Erdős-Straus Type A/B automated research harness

This directory operationalizes the research program recorded in docs/wellsprings/WS-CAND-003-erdos-straus-type-ab-shadow-structure.md.

Public library: every public note here is hosted at

freecomputation.org/research.html. The program overview is freecomputation.org/research-erdos-straus.html.

Research map

Current synthesis: DIAMOND.md records how the minimal Type A/B depth invariant, shadow graph, exact-depth spectrum, exact survivor hazard, prime-modulus backbone, composite rescue core, and Direct-Shadow Completeness program fit into one theorem architecture.

Moving frontier: CURRENT-FRONTIER.md is the shortest current-state record.

Two-target corridor (2026-08-15): TWO-P-PLUS-ONE-FILTER.md, HARD-Q7-TYPE-I-NO-RESCUE.md, Q11-TYPE-I-COMPANION.md, K15-TWO-TARGET-FILTER.md, and K19-TWO-TARGET-FILTER.md make the original-ES corridor combined-exact at 3,7,11,15,19 and add the linear form 2p+1.

Independent covering obstruction: HARD-SMOOTH-TYPEII-OBSTRUCTION.md proves that no {2,3,5,7}-smooth forced multiplier can produce a Type-II hit at the aligned shift k≡-p (mod 4M).

Public hunt (2026-08-15): ES-HUNT.md is the operator-facing record. bb.kernel (also B-BervigES.kernel) is the Python reference; CC.kernel is the fast hard-prime engine. Findings live in findings/. Letter numbers are content-addressed and identical on every machine that finds the same letter. From the repo root: ./centl es go (or ./centl es for the menu). --from N and --random start another hunt without replacing the first. ./centl es hunts lists them. Erdős–Straus remains open.

Shared-factor CN (2026-08-15): CN-SHARED-THEOREM.md proves lift-room, the odd totient-ratio lemma, the C2-thin reduction to complementary q=3, and the 205 → 10 absorption theorem. Unrestricted C2-shared is false. Directly novel admissible complementary covers are zero through k ≤ 1500; the replayable certificate is CN-SHARED-CERTIFICATE-2026-08-15.md.

Durable checkpoint: RESEARCH-BACKUP-2026-08-14.md freezes independently verified workflow/artifact provenance so the research state is recoverable from the repository independently of chat or local scratch data.

Core linked records:

• THEORY.md — foundational shadow and modulus-ancestry results;
• RESULTS-2026-08-14.md — automated frontier, shadow map, independent verification, and CENTL certification;
• DEPTH-SPECTRUM.md — exact-depth realization and structural-gap versus latency-gap distinction;
• DIRECT-SHADOW-COMPLETENESS.md — first candidatewise completeness attack through k=600;
• DIRECT-SHADOW-K1000.md — independently verified extension through k=1000;
• DIRECT-SHADOW-K1200.md — exact extension through k=1200, with 41,470/41,470 directly novel candidates carrying reduced avoiding progressions;
• FIBER-SELECTOR-K1200.md — full theorem-driven replay of the frozen k<=1200 bundle: 26,044 fiber-empty candidates plus 15,426/15,426 nonempty kernels solved by 0,±1,...,±64, with maximum radius 54;
• SHADOW-COVER-GEOMETRY.md — dense-cover diagnostics showing the phenomenon is not explained by a trivial global union bound;
• ODD-COVERING-BRIDGE.md — bridge to odd covering-system structure;
• SHADOW-KERNEL.md — exact prime-power local-load peeling theorem and small-prime kernel reduction;
• FIBER-SHADOW-KERNEL.md — sharper exact fiber-load elimination theorem;
• SMALL-SELECTOR-HYPOTHESIS.md — bounded integer-selector attack on residual fiber kernels;
• QUADRATIC-TRAP-SIGNATURE.md — exact Jacobi -1 signature of every Type A/B trap and quadratic character shield;
• CHARACTER-SHIELD-COMPLETENESS.md — proved collapse of collective scalar-character obstruction to fixed negative earlier layers;
• QUADRATIC-SIGNATURE-QUOTIENT.md and QUADRATIC-SIGNATURE-COSET.md — complete vector-valued Legendre-signature quotient/coset of every Type A/B trap set;
• MULTIPLICATIVE-TRAP-COSET.md and MULTIPLICATIVE-TRAP-QUOTIENT.md — multiplicative envelope, quotient shield, and exact two-box trap geometry;
• DYADIC-TRAP-LATTICE.md — exact power-of-two trap saturation and infinite Mersenne-type shadow lattice;
• SQUARE-LIFT-CORE.md — fixed-character residual primes outside the target modulus can enter only through even powers;
• SQUAREFREE-LIFT-CORE.md — squarefree-ancestor localization and exact projection-excess reduction;
• SQUARE-LIFT-RECIPROCITY.md — quadratic reciprocity forces every square-lift divisor onto the Jacobi-positive side of its squarefree ancestor, yielding infinite shadow families;
• JACOBI-SATURATION.md — classification of exact Jacobi-saturated Type A/B layers: only k=1,2,4;
• SQUARE-LIFT-SIGNATURE.md — exact projected local-signature space and automatic ancestor shadowing whenever kappa(a)=1;
• RECIPROCITY-DEFECT-QUOTIENT.md — defect quotient R_a, its dimension kappa(a)-1, and the prime-factor defect conservation law;
• SQUARE-LIFT-SIGNATURE-CLASSIFICATION.md — if-and-only-if theorem: kappa(a)=1 exactly when every square lift is ancestor-shadowed at quadratic-signature resolution; kappa(a)>1 produces infinitely many signature exceptions;
• RECIPROCITY-MATRIX.md — binary Type A/B reciprocity matrix with canonical left Jacobi null vector and right exponent-parity null vector;
• QUADRATIC-FIELD-BRIDGE.md — norm-form, split-prime, ideal-class conservation, and explicit genus/Rédei prior-art boundary for the square-lift theory;
• PRIME-MODULUS-BACKBONE.md — infinite exact-depth prime-modulus backbone;
• SURVIVOR-DENSITY.md — exact finite-depth density, mass, and conditional hazard;
• COMPOSITE-CORE.md — zero-density prime-modulus survivor core and composite-rescue reduction;
• PRIOR-ART.md — literature and priority boundary;
• CRYPTOLOGY.md, CRYPTOLOGY-THEORY.md, and CRYPTOLOGY-RESULTS-2026-08-14.md — controlled cryptology side investigation;
• ES-HUNT.md — public infinite hunt, seeds, and letter numbers;
• ../../docs/wellsprings/WS-CAND-003-erdos-straus-type-ab-shadow-structure.md — formal Wellspring Candidate record.

Automation

The main workflow .github/workflows/erdos-straus-research.yml regenerates the finite Type A/B research corpus, independently verifies certificates, feeds exact identities into CENTL, hashes outputs, and uploads the evidence bundle.

The candidatewise falsification workflow .github/workflows/erdos-straus-direct-shadow-completeness.yml performs the stronger theorem attack:

1. enumerate every directly novel hard-compatible candidate through the configured depth;
2. search for an integer avoiding every earlier Type A/B layer;
3. search for a reduced avoiding progression, yielding infinitely many exact-depth primes by Dirichlet;
4. independently recompute and verify every witness;
5. analyze the prime-power coordinate core;
6. apply exact coarse local-load peeling;
7. apply exact fiber-load peeling;
8. test a bounded fixed selector menu on the residual fiber kernel without consulting the stored witness;
9. apply quadratic-character analysis;
10. certify selected CRT progression identities with CENTL;
11. freeze hashes and upload the complete certificate bundle.

The current workflow defaults are k<=1500, s<=3,000,000, and residual selector menu 0, ±1, ..., ±64.

Additional theorem falsifiers and proof-mining analyzers now include:

• trap_coset_analyzer.py — exact finite regression of the multiplicative trap-coset theorem;
• quadratic_signature_shield_analyzer.py — exact affine local Legendre-signature shield;
• square_lift_core_analyzer.py — parity check and finite classification of the post-character square-lift core;
• squarefree_lift_core_analyzer.py — exact squarefree-ancestor projection-excess localization on a candidate bundle.

Latest frozen finite result

The completed candidatewise run through k<=1200 produced:

admissible candidates:             57,367
directly shadowed candidates:      15,897
directly novel candidates:         41,470
integer avoiding witnesses:        41,470
reduced avoiding witnesses:        41,470
unresolved integer candidates:          0
unresolved reduced candidates:          0
independent verifier:              VERIFIED

Every directly novel hard-compatible candidate in this finite range therefore has an explicit reduced avoiding progression. This is an exact finite theorem-certificate statement, not a universal proof of DSC-P.

A separate full replay on the same frozen candidate bundle independently resolves all 41,470 candidates by exact fiber peeling followed, when necessary, by a bounded residual selector. No stored witness is used to decide those two stages.

Exact structural tools

The project now contains a hierarchy of exact sufficient mechanisms and envelopes:

• prime-power peeling: a coordinate whose local load is below one can be eliminated while preserving satisfiability;
• fiber peeling: replacing full forbidden-set size by the maximum relevant fiber width gives a strictly sharper local elimination theorem;
• bounded residual selectors: after fiber peeling, the k<=1200 replay solved every nonempty kernel with the fixed menu 0,±1,...,±64;
• Jacobi character shield: every Type A/B trap modulo m_k=4k-1 has Jacobi symbol -1;
• character obstruction completeness: collective scalar-character inconsistency adds no obstruction beyond one fixed-negative earlier layer;
• full quadratic-signature coset: the complete vector of local Legendre signs of T_k is the affine space eta_k+V_k;
• multiplicative trap coset: the exact trap set sits inside one proper coset -H_k in the unit group;
• squarefree-lift localization: every character-fixed layer has a squarefree ancestor modulus and only its projection excess can remain exactly active;
• square-lift reciprocity: every divisor prime of a lift splits in the ancestor quadratic field;
• signature-shadow classification: kappa(a)=1 is exactly the universal square-lift signature-shadow regime;
• reciprocity defect conservation: higher-quotient lift defects must cancel according to prime-exponent parity;
• reciprocity matrix: the Legendre-symbol matrix has canonical conservation laws on both its left and right null spaces;
• quadratic-field norm bridge: square-lift depths are norms of (1+s sqrt(-d))/2, giving a principal-ideal conservation law stronger than its quadratic-character projection.

Each stage deliberately preserves the claim boundary: failure of a coarse shield or a bounded selector does not imply exact Type A/B coverage. It only says finer geometry remains.

Default main research contract

The main finite research run uses:

• prime limit 10,000,000;
• Mordell-hard classes modulo 840: 1, 121, 169, 289, 361, 529;
• Type A/B depth search through k=3000;
• the checked-in thirteen-record frontier as a regression fixture;
• exact direct-shadow analysis through k=3000;
• explicit non-union-shadow witnesses wherever a first-hit prime is present.

What the harness does not prove

It does not prove the Erdős-Straus conjecture. It does not prove López's universal Type A/B coverage conjecture. It does not yet prove universal Direct-Shadow Completeness. It does not establish literature priority.

Finite candidatewise results are theorem-certificate statements for their stated ranges. Kernel, selector, quadratic-signature, multiplicative-coset and square-lift tools are exact sufficient structures or envelopes, but a residual from any one of them is not evidence of a counterexample.

Research direction

The active theorem program is now:

Type A/B witnesses -> C_AB -> shadow graph -> exact-depth spectrum -> exact survivor process -> direct-shadow completeness -> fiber kernel -> bounded selector -> scalar character saturation -> local quadratic signatures -> reciprocity matrix/defect quotient -> multiplicative quotient -> exact square-lift/ray-class residue core -> composite rescue.

The immediate goal is to convert the increasingly tiny residual core into a universal local escape theorem. If achieved, the direct-shadow graph would become a complete obstruction theory for exact Type A/B first-hit realizability.

Chat discussion is exploratory. The repository is canonical.