"""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), , <1,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/ 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/ 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 /<1,n>- normalised frame carriers ((-1,-1,+1)) are ANCHOR-REJECTED, the -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), , " "<1,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 /)) = 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: " " = 8, <1,n> = 2, = 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/ 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/ " "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/, 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())