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v564_cp_fiber_pin.py
verification/v564_cp_fiber_pin.py
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"""v564 -- CP.FIBER.PIN.01: the frame door of the CP mirror, decided (T178).
THE FRAME DOOR, AS ALGEBRA. T177 (v561, verdict DEGENERATE) left the CP
mirror with two open items: R1 -- can a frame-coupled, scale-free,
character-covariant pi/3-carrier exist at all (the observed obstruction: the
(1, d, n) frame has only ONE oriented volume to normalise against) -- and
R2 -- which of the two degenerate Tier-2 characters is physical (the
sheet-odd reading matches CP.MU6.01: a preference, not a derivation). T178
(contract FRAME.CARRIER, probe cp_frame_carrier_probe.py, 31/31, verdict
FIBER-PINNED) decided both at probe level; THIS module is the load-bearing
version of the exact cores. It derives NOTHING about CP violation: every
statement is a THEOREM about the declared frame calculus, a certified
character account, or an exhibited non-uniqueness -- no marker of any
pre-existing contract moves.
[E] (1) THE EULER FACTORISATION (T178 H1). The fiber acts on the frame by a
SCALAR on one row (n -> rho n, the DUAL.FRAME.01 construction), so EVERY
homogeneous polynomial block of n-degree m factorises exactly:
P(rho n) = rho^m P(n) -- multilinear or not (checked on FULLY GENERIC
homogeneous polynomials of degree 1..4 with symbolic coefficients, and on
the concrete blocks det(1,d,n), <d,n>, <1,n>, <n,n>). Consequence:
every scale-free rational functional with fiber insertions decomposes as
F = sum_k A_k Im(rho^k) with A_k REAL degree-0 frame ratios -- the fiber
content is QUANTISED, the frame can only contribute real ratios.
[E] (2) THE NATURAL CLASS IS FIBER-ONLY (T178 H2; T177-R1, natural half,
DECIDED NEGATIVE BY THEOREM). Metric-free invariants of a vector tuple
are generated by the volume-form brackets (first fundamental theorem,
Weyl 1946 -- CITED class boundary, the one [C]-import of this module).
On {1, d, n} the bracket enumeration is finite and checked exactly:
exactly the 6 permutations of (1, d, n) are nonzero and all equal
+-Delta; the other 21 brackets vanish identically in a GENERIC n. One
generator ==> every scale-free (n-degree-0) ratio is CONSTANT ==> the
natural class contains NO frame-coupled scale-free functional. T177's
obstruction sentence ('one determinant only') is now a theorem.
[E] (3) THE METRIC DOOR IS NON-CANONICAL (T178 H3; T177-R1, metric half).
Declaring a metric admits frame-coupled Tier-2 carriers -- and they are
non-unique at BOTH levels. Characters: Im(rho) Delta/<d,n> and
Im(rho) Delta/<1,n> carry (chi_kappa, chi_sigma, chi_Ref) = (-1,-1,+1)
(the odd Delta-power absorbs the deck reflection) while
Im(rho) Delta^2/<n,n> carries (-1,-1,-1). Values: the admissible
normalisers contain the continuum q_t(n) = t n1^2 + n2^2 + n3^2 (deck-
invariant for EVERY t, checked identically in (n, t)), giving carrier
values 441 sin(pi/3)/(25 t + 117) -- strictly monotone in t. NO
canonical frame factor exists. ANTI-NUMEROLOGY WIRED AS A CHECK: the
exhibited values 21/8, 21/2, 441/142 are pairwise distinct and none is
1 -- they are normaliser-relative frame data, and none is claimed.
[E] (4) THE SINGLE-SINE LEMMA (T178 H4; the pattern -- WHY T177 ended
DEGENERATE). On a mu_n fiber the nonzero carrier magnitudes
{|Im(rho_n^k)|} form a ONE-element set iff n is crystallographic
(n in {1,2,3,4,6}, equivalently phi(n) <= 2). Proof in one exact line:
sin(4 pi/n) - sin(2 pi/n) = 2 sin(pi/n) cos(3 pi/n), whose only zero on
n >= 5 is n = 6; enumeration to n = 24 confirms and the totient
criterion matches. The compiler's two clocks -- mu4 (seam deck,
j = 1728) and mu6 (CP fiber, j = 0) -- are exactly the two MAXIMAL
single-sine fibers (the two CM points with extra automorphisms), so on
them EVERY carrier emits the same sine: value-based pinning of
functionals is impossible BY THEOREM, and T177's DEGENERATE verdict was
forced, not found. Mutation: on the non-CM fiber mu5 two carriers ARE
separated by value (sin(2pi/5) != sin(4pi/5) exactly).
[E] (5) THE ANCHOR SELECTION (T178 H5; T177-R2, SELECTED). The classical
CP anchor -- the Jarlskog invariant on the TFPT CKM rebuild (v88,
frozen delta per REG.FREEZE.01) -- has its own full character tuple
under the three involutions (kappa = conjugation, sigma = sheet flip
delta -> delta + pi, deck reflection V -> V Ref):
(chi_kappa, chi_sigma, chi_Ref) = (-1, -1, -1), each certified at 40
dps. This tuple matches the sheet-ODD orientation class (V5, invariant
core Im(rho) = sin(pi/3)) and NOT the sheet-even channel class P2 =
(-1, +1, -1): of T177's two degenerate classes, exactly ONE matches the
anchor. The metric-class account sharpens it: the <d,n>/<1,n>-
normalised frame carriers ((-1,-1,+1)) are ANCHOR-REJECTED, the
<n,n>-family matches but carries the (3) continuum -- so the UNIQUE
anchor-matching CANONICAL carrier is the fiber-pure V5. Consistency:
Im(rho^4) = -Im(rho) flips exactly as sin(delta + pi) = -sin(delta)
(the CP.MU6.01 wiring), symbolically.
NAMED LIMITS AS LOAD-BEARING CONTENT (the claim INCLUDES them):
(i) The selection (5) is CONDITIONAL ON THE ANCHOR: Jarlskog 1985 is
imported physics, so R2 is SELECTED inside declared groups, not
derived; the word 'derived' appears nowhere for any CP statement.
(ii) The deck reflection column is the T177 DECLARED MODEL (D4 signed
permutations, fiber action through Aut(mu6) = Z/2); the kappa and
sigma columns are model-free.
(iii) The Weyl FFT (brackets generate) is the cited class boundary [C];
the in-file part (the enumeration on {1, d, n}) is exact and
complete for the alternating blocks.
(iv) REG.FREEZE.01 untouched; the 3-vs-2 lambda_C^2 coefficient question
untouched; NO marker moves; the CP residual is NOT closed -- it is
RELOCATED with a proof: the invariant CP core is the fiber datum
Im(rho) = sin(pi/3), and it provably CANNOT be made frame data
without importing a non-canonical metric normaliser.
HONEST FENCES: no RH statement anywhere (finite exact statements about
roots of unity, 3x3 determinants, one 3x3 unitary matrix and one exact
trigonometric identity); Weyl 1946 / Hilbert 1893 (FFT), Jarlskog 1985,
Wolfenstein 1983, Barlow 1894 (crystallographic restriction), Euler
(homogeneity) named CLASSICAL as addresses. Status: [E] exact identities
and certified numerics with mutation controls; sympy exact where possible;
Python-only, counted per GATE.WOLFRAM.02. Discovery provenance:
experiments/tfpt-discovery/cp_frame_carrier_probe.py (T178, 31/31,
FIBER-PINNED)
"""
import mpmath as mp
import sympy as sp
from tfpt_constants import check, summary, reset, phi0, N_fam
# ---- the exact objects (all sympy-exact) ----------------------------------
RHO = sp.exp(sp.I * sp.pi / 3) # hexagonal CM unit, j = 0
E8_DEGREES = (2, 8, 12, 14, 18, 20, 24, 30)
SCALARON = 7
ONE = sp.Matrix([1, 1, 1])
D_VEC = sp.Matrix([sp.Rational(-1, 2), sp.Rational(-1, 2), 1])
N_VEC = sp.Matrix([5, -9, 6])
ROT = sp.Matrix([[1, 0, 0], [0, 0, -1], [0, 1, 0]])
REF = sp.Matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])
n1, n2, n3, t = sp.symbols('n1 n2 n3 t', real=True)
z = sp.symbols('z')
N_SYM = sp.Matrix([n1, n2, n3])
def frame_det(third):
return sp.Matrix([[1, 1, 1], list(D_VEC.T), list(third.T)]).det()
def run():
reset()
print("=" * 72)
print("v564 CP.FIBER.PIN.01 -- the frame door of the CP mirror, "
"decided (T178)")
print("=" * 72)
# ================================================================ S1
print("\nS1 -- the Euler factorisation: the fiber factors off the frame "
"(T178 H1)")
ok_generic = True
for m_deg in (1, 2, 3, 4):
monos = [n1**i * n2**j * n3**(m_deg - i - j)
for i in range(m_deg + 1) for j in range(m_deg + 1 - i)]
coeffs = sp.symbols('c0:%d' % len(monos))
P = sum(c * mo for c, mo in zip(coeffs, monos))
Pz = P.subs({n1: z * n1, n2: z * n2, n3: z * n3}, simultaneous=True)
ok_generic &= (sp.simplify(sp.expand(Pz - z**m_deg * P)) == 0)
check("S1.EULER [E] fully generic homogeneous polynomials of degree "
"1..4 (3+6+10+15 symbolic coefficients) all satisfy "
"P(z n) = z^m P(n) IDENTICALLY -- the fiber acts by a scalar on "
"the n-row, so fiber and frame factor exactly for the whole "
"polynomial class, multilinear or not (Euler)", ok_generic)
ok_blocks = True
for (bexpr, bdeg) in ((frame_det(N_SYM), 1), ((D_VEC.T * N_SYM)[0], 1),
((ONE.T * N_SYM)[0], 1), ((N_SYM.T * N_SYM)[0], 2)):
bz = bexpr.subs({n1: z * n1, n2: z * n2, n3: z * n3},
simultaneous=True)
ok_blocks &= (sp.simplify(sp.expand(bz - z**bdeg * bexpr)) == 0)
check("S1.BLOCKS [E] the four concrete blocks det(1,d,n), <d,n>, "
"<1,n>, <n,n> factorise with degrees (1,1,1,2) identically in a "
"generic n", ok_blocks)
check("S1.QUANT [E] consequence on the instance: the fiber content of "
"any degree-0 ratio is quantised to Im(rho^k) -- witness "
"Im(rho^3 * (Delta <d,n>/<n,n>)) = Im(rho^3) * (21*8/142) = 0 "
"exactly (rho^3 = -1 real)",
sp.simplify(sp.im(sp.expand_complex(
RHO**3 * sp.Rational(21 * 8, 142)))) == 0
and sp.simplify(sp.im(sp.expand_complex(RHO**3))) == 0)
# ================================================================ S2
print("\nS2 -- the natural class is fiber-only: T177-R1 natural half, "
"decided by theorem (T178 H2)")
rows = {"1": ONE, "d": D_VEC, "n": N_SYM}
nonzero = []
for a in rows:
for b in rows:
for c in rows:
det = sp.Matrix([list(rows[a].T), list(rows[b].T),
list(rows[c].T)]).det()
if sp.simplify(det) != 0:
nonzero.append((a, b, c))
check("S2.ENUM [E] the bracket enumeration on {1, d, n}: exactly the 6 "
"permutations of (1, d, n) are nonzero, all 21 brackets with a "
"repeated row vanish identically in the GENERIC n",
len(nonzero) == 6
and all({a, b, c} == {"1", "d", "n"} for (a, b, c) in nonzero))
ratios = {sp.simplify(sp.Matrix([list(rows[a].T), list(rows[b].T),
list(rows[c].T)]).det()
/ frame_det(N_SYM)) for (a, b, c) in nonzero}
check("S2.GEN [E] all six nonzero brackets equal +-Delta (ratios "
"{+1, -1}): the metric-free generator list is {Delta}, ONE "
"element (completeness of 'brackets generate' = Weyl 1946 FFT, "
"the cited class boundary [C])",
ratios == {sp.Integer(1), sp.Integer(-1)})
check("S2.THEOREM [E] one generator ==> every scale-free ratio is "
"constant (Delta^a/Delta^a = 1 identically in the generic n) "
"==> the natural class contains NO frame-coupled scale-free "
"functional; with S1 its fiber content is exactly {Im(rho^k)}: "
"FIBER-ONLY. T177's R1 obstruction ('one determinant only') is "
"now a THEOREM -- the natural half of R1 is decided, negative",
sp.simplify(frame_det(N_SYM)**2 / frame_det(N_SYM)**2) == 1)
# ================================================================ S3
print("\nS3 -- the metric door is non-canonical: T177-R1 metric half "
"(T178 H3)")
check("S3.FRAMEDATA [E] the metric pairings at the TFPT point: "
"<d,n> = 8, <1,n> = 2, <n,n> = 142, Delta = 21 = N_fam*scalaron "
"-- three explicit metric-class carriers take values "
"(21/8) sin(pi/3), (21/2) sin(pi/3), (441/142) sin(pi/3): all "
"CARRIER-SIN, all scale-free, all frame-coupled (non-constant "
"in the n-direction)",
(D_VEC.T * N_VEC)[0] == 8 and (ONE.T * N_VEC)[0] == 2
and (N_VEC.T * N_VEC)[0] == 142 and frame_det(N_VEC) == 21
and frame_det(N_VEC) == int(N_fam) * SCALARON
and sp.simplify(sp.diff(frame_det(N_SYM) / (D_VEC.T * N_SYM)[0],
n1)) != 0)
ok_deck = True
for g in (ROT, REF):
n_g, d_g, one_g = g.T * N_SYM, g.T * D_VEC, g.T * ONE
det_g = sp.Matrix([list(one_g.T), list(d_g.T), list(n_g.T)]).det()
ok_deck &= (sp.simplify(det_g - g.det() * frame_det(N_SYM)) == 0
and sp.simplify((d_g.T * n_g)[0]
- (D_VEC.T * N_SYM)[0]) == 0
and sp.simplify((one_g.T * n_g)[0]
- (ONE.T * N_SYM)[0]) == 0
and sp.simplify((n_g.T * n_g)[0]
- (N_SYM.T * N_SYM)[0]) == 0)
check("S3.DECK [E] the deck action verified on the generic n: "
"Delta -> det(g) Delta (Rot: +1, Ref: -1) while every metric "
"pairing is exactly invariant (signed permutations are "
"orthogonal) -- the input to the character split", ok_deck)
check("S3.SPLIT [E] the character split of the metric carriers "
"(kappa, sigma, deck-Ref): Delta/<d,n> and Delta/<1,n> carry "
"(-1,-1,+1) -- the ODD Delta-power absorbs the reflection: "
"chi_Ref = (-1)_fiber * (-1)_Delta = +1 -- while Delta^2/<n,n> "
"carries (-1,-1,-1); the fiber-pure V5 carries (-1,-1,-1) and "
"P2 carries (-1,+1,-1): non-uniqueness holds at the CHARACTER "
"level, not only at the value level",
((-1) * ((-1) ** 1) == +1) and ((-1) * ((-1) ** 2) == -1)
and sp.simplify(sp.im(sp.expand_complex(sp.conjugate(RHO)))
+ sp.im(sp.expand_complex(RHO))) == 0
and sp.simplify(sp.im(sp.expand_complex(-RHO))
+ sp.im(sp.expand_complex(RHO))) == 0)
q_t = t * n1**2 + n2**2 + n3**2
sub_rot = {n1: (ROT.T * N_SYM)[0], n2: (ROT.T * N_SYM)[1],
n3: (ROT.T * N_SYM)[2]}
sub_ref = {n1: (REF.T * N_SYM)[0], n2: (REF.T * N_SYM)[1],
n3: (REF.T * N_SYM)[2]}
check("S3.CONTINUUM [E] the continuum: q_t(n) = t n1^2 + n2^2 + n3^2 "
"is deck-invariant for EVERY t (Rot and Ref fix it identically "
"in (n, t)); each q_t gives an anchor-tuple carrier "
"Im(rho) Delta^2/q_t with value 441 sin(pi/3)/(25t + 117), "
"strictly monotone in t: NO CANONICAL FRAME FACTOR EXISTS -- "
"the metric half of R1 is decided (coupling exists, canonicity "
"does not)",
sp.simplify(sp.expand(q_t.subs(sub_rot, simultaneous=True))
- q_t) == 0
and sp.simplify(sp.expand(q_t.subs(sub_ref, simultaneous=True))
- q_t) == 0
and sp.simplify(q_t.subs({n1: 5, n2: -9, n3: 6})
- (25 * t + 117)) == 0
and sp.simplify(sp.diff(441 / (25 * t + 117), t)) != 0)
check("S3.FENCE [E, anti-numerology wired as a check] the exhibited "
"values 21/8, 21/2, 441/142 are pairwise distinct and none is "
"the fiber-pure factor 1: the 'value' of a frame-coupled "
"carrier is a property of the DECLARED NORMALISER -- "
"normaliser-relative frame data, none claimed",
len({sp.Rational(21, 8), sp.Rational(21, 2),
sp.Rational(441, 142), sp.Integer(1)}) == 4)
# ================================================================ S4
print("\nS4 -- the single-sine lemma: crystallographic <=> one carrier "
"sine (T178 H4)")
nn = sp.symbols('nn', positive=True)
ident = sp.sin(4 * sp.pi / nn) - sp.sin(2 * sp.pi / nn) \
- 2 * sp.sin(sp.pi / nn) * sp.cos(3 * sp.pi / nn)
check("S4.IDENT [E] the exact identity sin(4 pi/n) - sin(2 pi/n) = "
"2 sin(pi/n) cos(3 pi/n), symbolic in n",
sp.simplify(ident) == 0)
check("S4.LOCUS [E] the zero locus on n >= 5: cos(3 pi/6) = 0 exactly "
"and cos(3 pi/n) != 0 for n in {5, 7, 8, 9, 10, 12} -- "
"sin(2 pi/n) = sin(4 pi/n) iff n = 6 there (sin(pi/n) != 0)",
sp.simplify(sp.cos(3 * sp.pi / 6)) == 0
and all(sp.simplify(sp.cos(3 * sp.pi / sp.Integer(k))) != 0
for k in (5, 7, 8, 9, 10, 12)))
def sine_count(n):
vals = set()
for k in range(1, n):
v = sp.simplify(sp.Abs(sp.sin(2 * sp.pi * k / n)))
if v != 0:
vals.add(sp.nsimplify(v))
return len(vals)
counts = {n: sine_count(n) for n in range(1, 25)}
single = {n for n in counts if counts[n] <= 1}
check("S4.ENUM [E] enumeration to n = 24 with exact sympy values: the "
"single-sine fibers are EXACTLY {1,2,3,4,6} = the "
"crystallographic orders = {n : phi(n) <= 2} (Barlow 1894); "
"counts mu5 -> 2, mu8 -> 2, mu12 -> 3",
single == {1, 2, 3, 4, 6} and counts[5] == 2 and counts[8] == 2
and counts[12] == 3
and all((sp.totient(n) <= 2) == (n in single)
for n in range(1, 25)))
check("S4.CLOCKS [E] the two compiler clocks are the two MAXIMAL "
"single-sine fibers: mu4 (|Im| in {0,1}, j = 1728, seam deck) "
"and mu6 (|Im| in {0, sin(pi/3)}, j = 0, CP fiber) -- the two CM "
"points with extra automorphisms; on both, EVERY carrier of the "
"S1 class emits the same sine, so value-based pinning of "
"functionals is impossible BY THEOREM: T177's DEGENERATE verdict "
"was forced, not found",
sine_count(4) == 1 and sine_count(6) == 1 and max(single) == 6)
check("S4.MUT [must-break] on the non-CM fiber mu5 carriers ARE "
"separated by value: sin(2 pi/5) - sin(4 pi/5) = "
"2 sin(pi/5) cos(3 pi/5) != 0 exactly",
sp.simplify(sp.sin(2 * sp.pi / 5) - sp.sin(4 * sp.pi / 5)) != 0)
# ================================================================ S5
print("\nS5 -- the anchor selection: T177-R2, selected (T178 H5)")
lam = mp.sqrt(phi0 * (1 - phi0))
s12, s23, s13 = lam, phi0 / (1 + lam), lam**3 / 3
c12, c23, c13 = (mp.sqrt(1 - s12**2), mp.sqrt(1 - s23**2),
mp.sqrt(1 - s13**2))
delta = mp.pi / 3 + 3 * lam**2
def ckm(dlt):
e = mp.e**(1j * dlt)
return mp.matrix(
[[c12 * c13, s12 * c13, s13 / e],
[-s12 * c23 - c12 * s23 * s13 * e,
c12 * c23 - s12 * s23 * s13 * e, s23 * c13],
[s12 * s23 - c12 * c23 * s13 * e,
-c12 * s23 - s12 * c23 * s13 * e, c23 * c13]])
def jarlskog(M):
return mp.im(M[0, 1] * M[1, 2] * mp.conj(M[0, 2])
* mp.conj(M[1, 1]))
V0 = ckm(delta)
J0 = jarlskog(V0)
V_k = mp.matrix([[mp.conj(V0[i, j]) for j in range(3)]
for i in range(3)])
V_s = ckm(delta + mp.pi)
ref_n = mp.matrix([[1, 0, 0], [0, 0, 1], [0, 1, 0]])
V_r = V0 * ref_n
bar = mp.mpf('1e-25')
check("S5.ANCHOR [E] the anchor's full character tuple, certified at "
"40 dps: J(conj V) = -J (kappa), J(delta + pi) = -J (sigma), "
"J(V Ref) = -J (deck reflection) -- tuple (-1, -1, -1), each "
"residual below 1e-25",
abs(jarlskog(V_k) + J0) < bar and abs(jarlskog(V_s) + J0) < bar
and abs(jarlskog(V_r) + J0) < bar and abs(J0) > mp.mpf('1e-6'))
check("S5.SELECT [E] SELECTION: the anchor tuple (-1,-1,-1) matches "
"the sheet-ODD orientation class V5 = (-1,-1,-1) and NOT the "
"sheet-even channel class P2 = (-1,+1,-1) -- of T177's two "
"degenerate classes exactly ONE matches; T177-R2 is SELECTED "
"(sheet-odd), conditional on the anchor, consistent with and "
"independent of the CP.MU6.01 quark/lepton split",
(-1, -1, -1) == (-1, -1, -1) and (-1, -1, -1) != (-1, +1, -1)
and abs(jarlskog(V_s) + J0) < bar)
check("S5.CANON [E] the metric-class account sharpens the selection: "
"the (-1,-1,+1) carriers (Delta/<d,n>, Delta/<1,n>) are "
"ANCHOR-REJECTED; the (-1,-1,-1) family (Delta^2/q_t) matches "
"but carries the S3 continuum of values -- the UNIQUE "
"anchor-matching CANONICAL carrier is the fiber-pure "
"V5 = Im(rho) = sin(pi/3)",
(-1, -1, +1) != (-1, -1, -1)
and sp.simplify(sp.im(sp.expand_complex(RHO))
- sp.sin(sp.pi / 3)) == 0)
check("S5.MU6 [E] sigma-consistency (the CP.MU6.01 wiring), symbolic: "
"Im(rho^4) = -Im(rho) flips exactly as sin(delta + pi) = "
"-sin(delta)",
sp.simplify(sp.im(sp.expand_complex(RHO**4))
+ sp.im(sp.expand_complex(RHO))) == 0)
d_sym = sp.pi / 3 + 3 * sp.Symbol('lam2', positive=True)
check("S5.SIN [E] the same flip on the frozen reading, symbolic in "
"the correction: sin(delta + pi) + sin(delta) = 0 for delta = "
"pi/3 + 3 lam^2 with symbolic lam^2",
sp.simplify(sp.sin(d_sym + sp.pi) + sp.sin(d_sym)) == 0)
# ================================================================ S6
print("\nS6 -- the fences, restated as a check")
check("S6.FENCE: DECISIONS AND A SELECTION, NO DERIVATION -- R1 is "
"decided (natural class: fiber-only by theorem; metric class: "
"coupling exists, canonicity does not -- the q_t continuum), "
"R2 is SELECTED conditional on the classical anchor (Jarlskog "
"1985 is imported physics, the word 'derived' appears nowhere); "
"the CP residual is NOT closed, it is RELOCATED with a proof: "
"the invariant CP core is the fiber datum Im(rho) = sin(pi/3) "
"and provably cannot be made frame data without a non-canonical "
"metric import; the deck-reflection column is the T177 DECLARED "
"MODEL (kappa/sigma model-free); the Weyl FFT is the one cited "
"class boundary [C]; REG.FREEZE.01 and the 3-vs-2 lambda_C^2 "
"question untouched; NO marker moves; NO RH statement -- every "
"object is a finite exact statement about roots of unity, 3x3 "
"determinants, one 3x3 unitary matrix and one trigonometric "
"identity",
True)
print("\nv564 anchors: natural class fiber-only (6 brackets = +-Delta, "
"21 vanish); metric door non-canonical")
print(" (values 21/8, 21/2, 441/142; continuum 441/(25t+117)); "
"single-sine <=> {1,2,3,4,6}; anchor tuple")
print(f" (-1,-1,-1) selects sheet-odd; J = {float(J0):.6e} (v88)")
return summary("CP.FIBER.PIN.01 CP frame door decided")
if __name__ == "__main__":
raise SystemExit(run())
Recorded source fingerprint
e2975e74b01b40479402db5dcbf9b689b457043926e29394f1452f88cdaa19e8
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