Reduction of the Wagstaff Chebyshev sufficiency conjecture to the Single-Pass Conjecture

LaTeX source, reproducibility scripts, and certified data for the paper "Reduction of the Wagstaff Chebyshev sufficiency conjecture to the Single-Pass Conjecture" by Alexey Dolotov (v4.0; formerly "... to the Pell Primitive Pair Conjecture", v3.6 — see What changed in v4.0 below; the last pre-restructure version, v3.8, is archived in paper/archive/).

Erratum (2026-07-15)

The v3.6 abstract and introduction call the conjecture "equivalently the open 'only if' direction of the Vrba–Reix Wagstaff primality test". This is incorrect as stated. The literal Vrba–Reix prize iteration (S_0 = 1/4 (mod 2^p+1), S_{i+1} = S_i^2 - 2) is a Lucas-type iteration in Q(sqrt(-7)) — its base is theta = (1+3*sqrt(-7))/8 = (pibar/pi)^2 with 2 = pi*pibar — while Condition (II) lives in Z[sqrt(2)]. The two return conditions decouple at the level of individual prime factors in both directions, so the two sufficiency statements are logically incomparable: proving either would not formally settle the other. They agree empirically at every tested exponent. The corrected framing ("the Chua-form counterpart of the Vrba–Reix test") appears from v3.7 onwards, and a companion note (in preparation) works out the exact relation. The mathematical content of the reduction is unaffected.

Abstract (v4.0)

For W_p = (2^p + 1)/3 and omega_3 = 3 + 2*sqrt(2), the congruence omega_3^{(W_p+1)/2} = -1 (mod W_p) (Condition II) holds for every prime W_p; whether it implies primality is the open Wagstaff Chebyshev sufficiency conjecture — the Chua-form counterpart of the Vrba–Reix Wagstaff primality test (open since 2008, standing 500-euro reward; see the Erratum above on the exact relation).

Call a prime r with ord_r(2) = 2p a local passer at p if omega_3^{(W_p+1)/2} = -1 in Z[sqrt(2)]/(r), and write P_p for the set of local passers; every member of P_p divides W_p, and P_p = {W_p} whenever W_p is prime. The Single-Pass Conjecture (SPC) asserts |P_p| <= 1 for every prime p >= 5. The headline theorem is that SPC implies the sufficiency conjecture, unconditionally and with zero computational inputs (the only outside ingredient is a Nagell–Ljunggren result: W_p is never a perfect power, so a composite W_p has two distinct prime factors, and under a global pass both would lie in P_p).

Double passes decompose by residues mod 8: the inert pair (above the Platinum cutoff this is the Pell Primitive Pair Conjecture, PPPC), the mixed pair (MPC), and the split pair (subsumed by general-d NCT, the nonexistence-of-compatible-triples conjecture). The working chain proves the sufficiency conjecture from PPPC + MPC^> + X1 (MPC^> is MPC restricted above the survey cutoff; X1 is a single-exponent statement at p = 10,916,765,939), taking two stated computational inputs (the Platinum Lemma — a 684,965,381-row enumeration of all inert factors r <= 10^12 — and the witness discharges); that conjunction is in turn implied by SPC. Proved unconditionally along the way: NCT for every odd d <= 400 (28 fixed-d certificates in (100,400]), a complete danger-triple census through d = 400 (exactly two real triples exist, both at exponents closed by secondary factors), the Order-Pinning and Multi-Factor Pinning theorems, a Pair Separation theorem, diagonal vanishing through d = 400, quartic residuosity of 2 at primitive Pell divisors, and an exact-AP characterization. The reduction is conditional; the sufficiency conjecture remains open. Across every record, exactly four composite-W_p exponents with P_p nonempty are known — all singletons, all inert — and no configuration with |P_p| >= 2 has ever been observed.

This is the companion to the Brillhart–Lehmer–Selfridge primality-proofs paper (see Cite this work).

What changed in v4.0

• v4.0 (2026-07-16). The paper is restructured around the Single-Pass Conjecture: new section "Local passes and the Single-Pass Conjecture" (the passer set P_p, SPC, and the input-free headline reduction), a branch-decomposition proposition, and a rewritten reduction section. The working chain PPPC + MPC^> + X1 keeps hypotheses and computational inputs identical to v3.8, with a much shorter proof. The second-moment/bucket bookkeeping of earlier versions is removed; its surviving content is restated in sharper form (quantitative bound N_in <= Pi + 1; input-free pair-level form; triple-count erosion with a Wieferich-paired corner; Wieferich/multiplicity constraints). NCT is now closed for every odd d <= 400 (v3.6 had fifteen of the nineteen values in (200,400]), and the complete danger census through d = 400 is new. The last pre-restructure version is archived at paper/archive/wagstaff_chebyshev_reduction_v3.8.{tex,pdf}.
• v3.6 (2026-06-25). Previous bundle cut ("... to the Pell Primitive Pair Conjecture"), plus the 2026-07-15 Erratum above.

Layout

.
├── paper/
│   ├── wagstaff_chebyshev_reduction_v4.0.tex   LaTeX source (68 pp)
│   ├── wagstaff_chebyshev_reduction_v4.0.pdf
│   ├── archive/                                v3.8 (last pre-SPC version)
│   └── Makefile                                pdflatex targets
├── scripts/                                    reproducibility scripts
│   └── README.md                               script -> paper statement map
│       NCT fixed-d certificates: cp350/cp351/cp353/cp356/cp358/cp367/
│         cp372/cp375/cp376 per-d closures; cp365 consolidated artifact;
│         cp352_verify_nct_recompute.py independent re-check;
│         cp356_aprcl_recert.py, cp358_aprcl_range.py APR-CL re-cert
│       Danger census d <= 400: cp412_danger_census_d400.py +
│         cp412b_t1_recheck.py, cp412b_t1_aprcl_sweep.py (adversarial pass)
│       Witness discharges / X1: cp352_verify_discharges_recompute.py
│       Diagonal + corner + falsification: cp354_samed_diagonal_certificates.py,
│         cp354_factordb_v2d_diagonal.py, cp354_diag_corner_gcd.py,
│         cp358_corner_sweep_d1000.py, cp358_falsification_prewindow.py
│       Heuristic constants (M1–M3): cp395_pppc_heuristic_constants.py
│       Self-supporting ladder: cp410_twin_ladder.py
│       Cross-case: platinum_lemma.py, multi_factor_pinning.py,
│         exact_ap_density.py; second_moment_reduction.py (historical)
│       Vrba–Reix agreement: cp361_vrba_reix_check.py
│       Independent verification: cp362_verify_preliminaries.py,
│         cp363_verify_inert_foundations.py
│       Survey pipeline: survey.py, build_clean.py, audit.py, verify_sample.py
├── data/
│   ├── nct_certificates.json      static fixed-d certificates: pinned Psi_{4d}
│   │                              factorizations + APR-CL + dispositions for
│   │                              all 28 closures in (100,400]
│   ├── cp412_danger_census.json   complete danger-triple census 43 < d <= 400
│   │                              (34 rungs, 340 tests, 226 APR-CL primes)
│   ├── danger_triple_data.json    V_{114}, V_{134}, V_{662} factorizations
│   ├── sample_1000.csv            first 1000 rows of the clean survey CSV
│   ├── SHA256SUMS                 hashes
│   └── README.md                  data dictionary
├── reproducibility.md             end-to-end reproduction walkthrough
└── CITATION.cff   .zenodo.json   LICENSE   README.md

Cite this work

• Paper (arXiv): pending submission — arXiv ID will be inserted here
• This bundle (Zenodo): pending — deposited from the author's account after each release; DOI will be inserted here
• Inert-factor survey CSV (Zenodo): 10.5281/zenodo.19496206 (802 MB, 15,587,021 rows — the G_r/4 > 43 slice of the 684,965,381-factor Platinum enumeration, the only factors at which a local pass is possible; see reproducibility.md, Step 4)
• Companion BLS primality-proofs paper (Zenodo): 10.5281/zenodo.19645478

A machine-readable citation is in CITATION.cff.

Reproduce the central claims

NCT closures (no FactorDB needed). data/nct_certificates.json pins, for every fixed-d closure in (100, 400] (28 values — exactly the admissible odd d there; with the d <= 99 theorem this closes NCT for every odd d <= 400), the complete factorization of the Pell primitive part Psi_{4d} with each factor APR-CL-certified and its certificate disposition; product identities and primality re-check offline. Re-generate from scratch with python3 scripts/cp365_nct_certificate_bundle.py (PARI/GP gp required, FactorDB for the non-embedded values; overwrites the JSON in place; expected: DONE: 28/28 CLOSED; 182 factors).

Danger census d <= 400. data/cp412_danger_census.json pins the complete census of danger triples for admissible 43 < d <= 400: 34 rungs, complete V_{2d} factorizations, 340 (rung, prime) tests, 226 distinct primes all APR-CL-certified; exactly two real triples (both at exponents closed by secondary factors) plus two phantoms at prime W_p. The cp412* scripts are the pinned provenance (they were run in the research tree; the artifact is self-contained, and every listed prime can be re-checked with gp's isprime(., 2)).

Witness discharges. python3 scripts/cp352_verify_discharges_recompute.py re-verifies end-to-end (self-contained, no network) that the two discharged witness exponents carry an explicit prime factor at which Condition (II) fails, and that the three Platinum witnesses are genuine local passes.

Vrba–Reix global agreement. python3 scripts/cp361_vrba_reix_check.py checks that, at every tested exponent, the S^2-2 return tests and Condition (II) give the same global verdict and track primality of W_p. This is empirical agreement of global outcomes, not equivalence of the tests — the per-factor conditions differ (see the Erratum above).

Survey CSV (Platinum enumeration). Download the 802 MB CSV from Zenodo (10.5281/zenodo.19496206), place it at data/inert_factors.csv, verify (cd data && shasum -a 256 -c SHA256SUMS), audit with python3 scripts/audit.py, spot-check with python3 scripts/verify_sample.py data/sample_1000.csv.

See reproducibility.md for the full procedure.

License

MIT (code) / CC-BY-4.0 (paper, data). See LICENSE.