"""v562 -- PRIME.DENSE.LIMIT.01: the dense-limit identities (T176). THE ENDGAME OF THE PHASE-2 MEASUREMENT PROGRAMME, ITS EXACT CORES AS ALGEBRA. T176 (contract DENSE.LIMIT, probe dense_limit_probe.py, 24/24, 118.7 s, verdict SITS-AT-ZERO) raised the von-Mangoldt sieve by a factor 20.8 (ATOM_MAX 1.2e6 -> 2.5e7, density ceiling 361 -> 6120) and extended the deficit(dens) curve by two preregistered ratio-4 bins; both new bins are CONSISTENT WITH ZERO (pooled +0.0496 +- 0.0372, 1.3 sigma), the plateau window narrows by a factor 3.6 (0.264 -> 0.074 at 2 sigma), and the phase-2 measurement programme CLOSES AS PLANNED. THIS module is the load-bearing version of the exact, theorem-shaped cores of that endgame -- per instance, machine-checkable, each with a mutation control, on a SMALL declared surface (the big sieve is NOT rebuilt here: the identities are sieve-size-free, which is their point): [E] (1) R1 CLOSED -- THE FEYNMAN-HELLMANN IDENTITY (T176 P2, the strongest). R = (lam_min/lam_max)/DEL with DEL = 1 - r_12^2, so dlog R/ddelta = v^T dA~ v / lam_min - w^T dA~ w / lam_max - dDEL/DEL, where dA~ is the odd-Toeplitz image of the EXACT lag derivative dc/ddelta (the assembly moves 0.5 mu_p per unit phase between two adjacent lag cells and NOTHING else, so dc/ddelta is exact and piecewise constant -- THEOREM T3 of T176) and v, w are the extreme eigenvectors of the normalised low block (Hellmann 1937, Feynman 1939: dlambda = v^T dA v for a simple eigenvalue of a symmetric matrix). Verified per instance against the full nonlinear rebuild on a DECLARED central-difference step ladder whose U-SHAPE is exhibited: the residual has an interior minimum (truncation error above, round-off amplification below), which a wrong formula cannot have -- a wrong formula has a floor. T175's intervention regression (dlog R/ddelta = 879, measured) is thereby retired as a separate claim: it is an identity about the assembly, not a fit. [E] (2) THE PHASE POLE (T176 P3; replaces the 300x harmonic anomaly). On the same instances the full phase period is swept on a declared grid: positive definiteness of the low block survives only a small fraction of the period, so lam_min changes sign INSIDE the period and log R has a POLE there; the bisected loss point delta_PD satisfies delta_PD x |dlog R/ddelta| = O(1) (a pole at the natural scale), and the two lowest modes of the normalised low block are ALREADY nearly degenerate at delta = 0 -- the intervention does not create a crossing, it walks along one. CONSEQUENCE, wired as the reading: a first-harmonic regression of log R over the period estimates a Fourier coefficient of a function with a pole in the interval, which is why T175's harmonic came out ~300x below the local derivative -- NOTHING is wrong with either number; they measure different things. [E] (3) SIEVE CONSISTENCY (T176 P4). A bigger sieve adds atoms ABOVE the old cap and nothing else, so every cell whose comb was already complete under the old cap is UNCHANGED BY CONSTRUCTION: the lag vectors agree bit for bit, R agrees exactly, and the bin deficit computed under the old and the new table is IDENTICAL on the shared anchors -- the audit that lets any later probe raise ATOM_MAX without re-deriving the chain (T176 re-measured all six T175 bins at 0.0 sigma; here the same statement is checked at construction level). Must-break: an anchor whose zone runs past the old cap has a truncated comb there, and its cell CHANGES under the bigger sieve -- the no-op is a theorem about complete combs, not about sieves. [E] (4) THE CURVE ESTIMATOR AS A PROCEDURE + THE LADDER CAVEAT AS CONTENT (T176 P1 + the anti-promotion). The cluster-robust bin statistic of T175/T176 -- one OLS slope of log|R| on log h per anchor from its cells inside a declared density bin, then the unweighted mean of the anchor slopes with the cluster-robust s.e. of that mean -- is DETERMINISTIC and exactly reproducible (two independent evaluation orders agree to round-off). AND THE ANTI-PROMOTION IS WIRED AS A CHECK: the SAME bin deficit computed on the fine 2^(1/4) rung ladder and on its coarse sqrt(2) sub-ladder DIFFERS (log R is curved in log h -- T176 measured a2 = -0.0842 +- 0.0085 -- so a ratio-4 bin with ~12 rungs samples the curvature differently from one with ~6), i.e. THE ESTIMATOR IS LADDER-DEPENDENT: any ledger row quoting a bin deficit must quote the rung ladder with it, or the number is not reproducible. T176 measured the shift at up to 0.20 on its surface; the module exhibits the mechanism per instance. NAMED LIMITS AS LOAD-BEARING CONTENT (the claim INCLUDES them): (i) THE VERDICT IS A MEASUREMENT, NOT A CLAIM OF THIS MODULE. T176's SITS-AT-ZERO -- the extended curve +0.8886 -> +0.5148 -> +0.1948 -> +0.3922 -> +0.1520 -> +0.0579 -> +0.0376 +- 0.0410 -> +0.1046 +- 0.0881 over 4.0 decades of comb density, pooled +0.0496 +- 0.0372 over the two new bins, density slope 8.0 sigma, plateau window 0.264 -> 0.074 (2 sigma) -- is a sandbox MEASURED number on T176's 2.5e7 surface WITH ITS RUNG LADDER (the 2^(1/4) ladder containing T175's seven rungs), and it is NOT consumed here. THE LADDER CAVEAT IS PART OF THE QUOTE. (ii) THE LIMIT STAYS OPEN AND TYPED OPEN. A true zero at finite density, a power-law approach to zero and a plateau below 0.074 remain INDISTINGUISHABLE under any finite sieve reachable by resources (the window narrows like the square root of the anchor cap: one more decade would cost ATOM_MAX ~ 2.5e9). No limit is claimed, the open quantifier at link 16 is untouched, and the phase-2 CLOSURE is a closure of the MEASUREMENT PROGRAMME (the planned instrument is exhausted), not of the question. (iii) The curvature map stays sandbox: both channels (log h +0.1529 +- 0.0258, log dens -0.0434 +- 0.0060) and the failure of all six preregistered reparametrisations (the data wants dens^{+0.284} h -- no candidate) are T176 MEASURED/FIT items on its surface, not consumed here. HONEST FENCES: NO RH statement anywhere -- RH_DELTA = 0.5 is a yardstick in exactly one role; the arithmetic input is a FINITE von Mangoldt table below a declared cap (Chebyshev 1852 / Rosser-Schoenfeld 1962, verified on the table), a bigger sieve is a bigger finite table and nothing else; no zero of any L-function is read and no L-function is evaluated (AST firewall); the Weil fence is hard (positivity of a finite A_h is never routed through the Weil criterion; Weil 1952 cited, never used); Feynman-Hellmann (Hellmann 1937, Feynman 1939), Schur 1917, Kac-Murdock-Szego 1953, Dirichlet 1829, Chebyshev 1852 named CLASSICAL. Status: [E] per-instance identities and certified inequalities with mutation controls; deterministic (fixed seeds, no wall-clock dependence); Python-only, counted per GATE.WOLFRAM.02. Discovery provenance: experiments/tfpt-discovery/dense_limit_probe.py (T176, 24/24, SITS-AT-ZERO; the phase-2 measurement programme closes as planned) """ from __future__ import annotations import ast import math import time import numpy as np from tfpt_constants import check, summary, reset # ---------------------------------------------------------------- budgets ATOM_MAX = 320000 # the NEW (small) table cap of this module ATOM_MAX_OLD = 120000 # the OLD cap of the consistency check (S3) GL_N = 48 CHUNK = 16384 EULER = 0.5772156649015328606065120900824 LOG_PI = math.log(math.pi) KB_MAX = 32 # parity modes carried SCHUR_KB = 16 # the low block of the T152..T176 chain EPSM = float(np.finfo(float).eps) # --- the declared surface (fixed BEFORE any number) --------------------------- ANCHOR_LO = 40 # every anchor n has a COMPLETE comb: n^2 <= cap H_FINE = (128, 152, 181, 215, 256, 304) # the 2^(1/4) ladder (T176's ratio) H_COARSE = (128, 181, 256) # the sqrt(2) sub-ladder (T175's) DENS_BIN = (6.4, 102.4) # ONE declared T175/T176 ratio-4x2 bin (S4) N_FH_CELLS = 6 # declared FH instances (mid-rung, spread anchors) N_POLE_CELLS = 4 # declared pole instances (densest anchors) FD_GRID = (1.0e-8, 1.0e-7, 1.0e-6, 1.0e-5, 1.0e-4) # DECLARED step ladder PHASE_FAM = (2, 3, 5, 7, 11, 13) # T175/T176's PRIMARY family N_PER = 64 # declared period grid (no harmonic hunting) N_BIS = 40 # declared bisection depth # --- the Chebyshev input, preregistered (T162..T176) -------------------------- KAPPA_X0 = 100.0 KAPPA_REF = 0.038821 TOL_KAPPA = 1.0e-6 B_PSI = 1.03883 # psi(x) <= B_PSI x (Chebyshev 1852; RS 1962) # --- the ledger exponents / T176 numbers (YARDSTICKS + quotes, not claims) ---- RH_DELTA = 0.5 # RH-equivalent strength, a YARDSTICK only T176_POOLED = 0.0496 # T176 pooled new-bin deficit -- sandbox MEASURED, T176_POOLED_SE = 0.0372 # quoted for documentation, NOT consumed T176_PLATEAU = 0.0744 # T176 2-sigma plateau window -- quoted only T175_DLOGR = 879.0 # T175's measured intervention slope -- quoted # --- preregistered tolerances / bars (declared BEFORE any number) -------------- TOL_EXACT = 1.0e-12 # small-matrix / assembly identities ROUND_BAR = 1.0e-9 # the probe's declared round-off horizon of the # full h x h forms (T176) -- used where a small # increment is read off an O(1) assembly TOL_FH = 1.0e-5 # FH vs rebuild at the U minimum (T176: 3.4e-6) U_SHAPE_FAC = 3.0 # both ladder ends must sit >= this over the min BAR_MUT_FH = 1.0e-2 # a mutated formula must miss by >= this (rel.) BAR_POLE_FRAC = 0.90 # PD must fail on > 10% of the period grid POLE_PROD_LO, POLE_PROD_HI = 0.02, 50.0 # declared O(1) band for the product BAR_NEAR = 0.35 # (lam2-lam1)/lam2 at delta = 0 must sit below BAR_LADDER = 1.0e-3 # fine vs coarse ladder must differ by >= this BAR_TRUNC = 1.0e-3 # a truncated-comb cell must MOVE under the sieve COND_BAR = 1.0e12 # the T159 numerical horizon on cond(B_LL) SEED = 562 def sym(A): return 0.5 * (A + A.T) def ast_zero_firewall(src_path: str) -> bool: with open(src_path, "r", encoding="utf-8") as fh: tree = ast.parse(fh.read()) hits = [] for node in ast.walk(tree): if isinstance(node, ast.Call): f = node.func if isinstance(f, ast.Attribute) and f.attr in ( "zetazero", "nzeros", "second_sheet_zero", ): hits.append(f.attr) if isinstance(f, ast.Name) and f.id in ("zetazero", "nzeros"): hits.append(f.id) if isinstance(node, ast.Attribute) and node.attr in ("zetazero",): hits.append(node.attr) return not hits # ---------------------------------------------------------------- atoms def von_mangoldt_table(n_max): sieve = np.ones(n_max + 1, dtype=bool) sieve[:2] = False for i in range(2, int(math.isqrt(n_max)) + 1): if sieve[i]: sieve[i * i::i] = False lam = np.zeros(n_max + 1) for p in np.nonzero(sieve)[0]: p = int(p) lp = math.log(p) q = p while q <= n_max: lam[q] = lp q *= p return lam LAM_TAB = von_mangoldt_table(ATOM_MAX) _NN = np.nonzero(LAM_TAB > 0.0)[0] U_ALL = np.log(_NN.astype(float)) MU_ALL = 2.0 * LAM_TAB[_NN] / np.sqrt(_NN.astype(float)) # the OLD table is a PREFIX of the new one: same atoms below the old cap _OLD_CUT = int(np.searchsorted(_NN, ATOM_MAX_OLD, side="right")) _GLX, _GLW = np.polynomial.legendre.leggauss(GL_N) def chebyshev_kappa(): nn = _NN.astype(float) psi = np.cumsum(LAM_TAB[_NN]) keep = nn >= KAPPA_X0 return float(np.max(np.abs(psi[keep] - nn[keep]) / nn[keep])) def atoms_in(alpha, cut=None): n = int(np.searchsorted(U_ALL, 2.0 * alpha + 1.0e-14, side="right")) return min(n, cut) if cut is not None else n # --------------------------------- the archimedean kernel (Weil 1952, cited) def _arch_A_far(s, D): s = np.asarray(s, dtype=float).reshape(-1, 1) out = np.zeros(s.shape[0]) for lo, hi in ((s - D, s), (s, s + D)): mid = 0.5 * (lo + hi) half = 0.5 * (hi - lo) w = mid + half * _GLX[None, :] val = ((1.0 - np.abs(s - w) / D) * np.exp(-0.5 * w) / (-np.expm1(-2.0 * w))) out -= half[:, 0] * (val @ _GLW) return out def _arch_integrand(w, s, D): tri_s = max(0.0, 1.0 - abs(s) / D) S = 0.5 * (np.maximum(0.0, 1.0 - np.abs(s - w) / D) + np.maximum(0.0, 1.0 - np.abs(s + w) / D)) return ((tri_s * np.exp(-2.0 * w) - S * np.exp(-0.5 * w)) / (-np.expm1(-2.0 * w))) def _arch_A_near(s, D): s = abs(float(s)) tri_s = max(0.0, 1.0 - s / D) W = s + D pts = sorted({0.0, s, D - s, W}) pts = [p for p in pts if 0.0 <= p <= W] tot = 0.0 for lo, hi in zip(pts[:-1], pts[1:]): if hi <= lo: continue mid = 0.5 * (lo + hi) half = 0.5 * (hi - lo) w = mid + half * _GLX tot += half * float(np.dot(_GLW, _arch_integrand(w, s, D))) return (-(EULER + LOG_PI) * tri_s + 2.0 * tot + tri_s * (-math.log1p(-math.exp(-2.0 * W)))) def arch_A(sv, D): sv = np.abs(np.asarray(sv, dtype=float)) out = np.empty(sv.shape[0]) far = sv >= D if far.any(): out[far] = _arch_A_far(sv[far], D) for i in np.nonzero(~far)[0]: out[i] = _arch_A_near(sv[i], D) return out def arch_lags(M, D): out = np.empty(M) for a in range(0, M, CHUNK): b = min(M, a + CHUNK) out[a:b] = arch_A(np.arange(a, b) * D, D) return out # --------------------------------- parity geometry (KMS 1953; Dirichlet 1829) def parity_mu(m): N = 2 * m + 1 kk = np.arange(1, m + 1, dtype=float) return 4.0 * np.sin(math.pi * kk / N) ** 2 def parity_basis(m, kb=None): N = 2 * m + 1 jj = np.arange(m) kk = np.arange(1, (kb or m) + 1) return (2.0 / math.sqrt(N)) * np.sin( 2.0 * math.pi * np.outer(kk, jj + 1.0) / N) def apply_LP(V, m): out = 2.0 * np.asarray(V, dtype=float).copy() out[:, 1:] -= V[:, :-1] out[:, :-1] -= V[:, 1:] out[:, -1] += V[:, -1] return out _IDX = {} def toeplitz_index(M): if M not in _IDX: h = M // 2 rr = np.arange(h) _IDX[M] = (np.abs(rr[:, None] - rr[None, :]), (M - 1) - rr[:, None] - rr[None, :]) return _IDX[M] def odd_toeplitz(c, M): it, ih = toeplitz_index(M) return np.asarray(c)[it] - np.asarray(c)[ih] _TB = {} def basis_of(hz): if hz not in _TB: _TB[hz] = (parity_basis(hz, KB_MAX), parity_mu(hz)[:KB_MAX]) return _TB[hz] # --------------------------------- THE T176 ASSEMBLY, verbatim (bincount) def atom_lags(alpha, M, u, mu): """T176's production assembly: a linear spline of total mass one around u_n = log n, split between cells i0 and i0 + 1 with weights 1 - f and f, f = frac(u/D) the PLACEMENT PHASE, plus the reflected arm below D.""" D = 2.0 * alpha / M q = u / D i0 = np.floor(q).astype(np.int64) f = q - i0 c = np.bincount(i0, weights=-mu * 0.5 * (1.0 - f), minlength=M)[:M].copy() idx = i0 + 1 ok = idx < M c += np.bincount(idx[ok], weights=-mu[ok] * 0.5 * f[ok], minlength=M)[:M] refl = u < D if refl.any(): c[0] -= 0.5 * float(np.sum(mu[refl] * (1.0 - u[refl] / D))) return c, D def gap_of(Ahat, prof, kb=SCHUR_KB): isq = 1.0 / np.sqrt(np.asarray(prof, dtype=float)[:kb]) ev = np.linalg.eigvalsh(sym(Ahat[:kb, :kb] * np.outer(isq, isq))) return float(ev[0]), float(ev[-1]) def del_of(Ahat): a11, a12, a22 = float(Ahat[0, 0]), float(Ahat[0, 1]), float(Ahat[1, 1]) det = a11 * a22 - a12 * a12 return det / (a11 * a22) def build_cell(n_zone, h, cut=None): """ONE window; the builder sees (n_zone, alpha, M) and nothing else (T174 P12). `cut` truncates the atom table at the OLD cap (S3 only).""" alpha = math.log(float(n_zone)) M = 2 * h ka = atoms_in(alpha, cut) c_at, D = atom_lags(alpha, M, U_ALL[:ka], MU_ALL[:ka]) c = arch_lags(M, D) + c_at Tb, mu = basis_of(h) Ah = sym(Tb @ (odd_toeplitz(c, M) @ Tb.T)) lmin, lmax = gap_of(Ah, mu) dl = del_of(Ah) return dict(n=n_zone, alpha=alpha, h=h, M=M, D=D, n_atom=ka, lmin=lmin, lmax=lmax, GAP=lmin / lmax, DEL=dl, R=(lmin / lmax) / max(abs(dl), 1.0e-300), pd=bool(lmin > 0.0), cond=abs(lmax / max(abs(lmin), 1.0e-300)), dens=float(ka) / float(M), complete=float(n_zone) ** 2 <= float(ATOM_MAX) + 0.5) # --------------------------------- the T176 phase machinery, verbatim def dc_dphase(cell): """THE EXACT LAG DERIVATIVE (T176 THEOREM T3): shifting the placement phase f -> f + delta of a family atom moves 0.5 mu_p delta from cell i0 + 1 to cell i0 and NOTHING else; dc/ddelta is exact and piecewise constant in delta between wraps.""" dc = np.zeros(cell["M"]) used = 0 for p in PHASE_FAM: q = math.log(float(p)) / cell["D"] i0 = int(math.floor(q)) if i0 + 1 >= cell["M"]: continue mp = 2.0 * math.log(float(p)) / math.sqrt(float(p)) dc[i0] += 0.5 * mp dc[i0 + 1] -= 0.5 * mp used += 1 return dc, used def comb_with_family_shift(cell, delta): """The lag vector with the six family atoms displaced by delta (exact rebuild; T175's intervention, T176's E3 control).""" cat, _ = atom_lags(cell["alpha"], cell["M"], U_ALL[:cell["n_atom"]], MU_ALL[:cell["n_atom"]]) if delta != 0.0: for p in PHASE_FAM: q = math.log(float(p)) / cell["D"] i0, f = int(math.floor(q)), math.fmod(q, 1.0) if i0 + 1 >= cell["M"]: continue f2 = math.fmod(f + delta, 1.0) mp = 2.0 * math.log(float(p)) / math.sqrt(float(p)) cat[i0] += 0.5 * mp * (f2 - f) cat[i0 + 1] -= 0.5 * mp * (f2 - f) return cat def logR_at(cell, delta): cat = comb_with_family_shift(cell, delta) Tb, mm = basis_of(cell["h"]) Ah = sym(Tb @ (odd_toeplitz(arch_lags(cell["M"], cell["D"]) + cat, cell["M"]) @ Tb.T)) lmin, lmax = gap_of(Ah, mm) if lmin <= 0.0: return None return math.log((lmin / lmax) / max(abs(del_of(Ah)), 1.0e-300)) def fh_closed(cell, swap_vectors=False, drop_del=False): """dlog R/ddelta IN CLOSED FORM (Feynman-Hellmann). The two mutation knobs build the WRONG formula on purpose (S1.MUT).""" cat = comb_with_family_shift(cell, 0.0) Tb, mm = basis_of(cell["h"]) Ah = sym(Tb @ (odd_toeplitz(arch_lags(cell["M"], cell["D"]) + cat, cell["M"]) @ Tb.T)) dc, used = dc_dphase(cell) dAh = sym(Tb @ (odd_toeplitz(dc, cell["M"]) @ Tb.T)) isq = 1.0 / np.sqrt(np.asarray(mm, dtype=float)[:SCHUR_KB]) S = np.outer(isq, isq) At = sym(Ah[:SCHUR_KB, :SCHUR_KB] * S) dAt = sym(dAh[:SCHUR_KB, :SCHUR_KB] * S) ev, V = np.linalg.eigh(At) v, w = V[:, 0], V[:, -1] if swap_vectors: v, w = w, v dlmin = float(v @ (dAt @ v)) dlmax = float(w @ (dAt @ w)) a11, a12, a22 = float(Ah[0, 0]), float(Ah[0, 1]), float(Ah[1, 1]) d11, d12, d22 = float(dAh[0, 0]), float(dAh[0, 1]), float(dAh[1, 1]) DEL = 1.0 - a12 * a12 / (a11 * a22) dDEL = (-2.0 * a12 * d12 / (a11 * a22) + a12 * a12 / (a11 * a22) * (d11 / a11 + d22 / a22)) t_del = 0.0 if drop_del else (-dDEL / DEL) return dict(dlogR=dlmin / ev[0] - dlmax / ev[-1] + t_del, t_min=dlmin / ev[0], t_max=-dlmax / ev[-1], t_del=-dDEL / DEL, lmin=float(ev[0]), lam2=float(ev[1]), used=used) def sweeper(cell): """delta -> spectrum of the normalised low block, arithmetic that does not depend on delta computed ONCE (T176 E3.ii, verbatim logic).""" cat = comb_with_family_shift(cell, 0.0) base = arch_lags(cell["M"], cell["D"]) + cat Tb, mm = basis_of(cell["h"]) isq = 1.0 / np.sqrt(np.asarray(mm, dtype=float)[:SCHUR_KB]) S = np.outer(isq, isq) fam = [] for p in PHASE_FAM: q = math.log(float(p)) / cell["D"] i0, f = int(math.floor(q)), math.fmod(q, 1.0) if i0 + 1 < cell["M"]: fam.append((i0, f, 2.0 * math.log(float(p)) / math.sqrt(float(p)))) def at(delta): c = base.copy() for i0, f, mp in fam: df = math.fmod(f + delta, 1.0) - f c[i0] += 0.5 * mp * df c[i0 + 1] -= 0.5 * mp * df Ah = sym(Tb @ (odd_toeplitz(c, cell["M"]) @ Tb.T)) At = sym(Ah[:SCHUR_KB, :SCHUR_KB] * S) return np.linalg.eigvalsh(At) return at # --------------------------------- the T175/T176 curve estimator, verbatim def ols_slope(lx, ly): lx = np.asarray(lx, dtype=float) ly = np.asarray(ly, dtype=float) xc = lx - lx.mean() return float(np.sum(xc * ly) / np.sum(xc * xc)) def bin_deficit(cells, lo, hi, min_rung=3): """T175/T176's estimator, VERBATIM: one OLS slope of log|R| on log h per anchor from its cells INSIDE the bin, then the unweighted mean of the anchor slopes with the CLUSTER-ROBUST s.e. of that mean.""" byn = {} for r in cells: if lo <= r["dens"] < hi: byn.setdefault(r["n"], []).append(r) es = [] for n in sorted(byn): row = sorted(byn[n], key=lambda r: r["h"]) if len(row) < min_rung: continue es.append(-ols_slope([math.log(float(r["h"])) for r in row], [math.log(abs(r["R"])) for r in row])) if len(es) < 2: return None e = np.array(es) return (float(np.mean(e)), float(np.std(e, ddof=1) / math.sqrt(len(e))), len(e)) def run(): reset() t0 = time.time() rng = np.random.default_rng(SEED) print("=" * 72) print("v562 PRIME.DENSE.LIMIT.01 -- the dense-limit identities (T176)") print("=" * 72) # ================================================================ S0 print("\nS0 -- firewall, the one arithmetic input, the declared surface") check("S0.AST: no Riemann-zero / zetazero loader in this module", ast_zero_firewall(__file__)) kappa = chebyshev_kappa() zone_max = int(math.isqrt(ATOM_MAX)) anchors = [int(n) for n in _NN if ANCHOR_LO <= n <= zone_max and n * n <= ATOM_MAX] # the declared sub-selection: every 6th prime-power anchor (cost only) anchors = anchors[::6] CELLS = [build_cell(n, h) for n in anchors for h in H_FINE] n_pd = sum(r["pd"] for r in CELLS) check(f"S0.SURFACE: kappa = {kappa:.6f} reproduced to " f"{abs(kappa - KAPPA_REF):.1e} <= {TOL_KAPPA:.0e} at every jump " f"point of psi(x)/x on the table in [{KAPPA_X0:.0f}, {ATOM_MAX}] " f"(Chebyshev 1852 / Rosser-Schoenfeld 1962 -- the ONE " f"unconditional arithmetic input; psi(x)/x <= {B_PSI} holds), and " f"the declared surface is {len(anchors)} prime-power anchors in " f"[{ANCHOR_LO}, {zone_max}] (every 6th, declared; comb COMPLETE " f"on every cell, X = n^2 <= {ATOM_MAX}) x the declared 2^(1/4) " f"ladder h = {H_FINE} = {len(CELLS)} cells, {n_pd}/{len(CELLS)} " f"positive definite with cond(B_LL) <= " f"{max(r['cond'] for r in CELLS):.1e} <= {COND_BAR:.0e} (the " f"declared T159 horizon), dens spanning " f"{min(r['dens'] for r in CELLS):.2f}.." f"{max(r['dens'] for r in CELLS):.1f} atoms per lag cell", abs(kappa - KAPPA_REF) <= TOL_KAPPA and all(r["pd"] and r["complete"] for r in CELLS) and max(r["cond"] for r in CELLS) <= COND_BAR) # ================================================================ S1 print("\nS1 -- R1 CLOSED: the Feynman-Hellmann identity (T176 P2)") byn = {} for r in CELLS: byn.setdefault(r["n"], []).append(r) rows = [sorted(v, key=lambda r: r["h"]) for _, v in sorted(byn.items())] mids = [row[len(row) // 2] for row in rows if len(row) >= 4] fh_cells = mids[:: max(1, len(mids) // N_FH_CELLS)][:N_FH_CELLS] # (a) the exact lag derivative: c(delta) = c(0) + delta dc between wraps c0 = comb_with_family_shift(fh_cells[0], 0.0) dtest = 1.0e-5 c1 = comb_with_family_shift(fh_cells[0], dtest) dc0, used0 = dc_dphase(fh_cells[0]) e_lin = (float(np.max(np.abs(c1 - c0 - dtest * dc0))) / max(float(np.max(np.abs(dtest * dc0))), 1.0e-300)) check(f"S1.DCEXACT [THEOREM] T176 T3, the exactness that makes first " f"order an identity: the assembly is PIECEWISE LINEAR in the " f"placement phase -- c(delta) = c(0) + delta dc/ddelta EXACTLY " f"between wraps (checked at delta = {dtest:.0e} on the declared " f"cell n = {fh_cells[0]['n']}, h = {fh_cells[0]['h']}: residual " f"{e_lin:.1e} <= {ROUND_BAR:.0e} relative to the increment, " f"{used0} family atoms in window; the increment is {dtest:.0e} " f"of an O(1) lag vector, so the honest bar is the probe's " f"declared round-off horizon of the full assembly, not " f"double-precision exactness) -- the assembly moves 0.5 mu_p " f"per unit phase between two adjacent lag cells and NOTHING " f"else", e_lin <= ROUND_BAR and used0 == len(PHASE_FAM)) # (b) the identity, against the rebuild, with the U-shape exhibited med_by_step = {d: [] for d in FD_GRID} n_ok = 0 for cell in fh_cells: f = fh_closed(cell) if abs(f["dlogR"]) < 1.0e-30: continue got = {} for d in FD_GRID: p, m2 = logR_at(cell, +d), logR_at(cell, -d) if p is None or m2 is None: continue got[d] = (p - m2) / (2.0 * d) if len(got) < len(FD_GRID): continue n_ok += 1 for d, qd in got.items(): med_by_step[d].append(abs(qd - f["dlogR"]) / abs(f["dlogR"])) med = [(d, float(np.median(med_by_step[d]))) for d in FD_GRID] best = min(med, key=lambda t: t[1]) print(" FD ladder (median |quotient - closed| / |closed|): " + ", ".join(f"{d:.0e} -> {m:.2e}" for d, m in med)) check(f"S1.FH [THEOREM, per instance] R1 CLOSED: dlog R/ddelta = " f"v^T dA~ v/lam_min - w^T dA~ w/lam_max - dDEL/DEL " f"(Feynman-Hellmann; Hellmann 1937, Feynman 1939) reproduces the " f"full nonlinear rebuild on {n_ok}/{len(fh_cells)} declared " f"instances at the minimum of the declared step ladder: best " f"median residual {best[1]:.2e} <= {TOL_FH:.0e} at delta = " f"{best[0]:.0e}, WITH THE U-SHAPE EXHIBITED (round-off limited " f"at {FD_GRID[0]:.0e}: {med[0][1]:.2e}; truncation limited at " f"{FD_GRID[-1]:.0e}: {med[-1][1]:.2e}; both >= {U_SHAPE_FAC:.0f}x " f"the minimum) -- the residual is the DIFFERENCE QUOTIENT, not " f"the identity; a wrong formula has a floor, not a minimum. " f"T175's measured intervention slope ({T175_DLOGR:.0f}) is " f"retired as a separate claim: it is an identity about the " f"assembly (T176 C1)", n_ok == len(fh_cells) and best[1] <= TOL_FH and med[0][1] >= U_SHAPE_FAC * best[1] and med[-1][1] >= U_SHAPE_FAC * best[1]) # (c) the decomposition, reported (T176: dDEL/DEL carries the bulk) shares = [] for cell in fh_cells: f = fh_closed(cell) tot = abs(f["t_min"]) + abs(f["t_max"]) + abs(f["t_del"]) shares.append(abs(f["t_del"]) / max(tot, 1.0e-300)) check(f"S1.DECOMP [E] the decomposition that corrected the expected " f"story: of the three closed terms the near-degeneracy channel " f"dDEL/DEL carries the LARGEST share on every declared instance " f"(shares {min(shares):.2f}..{max(shares):.2f}, all > 1/3; T176 " f"measured 84% median on its surface) -- the phase response of R " f"lives in the 2x2 near-degeneracy channel, which is exactly the " f"channel T174 proved does NOT cancel", min(shares) > 1.0 / 3.0) # (d) mutations: the wrong formula has a floor, not a U worst_mut = 0.0 cell = fh_cells[0] f_true = fh_closed(cell) d_ref = best[0] p, m2 = logR_at(cell, +d_ref), logR_at(cell, -d_ref) quot = (p - m2) / (2.0 * d_ref) for kw in (dict(swap_vectors=True), dict(drop_del=True)): f_bad = fh_closed(cell, **kw) worst_mut = max(worst_mut, 0.0 if abs(quot) < 1e-300 else abs(f_bad["dlogR"] - quot) / abs(quot)) e_true = abs(f_true["dlogR"] - quot) / abs(quot) check(f"S1.MUT [must-break] two mutated formulas miss the measured " f"quotient at the optimal step by >= {BAR_MUT_FH:.0e} relative " f"(swapped eigenvectors and dropped dDEL/DEL term: worst miss " f"{worst_mut:.2e}) while the true closed form sits at " f"{e_true:.2e}: the identity is a statement about THE formula, " f"not about any first-order expression", worst_mut >= BAR_MUT_FH and e_true <= TOL_FH * 10.0) # ================================================================ S2 print("\nS2 -- the phase pole (T176 P3): the 300x anomaly explained") dense_mids = sorted(mids, key=lambda r: -r["dens"])[:N_POLE_CELLS] frac_pd, prods, nears, dpds = [], [], [], [] for cell in dense_mids: at = sweeper(cell) f = fh_closed(cell) dd = np.arange(N_PER) / float(N_PER) lm = np.array([at(float(d))[0] for d in dd]) frac_pd.append(float(np.mean(lm > 0.0))) nears.append((f["lam2"] - f["lmin"]) / f["lam2"]) lo, hi = 0.0, None for s in (1.0e-6, 1.0e-5, 1.0e-4, 1.0e-3, 1.0e-2, 0.1, 0.5): if at(s)[0] <= 0.0: hi = s break lo = s if hi is None: continue for _ in range(N_BIS): mid = 0.5 * (lo + hi) if at(mid)[0] > 0.0: lo = mid else: hi = mid dpd = 0.5 * (lo + hi) dpds.append(dpd) prods.append(dpd * abs(f["dlogR"])) med_frac = float(np.median(frac_pd)) med_prod = float(np.median(prods)) check(f"S2.POLE [E, per instance] THE POLE: on the {len(dense_mids)} " f"densest declared instances the full phase period is swept on " f"the declared {N_PER}-point grid (no harmonic hunting) and " f"positive definiteness survives only a fraction " f"{min(frac_pd):.2f}..{max(frac_pd):.2f} of the period (median " f"{med_frac:.2f} < {BAR_POLE_FRAC}) -- lam_min CHANGES SIGN " f"inside the period, so log R has a POLE there and is NOT " f"defined on the full period; bisected to {N_BIS} levels the " f"loss point sits at delta_PD = {min(dpds):.1e}..{max(dpds):.1e} " f"with delta_PD x |dlog R/ddelta| = {med_prod:.2f} median, " f"inside the declared O(1) band [{POLE_PROD_LO}, {POLE_PROD_HI}] " f"-- a pole at the natural scale. THE READING (T176): a " f"first-harmonic regression over the period estimates a Fourier " f"coefficient of a function with a pole in the interval, which " f"is why T175's harmonic came out ~300x below the local " f"derivative -- neither number is wrong; they measure different " f"things. The 300x anomaly is RETIRED, replaced by this pole", len(dpds) == len(dense_mids) and med_frac < BAR_POLE_FRAC and POLE_PROD_LO <= med_prod <= POLE_PROD_HI) check(f"S2.NEAR [E, per instance] THE CROSSING IS ALREADY THERE: the " f"two lowest modes of the normalised low block sit within " f"{min(nears):.3f}..{max(nears):.3f} of each other at delta = 0 " f"(relative spacing (lam_2 - lam_1)/lam_2 < {BAR_NEAR} on every " f"instance; T176 measured 13.4% median) -- the intervention does " f"not CREATE a near-degeneracy, it walks along one, which closes " f"the loop with S1.DECOMP: the response lives in dDEL/DEL, and " f"DEL = 1 - r_12^2 IS the near-degeneracy of the bottom two " f"directions", max(nears) < BAR_NEAR) # control: the bisection brackets a true sign change at0 = sweeper(dense_mids[0]) d0 = dpds[0] e_base = (abs(at0(0.0)[0] - dense_mids[0]["lmin"]) / abs(dense_mids[0]["lmin"])) check(f"S2.CTRL: the bisected loss point is a genuine sign change -- " f"lam_min(0.98 delta_PD) = {at0(0.98 * d0)[0]:+.2e} > 0 and " f"lam_min(1.02 delta_PD) = {at0(1.02 * d0)[0]:+.2e} < 0 on the " f"first declared instance, and the sweeper's delta = 0 spectrum " f"reproduces the cell's lam_min to {e_base:.1e} <= " f"{ROUND_BAR:.0e} relative (same assembly, same block)", at0(0.98 * d0)[0] > 0.0 and at0(1.02 * d0)[0] < 0.0 and e_base <= ROUND_BAR) # ================================================================ S3 print("\nS3 -- sieve consistency (T176 P4): the audit at construction " "level") old_ok = [n for n in anchors if n * n <= ATOM_MAX_OLD] worst_c, worst_R = 0.0, 0.0 for n in old_ok: for h in H_COARSE: alpha = math.log(float(n)) ka_new = atoms_in(alpha) ka_old = atoms_in(alpha, _OLD_CUT) c_new, _ = atom_lags(alpha, 2 * h, U_ALL[:ka_new], MU_ALL[:ka_new]) c_old, _ = atom_lags(alpha, 2 * h, U_ALL[:ka_old], MU_ALL[:ka_old]) worst_c = max(worst_c, float(np.max(np.abs(c_new - c_old)))) if ka_new != ka_old: worst_c = np.inf cells_old = [build_cell(n, h, cut=_OLD_CUT) for n in old_ok for h in H_FINE] cells_new = [build_cell(n, h) for n in old_ok for h in H_FINE] for a, b in zip(cells_old, cells_new): worst_R = max(worst_R, abs(a["R"] / b["R"] - 1.0)) lo_b = min(r["dens"] for r in cells_new) hi_b = max(r["dens"] for r in cells_new) + 1.0e-9 b_old = bin_deficit(cells_old, lo_b, hi_b) b_new = bin_deficit(cells_new, lo_b, hi_b) check(f"S3.UNCHANGED [THEOREM] SIEVE CONSISTENCY AT CONSTRUCTION " f"LEVEL: a bigger sieve adds atoms ABOVE the old cap and nothing " f"else, so on the {len(old_ok)} anchors whose comb is complete " f"under the OLD cap {ATOM_MAX_OLD} the lag vectors under old and " f"new table agree BIT FOR BIT (worst |dc| = {worst_c:.1e} == 0), " f"R agrees to {worst_R:.1e} on {len(cells_new)} cells, and the " f"cluster-robust bin deficit is IDENTICAL: {b_old[0]:+.6f} +- " f"{b_old[1]:.6f} (old, {b_old[2]} anchors) vs {b_new[0]:+.6f} +- " f"{b_new[1]:.6f} (new) -- T176's cell-by-cell audit (all six " f"T175 bins at 0.0 sigma under the 20.8x sieve) is this " f"construction fact plus statistics, and it is what lets any " f"later probe raise ATOM_MAX without re-deriving the chain", worst_c == 0.0 and worst_R == 0.0 and b_old[0] == b_new[0] and b_old[1] == b_new[1]) n_tr = max(n for n in anchors if n * n > ATOM_MAX_OLD) r_tr_old = build_cell(int(n_tr), H_COARSE[0], cut=_OLD_CUT) r_tr_new = build_cell(int(n_tr), H_COARSE[0]) move_tr = abs(r_tr_old["R"] / r_tr_new["R"] - 1.0) check(f"S3.MUT [must-break] the no-op is a theorem about COMPLETE " f"combs, not about sieves: the deepest declared anchor n = " f"{int(n_tr)} (X = {int(n_tr) ** 2} > {ATOM_MAX_OLD} = old cap) " f"has a TRUNCATED comb under the old table ({r_tr_old['n_atom']} " f"atoms vs {r_tr_new['n_atom']}) and its R moves by " f"{move_tr:.2e} >= {BAR_TRUNC:.0e} under the bigger sieve -- " f"exactly where the construction says it must", move_tr >= BAR_TRUNC) # ================================================================ S4 print("\nS4 -- the curve estimator as a procedure + the ladder caveat " "(T176 P1 + the anti-promotion)") b_fine = bin_deficit(CELLS, DENS_BIN[0], DENS_BIN[1]) order = rng.permutation(len(CELLS)) b_perm = bin_deficit([CELLS[i] for i in order], DENS_BIN[0], DENS_BIN[1]) coarse_cells = [r for r in CELLS if r["h"] in H_COARSE] b_coarse = bin_deficit(coarse_cells, DENS_BIN[0], DENS_BIN[1]) d_ladder = abs(b_fine[0] - b_coarse[0]) check(f"S4.ESTIMATOR [E] THE PROCEDURE IS DETERMINISTIC AND " f"REPRODUCIBLE: the cluster-robust bin statistic of T175/T176 " f"(one OLS slope of log|R| on log h per anchor inside the " f"declared bin dens in [{DENS_BIN[0]}, {DENS_BIN[1]}), then the " f"unweighted anchor mean with the cluster-robust s.e.) gives " f"{b_fine[0]:+.6f} +- {b_fine[1]:.6f} on {b_fine[2]} anchor " f"clusters, and recomputing under a permuted cell order " f"reproduces it EXACTLY ({abs(b_fine[0] - b_perm[0]):.1e} == 0): " f"no Monte Carlo, no tuning, no wall clock -- the T176 curve is " f"this procedure applied to its declared bins", b_fine is not None and b_fine[0] == b_perm[0] and b_fine[1] == b_perm[1]) check(f"S4.LADDER [E, THE ANTI-PROMOTION WIRED AS A CHECK] the " f"estimator is LADDER-DEPENDENT, exhibited per instance: the " f"SAME declared bin on the fine 2^(1/4) ladder gives " f"{b_fine[0]:+.4f} +- {b_fine[1]:.4f} and on the coarse sqrt(2) " f"sub-ladder {b_coarse[0]:+.4f} +- {b_coarse[1]:.4f} -- a shift " f"of {d_ladder:.4f} >= {BAR_LADDER:.0e} (log R is CURVED in " f"log h: T176 measured a2 = -0.0842 +- 0.0085, so ladders with " f"different rung counts sample the curvature differently; T176 " f"saw shifts up to 0.20). CONSEQUENCE, load-bearing: any ledger " f"row quoting a bin deficit MUST quote the rung ladder with it, " f"or the number is not reproducible -- the PRIME.DENSE.LIMIT.01 " f"row does exactly that", d_ladder >= BAR_LADDER) # the scramble control at the densest cells (T176 E3.iii, 194/194) n_kill, n_draw = 0, 0 for cell in dense_mids[:2]: for _ in range(3): ka = cell["n_atom"] D, M = cell["D"], cell["M"] q = U_ALL[:ka] / D i0 = np.floor(q).astype(np.int64) f = rng.random(ka) mu = MU_ALL[:ka] c = np.bincount(i0, weights=-mu * 0.5 * (1.0 - f), minlength=M)[:M].copy() ok = i0 + 1 < M c += np.bincount(i0[ok] + 1, weights=-mu[ok] * 0.5 * f[ok], minlength=M)[:M] refl = U_ALL[:ka] < D if refl.any(): c[0] -= 0.5 * float(np.sum(mu[refl] * (1.0 - U_ALL[:ka][refl] / D))) Tb, mm = basis_of(cell["h"]) Ah = sym(Tb @ (odd_toeplitz(arch_lags(M, D) + c, M) @ Tb.T)) lmin, _ = gap_of(Ah, mm) n_draw += 1 if lmin <= 0.0: n_kill += 1 check(f"S4.SCRAMBLE [must-break] the phase scramble at the densest " f"declared cells (every atom keeps its lag CELL, occupancy " f"preserved exactly, only the sub-cell placement is drawn " f"uniform): positive definiteness is destroyed on {n_kill}/" f"{n_draw} draws (T176: 194/194 at its density) -- the " f"dense-limit deficit is a property of WHERE the atoms sit, not " f"of how many there are; a one-sided destruction control, never " f"a calibrated null", n_kill == n_draw) # KMS exactness control at a ladder rung (T176 stress 2) m2 = H_FINE[3] Tb2, mu2 = basis_of(m2) e_ort = float(np.max(np.abs(Tb2 @ Tb2.T - np.eye(Tb2.shape[0])))) e_lp = float(np.max(np.abs(apply_LP(Tb2, m2) - mu2[:, None] * Tb2))) check(f"S4.CTRL: the KMS sine modes at the ladder rung m = {m2} are " f"orthonormal to {e_ort:.1e} and satisfy L_P t_k = mu^P_k t_k to " f"{e_lp:.1e} <= {TOL_EXACT:.0e} with the Dirichlet end x(-1) = 0 " f"and the parity end x(m) = -x(m-1) (Kac-Murdock-Szego 1953; " f"T176 stress 2 measured 1.9e-15) -- the classical spine the " f"identities stand on", e_ort <= TOL_EXACT and e_lp <= TOL_EXACT) # ================================================================ S5 print("\nS5 -- the fences, restated as a check") check("S5.FENCE: per-instance identities and certified inequalities on " "a SMALL declared surface only -- the big sieve is NOT rebuilt " "here because the identities are sieve-size-free, which is their " "point; THE VERDICT IS A MEASUREMENT AND NOT A CLAIM OF THIS " f"MODULE -- T176's SITS-AT-ZERO (pooled {T176_POOLED:+.4f} +- " f"{T176_POOLED_SE:.4f} over the two new bins, plateau window " f"{T176_PLATEAU:.4f} at 2 sigma, factor 3.6 narrower than T175) " "stays a sandbox MEASURED number on T176's 2.5e7 surface WITH " "ITS RUNG LADDER -- the ladder caveat is part of the quote " "(S4.LADDER); THE LIMIT STAYS OPEN AND TYPED OPEN: true zero at " "finite density, power-law approach and low plateau remain " "indistinguishable under any resource-reachable sieve (the " "window narrows like the square root of the anchor cap -- one " "more decade costs ATOM_MAX ~ 2.5e9), no limit is claimed, and " "the open quantifier at link 16 is untouched; THE PHASE-2 " "CLOSURE IS A CLOSURE OF THE MEASUREMENT PROGRAMME (the planned " "instrument is exhausted), NOT OF THE QUESTION -- the work " "continues in the classification papers and the backflow lines; " "NO RH statement is made, assumed, approached or weakened, no " "zero of any L-function is read, no L-function is evaluated " "(AST firewall), RH_DELTA = 0.5 is a YARDSTICK in exactly one " "role; THE WEIL FENCE: positivity of a finite A_h is never " "routed through the Weil criterion, the audited chain is the " "R4-free R1 form; no marker of any pre-existing contract moves; " "Feynman-Hellmann (Hellmann 1937, Feynman 1939), Schur 1917, " "KMS 1953, Dirichlet 1829, Chebyshev 1852 / Rosser-Schoenfeld " "1962 named CLASSICAL; Weil 1952 CITED, never used as a " "criterion; zero-firewall AST-checked", True) elapsed = time.time() - t0 print(f"\nv562 runtime: {elapsed:.1f}s") print(f" FH identity: best median residual {best[1]:.2e} at delta = " f"{best[0]:.0e} (U-shape exhibited); dDEL/DEL shares " f"{min(shares):.2f}..{max(shares):.2f}") print(f" pole: PD survives {med_frac:.2f} of the period median, " f"delta_PD x slope = {med_prod:.2f}; near-degeneracy at delta=0: " f"{min(nears):.3f}..{max(nears):.3f}") print(f" sieve consistency: bit-for-bit on {len(old_ok)} shared " f"anchors; ladder shift {d_ladder:.4f} (the anti-promotion); " f"scramble kills {n_kill}/{n_draw}") return summary("PRIME.DENSE.LIMIT.01 dense-limit identities") if __name__ == "__main__": raise SystemExit(run())