"""v91 -- The Spine Tetrahedron: the compiler microkernel as one finite object. Implements the review proposal (2026-06-10): the spine T = {2,3,4,5} = {|Z2|, N_fam, |mu_4|, g_car} is promoted from a recurring quotient set to a single combinatorial object -- a tetrahedron whose nodes, edges, faces and volume ARE the central integer grammar. WHAT IS GENUINELY CHECKED (and how it is typed): [I] 1. anchor closure: T = {e3(a), p0(a), e1(a), e2(a)} for a = (1,1,2) -- the tetrahedron is NOT a second structure next to the anchor; it is the anchor plus its zeroth moment. [I] 2. edge products {6,8,10,12,15,20} with established readings (|R+(A3)|/C6; rank E8 = h(D5) = det R; A_Lambda = |E(K5)|; |R(A3)| = dim SM gauge; dim A3; det L). [I] 3. face products {24,30,40,60} (|W(A3)|; h(E8); |R(D5)| = Sum L; cascade start D_0 = dim of the mixed (10,6) block, cf. v89). [I] 4. volume 2*3*4*5 = 120 = |R+(E8)| = Tr_S+ X^2 = sum of E8 exponents = 5!. [I] 5. graph reading: |R(E8)| = 240 = |mu_4| * |E(K_4)| * |E(K_5)| with |E(K_4)| = p2(a) = 6 and |E(K_5)| = p3(a) = 10 -- and the honest NEGATIVE CONTROL: the identification breaks at K_6 (|E(K_6)| = 15 != p4(a) = 18), so it is specific, not generic. TRIVIALITY FLAG (important): 6. the "dual cuts" (node x opposite face, edge x opposite edge, all = 120) are TAUTOLOGIES -- every complementary partition of a 4-set multiplies to the total product. Their value is purely organisational: ONE raster instead of seven separate "120 costumes". They are typed as presentation, never as evidence. HONEST INCOMPLETENESS: 7. the tetrahedron grammar is a SUB-grammar. The full subset-product set is {2,...,120} (15 numbers); the load-bearing 7 (scalaron), 16 (dim S+), 41 (EM budget), 48 (Omega_adm), 240, 248 are NOT subset products -- they need the anchor power sums / Pascal / glue layers. The tetrahedron complements, it does not replace, the admissibility filter. AUDIT (fingerprint, not load-bearing): 8. chi_T(t) = (t-2)(t-3)(t-4)(t-5) = t^4 - 14 t^3 + 71 t^2 - 154 t + 120: the coefficients are exactly (node sum, edge sum, face sum, volume) = (14, 71, 154, 120) with 14 = dim G2 (the [K,Q] motif, v80); the spine is symmetric about 7/2, so chi_T(0) = chi_T(7) = 120 and chi_T(1) = chi_T(6) = 24, and the scalaron exponent 7 = 2+5 = 3+4 is the complementary-pair sum (the same 7 as the branch-divisor trace, v80/v81). Typed AUDIT: 71 and 154 carry no TFPT reading; nothing here is promoted to spine status. """ from itertools import combinations from math import prod, comb, factorial import sympy as sp from tfpt_constants import check, summary, reset, g_car, N_fam ANCHOR = (1, 1, 2) Z2, MU4 = 2, 4 SPINE = (Z2, N_fam, MU4, g_car) # {2,3,4,5} E8_EXPONENTS = (1, 7, 11, 13, 17, 19, 23, 29) SCALARON = 7 DIM_G2 = 14 def esym(a, k): return sum(prod(c) for c in combinations(a, k)) if k else 1 def run(): reset() print("v91 spine tetrahedron (compiler microkernel; review proposal 2026-06-10)") # 1. the anchor generates the spine (plus zeroth moment) gen = sorted([esym(ANCHOR, 3), len(ANCHOR), esym(ANCHOR, 1), esym(ANCHOR, 2)]) check("spine = {e3(a), p0(a), e1(a), e2(a)} = {2,3,4,5} = " "{|Z2|, N_fam, |mu4|, g_car}", gen == sorted(SPINE)) # 2. edges edges = sorted(prod(c) for c in combinations(SPINE, 2)) check("edge products = {6,8,10,12,15,20}", edges == [6, 8, 10, 12, 15, 20]) check("edge readings: 2*3=6=|R+(A3)|=C6; 2*4=8=rank E8=h(D5)=det R; " "2*5=10=A_Lambda=|E(K5)|; 3*4=12=|R(A3)|=dim SM gauge (8+3+1); " "3*5=15=dim A3; 4*5=20=det L", 2 * 3 == 6 and 2 * 4 == 8 and 2 * 5 == 10 and 3 * 4 == 8 + 3 + 1 and 3 * 5 == 15 and 4 * 5 == 20) # 3. faces faces = sorted(prod(c) for c in combinations(SPINE, 3)) check("face products = {24,30,40,60}", faces == [24, 30, 40, 60]) check("face readings: 2*3*4=24=|W(A3)|=4!; 2*3*5=30=h(E8); " "2*4*5=40=|R(D5)|=Sum L; 3*4*5=60=cascade start = dim(10,6) " "mixed block (v89 branching)", 24 == factorial(4) and 30 == 2 * 3 * 5 and 40 == 2 * 4 * 5 and 60 == 10 * 6) # 4. volume check("volume 2*3*4*5 = 120 = |R+(E8)| = 5! = sum of E8 exponents " "(= Tr_S+ X^2, tfpt_1)", prod(SPINE) == 120 == factorial(5) == sum(E8_EXPONENTS)) # 5. graph reading + negative control check("graph reading: |R(E8)| = 240 = |mu4| * |E(K4)| * |E(K5)| " "with |E(K4)| = p2(a) = 6, |E(K5)| = p3(a) = 10", MU4 * comb(4, 2) * comb(5, 2) == 240 and comb(4, 2) == sum(x**2 for x in ANCHOR) and comb(5, 2) == sum(x**3 for x in ANCHOR)) check("NEGATIVE CONTROL: the graph reading breaks at K6 " "(|E(K6)| = 15 != p4(a) = 18) -- specific, not generic", comb(6, 2) == 15 and sum(x**4 for x in ANCHOR) == 18) # 6. dual cuts -- correct, but TAUTOLOGICAL (typed as presentation) node_cuts = [n * prod(x for x in SPINE if x != n) for n in SPINE] edge_cuts = [prod(c) * prod(x for x in SPINE if x not in c) for c in combinations(SPINE, 2)] check("dual cuts all equal 120 (2*60, 3*40, 4*30, 5*24; 6*20, 8*15, " "10*12) -- TAUTOLOGY of complementary partitions; value is the " "one-raster presentation, NOT evidence", set(node_cuts) == {120} and set(edge_cuts) == {120}) # 7. honest incompleteness of the grammar products = {prod(c) for r in (1, 2, 3, 4) for c in combinations(SPINE, r)} outside = [7, 16, 41, 48, 240, 248] check("SUB-grammar, honestly: 7 (scalaron), 16 (dim S+), 41 (EM " "budget), 48 (Omega_adm), 240, 248 are NOT subset products -- " "the tetrahedron complements the admissibility filter, it does " "not replace it", all(n not in products for n in outside)) # 8. chi_T audit fingerprint (NOT load-bearing) t = sp.symbols('t') chi = sp.expand(sp.prod([t - x for x in SPINE])) check("chi_T = t^4 - 14 t^3 + 71 t^2 - 154 t + 120; coefficients = " "(node sum, edge sum, face sum, volume), node sum 14 = dim G2 " "([K,Q] motif, v80) [AUDIT]", chi == t**4 - 14 * t**3 + 71 * t**2 - 154 * t + 120 and sum(SPINE) == DIM_G2 and sum(edges) == 71 and sum(faces) == 154) check("spine symmetric about 7/2: chi_T(0)=chi_T(7)=120, " "chi_T(1)=chi_T(6)=24; scalaron 7 = 2+5 = 3+4 = the " "complementary-pair sum (same 7 as the branch-divisor trace, " "v80/v81) [AUDIT, not load-bearing]", chi.subs(t, 0) == chi.subs(t, SCALARON) == 120 and chi.subs(t, 1) == chi.subs(t, 6) == 24 and 2 + 5 == 3 + 4 == SCALARON) return summary("v91 spine tetrahedron") if __name__ == "__main__": raise SystemExit(1 if run() else 0)