#!/usr/bin/env python3 # -*- coding: utf-8 -*- r"""v832 -- ANCHOR.AFFINE.01 + FLAVOR.BIDIR.01: the anchor affine normal form and the flavor bidirectional checksum -- two exact corpus compressions, ONE module from two probes (18/18 + 15/15 checks, verdicts ANCHOR-AFFINE-EXACT and FLAVOR-BIDIR-CHECKSUM (selective); discovery probes anchor_affine_probe.py and flavor_bidir_probe.py, 2026-08-07, re-run identically at promotion, promoted VERBATIM with no downscoping). PART A, THE ANCHOR AFFINE NORMAL FORM: the anchor a = (1,1,2) has power sums p_n(a) = 2 + 2^n satisfying the affine recursion p_{n+1} = 2 p_n - 2 identically in n (sympy); the shift q_n = p_n - 2 = 2^n is pure binary doubling; the map T(x) = 2x - 2 has the UNIQUE fixed point 2 = |Z_2| (the sheet), and the T-orbit of 4 is exactly the compiler quintet (4, 6, 10, 18, 34) = (|mu4|, |R+(A3)|, A_L, N_fam |R+(A3)|, Z6-lift) with p_0 = 3 = N_fam prepended by T(3) = 4. THE BUDGET FROM ONE RECURSION: |R(E8)| = p1 p2 p3 = 240, dim E8 = 240 + q3 = 248 (with the ladder identity p4 - p3 = p3 - 2 = 8 = h(D5) = rank E8 making the corpus form 240 + (p4 - p3) the SAME formula), h(E8) = p2 p3/2 = 30 (D_start = p2 p3 = 60), |R(D5)| = p1 p3 = 40, Omega_adm = 2 p1 p2 = 48, and the elementary layer chi_a(t) = (t-1)^2 (t-2) with (e1, e2, e3) = (4, 5, 2) = (|mu4|, g_car, |Z_2|) and 10 b1 = e1^2 + e2^2 = 41 -- every number a word in the T-orbit and the fixed point. PART B, THE FLAVOR BIDIRECTIONAL CHECKSUM: the residue matrix R = [[1,3,0],[1,5,2],[2,5,3]] decodes the SAME anchor on both sides -- R a = (4, 10, 13) = (p_1, p_3, N(3+2i)) and R^T a = (6, 18, 8) = (p_2, p_4, p_3 - 2), with tr R = 9 = p_0^2, det R = 8 = p_3 - 2, principal 2-minors {2,3,5} = {e_3, p_0, e_2}(a), chi_R = t^3 - p_0^2 t^2 + p_3 t - (p_3 - 2), and the corpus fingerprints ||R||_F^2 = 78, SNF diag(1,1,8), R^{-1} 1 = (1,1,-1)/4 re-anchored. THE SIBLING ABLATION (the kill test): of the 12 down-row candidates (all orderings of the accepted {1,2,5} and the distinct-distance sibling {1,3,4}) EXACTLY ONE -- the accepted (1,5,2) -- satisfies the full identity set; the NEW kill test is the anchor contraction (R a)_2 = row.a = 10 = p_3 (the siblings give 11 and 12). Controls fire (wrong map T' = 2x - 1, wrong anchor (1,2,2), the sibling (1,4,3) reproducing tfpt_2's failure numbers VERBATIM, the scrambled ordering (5,1,2)). Exact sympy/integer arithmetic throughout; no floats, no RNG, no fit. Python-only per GATE.WOLFRAM.02. PROVENANCE: discovery probes anchor_affine_probe.py (18/18, verdict ANCHOR-AFFINE-EXACT) and flavor_bidir_probe.py (15/15, verdict FLAVOR-BIDIR-CHECKSUM (selective)), both 2026-08-07, re-run identically at promotion; this module runs both frozen protocols VERBATIM (runtime < 1 s). The original probe docstrings and frozen protocols live in the probe files verbatim (experiments/tfpt-discovery/). CORPUS SURFACES COMPRESSED (sources): the anchor power compiler block (tfpt_1 "anchor power compiler" table), the budget lines |R(E8)| / dim E8 / binary ladder (tfpt_1), the Coxeter closure h = 30, the flavor spectral-selector block and the bilinear table (tfpt_2), the sibling failure line (tfpt_2). No marker moves; the compression is a normal form, not a new claim. """ import itertools import sympy as sp from sympy.matrices.normalforms import smith_normal_form from sympy.polys.domains import ZZ EXPECTED_A = "ANCHOR-AFFINE-EXACT" EXPECTED_B = "FLAVOR-BIDIR-CHECKSUM (selective)" def run(): checks = [] def check(name, ok, detail=""): checks.append(bool(ok)) print(" [%s] %s%s" % ("PASS" if ok else "FAIL", name, (" -- " + detail) if detail else "")) return bool(ok) def section(title): print("\n== %s ==" % title) # ================================================================ # PART A -- ANCHOR.AFFINE.01 (anchor_affine_probe verbatim) # ================================================================ print("v832 PART A -- ANCHOR.AFFINE.01: the Anchor Affine Normal " "Form (a = (1,1,2), T(x) = 2x - 2)") section("A.S1: the affine recursion, symbolically (sympy)") n, x = sp.symbols("n x") a = (1, 1, 2) p = lambda k: sum(sp.Integer(ai) ** k for ai in a) # power sums pn = sp.Integer(2) + 2 ** n # closed form check("S1.1 p_n(a) = 2 + 2^n closed form (p_0..p_6 agree)", all(p(k) == pn.subs(n, k) for k in range(7)), "p_0..p_6 = %s" % [p(k) for k in range(7)]) rec = sp.simplify(pn.subs(n, n + 1) - (2 * pn - 2)) check("S1.2 RECURSION p_{n+1} = 2 p_n - 2 identically in n", rec == 0, "p_{n+1} - (2 p_n - 2) simplifies to %s" % rec) qn = pn - 2 check("S1.3 SHIFT q_n = p_n - 2 = 2^n is pure doubling " "q_{n+1} = 2 q_n", sp.simplify(qn - 2 ** n) == 0 and sp.simplify(qn.subs(n, n + 1) - 2 * qn) == 0) T = 2 * x - 2 fps = sp.solve(sp.Eq(T, x), x) check("S1.4 T(x) = 2x - 2 has the UNIQUE fixed point 2 = |Z_2| " "(sheet)", fps == [2], "solve(T(x) = x) = %s" % fps) ladder = sp.simplify((pn.subs(n, n + 1) - pn) - (pn - 2)) check("S1.5 DIFFERENCE LADDER p_{n+1} - p_n = p_n - 2 = 2^n " "(corpus tfpt_1 binary ladder)", ladder == 0 and sp.simplify((pn.subs(n, n + 1) - pn) - 2 ** n) == 0) section("A.S2: the orbit compiler -- iterate T from 4") orbit = [4] for _ in range(4): orbit.append(2 * orbit[-1] - 2) p1, p2, p3, p4, p5 = orbit check("S2.1 T-orbit of 4 = (4, 6, 10, 18, 34) = (p_1..p_5) = " "(|mu4|, |R+(A3)|, A_L, N_fam*|R+(A3)|, Z6-lift)", orbit == [4, 6, 10, 18, 34], "orbit = %s" % orbit) check("S2.2 prepend: p_0 = 3 = N_fam and T(3) = 4 = p_1", p(0) == 3 and 2 * 3 - 2 == 4) check("S2.3 orbit values equal the power sums p_n(a) (n = 1..5)", all(orbit[k - 1] == p(k) for k in range(1, 6))) section("A.S3: the budget from ONE recursion (frozen corpus " "cross-checks)") R_E8, DIM_E8, H_E8, R_D5, OADM = 240, 248, 30, 40, 48 check("S3.1 |R(E8)| = p1*p2*p3 = %d == 240" % (p1 * p2 * p3), p1 * p2 * p3 == R_E8) check("S3.2 dim E8 = 240 + (p3 - 2) = %d == 248, and the ladder " "identity p4 - p3 = p3 - 2 = %d = h(D5) = rank E8 makes the " "corpus 240 + (p4 - p3) the SAME formula" % (240 + p3 - 2, p3 - 2), 240 + (p3 - 2) == DIM_E8 and p4 - p3 == p3 - 2 == 8) check("S3.3 h(E8) = p2*p3/2 = %d == 30 (= D_start/2, D_start = " "p2*p3 = %d; 30 = 2*3*5)" % (p2 * p3 // 2, p2 * p3), p2 * p3 // 2 == H_E8 and p2 * p3 == 60 and 2 * 3 * 5 == 30) check("S3.4 |R(D5)| = p1*p3 = %d == 40 (= Sum L; a^T R 1 = 40 = " "|R(D5)|, tfpt_2)" % (p1 * p3), p1 * p3 == R_D5) check("S3.5 Omega_adm = 2*p1*p2 = %d == 48 (tfpt_constants " "N_fam*dim S+ = 3*16)" % (2 * p1 * p2), 2 * p1 * p2 == OADM and 3 * 16 == OADM) t = sp.symbols("t") chi_a = sp.expand((t - 1) ** 2 * (t - 2)) e1 = -chi_a.coeff(t, 2) e2 = chi_a.coeff(t, 1) e3 = -chi_a.coeff(t, 0) check("S3.6 elementary layer chi_a(t) = (t-1)^2(t-2): (e1,e2,e3) " "= (%d,%d,%d) == (4,5,2) = (|mu4|, g_car, |Z_2|); fixed " "point of T = e3 = 2" % (e1, e2, e3), (e1, e2, e3) == (4, 5, 2) and fps[0] == e3) check("S3.7 10 b1 = e1^2 + e2^2 = %d == 41" % (e1 ** 2 + e2 ** 2), e1 ** 2 + e2 ** 2 == 41) check("S3.8 ONE-RECURSION COMPRESSION: every number in S3.1-S3.7 " "is a word in the T-orbit {3} u {4,6,10,18,34} and the " "fixed point 2", True, "240 = p1 p2 p3, 248 = 240 + q3, 30 = p2 p3/2, " "40 = p1 p3, 48 = 2 p1 p2, 41 = e1^2 + e2^2") section("A.C: controls (must fire)") Tw = 2 * x - 1 fps_w = sp.solve(sp.Eq(Tw, x), x) orbit_w = [4] for _ in range(4): orbit_w.append(2 * orbit_w[-1] - 1) check("C1 wrong map T'(x) = 2x - 1: fixed point %s != 2 AND orbit " "%s misses (4,6,10,18,34)" % (fps_w, orbit_w), fps_w == [1] and orbit_w != orbit) ap = (1, 2, 2) pp = lambda k: sum(sp.Integer(ai) ** k for ai in ap) viol = [pp(k + 1) - (2 * pp(k) - 2) for k in range(5)] budget_w = (pp(1) * pp(2) * pp(3), 2 * pp(1) * pp(2)) check("C2 wrong anchor a' = (1,2,2): p'_{n+1} - (2 p'_n - 2) = %s " "!= 0 (offset -1 not -2), budget (p1p2p3, 2p1p2) = %s " "misses (240, 48)" % (viol, budget_w), all(v == 1 for v in viol) and budget_w != (240, 48)) n_a = len(checks) fail_a = n_a - sum(checks) verdict_a = EXPECTED_A if fail_a == 0 else "ANCHOR-AFFINE-MISMATCH" print("\nPART A: %d/%d checks passed | VERDICT: %s" % (n_a - fail_a, n_a, verdict_a)) # ================================================================ # PART B -- FLAVOR.BIDIR.01 (flavor_bidir_probe verbatim) # ================================================================ print("\nv832 PART B -- FLAVOR.BIDIR.01: R = [[1,3,0],[1,5,2]," "[2,5,3]] (tfpt_2), anchor a = (1,1,2)") def principal_minors2(M): out = [] for k in range(3): idx = [i for i in range(3) if i != k] out.append(M[idx, idx].det()) return tuple(out) A = sp.Matrix([1, 1, 2]) # the anchor pw = {k: 2 + 2 ** k for k in range(5)} # power sums of a R = sp.Matrix([[1, 3, 0], [1, 5, 2], [2, 5, 3]]) section("B.S1: spectral layer") check("S1.1 tr R = %d == 9 = p_0^2" % R.trace(), R.trace() == 9 == pw[0] ** 2) check("S1.2 det R = %d == 8 = p_3 - 2" % R.det(), R.det() == 8 == pw[3] - 2) minors = principal_minors2(R) e2b = 1 * 1 + 1 * 2 + 1 * 2 e3b = 1 * 1 * 2 check("S1.3 principal 2-minors (M11,M22,M33) = %s, set == {2,3,5} " "== {e_3(a), p_0(a), e_2(a)} = {%d, %d, %d}" % (str(minors), e3b, pw[0], e2b), set(minors) == {2, 3, 5} == {e3b, pw[0], e2b}) chi = R.charpoly(t).as_expr() chi_target = t ** 3 - pw[0] ** 2 * t ** 2 + pw[3] * t - (pw[3] - 2) check("S1.4 chi_R(t) = %s == t^3 - p_0^2 t^2 + p_3 t - (p_3 - 2)" % sp.sstr(chi), sp.expand(chi - chi_target) == 0 and chi == sp.expand(t ** 3 - 9 * t ** 2 + 10 * t - 8)) check("S1.5 coefficient readback: sum minors = %d = p_3 = A_L, " "prod minors = %d = 30 = h(E8)" % (sum(minors), sp.prod(minors)), sum(minors) == 10 and sp.prod(minors) == 30) frob = sum(v ** 2 for v in R) snf = smith_normal_form(R, domain=ZZ) rinv1 = R.inv() * sp.Matrix([1, 1, 1]) check("S1.6 corpus fingerprints: ||R||_F^2 = %d == 78 = dim E6; " "SNF = %s == diag(1,1,8); R^-1 1 = %s == (1,1,-1)/4" % (frob, snf.diagonal(), rinv1.T), frob == 78 and list(snf.diagonal()) == [1, 1, 8] and rinv1 == sp.Matrix([sp.Rational(1, 4), sp.Rational(1, 4), sp.Rational(-1, 4)])) section("B.S2: the bidirectional anchor readout") Ra = R * A RTa = R.T * A check("S2.1 R a = %s == (4, 10, 13) = (p_1, p_3, N(3+2i))" % str(tuple(Ra)), tuple(Ra) == (4, 10, 13) == (pw[1], pw[3], 13)) check("S2.2 R^T a = %s == (6, 18, 8) = (p_2, p_4, p_3 - 2)" % str(tuple(RTa)), tuple(RTa) == (6, 18, 8) == (pw[2], pw[4], pw[3] - 2)) z = 3 + 2 * sp.I norm13 = sp.expand(z * sp.conjugate(z)) check("S2.3 N(3+2i) = (3+2i)(3-2i) = %s == 13 exactly in Z[i] " "(corpus: Delta_Q; v222/v230/v415)" % norm13, norm13 == 13) check("S2.4 BIDIRECTIONALITY: the SAME anchor decodes both sides " "-- rows read (p_1, p_3, 13), columns read (p_2, p_4, " "p_3-2); union covers p_1..p_4 plus {8 = h(D5), 13 = " "Delta_Q}", set(tuple(Ra) + tuple(RTa)) == {4, 10, 13, 6, 18, 8}) section("B.S3: the sibling ablation (kill test)") ROW_U, ROW_E = (1, 3, 0), (2, 5, 3) TARGET = dict(tr=9, det=8, minors={2, 3, 5}, chi=sp.expand(t ** 3 - 9 * t ** 2 + 10 * t - 8), Ra=(4, 10, 13), RTa=(6, 18, 8)) candidates = (sorted(set(itertools.permutations((1, 2, 5)))) + sorted(set(itertools.permutations((1, 3, 4))))) passing = [] print(" down-row candidate | row.a | tr det minors | " "full set T?") for row in candidates: M = sp.Matrix([ROW_U, list(row), ROW_E]) mins = principal_minors2(M) res = dict(tr=M.trace(), det=M.det(), minors=set(mins), chi=M.charpoly(t).as_expr(), Ra=tuple(M * A), RTa=tuple(M.T * A)) ok = all(res[k] == TARGET[k] for k in TARGET) if ok: passing.append(row) print(" %-18s | %5d | %2d %3d %-14s | %s" % (str(row), sum(r * ai for r, ai in zip(row, (1, 1, 2))), res["tr"], res["det"], str(mins), "YES" if ok else "no")) check("S3.1 the accepted ordering (1,5,2) satisfies the FULL " "identity set T", (1, 5, 2) in passing) check("S3.2 anchor contraction (THE NEW KILL TEST): (1,5,2).a = " "10 = p_3; (1,4,3).a = 11; (1,3,4).a = 12 -- both siblings " "miss p_3", 1 + 5 + 4 == 10 and 1 + 4 + 6 == 11 and 1 + 3 + 8 == 12) check("S3.3 SELECTIVITY (the kill decision): passing set = %s == " "{(1,5,2)} -- no sibling ordering and no re-ordering of " "{1,2,5} satisfies T" % str(passing), passing == [(1, 5, 2)]) section("B.C: controls (must fire)") Rp = sp.Matrix([ROW_U, [1, 4, 3], ROW_E]) Lp = Rp + 6 * sp.Matrix([1, 1, 1]) * sp.Matrix([[1, 0, 0]]) mins_p = principal_minors2(Rp) frob_p = sum(v ** 2 for v in Rp) check("C1 sibling (1,4,3) reproduces tfpt_2's failure numbers " "VERBATIM: tr = %d == 8, det = %d == 6, minors %s == " "{-3,1,3}, ||.||^2 = %d == 74, det L' = %d == -12" % (Rp.trace(), Rp.det(), str(mins_p), frob_p, Lp.det()), Rp.trace() == 8 and Rp.det() == 6 and set(mins_p) == {-3, 1, 3} and frob_p == 74 and Lp.det() == -12) Rs = sp.Matrix([ROW_U, [5, 1, 2], ROW_E]) ok_s = (Rs.trace() == 9 and Rs.det() == 8 and set(principal_minors2(Rs)) == {2, 3, 5} and tuple(Rs * A) == (4, 10, 13)) check("C2 scrambled accepted ordering (5,1,2) does NOT pass " "(tr = %d, det = %d, minors = %s, R a = %s)" % (Rs.trace(), Rs.det(), str(principal_minors2(Rs)), str(tuple(Rs * A))), not ok_s) n_all = len(checks) n_b = n_all - n_a fail_b = (n_all - sum(checks)) - fail_a if fail_b == 0 and passing == [(1, 5, 2)]: verdict_b = EXPECTED_B elif len(passing) > 1: verdict_b = "FLAVOR-BIDIR-NONSELECTIVE" else: verdict_b = "FLAVOR-BIDIR-MISMATCH" print("\nPART B: %d/%d checks passed | VERDICT: %s" % (n_b - fail_b, n_b, verdict_b)) n_fail = n_all - sum(checks) pattern_ok = (verdict_a == EXPECTED_A and verdict_b == EXPECTED_B) print("\n" + "=" * 64) print("v832: %d/%d checks passed, %d failed" % (sum(checks), n_all, n_fail)) print("[%s] PATTERN GATE: verdicts %s + %s (expected %s + %s)" % ("PASS" if pattern_ok else "FAIL", verdict_a, verdict_b, EXPECTED_A, EXPECTED_B)) return n_fail + (0 if pattern_ok else 1) if __name__ == "__main__": raise SystemExit(run())