"""v119 -- Second external-review validation (2026-06-11b): the anchor ratio triad, the 121 audit lemma, and the normal-form micro-identities. [I] exact identities + explicit sensitivity controls; presentational recommendations recorded, not checked. An external review proposed several sharpenings. Validation result: [I] 1. ANCHOR NORMAL-FORM MICRO-IDENTITIES (all exact). With p_n = 2 + 2^n and (e1, e2, e3) = (4, 5, 2): Omega_adm = 2 p1 p2 = 48, 10 b1 = 1 + p1 p3 = 41, Delta_Y = e2^2 = p0^2 + 2 rank = 25, h(E8) = e3 p0 e2 = 30, rank E8 = p0 + e2 = 8, |R(E8)| = p1 p2 p3 = 240, dim = 240 + (p4 - p3) = 248. [I] 2. THE ANCHOR RATIO TRIAD (named lemma). The three elementary symmetric values of the anchor, normalised by the family count: e3/p0 = 2/3 (the KOIDE BRANCH / sheet ratio), e1/p0 = 4/3 (the SEED GAIN: phi0 = (4/3)c3 + 48 c3^4, 4/3 = |mu_4|/N_fam, v106), e2/p0 = 5/3 (the CARRIER BRANCH: the canonical flow's critical points are exactly (2, 5) = (e3, e2), v99/v102, with inflection 7/2 = (e3+e2)/2). The recurring fractions 2/3, 4/3, 5/3 are not scattered numbers: they are the three normalised anchor coefficients. [I] 3. THE 121 AUDIT LEMMA (recorded as AUDIT, not load-bearing). With h = 1 + a = (2, 2, 3) in the canonical anchor ordering and the established residue operator R (det 8, SNF (1,1,8)): h^T R h = 121 = 11^2 = ||Pl(K)||_1^2 = (p3 + 1)^2 -- the quark-ratio 11 appears a SECOND, independent time as the quadratic anchor norm of R. SENSITIVITY CONTROL: over the 3 distinct orderings of h the form takes values {105, 121, 135}; only the canonical anchor ordering gives 121. BONUS: 1^T R 1 = 22 = 2 x 11 (the entry sum of R) and a^T R a = 40 = p1 p3 = 10 b1 - 1. This does NOT replace the oriented Plucker mechanism (v42/v49) -- it is a second checksum on the same integer. [I] 4. THE FLAVOR SURFACE CLAIM = ESTABLISHED v94/v95. The review's M(s,t) = R + Q diag(s,t,t) with R = M(0,0), K = M(1,-1), L = M(2,0), F = M(1,1) and the centered cross Q = U + V, {R, L} = C -+ U, {K, F} = C -+ V is exactly the sheet diamond (v94) / centered diamond (v95) -- re-verified here; the review's suggestion is presentational, not new mathematics. 5. ALREADY CLOSED TODAY (no new checks): the mu_4 projection lemma = v115/v116; the anchor triptych = v104/v105; "interior free, structure = certificate" = v113. Presentational recommendations (anchor-first entry, normal-form table, M(s,t) presentation) recorded in next.txt as editorial items. """ import sympy as sp from itertools import permutations from tfpt_constants import check, summary, reset R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) Q = sp.Matrix([[3, 1, 0], [3, 2, 0], [3, 2, 1]]) K = sp.Matrix([[4, 2, 0], [4, 3, 2], [5, 3, 2]]) L = sp.Matrix([[7, 3, 0], [7, 5, 2], [8, 5, 3]]) A = sp.Matrix([1, 1, 2]) ONE = sp.ones(3, 1) def p(n): return 2 + 2 ** n def run(): reset() print("v119 second review validation (triad, 121 audit, micro-identities)") e1, e2, e3 = 4, 5, 2 # 1. micro-identities check("ANCHOR NORMAL-FORM MICRO-IDENTITIES: Omega_adm = 2 p1 p2 = " "48, 10 b1 = 1 + p1 p3 = 41, Delta_Y = e2^2 = p0^2 + 2 rank " "= 25, h(E8) = e3 p0 e2 = 30, rank = p0 + e2 = 8, 240 = " "p1 p2 p3, 248 = 240 + (p4 - p3)", 2 * p(1) * p(2) == 48 and 1 + p(1) * p(3) == 41 and e2 ** 2 == 25 and p(0) ** 2 + 2 * 8 == 25 and e3 * p(0) * e2 == 30 and p(0) + e2 == 8 and p(1) * p(2) * p(3) == 240 and p(1) * p(2) * p(3) + p(4) - p(3) == 248) # 2. anchor ratio triad check("ANCHOR RATIO TRIAD: (e3, e1, e2)/p0 = (2/3, 4/3, 5/3) = " "(Koide branch / sheet ratio, seed gain |mu_4|/N_fam (v106), " "carrier branch); the canonical flow's critical points are " "exactly (2, 5) = (e3, e2) (v99/v102) with inflection 7/2 = " "(e3+e2)/2 -- the recurring fractions are the three " "normalised anchor coefficients", (sp.Rational(e3, 3), sp.Rational(e1, 3), sp.Rational(e2, 3)) == (sp.Rational(2, 3), sp.Rational(4, 3), sp.Rational(5, 3)) and (2, 5) == (e3, e2) and sp.Rational(e3 + e2, 2) == sp.Rational(7, 2)) # 3. the 121 audit lemma h = ONE + A val = (h.T * R * h)[0] perm_vals = sorted(set(int((sp.Matrix(q).T * R * sp.Matrix(q))[0]) for q in permutations([2, 2, 3]))) bmat = sp.Matrix.vstack(ONE.T * K, A.T * K) pl = sp.Matrix([bmat[:, [1, 2]].det(), -bmat[:, [0, 2]].det(), bmat[:, [0, 1]].det()]) check("THE 121 AUDIT LEMMA: (1+a)^T R (1+a) = 121 = 11^2 = " "||Pl(K)||_1^2 = (p3+1)^2 -- the quark-ratio 11 appears a " "second, independent time as the quadratic anchor norm of R", val == 121 and 121 == 11 ** 2 and sum(abs(x) for x in pl) == 11 and (p(3) + 1) ** 2 == 121) check("SENSITIVITY CONTROL (honesty): over the 3 distinct orderings " "of h = (2,2,3) the form takes values {105, 121, 135} -- only " "the canonical anchor ordering gives 121; recorded as AUDIT, " "not load-bearing (the oriented Plucker mechanism v42/v49 " "stays the derivation)", perm_vals == [105, 121, 135]) check("BONUS CHECKSUMS: 1^T R 1 = 22 = 2 x 11 (entry sum of R) and " "a^T R a = 40 = p1 p3 = 10 b1 - 1", (ONE.T * R * ONE)[0] == 22 and 22 == 2 * 11 and (A.T * R * A)[0] == 40 and 40 == p(1) * p(3)) # 4. flavor surface = established v94/v95 def msurf(s, t): return R + Q * sp.diag(s, t, t) sig = sp.diag(1, -1, -1) u_ax = 3 * ONE * sp.Matrix([[1, 0, 0]]) v_ax = Q - u_ax c_mid = R + u_ax check("FLAVOR SURFACE = ESTABLISHED v94/v95 (review claim " "re-verified, presentational): R = M(0,0), K = M(1,-1), " "L = M(2,0), F = M(1,1) on M(s,t) = R + Q diag(s,t,t); " "centered cross Q = U + V, R = C-U, L = C+U, K = C-V, " "F = C+V", msurf(0, 0) == R and msurf(1, -1) == K and msurf(2, 0) == L and msurf(1, 1) == R + Q and K == R + Q * sig and L == R + Q * (sp.eye(3) + sig) and R == c_mid - u_ax and L == c_mid + u_ax and K == c_mid - v_ax and R + Q == c_mid + v_ax) # 5. bookkeeping check("REVIEW BOOKKEEPING (recorded): mu_4 projection lemma = " "v115/v116 (closed today); anchor triptych = v104/v105; " "'interior free, structure = certificate' = v113; " "anchor-first presentation / normal-form table / M(s,t) " "presentation = editorial items (next.txt), not checks", True) return summary("v119 second review validation") if __name__ == "__main__": raise SystemExit(1 if run() else 0)