"""v570 -- PRIME.PENCIL.SEPFLOOR.01: the v569 open question, reduced and certified on the declared surface -- the spectator separation follows from two one-line theorems plus the scalar dominance det S >> det B, and the cheap witness route is honestly killed. PROVENANCE. v569 answered the Priority-5 question of the self-code review on the reachable surface (verdict ONE-MODE) and left ONE new named open question, strictly weaker than Paper-II Problem 7.1: does the spectator eigenvalue lam1 of the relative pencil Z = B^{-1} S stay uniformly separated from 1? There the separation was MEASURED (min |1 - lam1| = 3.53). This module CERTIFIES it from closed scalars and types what the h-uniform question reduces to. Audited by the discovery probe (separation_floor_probe.py, 9/9, verdict FLOOR-CERTIFIED). Construction base: the declared T170 frame-A scan of v563_paper2_readouts (imported READ-ONLY, bit for bit -- no new surface, no new instrument). THE REDUCTION (exact; eigenvalues real because Z is similar symmetric): (i) P := det Z = det S/det B > 0 and T := tr Z > 0 give both eigenvalues positive, hence lam1 >= sqrt(P) (lam1^2 - P = lam1 (lam1 - lam2) >= 0). (ii) P <= 0 gives lam2 <= 0 <= lam1, hence lam1 = T - lam2 >= T. The closed per-window floor F := sqrt(P) if P > 0 else T certifies the separation whenever F >= 1 + margin: the open question reduces to DETERMINANT/TRACE DOMINANCE of the comb block over the arch block. [E] 1. The two floor lemmas verified symbolically; T, P and the link 1 - T + P = det(B-S)/det B are closed objects of (B, S). [E] 2. THE CERTIFICATION (the central result): on ALL 70 declared windows the closed floor is sound (lam1 >= F) and clears the margin, min F = 2.130 >= 1.05 -- the v569 separation is no longer only a measured eigenvalue fact, it follows from det S/det B and tr Z alone. [MEASURED] 3. The dominance GROWS with depth: P = det S/det B in [4.5, 120.6], median 30.1, log-log slope +0.900 in h on the surface (reported with its ladder; no rate claim beyond the surface). [E] 4. The pair decomposition is exact (Cauchy-Binet) -- and the cheap witness route is DEAD, reported honestly: det S is itself a pair-level cancellation object (negative pair mass up to 13.7 x det S, median 8.0 x; the best single-pair witness delivers only 0.13--0.34 x det B against the >= 1.10 it would need). The h-uniform question is typed as pair-level size-vs-cancellation -- an 8:9 cancellation, not a 4.7-order one, but not free. [E] 5. MUST-BREAK: the position scramble (same masses, the v563 intervention) FLIPS the trace sign (T = -78.28 vs +32.15 real): the T > 0 gate detects the destroyed geometry loudly -- the dominance is arithmetic placement, not a build artefact. NAMED LIMITS AS LOAD-BEARING CONTENT (the claim INCLUDES them): (i) Declared finite surface only: the certification is per window on the reachable surface; whether the dominance persists as h grows is the SAME named open question as v569's, now in the sharper closed-scalar form det S >= (1 + c)^2 det B (or tr Z >= 1 + c). (ii) Paper-II Problem 7.1 (the uniform deficit quantifier) is UNTOUCHED; MEASURED quantities keep their ladders (the v562 anti-promotion applies); no rate, no bound, NO RH statement. Fences: elementary 2x2 spectral theory CLASSICAL; Cauchy-Binet CLASSICAL; the v563 fence chain (Schur 1917 etc.) unchanged. Python-only, counted per GATE.WOLFRAM.02. Discovery surface: experiments/tfpt-discovery/separation_floor_probe.py (2026-07-31, 9/9, FLOOR-CERTIFIED). """ import time import numpy as np import sympy as sp import v563_paper2_readouts as core # READ-ONLY import of the surface N_PASS = 0 N_FAIL = 0 # declared thresholds (frozen before the run, from the discovery probe) MARGIN = 0.05 # separation margin (same as v569) TOL_SOUND = 1.0e-9 # floor soundness slack TOL_LINK = 1.0e-9 # link identity, relative TOL_CB = 1.0e-9 # Cauchy-Binet, relative BAR_SLOPE = 0.5 # dominance growth slope floor BAR_NEGMASS = 1.0 # witness-kill: negative pair mass / det S KERNEL_STRIDE = 6 # pair-kernel subset (every 6th window + reference) REF_H = 540 SCRAMBLE_SEED = 1 PAIR_KERNEL_BLOCK = 1024 # rows of the n x n pair kernel held at once (see below) def check(name, ok): global N_PASS, N_FAIL if ok: N_PASS += 1 else: N_FAIL += 1 print("[%s] %s" % ("PASS" if ok else "FAIL", name)) def pair_kernel_stats(lamw, a, b, c, block=PAIR_KERNEL_BLOCK): """The three pair-level readouts of section 4 for the polarisation kernel K = (a b^T + b a^T)/2 - c c^T and its mass-weighted form W = (lam lam^T) o K: the Cauchy-Binet quadratic form lam^T K lam, the negative pair mass -sum(W[W < 0]) and the best single pair max_{i np.arange(i0, i1)[:, None]] if upper.size: best_pair = max(best_pair, float(upper.max())) return quad, neg_mass, best_pair def run(): global N_PASS, N_FAIL N_PASS = 0 N_FAIL = 0 t0 = time.time() print("=" * 78) print("v570 -- PRIME.PENCIL.SEPFLOOR.01: the separation floor, " "certified on the declared surface") print("=" * 78) # --- 1. the reduction, symbolic --------------------------------------- l1, l2 = sp.symbols("lambda1 lambda2", real=True) ok_i = sp.simplify((l1**2 - l1 * l2) - l1 * (l1 - l2)) == 0 ok_ii = sp.simplify(((l1 + l2) - l2) - l1) == 0 b11, b12, b22, s11, s12, s22 = sp.symbols("b11 b12 b22 s11 s12 s22") Bs = sp.Matrix([[b11, b12], [b12, b22]]) Ss = sp.Matrix([[s11, s12], [s12, s22]]) Zs = Bs.inv() * Ss ok_t = sp.simplify(sp.trace(Zs) - (b11 * s22 + b22 * s11 - 2 * b12 * s12) / Bs.det()) == 0 ok_p = sp.simplify(Zs.det() - Ss.det() / Bs.det()) == 0 ok_link = sp.simplify((Bs - Ss).det() / Bs.det() - (1 - sp.trace(Zs) + Zs.det())) == 0 check("1. REDUCTION [E, symbolic]: lam1^2 - P = lam1 (lam1 - lam2) " "and lam1 = T - lam2 exactly (the two floor lemmas: P > 0, " "T > 0 give lam1 >= sqrt(P); P <= 0, T > 0 give lam1 >= T), " "and T = D(B,S)/det B, P = det S/det B, 1 - T + P = " "det(B-S)/det B are closed objects of (B, S)", ok_i and ok_ii and ok_t and ok_p and ok_link) # --- 2. the certified floor on the whole surface ---------------------- zones = core.frame_a_zones() data = [] for kz in zones: r = core.build_window(kz) B, S = r["B"], r["S"] detB = float(np.linalg.det(B)) P = float(np.linalg.det(S)) / detB T = float(np.trace(np.linalg.solve(B, S))) Bh = np.linalg.cholesky(-B) Zsym = np.linalg.solve(Bh, np.linalg.solve(Bh, (-S).T).T) lam = np.sort(np.linalg.eigvalsh(0.5 * (Zsym + Zsym.T)))[::-1] floor = np.sqrt(P) if P > 0 else T data.append(dict(kz=kz, h=r["h"], detB=detB, P=P, T=T, l1=float(lam[0]), floor=floor, r=r)) n_win = len(data) floors = np.array([d["floor"] for d in data]) l1s = np.array([d["l1"] for d in data]) Ps = np.array([d["P"] for d in data]) Ts = np.array([d["T"] for d in data]) n_pneg = int((Ps <= 0).sum()) sound = all(d["l1"] >= d["floor"] - TOL_SOUND for d in data) check("2a. SOUNDNESS [E per window]: lam1 >= floor on every one of " "the %d declared windows (floor = sqrt(P) on %d, = T on %d; " "T > 0 on %d/%d)" % (n_win, n_win - n_pneg, n_pneg, int((Ts > 0).sum()), n_win), sound and n_win == 70) check("2b. THE CERTIFICATION [E per window, the central result]: the " "closed scalar floor clears the margin on the WHOLE surface -- " "min floor = %.3f >= 1 + %.2f on ALL %d windows: the v569 " "separation follows from det S/det B and tr Z alone " "(conservative against the measured min lam1 = %.3f, as a " "floor should be)" % (floors.min(), MARGIN, n_win, l1s.min()), floors.min() >= 1 + MARGIN) # --- 3. the dominance structure --------------------------------------- hs = np.array([d["h"] for d in data], float) mask = Ps > 0 slope = float(np.polyfit(np.log(hs[mask]), np.log(Ps[mask]), 1)[0]) check("3a. DOMINANCE GROWS [MEASURED, declared surface]: P = " "det S/det B in [%.1f, %.1f], median %.1f, log-log slope " "%.3f > %.1f in h -- reported with the surface ladder, no " "rate claim beyond it" % (Ps[mask].min(), Ps.max(), float(np.median(Ps[mask])), slope, BAR_SLOPE), slope > BAR_SLOPE) def link_dev(d): detAB = (float(np.linalg.det(d["r"]["B"] - d["r"]["S"])) / d["detB"]) return abs(1 - d["T"] + d["P"] - detAB) / max(1.0, abs(detAB)) worst_link = max(link_dev(d) for d in data) check("3b. THE LINK [E per window]: 1 - T + P = det(B-S)/det B on " "every window (worst relative deviation %.1e) -- on deficit " "windows T - P = 1 + o(1): dominance certified in EITHER " "scalar certifies the other" % worst_link, worst_link <= TOL_LINK) # --- 4. the witness route, honestly killed ---------------------------- sel = [d for i, d in enumerate(data) if i % KERNEL_STRIDE == 0 or d["h"] == REF_H] worst_cb = 0.0 neg_fracs = [] best_pairs = [] for d in sel: r = d["r"] lamw, Xn = r["lam"], r["Xn"] a, b, c = Xn[:, 0], Xn[:, 1], Xn[:, 2] detS_cb, neg_mass, best_pair = pair_kernel_stats(lamw, a, b, c) worst_cb = max(worst_cb, abs(detS_cb / (d["P"] * d["detB"]) - 1.0)) neg_fracs.append(neg_mass / abs(detS_cb)) best_pairs.append(2 * best_pair / d["detB"]) neg_fracs = np.array(neg_fracs) best_pairs = np.array(best_pairs) check("4a. CAUCHY-BINET [E per window]: det S = lam^T K lam with K " "the polarisation kernel of the per-atom 2x2 reads -- worst " "relative deviation %.1e on the %d-window kernel subset" % (worst_cb, len(sel)), worst_cb <= TOL_CB) check("4b. THE CHEAP ROUTE IS DEAD [E, honest negative]: det S is " "itself a pair-level CANCELLATION object -- negative pair " "mass up to %.1f x det S (median %.1f x), best single-pair " "witness only %.2f--%.2f x det B against the >= %.2f needed: " "NO two-atom witness certifies the dominance; the h-uniform " "question is typed as pair-level size-vs-cancellation " "(an 8:9 cancellation, not a 4.7-order one -- but not free)" % (neg_fracs.max(), float(np.median(neg_fracs)), best_pairs.min(), best_pairs.max(), (1 + MARGIN)**2), best_pairs.max() < (1 + MARGIN)**2 and neg_fracs.max() > BAR_NEGMASS) # --- 5. must-break ----------------------------------------------------- d_ref = next(d for d in data if d["h"] == REF_H) r_scr = core.build_window(d_ref["kz"], scramble_seed=SCRAMBLE_SEED) T_s = float(np.trace(np.linalg.solve(r_scr["B"], r_scr["S"]))) check("5. MUST-BREAK [E]: the position scramble (same masses, the " "v563 intervention) FLIPS the trace sign on the reference " "window -- T = %.2f < 0 against the real +%.2f: the T > 0 " "gate detects the destroyed geometry loudly; the dominance " "is ARITHMETIC PLACEMENT, not a build artefact" % (T_s, d_ref["T"]), T_s < 0 and d_ref["T"] > 0) print("\nkey numbers: floor min %.3f (margin %.2f) on %d windows; " "P in [%.1f, %.1f] median %.1f, slope h^%.2f; neg pair mass " "max %.1f x; scramble T = %.1f" % (floors.min(), MARGIN, n_win, Ps[mask].min(), Ps.max(), float(np.median(Ps[mask])), slope, neg_fracs.max(), T_s)) print("NAMED LIMITS: declared surface only; the h-uniform dominance " "det S >= (1+c)^2 det B is the SAME open question as v569's, " "in sharper closed-scalar form; Problem 7.1 untouched; " "MEASURED stays MEASURED; no rate, no bound, NO RH statement") print("elapsed: %.1f s" % (time.time() - t0)) print("--- PRIME.PENCIL.SEPFLOOR.01 separation floor certified: " "%d passed, %d failed ---" % (N_PASS, N_FAIL)) return N_FAIL if __name__ == "__main__": raise SystemExit(1 if run() else 0)