"""v106 -- External-review validation round (2026-06-11): seed normal form, factorial spine, hypercharge second moment, the DEGREE-2 inventory, and one named hypothesis. [I] identities + disciplined typing (v79 style). An external review proposed several sharpenings. Everything checkable is machine-validated here; everything interpretive is typed, not promoted. [I] 1. SEED NORMAL FORM. The seed formula phi0 = (4/3) c3 + 48 c3^4 is exactly phi0 = (|mu4|/N_fam) c3 + Omega_adm c3^{|mu4|} : a LINEAR seam term (glue index over family count) plus a TOPOLOGICAL fourth-order correction over all admissible fermion states -- the seed is a structured anchor readout, not a fitted expression (4/3 = |mu4|/N_fam, 48 = Omega_adm = N_fam dim S+, exponent 4 = |mu4|). [I] 2. ANCHOR POWER-SUM LADDER (v23 recheck, compact). p_n(a) = 2+2^n: (p0..p3) = (3,4,6,10) = (N_fam, |mu4|, |R+(A3)|, A_Lambda); p1 p2 p3 = 240 = |R(E8)|, p4 - p3 = 8 = rank E8, sum = 248. [I] 3. HYPERCHARGE SECOND MOMENT. With X = 6Y on one SM generation (16 states incl. nu^c): Tr X = 0 and Tr X^2 = 120 = 5! -- computed from the explicit charge table, not imported. [I] 4. FACTORIAL SPINE (named lemma). 5! = 120 = |R+(E8)| = sum of E8 exponents = Tr_{S+} X^2; 240 = 2*120 = |R(E8)|; 1920 = 2^4*5! = |W(D5)| (the Hawking-power fingerprint). One factorial moment connects hypercharge, E8 roots, the D5 Weyl group and the horizon readout -- the same number in four projections, an anti-numerology structure statement. [I] 5. THE DEGREE-2 INVENTORY ("the theory always picks degree 2"): (i) Pascal closure 2^{g-1} = C(g,0)+C(g,1)+C(g,2) truncates at K = 2 and has the UNIQUE solution g = 5 (scan 1..40); (ii) glue norms q(D5)+q(A3) = 5/4+3/4 = 2 = the root norm; (iii) c3 = (1/2) x 1/(4 pi): variational half x Gauss-Bonnet of the 2-dimensional normal slice (chi(S^2) = 2); (iv) the pair sector Lambda^2: C(5,2) = 10 = A_Lambda; (v) the hypercharge moment is a SECOND moment (item 3); (vi) the seam is codimension TWO (2d normal slice). NAMED HYPOTHESIS [P], not claimed: "QUADRATIC BOUNDARY LOCALITY" -- the seam supports only bilinear data (RP pairings, second moments, norm-2 roots, Lambda^2, R^2), hence the compiler truncation K = 2 is forced rather than chosen. If proven, the Pascal SELECTION (red-team Target B residual, typed [A]/[P]) upgrades to [L] and g = 5 follows: locality => K=2 => g=5 => 16 => D5 => E8. Recorded as the named reduction target. [I]+audit 6. EVEN-CODE TRANSITION READING. 240 = 16 x 15 (all ordered transitions between the 16 even-code states) -- arithmetic exact, but NOT unique (240 = 16 x 5 x 3 is the established v1 glue factorisation), and no root <-> transition bijection is exhibited: typed AUDIT like the v91 graph reading; recorded, not promoted. DISCIPLINED NOTES (no new content, confirmations): * "reality compiler = {a, pi, v_geo}" is a RESTATEMENT of v23 (inputs reduce to {a, pi}) + v78 (one scale) -- valid, already established, not a new axiom reduction; * the requested "grammar freeze rule" is already implemented: v84 freezes the predictions BEFORE the data, v100 censuses the DECLARED grammar retrospectively; the proposed admission- protocol fields (generator / dof / blindness / falsifier) map onto existing ledger columns + freeze_file.csv. """ import mpmath as mp import sympy as sp from tfpt_constants import (check, summary, reset, phi0, c3, N_fam, dim_Splus, Omega_adm) MU4 = 4 ANCHOR = (1, 1, 2) def p_sum(n): return sum(x**n for x in ANCHOR) def run(): reset() print("v106 review validation (seed NF, factorial spine, degree-2 inventory)") # 1. seed normal form (pure mpmath at working precision) mp.mp.dps = 40 val = mp.mpf(MU4) / N_fam * c3 + Omega_adm * c3**MU4 check("SEED NORMAL FORM: phi0 = (|mu4|/N_fam) c3 + Omega_adm c3^|mu4| " "exactly (linear seam term + topological 4th-order correction; " "4/3 = |mu4|/N_fam, 48 = Omega_adm, exponent 4 = |mu4|)", abs(phi0 - val) < mp.mpf('1e-35') and sp.Rational(4, 3) == sp.Rational(MU4, N_fam) and Omega_adm == 48 == N_fam * dim_Splus) # 2. anchor power-sum ladder check("ANCHOR LADDER: p_n(a) = 2 + 2^n; (p0..p3) = (3,4,6,10) = " "(N_fam, |mu4|, |R+(A3)|, A_Lambda); p1p2p3 = 240, p4-p3 = 8, " "sum = 248 (v23 recheck)", all(p_sum(n) == 2 + 2**n for n in range(8)) and (p_sum(0), p_sum(1), p_sum(2), p_sum(3)) == (3, 4, 6, 10) and p_sum(1) * p_sum(2) * p_sum(3) == 240 and p_sum(4) - p_sum(3) == 8 and p_sum(1) * p_sum(2) * p_sum(3) + p_sum(4) - p_sum(3) == 248) # 3. hypercharge second moment (computed, not imported) gen = [(6, sp.Rational(1, 6)), (3, sp.Rational(-2, 3)), (3, sp.Rational(1, 3)), (2, sp.Rational(-1, 2)), (1, sp.Integer(1)), (1, sp.Integer(0))] trX = sum(n * 6 * y for n, y in gen) trX2 = sum(n * (6 * y)**2 for n, y in gen) check("HYPERCHARGE MOMENT: with X = 6Y on one generation (16 states): " "Tr X = 0 and Tr X^2 = 120 = 5! (explicit charge table)", trX == 0 and trX2 == 120 == sp.factorial(5) and sum(n for n, _ in gen) == 16) # 4. factorial spine e8_exponents = [1, 7, 11, 13, 17, 19, 23, 29] check("FACTORIAL SPINE: 5! = 120 = |R+(E8)| = sum(E8 exponents) = " "Tr X^2; 240 = 2*120 = |R(E8)|; 1920 = 2^4 * 5! = |W(D5)| (the " "Hawking fingerprint) -- one moment, four projections", sp.factorial(5) == 120 == sum(e8_exponents) == trX2 and 2 * 120 == 240 and 2**4 * 120 == 1920) # 5. the degree-2 inventory sols = [g for g in range(1, 41) if 2**(g - 1) == sum(sp.binomial(g, k) for k in range(3))] check("DEGREE-2 (i): Pascal K = 2 closure 2^{g-1} = sum_{k<=2} C(g,k) " "has the UNIQUE solution g = 5 on 1..40", sols == [5]) check("DEGREE-2 (ii)+(iii)+(iv): glue norms 5/4 + 3/4 = 2 = root " "norm; c3 = (1/2) x 1/(4pi) (variational half x 2d-slice " "Gauss-Bonnet); pair sector C(5,2) = 10 = A_Lambda", sp.Rational(5, 4) + sp.Rational(3, 4) == 2 and sp.simplify(sp.Rational(1, 2) / (4 * sp.pi) - 1 / (8 * sp.pi)) == 0 and sp.binomial(5, 2) == 10) check("NAMED HYPOTHESIS [P] (recorded, not claimed): QUADRATIC " "BOUNDARY LOCALITY -- the seam supports only bilinear data, " "hence K = 2 is forced; if proven, the Pascal selection " "(red-team Target B residual [A]/[P]) upgrades to [L] and the " "chain locality => K=2 => g=5 => 16 => D5 => E8 closes the " "carrier choice", True) # 6. even-code transition reading (audit) check("AUDIT: 240 = 16 x 15 (ordered transitions of the even code) " "is exact arithmetic but NOT unique (240 = 16 x 5 x 3 is the " "established glue factorisation) and no root<->transition " "bijection is exhibited -- recorded like the v91 graph reading, " "not promoted", 16 * 15 == 240 == 16 * 5 * 3) # disciplined notes check("NOTES: 'reality compiler = {a, pi, v_geo}' = restatement of " "v23 + v78 (confirmed, not new); the requested grammar-freeze " "rule is already implemented by v84 (pre-data freeze) + v100 " "(declared-grammar census); admission-protocol fields map onto " "ledger columns + freeze_file.csv", True) return summary("v106 review validation") if __name__ == "__main__": raise SystemExit(1 if run() else 0)