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TNFR AI Agent
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feat(riemann): P43 - chi-twisted Paley-gap consistency diagnostic
L-track analogue of zeta-track P25 (paley_gap_coercivity.py). Transports the Paley-gap philosophy of Martinez Gamo (Zenodo 10.5281/zenodo.17665853 v2, Nov 2025) onto the chi-twisted side for every primitive real Dirichlet character chi_3, chi_4, chi_5. Compares three representations of -L'(s,chi)/L(s,chi): P32 closed-form weighted spectrum (tnfr_log_l_derivative), P34 chi-twisted weighted spectral trace (twisted_weighted_spectral_trace), and classical truncated Dirichlet sum (classical_log_l_derivative). Forms three absolute Paley-gap quantities g_P32, g_P34, g_cross = |Z_P34 - Z_P32|. Empirical results on (n_primes, k_max, N_max_cls) = (18, 5, 50000), sigma in [1.5, 4.0] with N=11: at J_0 = 0 the decoupled cross gap collapses to machine precision (chi_3 5.55e-17, chi_4 4.16e-17, chi_5 1.11e-16) -- Paley-style algebraic identity between P32 and P34 on the L-track, regression test. At J_0 = 1e-2 the coupling-induced cross gap jumps to O(1e-5) (chi_3 1.01e-5, chi_4 8.25e-6, chi_5 1.51e-5), twelve orders of magnitude above noise; clean structural-deformation signal free of classical-truncation noise which contaminates g_P32 at 1e-3 to 1e-2 level. Roadmap consolidation: original P43/P44 entries (stratified, adaptive coercivity) were absorbed into P42 (twisted_coercivity_uniform.py) since the zeta-track precedent puts P22+P23+P24 in one module. P43 is now reassigned to the P25 Paley-gap analogue. Honest scope: P43 is a consistency diagnostic at coupling zero and a deformation magnitude at coupling positive. Vanishing of g_cross at J_0 = 0 is necessary but not sufficient for any structural positivity claim and not connected to GRH localisation; it is a regression test selecting consistent realisations, not a coercivity certificate. Does NOT prove GRH for any L(s, chi), does NOT extend to complex characters, does NOT advance the P42 interval-coercivity certificate, does NOT advance G4 = RH. Files: src/tnfr/riemann/twisted_paley_gap_coercivity.py (new), src/tnfr/riemann/__init__.py (5 exports wired), examples/70_twisted_paley_gap_coercivity_demo.py (new), theory/TNFR_RIEMANN_RESEARCH_NOTES.md (sec 13vicies-secundo + 19.1 P43 row).
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‎examples/70_twisted_paley_gap_coercivity_demo.py‎

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"""P43 chi-twisted Paley-gap coercivity demo.
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L-track analogue of the zeta-track P25 demo
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(examples/52_paley_gap_coercivity_demo.py). Transports the
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Paley-gap philosophy of Mart\u00ednez Gamo, *Spectral note: Paley
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gap via lambda_2 (residue circulants)*, Zenodo
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10.5281/zenodo.17665853 v2 (November 2025), to the chi-twisted
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TNFR-Riemann program for primitive real Dirichlet characters
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chi_3, chi_4, chi_5.
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Computes three chi-twisted Paley-gap quantities over a real
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sigma-grid:
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g_P32(sigma) = |Z_P32(sigma, chi) - Z_classical(sigma, chi)|
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g_P34(sigma) = |Z_P34(sigma, chi) - Z_classical(sigma, chi)|
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g_cross(sigma) = |Z_P34(sigma, chi) - Z_P32(sigma, chi)|
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Two bundles per character are exercised:
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[A] Decoupled (coupling = 0) -- Paley-style identity expected:
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g_cross collapses to machine precision for every sigma and
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every chi. This is a regression test, not a discovery.
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[B] Weakly coupled (coupling = 1e-2) -- coupling-induced cross
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gap appears. This is the diagnostic signal: it quantifies
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how strongly inter-ladder coupling deforms the chi-twisted
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prime-ladder identity, free of classical-truncation noise.
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HONEST SCOPE: this is a consistency diagnostic, not a proof of
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uniform coercivity and not a proof of the Generalised Riemann
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Hypothesis for L(s, chi). The Zenodo source note itself disclaims
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primality proof status; P43 inherits the same scope at the L-track
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coercivity level. Gap G4-chi (GRH localisation on Re(s) = 1/2 for
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L(s, chi)) is NOT addressed by P43.
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"""
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from __future__ import annotations
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import sys
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import numpy as np
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from tnfr.riemann.dirichlet_l import (
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real_character_mod_3,
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real_character_mod_4,
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real_character_mod_5,
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)
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from tnfr.riemann.twisted_paley_gap_coercivity import sweep_twisted_paley_gap
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from tnfr.riemann.twisted_prime_ladder_hamiltonian import (
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build_twisted_prime_ladder_hamiltonian,
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)
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def _ensure_utf8_stdout() -> None:
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try:
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sys.stdout.reconfigure(encoding="utf-8") # type: ignore[attr-defined]
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except Exception:
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pass
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def _print_sweep_table(sweep) -> None:
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print(
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f" {'sigma':>8s} {'g_P32':>14s} "
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f"{'g_P34':>14s} {'g_cross':>14s}"
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)
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for s, g32, g34, gc in zip(
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sweep.sigmas, sweep.g_p32, sweep.g_p34, sweep.g_cross,
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):
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print(f" {s:>8.3f} {g32:>14.6e} {g34:>14.6e} {gc:>14.6e}")
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def main() -> int:
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_ensure_utf8_stdout()
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n_primes = 18
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max_power = 5
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sigmas = np.linspace(1.5, 4.0, 11)
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n_max_classical = 50_000
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coupling_b = 1.0e-2
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characters = [
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("chi_3", real_character_mod_3()),
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("chi_4", real_character_mod_4()),
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("chi_5", real_character_mod_5()),
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]
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print("=" * 78)
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print("P43 CHI-TWISTED PALEY-GAP COERCIVITY DEMO")
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print("(L-track analogue of P25; after Mart\u00ednez Gamo, "
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"Zenodo 17665853 v2, 2025)")
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print("=" * 78)
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print(f"n_primes = {n_primes}")
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print(f"max_power = {max_power}")
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print(
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f"sigma grid = [{sigmas[0]:.3f}, {sigmas[-1]:.3f}],"
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f" {sigmas.size} points"
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)
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print(f"n_max_classical = {n_max_classical}")
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print(f"coupling (B) = {coupling_b:.1e}")
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print()
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print(
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"HONEST SCOPE: consistency diagnostic only. Does NOT prove\n"
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"GRH for any L(s, chi); does NOT advance G4 = RH."
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)
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print()
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summary_rows = []
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for name, chi in characters:
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print("=" * 78)
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print(f"CHARACTER {name} (modulus q = {chi.modulus})")
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print("=" * 78)
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# --------------------------------------------------------------
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# [A] Decoupled bundle: Paley-style identity expected
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# --------------------------------------------------------------
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bundle_a = build_twisted_prime_ladder_hamiltonian(
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chi=chi,
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n_primes=n_primes,
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max_power=max_power,
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coupling=0.0,
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)
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sweep_a = sweep_twisted_paley_gap(
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bundle_a, chi, sigmas, n_max_classical=n_max_classical,
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)
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print(f"[A] {name} decoupled bundle (coupling = 0)")
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print(" " + sweep_a.summary())
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print()
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_print_sweep_table(sweep_a)
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print()
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# --------------------------------------------------------------
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# [B] Weakly coupled bundle: cross gap should appear
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# --------------------------------------------------------------
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bundle_b = build_twisted_prime_ladder_hamiltonian(
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chi=chi,
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n_primes=n_primes,
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max_power=max_power,
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coupling=coupling_b,
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)
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sweep_b = sweep_twisted_paley_gap(
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bundle_b, chi, sigmas, n_max_classical=n_max_classical,
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)
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print(f"[B] {name} weakly coupled bundle (coupling = {coupling_b:.1e})")
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print(" " + sweep_b.summary())
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print()
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_print_sweep_table(sweep_b)
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print()
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summary_rows.append(
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(
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name,
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chi.modulus,
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sweep_a.max_g_cross,
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sweep_b.max_g_cross,
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sweep_a.max_g_p32,
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sweep_b.max_g_p32,
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)
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)
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# ------------------------------------------------------------------
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# Cross-character summary
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# ------------------------------------------------------------------
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print("=" * 78)
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print("CROSS-CHARACTER SUMMARY (worst-case Paley-gaps)")
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print("=" * 78)
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print(
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f" {'chi':>6s} {'q':>3s} "
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f"{'max g_cross [A]':>18s} {'max g_cross [B]':>18s} "
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f"{'max g_P32 [A]':>14s} {'max g_P32 [B]':>14s}"
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)
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for name, q, gxa, gxb, gpa, gpb in summary_rows:
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print(
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f" {name:>6s} {q:>3d} "
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f"{gxa:>18.3e} {gxb:>18.3e} "
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f"{gpa:>14.3e} {gpb:>14.3e}"
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)
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print()
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# ------------------------------------------------------------------
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# Interpretation
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# ------------------------------------------------------------------
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print("-" * 78)
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print("INTERPRETATION (chi-twisted Paley-gap philosophy)")
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print("-" * 78)
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print(
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" - g_P32 is the truncation gap of the P32 chi-twisted\n"
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" prime-ladder closed form vs the classical chi-twisted sum;\n"
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" it shrinks as (n_primes, max_power, n_max_classical) grow.\n"
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" - g_P34 is the same fidelity check applied to the P34\n"
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" chi-twisted spectral trace.\n"
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" - g_cross = |Z_P34 - Z_P32| is the structural Paley-gap on\n"
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" the L-track: at coupling = 0 it must vanish to machine\n"
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" precision (Paley-style algebraic identity between P32 and\n"
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" P34), and at coupling > 0 it measures pure coupling-induced\n"
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" deformation, free of classical-truncation noise.\n"
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" - Honest scope: consistency diagnostic only. Does NOT prove\n"
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" GRH for any L(s, chi); does NOT advance G4 = RH. The\n"
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" Zenodo source disclaims primality proof status; P43\n"
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" inherits the same scope at the L-track coercivity level."
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)
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return 0
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if __name__ == "__main__":
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raise SystemExit(main())

‎src/tnfr/riemann/__init__.py‎

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the worst segments. Implemented for primitive real characters
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(chi_3, chi_4, chi_5). Diagnostic only; does NOT prove GRH for
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any L(s, chi) and does NOT advance G4 = RH.
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twisted_paley_gap_coercivity
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P43 chi-twisted Paley-gap consistency diagnostic. Structural
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analogue of P25 on the chi-twisted bundle: compares three
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representations of -L'(s,chi)/L(s,chi) -- the P32 closed-form
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weighted spectrum, the P34 chi-twisted spectral trace, and the
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classical truncated Dirichlet series sum chi(n) Lambda(n)/n^s --
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via three absolute Paley-gap quantities g_P32, g_P34, g_cross.
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The cross gap vanishes identically at coupling = 0 (Paley-style
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identity between closed-form and self-adjoint realisations);
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non-zero coupling exposes the deformation magnitude. Implemented
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for primitive real characters (chi_3, chi_4, chi_5). Diagnostic
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only; does NOT prove GRH for any L(s, chi) and does NOT advance
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G4 = RH.
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oscillatory_correction
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P31 Branch B1 retry: prime-ladder oscillatory correction of the
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P28 / P30 smooth targets. Reconstructs S(T) = pi^{-1} arg zeta(1/2
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TwistedUniformCoercivityCertificate,
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verify_twisted_uniform_coercivity_empirical,
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)
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from .twisted_paley_gap_coercivity import (
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# P43: chi-twisted Paley-gap consistency diagnostic
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TwistedPaleyGapSweep,
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sweep_twisted_paley_gap,
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twisted_paley_gap_cross,
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twisted_paley_gap_p32,
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twisted_paley_gap_p34,
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)
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from .spectral_emergence import (
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# P29: Spectral universality emergence under canonical UM+RA coupling
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CANONICAL_COUPLING_LAWS,
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# P42: chi-twisted uniform-coercivity certificate (diagnostic)
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"TwistedUniformCoercivityCertificate",
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"verify_twisted_uniform_coercivity_empirical",
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# P43: chi-twisted Paley-gap consistency diagnostic
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"TwistedPaleyGapSweep",
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"sweep_twisted_paley_gap",
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"twisted_paley_gap_cross",
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"twisted_paley_gap_p32",
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"twisted_paley_gap_p34",
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# P29: Spectral universality emergence under canonical UM+RA coupling
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"CANONICAL_COUPLING_LAWS",
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"InterPrimeCoupling",

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