"""v218 -- The Diamond Axis Geometry: the Sheet Diamond is a discrete geometry with TWO axes, and the missing corner F is the natural transfer completion. Sharpens v94 (sheet diamond) / v95 (centered cross C,U,V) with three exact [E] lemmas plus honestly-typed audit/heuristic blocks. NO new numbers -- it organises the existing operators {Q,K,R,C,L,F} more strictly. [I] 1. DIAMOND AXIS CURVATURE LEMMA. Along the two canonical axes of the centered cross (C the center, U winding, V sheet, v95) the determinant is LINEAR on the winding axis and QUADRATIC on the sheet axis: det(C + x U) = 14 + 6 x (flat: slope 6 = |R^+(A_3)|) det(C + y V) = 14 + 14 y + 4 y^2 (2nd diff = 8 = rank E_8) det B(C + x U) = 28 + 12 x = 2 det(C + x U) det B(C + y V) = 28 + 21 y + 3 y^2 (2nd diff = 6 = |R^+(A_3)|) So winding flows linearly (A_3-driven), the sheet split CURVES, and the two curvatures are exactly rank E_8 = 8 (determinant) and |R^+(A_3)| = 6 (anchor block) -- a discrete differential geometry of the diamond. [I] 2. PLUECKER TRANSFER LADDER. The anchor-Pluecker coordinates lift K->C->F in two exact steps: Pl(K) = (-1, 6, 4), Pl(C) = (0,14,14), Pl(F) = (1,22,30), Pl(C) - Pl(K) = (1, 8, 10) = (N_Phi, rank E_8, A_Lambda), Pl(F) - Pl(C) = (1, 8, 16) = (N_Phi, rank E_8, dim S^+). Each step adds (1, 8) in the first two coordinates; the third adds first the decuple A_Lambda = 10, then the full spinor generation dim S^+ = 16. The Pluecker identity is [E]; the F_transfer source->pole reading (a two-stage sheet lift) stays [C]. [I] 3. SPECTRAL RAMIFICATION LADDER. For Q,K,C,F the characteristic polynomial splits as (t - r)(t^2 - b t + c) with r rational, and the FULL cubic discriminant factors as Disc(chi) = q(r)^2 * Disc(q): squares q(r)^2 in {1^2, 3^2, 4^2, 6^2} = (N_Phi, N_fam, |mu4|, |R^+(A_3)|)^2, kernels Disc(q) in {13, 48, 65, 105} = (Delta_Q, Omega_adm, g_car*Delta_Q, N_fam*g_car*7), with Delta_Q = 13 = b^2 + s^2 = |R(A_3)| + 1 the carrier-split norm (v14). The missing corner F carries the kernel 105 = 3*5*7 = family*carrier*scalaron. [I] 4. ANCHOR DEFECT SPECTRUM (audit). One anchor operation Delta_a(M) = a^T M a - 1^T M 1 reads six known blocks across the whole diamond: Delta_a(Q,K,R,C,L,F) = (8, 16, 18, 21, 24, 26) = (rank E_8, dim S^+, N_fam|R^+(A_3)|, N_fam*7, |W(A_3)|, 2*13), monotone in the SAME ordering as the determinant ladder. NOTE the cleaned notation: 26 = 2*13 uses the SPECTRAL Delta_Q = 13, NOT the anchor defect Delta_a(Q) = 8. AUDIT-typed: spine only if the six operators are shown canonical and complete. [I] 5. PAIR-SUM ATLAS + DETERMINANT STAIRCASE (audit fingerprints). The four corner determinants (det R,K,L,F) = (8,4,20,32) have six pair sums {12,24,28,36,40,52} and total 64 = dim S^+ * |mu4|; the six diamond determinants (3,4,8,14,20,32) sum to 81 = N_fam^4 with steps (+1,+4,+6,+6,+12). The spine reads are 12 = dim g_SM = |R(A_3)|, 24 = |W(A_3)|, 36 = N_fam|R(A_3)|, 40 = |R(D_5)|, 64 = dim S^+|mu4|; but 28 = 2 dim G_2 and 52 = dim F_4 are EXOTIC Lie labels -- audit only, NOT spine (same typing as the G_2/F_4 shadows in v94/v95). [C] 6. F AS THE TRANSFER COMPLETION CORNER (heuristic, NOT a hard rule). F is where the diamond tips from compiler into transfer: det F = 32 = 2^g_car, a^T F 1 = 53 = sum Pl(F), det L + det F = 52, det R + det F = 40. The search principle "every F_transfer kernel must cast an F / Pluecker / (2/3)^6 shadow" is recorded as a HEURISTIC only -- promoting it to a hard admissibility filter is explicitly rejected (cf. the v94 firewall rejection of the over-strong adjacent-quotient rule). Python-only here; the exact identities (axis curvatures, the Pluecker ladder, the discriminant factorisations) are mirrored in wolfram/tfpt_readouts_extension.wl. """ import sympy as sp from tfpt_constants import (check, summary, reset, g_car, N_fam, dim_Splus, Omega_adm) R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) ONE = sp.Matrix([1, 1, 1]) A = sp.Matrix([1, 1, 2]) C = R + Q * sp.diag(1, 0, 0) U = Q * sp.diag(1, 0, 0) V = Q * sp.diag(0, 1, 1) K = C - V L = C + U F = C + V # Lie-theoretic block constants (same convention as v94/v95) RANK_E8 = 8 R_A3 = 12 # |R(A_3)| (all roots of A_3 = su(4)) RP_A3 = 6 # |R^+(A_3)| (positive roots) W_A3 = 24 # |W(A_3)| = 4! R_D5 = 40 # |R(D_5)| (roots of D_5) DIM_GSM = 12 # dim su(3)+su(2)+u(1) A_LAMBDA = 10 # |E(K_5)| decuple SCALARON = 7 # scalaron exponent DELTA_Q = 13 # carrier-split norm b^2+s^2 = |R(A_3)|+1 (v14) MU4 = 4 N_PHI = 1 def anchor_block(M): return sp.Matrix([[(ONE.T * M * ONE)[0], (ONE.T * M * A)[0]], [(A.T * M * ONE)[0], (A.T * M * A)[0]]]) def pi_left(M): blk = sp.Matrix.vstack((ONE.T * M), (A.T * M)) return [blk[:, [i, j]].det() for i, j in ((0, 1), (0, 2), (1, 2))] def defect(M): return (A.T * M * A)[0] - (ONE.T * M * ONE)[0] def ramification(M): """(|q(r)|, Disc(q)) for chi_M = (t - r)(t^2 - b t + c), r the rational root.""" t = sp.symbols('t') p = M.charpoly(t).as_expr() r = next(rt for rt in sp.roots(p, t) if rt.is_rational) q = sp.factor(sp.cancel(p / (t - r))) return abs(q.subs(t, r)), sp.discriminant(q, t) def run(): reset() print("v218 diamond axis geometry: U-axis flat (A3), V-axis curved (E8), F the transfer corner") x, y = sp.symbols('x y') # ---- LEMMA 1: axis curvature ---- dU = sp.expand((C + x * U).det()) dV = sp.expand((C + y * V).det()) bU = sp.expand(anchor_block(C + x * U).det()) bV = sp.expand(anchor_block(C + y * V).det()) sheet = [(C + i * V).det() for i in (-1, 0, 1)] bsheet = [anchor_block(C + i * V).det() for i in (-1, 0, 1)] second_diff = sheet[2] - 2 * sheet[1] + sheet[0] bsecond_diff = bsheet[2] - 2 * bsheet[1] + bsheet[0] check("LEMMA 1 AXIS CURVATURE [I]: det(C+xU) = 14+6x is LINEAR (winding flat, " "slope 6 = |R^+(A_3)|); det(C+yV) = 14+14y+4y^2 is QUADRATIC with 2nd " "difference = %d = rank E_8; det B(C+xU) = 28+12x = 2 det(C+xU); the " "anchor-block sheet 2nd difference = %d = |R^+(A_3)| -- two curvatures " "(8 det, 6 anchor) = (rank E_8, |R^+(A_3)|)" % (second_diff, bsecond_diff), dU == 6 * x + 14 and dV == 4 * y**2 + 14 * y + 14 and bU == 12 * x + 28 and sp.expand(bU - 2 * dU) == 0 and bV == 3 * y**2 + 21 * y + 28 and second_diff == RANK_E8 and bsecond_diff == RP_A3) # ---- LEMMA 2: Pluecker transfer ladder ---- plK, plC, plF = pi_left(K), pi_left(C), pi_left(F) step1 = [plC[i] - plK[i] for i in range(3)] step2 = [plF[i] - plC[i] for i in range(3)] check("LEMMA 2 PLUECKER TRANSFER LADDER [I]: Pl(K)=%s -> Pl(C)=%s -> Pl(F)=%s " "lifts in two steps: C-K = %s = (N_Phi, rank E_8, A_Lambda), F-C = %s = " "(N_Phi, rank E_8, dim S^+); each step adds (1,8), the third adds the " "decuple 10 then the full generation 16 (identity [I]; source->pole " "reading [C])" % (plK, plC, plF, step1, step2), plK == [-1, 6, 4] and plC == [0, 14, 14] and plF == [1, 22, 30] and step1 == [N_PHI, RANK_E8, A_LAMBDA] and step2 == [N_PHI, RANK_E8, dim_Splus]) # ---- LEMMA 3: spectral ramification ladder ---- sq, ker = {}, {} for nm, M in (('Q', Q), ('K', K), ('C', C), ('F', F)): s_, k_ = ramification(M) sq[nm], ker[nm] = s_, k_ squares = [sq['Q'], sq['K'], sq['C'], sq['F']] kernels = [ker['Q'], ker['K'], ker['C'], ker['F']] check("LEMMA 3 SPECTRAL RAMIFICATION [I]: for Q,K,C,F the cubic discriminant " "factors as q(r)^2 * Disc(q); squares |q(r)| = %s = (N_Phi, N_fam, " "|mu4|, |R^+(A_3)|); kernels Disc(q) = %s = (Delta_Q, Omega_adm, " "g_car*Delta_Q, N_fam*g_car*7), Delta_Q = 13 = |R(A_3)|+1 (v14); F " "carries 105 = 3*5*7 = family*carrier*scalaron" % (squares, kernels), squares == [N_PHI, N_fam, MU4, RP_A3] and kernels == [DELTA_Q, Omega_adm, g_car * DELTA_Q, N_fam * g_car * SCALARON]) # ---- AUDIT 4: anchor defect spectrum (cleaned notation) ---- spectrum = [defect(M) for M in (Q, K, R, C, L, F)] check("AUDIT ANCHOR DEFECT SPECTRUM [I]: Delta_a(Q,K,R,C,L,F) = %s = " "(rank E_8, dim S^+, N_fam|R^+(A_3)|, N_fam*7, |W(A_3)|, 2*13), " "monotone in the determinant ordering; 26 = 2*13 uses the SPECTRAL " "Delta_Q=13, NOT the anchor defect Delta_a(Q)=8 (cleaned). AUDIT: " "spine only if the six operators are canonical/complete" % spectrum, spectrum == [RANK_E8, dim_Splus, N_fam * RP_A3, N_fam * SCALARON, W_A3, 2 * DELTA_Q] and spectrum == sorted(spectrum)) # ---- AUDIT 5: pair-sum atlas + determinant staircase ---- corners = {'R': R.det(), 'K': K.det(), 'L': L.det(), 'F': F.det()} import itertools pairsums = sorted(corners[i] + corners[j] for i, j in itertools.combinations(['R', 'K', 'L', 'F'], 2)) total4 = sum(corners.values()) seq = [Q.det(), K.det(), R.det(), C.det(), L.det(), F.det()] steps = [seq[i + 1] - seq[i] for i in range(5)] spine_sums = {12: DIM_GSM, 24: W_A3, 36: N_fam * R_A3, 40: R_D5} exotic = {28: 2 * 14, 52: 52} # 2 dim G_2, dim F_4 -- audit only check("AUDIT PAIR-SUM ATLAS + STAIRCASE [I]: corner-det pair sums = %s, total " "= %d = dim S^+ * |mu4|; det staircase (3,4,8,14,20,32) sum 81 = " "N_fam^4, steps %s. SPINE: 12=dim g_SM, 24=|W(A_3)|, 36=N_fam|R(A_3)|, " "40=|R(D_5)|, 64=dim S^+|mu4|; but 28=2 dim G_2 and 52=dim F_4 are " "AUDIT-only exotic labels, NOT spine (same typing as v94/v95)" % (pairsums, total4, steps), pairsums == [12, 24, 28, 36, 40, 52] and total4 == dim_Splus * MU4 and sum(seq) == N_fam**4 and steps == [1, 4, 6, 6, 12] and set(spine_sums) <= set(pairsums) and set(exotic) <= set(pairsums)) # ---- HEURISTIC 6: F as the transfer completion corner ---- check("HEURISTIC F = TRANSFER COMPLETION CORNER [C]: det F = 32 = 2^g_car, " "a^T F 1 = 53 = sum Pl(F), det L + det F = 52, det R + det F = 40. The " "rule 'every F_transfer kernel casts an F / Pluecker / (2/3)^6 shadow' " "is a HEURISTIC search principle ONLY -- NOT promoted to a hard " "admissibility filter (cf. the v94 firewall rejection)", F.det() == 2**g_car and (A.T * F * ONE)[0] == 53 and sum(pi_left(F)) == 53 and L.det() + F.det() == 52 and R.det() + F.det() == 40) return summary("v218 diamond axis geometry (U=A3 flow, V=E8 curvature, F=transfer corner)") if __name__ == "__main__": raise SystemExit(1 if run() else 0)