDICT (frozen rule)
# ==================================================================
section("S8: VERDICT (frozen enum: CLOCK-FIBER-EXACT / "
"CLOCK-FIBER-PARTIAL / CLOCK-FIBER-DEAD / TEST-VOID)")
deploy_ok = all(ok for name, ok in CHECKS
if name.startswith(("S1.", "S3.")))
controls_ok = cc1 and cc2 and cc3
both_descend = lc_ok and w_descends
z6_compatible = False
simply_transitive = False
if both_descend:
lw = line_perm_of_rootperm(P_w)
# Z6 squares: images of c^2 (Z6 side of Z12) and w^5 (Z6 side of
# Z30) must generate the same line action
lc2 = line_perm_of_rootperm(perm_pow(P_c, 2))
lw5 = line_perm_of_rootperm(perm_pow(P_w, 5))
z6_compatible = (lc2 is not None and lw5 is not None
and orbit_census([lc2], 60) == orbit_census([lw5], 60))
# joint line group
jseen = {ident60}
jdq = deque([ident60])
while jdq:
p = jdq.popleft()
for g in (lc, lw):
q = tuple(p[g[i]] for i in range(60))
if q not in jseen:
jseen.add(q)
jdq.append(q)
joint_line_census = orbit_census([lc, lw], 60)
simply_transitive = (len(jseen) == 60
and joint_line_census == Counter({60: 1}))
if not (deploy_ok and controls_ok):
VERDICT = "TEST-VOID"
elif not both_descend or kill_orders:
VERDICT = "CLOCK-FIBER-DEAD"
elif z6_compatible and simply_transitive:
VERDICT = "CLOCK-FIBER-EXACT"
elif z6_compatible:
VERDICT = "CLOCK-FIBER-PARTIAL"
else:
VERDICT = "CLOCK-FIBER-DEAD"
print("VERDICT: %s" % VERDICT)
print(" seam clock line order: %d (Z12 acts through Z3 = <sigma> "
"only; line-trivial powers %s)" % (lc_ord, lc_kernel))
print(" Coxeter descent powers: %s -> line shadow orders %s"
% (descent_powers, line_shadow))
print(" full line-compatible group: order %d, max element order %d, "
"orders 15/30/60 present: %s/%s/%s"
% (full_order, maxord, has15, has30, has60))
print(" abstract fiber product: Z12 x_{Z6} Z30 = Z60 (cyclic, order "
"60); realized on the lines: %s" % simply_transitive)
print(" mu4 layer 240 = 4 x 60: exact (S1.2)")
print(" joint <c, w> census on the 240 roots: %s" % dict(joint_census))
n_pass = sum(1 for _, ok in CHECKS if ok)
print("%d/%d checks passed" % (n_pass, len(CHECKS)))
print("FROZEN_SPEC SHA-256: %s" % SPEC_SHA)
print("elapsed: %.1f s" % (time.time() - T0))
print("ALL CHECKS PASSED" if n_pass == len(CHECKS) else
"SOME CHECKS FAILED")
return (0 if n_pass == len(CHECKS) else 1), list(CHECKS)
def run():
"""run_all entry point (combined adjudication): part 1 must be 21/21
(ONE-OBJECT-PARTIAL: the glue census forces d = 4, the c_3 link stays
a corpus-legitimate rewriting), part 2 must be 23/23 with the
three-level kill (CLOCK-FIBER-DEAD: no realization of
Z12 x_{Z6} Z30 on the 60 lines exists)."""
rc1 = _run_part1()
fails1 = [n for (n, ok) in CHECKS if not ok]
part1_ok = (rc1 == 0 and not fails1 and len(CHECKS) == 21)
print("\n[%s] PART-1 PATTERN GATE: expected 21/21 "
"(ONE-OBJECT-PARTIAL) -- fails: %s"
% ("PASS" if part1_ok else "FAIL", fails1 or "none"))
rc2, chks2 = _run_part2()
fails2 = [n for (n, ok) in chks2 if not ok]
part2_ok = (rc2 == 0 and not fails2 and len(chks2) == 23)
print("\n[%s] PART-2 PATTERN GATE: expected 23/23 "
"(CLOCK-FIBER-DEAD) -- fails: %s"
% ("PASS" if part2_ok else "FAIL", fails2 or "none"))
ok = part1_ok and part2_ok
print("\nCOMBINED ADJUDICATION: %s -- ONE-OBJECT-PARTIAL + "
"CLOCK-FIBER-DEAD: the exact discriminant-form census over "
"d = 2..12 admits an even unimodular diagonal glue of "
"A_{d-1} (+) D_{d+1} ONLY at d = 4 (v92's two Z4 anti-isometry "
"glues k -> k and k -> 3k reproduced; explicit E8 certificate "
"with 240 integral norm-2 vectors, coset counts [52, 64, 60, "
"64]; controls open d = 9 without evenness and d in {7, 8, 9, "
"12} without unimodularity), forcing g_car = d+1 = 5, N_fam = "
"d-1 = 3, |mu4| = 4 = d as readouts of ONE four-point object; "
"c_3 = 1/(2 pi d)|_{d=4} = 1/(8 pi) stays a corpus-legitimate "
"REWRITING, not a new P1 derivation. The 60-line clock "
"synchronization hypothesis L_60 = Z12 x_{Z6} Z30 is DEAD at "
"three independent levels: the seam clock acts only through "
"Z3, the Coxeter clock does not descend (powers {0, 15} only), "
"and the FULL line-compatible group (order 23040, max element "
"order 12) contains NO element of order 15, 30 or 60 -- while "
"the abstract fiber product IS cyclic Z60 exactly (CC1-CC3 "
"fire). NO RH claim."
% ("PASS" if ok else "FAIL"))
return 0 if ok else 1
if __name__ == "__main__":
raise SystemExit(run())
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