"""v170 -- E8 Slice Compression: the seven E8 audit slices are one projection of two small alphabets (anchor power-sums + flavor-operator determinants). This is a compression/anti-numerology lemma, not new physics: every block of the E8 audit atlas is shown to be a product/sum of ALREADY-typed invariants, so the atlas stops being seven separate trophy lists and becomes a closed grammar over two alphabets. Exact integer/lattice identities; the physical reading is AUDIT. ALPHABETS. Anchor a=(1,1,2): power sums p_n(a)=1+1+2^n=2+2^n give P=(p0,p1,p2,p3)=(3,4,6,10). The six Sheet-Diamond operators Q,K,R,C,L,F give D=(det)=(3,4,8,14,20,32). Delta = p0+p3 = 13 = Delta_Q. [I] 1. K4 POWER-SUM GRAPH. The six edge products of P are all carrier blocks: p0p1=12=|R(A3)| (SM gauge dim), p0p2=18=N_fam|R+(A3)|, p0p3=30=h(E8), p1p2=24=|W(A3)|, p1p3=40=|R(D5)| (flavour budget), p2p3=60=D_start; triples p0p1p3=120=|R+(E8)|, p1p2p3=240=|R(E8)|, p0p1p2p3=720=6!. [I] 2. DETERMINANT HEXAGON. The six natural Sheet-Diamond operators have determinant sum sum det = 3+4+8+14+20+32 = 81 = N_fam^4 -- the total determinant charge of the flavour operator space. [I] 3. PRODUCT ATLAS. Determinant pair-products land on E8 slice blocks: detQ detR=24=|W(A3)|, detQ detL=60, detK detL=80 (A8 adjoint block), detK detF=128 (D8 spinor), detR detC=112 (E7xA1 off-block). [I] 4. E8 SLICE COMPRESSION (the lemma). All seven 248-slices are written from P, D, Delta and the symmetric data (e1=4,e2=5,dim S+=16): D8 248 = p0p1p3 + p1 dF (120+128) D5xA3 248 = (p1p3+e2)+(p0p1+p0)+p2p3+2 e1 dimS+ (45+15+60+64+64) E6xA2 248 = p2 Delta + dR + 81 + 81 (78+8+81+81) E7xA1 248 = (p0p1p3+Delta)+p0+dR dC (133+3+112) F4xG2 248 = (p1^2+p2^2)+dC+Delta dC (52+14+182) A4xA4 248 = 2 p1p2 + 4 e2 p3 (24+24+200) A8 248 = dK dL + 2 p2 dC (80+84+84) [I] 5. ROW-BUDGET CROSS. One affine map generates every flavour row budget: row(C+sU+tV) = (7,11,13) + s(3,3,3) + t(1,2,3) -- democratic family winding (3,3,3) plus the A3 exponent grading (1,2,3). [I] 6. dim E6 = |R+(A3)| (N_fam+A_Lambda). 78 = p2 Delta = 6*13, binding the flavour-norm |R|_F^2=78 to the hexagon p2=6 and the quark denominator Delta=13. Not a decorative Lie hit. Status [E] (exact integer/lattice identities); physical reading AUDIT -- it bridges the Sheet Diamond and the E8 slice atlas, and the blocks are built only from admissible invariant classes (power sums, determinants, branching dims). """ import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam # anchor power sums p_n = 2 + 2^n P = [2 + 2**n for n in range(4)] # (3,4,6,10) p0, p1, p2, p3 = P DELTA = p0 + p3 # 13 = Delta_Q E1, E2, DIMSP = 4, 5, 16 # |mu4|, g_car-split e2, dim S+ # Sheet-Diamond operators (canonical TFPT matrices; R as in v4_flavor_matrix) R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) L = sp.Matrix([[7, 3, 0], [7, 5, 2], [8, 5, 3]]) K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]]) Q = L - K C = (R + L) / 2 F = R + Q DET = {n: int(m.det()) for n, m in [("Q", Q), ("K", K), ("R", R), ("C", C), ("L", L), ("F", F)]} def _rows(m): return tuple(int(sum(m.row(i))) for i in range(3)) def run(): reset() print("v170 E8 Slice Compression (two alphabets: power sums + diamond dets)") # 1. K4 power-sum graph edges = {(p0*p1, 12), (p0*p2, 18), (p0*p3, 30), (p1*p2, 24), (p1*p3, 40), (p2*p3, 60)} check("K4 POWER-SUM GRAPH: p_n=2+2^n give P=(3,4,6,10); the six edge products " "(12,18,30,24,40,60) = (|R(A3)|, N_fam|R+(A3)|, h(E8), |W(A3)|, |R(D5)|, " "D_start); triples 120=|R+(E8)|, 240=|R(E8)|, 720=6!", P == [3, 4, 6, 10] and all(a == b for a, b in edges) and p0*p1*p3 == 120 and p1*p2*p3 == 240 and p0*p1*p2*p3 == 720, exact=True) # 2. determinant hexagon = N_fam^4 check("DETERMINANT HEXAGON: det(Q,K,R,C,L,F) = (3,4,8,14,20,32), sum = 81 = " "N_fam^4 -- the total determinant charge of the flavour operator space", DET == {"Q": 3, "K": 4, "R": 8, "C": 14, "L": 20, "F": 32} and sum(DET.values()) == 81 == N_fam**4, exact=True) # 3. product atlas prod = {DET["Q"]*DET["R"]: 24, DET["Q"]*DET["L"]: 60, DET["K"]*DET["L"]: 80, DET["K"]*DET["F"]: 128, DET["R"]*DET["C"]: 112} check("PRODUCT ATLAS: det pair-products land on E8 slice blocks -- " "detQ detR=24=|W(A3)|, detQ detL=60, detK detL=80 (A8 adj), " "detK detF=128 (D8 spinor), detR detC=112 (E7xA1 off-block)", all(a == b for a, b in prod.items()), exact=True) # 4. E8 slice compression -- all seven slices = 248 slices = { "D8": p0*p1*p3 + p1*DET["F"], "D5xA3": (p1*p3 + E2) + (p0*p1 + p0) + p2*p3 + 2*E1*DIMSP, "E6xA2": p2*DELTA + DET["R"] + 81 + 81, "E7xA1": (p0*p1*p3 + DELTA) + p0 + DET["R"]*DET["C"], "F4xG2": (p1**2 + p2**2) + DET["C"] + DELTA*DET["C"], "A4xA4": 2*p1*p2 + 4*E2*p3, "A8": DET["K"]*DET["L"] + 2*p2*DET["C"], } check("E8 SLICE COMPRESSION (the lemma): all SEVEN E8 audit slices = 248 from " "P, D, Delta and (e1,e2,dim S+) -- the atlas is one projection of two " "alphabets, not seven separate narratives: %s" % ", ".join(f"{k}={v}" for k, v in slices.items()), all(v == 248 for v in slices.values()), exact=True) # 5. row-budget cross U = sp.Matrix([[3, 0, 0], [3, 0, 0], [3, 0, 0]]) # N_fam * 1 e1^T (rows (3,3,3)) V = Q * sp.diag(0, 1, 1) # rows (1,2,3) affine_ok = (_rows(C) == (7, 11, 13) and _rows(U) == (3, 3, 3) and _rows(V) == (1, 2, 3) and _rows(Q) == (4, 5, 6) and _rows(K) == (6, 9, 10) and _rows(R) == (4, 8, 10) and _rows(L) == (10, 14, 16) and _rows(F) == (8, 13, 16)) check("ROW-BUDGET CROSS: one affine map row(C+sU+tV) = (7,11,13)+s(3,3,3)+" "t(1,2,3) generates every flavour row budget -- democratic winding " "(3,3,3) + A3 exponent grading (1,2,3)", affine_ok, exact=True) # 6. dim E6 = |R+(A3)|(N_fam + A_Lambda) check("dim E6 = |R+(A3)|(N_fam+A_Lambda): 78 = p2*Delta = 6*13 binds the " "flavour norm |R|_F^2=78 to the hexagon p2=6 and the quark denominator " "Delta=13 -- not a decorative Lie hit", p2*DELTA == 78 and p2 == 6 and DELTA == 13, exact=True) return summary("v170 E8 Slice Compression") if __name__ == "__main__": raise SystemExit(1 if run() else 0)