Erdős-Straus Type A/B shadow research

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Status: Wellspring Candidate / active theorem program. The Erdős-Straus conjecture remains open. López Type A/B coverage remains open. Universal Direct-Shadow Completeness is false. This page reports exact theorems and finite certificates inside the Type A/B shadow program, not a solution of the conjecture.

August 15, 2026 — public hunt

The theorem notes now have an operator-facing hunt. Two kernels coexist: bb.kernel (Python reference, also B-BervigES.kernel) and CC.kernel (C attack engine). From the CENTL root the commands are ./centl es for the menu and ./centl es go for the infinite hunt. A start factor of 0, a chosen bound, or --random chooses where a hunt begins. The seed is a resume cursor, not a random-number seed. Ctrl+C stops the engine and writes the cursor. The next go continues from there.

The hunt is the sequence of intervals (s, s+Δ] with no last interval. It collects letters instead of stopping on the first one. Every letter carries a number that is the first 128 bits of SHA-256 of a fixed description of what was found. Two people who find the same letter on different machines compute the same number and the same file name. The start factor is not part of that identity. A cleared window is a finite certificate, not a proof of the conjecture.

Primary sources:

• ES-HUNT.md
• findings/START-HERE.md
• findings/HOW-GRADES-WORK.md
• findings/seeds/README.md
• bb.kernel
• CC.kernel
• CC.kernel/PROOF.md

August 15, 2026 — two-target corridor

The original-ES lane is the exact two-target signed-box criterion. Combined Type I / Type II failure is now classified at the first corridor positions:

• Linear form 2p+1: a divisor 7 mod 8 is an explicit certificate. A counterexample must keep 2p+1 in the same {1,3} mod 8 semigroup already forced on p+2.
• q=3 and q=7: for Mordell-hard primes, Type I never rescues a Type-II miss. Combined failure equals the existing Type-II miss.
• q=11: Type I does rescue an explicit list of Type-II misses, including a thin three-class residue set when the quadratic box is only {1,3,4}.
• k=15: both targets live outside the two-primary subgroup {1,2,4,8}. Combined miss is that subgroup trap, or one thin packet involving primes 11 mod 15.
• k=19: Type II misses on a QR-trap. Type I rescues the nonresidue classes of p when the QR box is full. On the hard class p ≡ 121 (mod 840) the forced factors 5 and 7 fill the QR subgroup, so a nonresidue factor of (p+19)/4 is an immediate Type-II hit. Combined miss on that class is exactly a QR-supported C with p itself a residue modulo 19.

Through 2·106, every hard prime that misses 3,7,11 is solved by some later shift at most 59. That is a finite census, not a bound. A separate covering attack is now closed in the negative: no {2,3,5,7}-smooth forced multiplier can produce a Type-II hit at the aligned shift k ≡ −p (mod 4M). Any uniform Type-II arithmetic-progression cover of a hard class must import an external prime ℓ ≥ 11. The next exact corridor target is the Type-I companion at q=23.

Primary sources:

• TWO-P-PLUS-ONE-FILTER.md
• HARD-Q7-TYPE-I-NO-RESCUE.md
• Q11-TYPE-I-COMPANION.md
• K15-TWO-TARGET-FILTER.md
• K19-TWO-TARGET-FILTER.md
• CURRENT-FRONTIER.md
• ERDOS-STRAUS-WALL.md

August 15, 2026 — ancestry and covering-core

The Type A/B shadow program now has a general ancestry skeleton, several exact quotient classifications, and a universal local theorem for the single-active Class-C branch:

• Divisor-child ancestry theorem: for ancestry shift s, if a | gcd(j,s) and the child depth is K = a·p with p prime, then the entire child trap set is shadowed by the ancestor.
• Asymptotic ancestry skeleton: for j ≥ s+1, a full ancestry shadow has at most one prime factor outside the shift support. For odd s, every nonsmooth full-shadow child has the form K = a·p with a | gcd(j,s).
• Odd-prime-shift rigidity above the small-ancestor window: if r is an odd prime and j ≥ r+1, full shadow holds iff K is prime or K = r·p. Small j ≤ r can contain additional multiplicative exceptions and is a separate classification problem.
• Exact quotient classifications: quotients 13, 17, 21 and 29 are classified completely.
• Universal C1 active-row escape: if a directly novel candidate has exactly one active fixed-negative layer, that layer always admits an exact reduced local escape. Full C1 remains open because nonfixed exact rows can survive simultaneously.
• Shared-factor reduction: lift-room plus an odd totient-ratio lemma collapse C2-shared to complementary q=3 covers. Unrestricted complementary covers exist, so C2-shared is not a theorem for arbitrary (L,r). On admissible candidates the only family through k ≤ 1500 is the j=205 child of layer 10, which is directly shadowed and therefore not a DSC-P counterexample.

Primary sources:

• ANCESTRY-DIVISOR-CHILD-THEOREM.md
• ANCESTRY-ASYMPTOTIC-SKELETON.md
• ODD-PRIME-SHIFT-ASYMPTOTIC-RIGIDITY.md
• SINGLE-ACTIVE-REDUCED-ESCAPE-THEOREM.md
• ERDOS-STRAUS-WALL.md (honest wall statement)

What the program is

Miguel Angel López introduced Type A/B solution forms and conjectured that every prime has such a solution. FCF’s contribution sits on top of that framework: a minimal first-hit depth C_AB(p), exact layer cardinality, a shadow graph on congruence layers, finite Direct-Shadow certificates, fiber peeling, character shields, multiplicative two-box geometry, an ancestry skeleton, and a residual Class-C attack aimed at universal DSC-P.

Define m(k)=4k−1 and the Type A/B trap set

T_k = {−d, −4d mod (4k−1) : d divides k}.

The Type A/B witness depth is

C_AB(p) = min { k ≥ 1 : p mod (4k−1) is in T_k }.

Finite DSC certificates

Candidatewise Direct-Shadow attack through k ≤ 1200 (independently verified):

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

Through k ≤ 1500: 53,240 / 53,240 directly novel candidates have reduced avoiding progressions. The same finite range is independently closed by fiber peeling plus a bounded residual selector; the largest selector radius used is 54 and the largest observed residual prime is 31.

These are exact finite theorem-certificate statements for their ranges, not universal DSC-P.

Certified frontier through 10,000,000

Hard classes mod 840: 1, 121, 169, 289, 361, 529. Every selected prime through 10,000,000 received a Type A/B witness by depth 2622.

Ancestry rigidity

For ancestry quotient Q = 4s+1, child depth K = Qj − s, the following unrestricted trap-set classifications are exact:

Important scope: there is no all-j theorem saying every odd-prime shift has only the r·p composite shape. The correct uniform theorem applies for j ≥ r+1. The small strip j ≤ r has genuine extra examples, such as r=17, j=2, K=121, whose divisors all land in the fixed ancestor trap image.

C1 and the residual wall

For a directly novel candidate with exactly one active fixed-negative layer, the active quotient is universally of the form p or p². The exact compatible-trap pullback is injective into the parameter line.

The active row itself is now completely understood locally:

• Class A: p | L, so reducedness is fixed and direct novelty leaves an exact safe class.
• Class B: necessarily q=p² with p≥11. The exact forbidden fiber obeys |R| ≤ (p²+3)/2, strictly smaller than the p²−p reduced parameter classes. A reduced exact escape is therefore forced.

Hence a unique active fixed-negative row is never by itself a reduced covering obstruction.

Full C1 is still open because the final exact residual system is usually dominated by nonfixed earlier rows. In the independently verified k≤1500 C1 census, 2,770 candidates have one active fixed-negative layer; 1,480 have nonempty final fiber kernels; the active row survives into those kernels only 18 times, while nonfixed residual rows account for 69,672 edge incidences. Every one of the 1,480 finite residual systems nevertheless has a reduced avoiding selector.

Hard-class single-active finite result

A separate two-construction falsification run through k≤100,000 examined 8,021,288 hard-compatible target candidates. Among 419,123 candidates with exactly one active fixed-negative layer, the observed quotient was always 3, 5, or 9, and every case was Class A. This is an exact finite certificate, not a universal theorem.

What solving Erdős-Straus still requires

1. Universal DSC-P (every directly novel candidate is reduced-realizable).
2. López Type A/B coverage for every prime (density one is not enough).
3. Composite n arguments wired to the same structure.

See ERDOS-STRAUS-WALL.md.

Research map

• ES-HUNT.md — public hunt, seeds, letter numbers
• DIAMOND.md — full synthesis
• CURRENT-FRONTIER.md — moving edge
• CLASS-C-RESIDUAL-CORE.md — residual node
• MULTIPLICATIVE-TRAP-QUOTIENT.md — two-box geometry
• WS-CAND-003 — formal Wellspring Candidate record

Primary prior work: López, A Complete Congruence System for the Erdos-Straus Conjecture.

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