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Research proportions — Type A/B program

Theorem · hosted from the CENTL repository

Research library · Theorem

Theorem

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Source in the repository

Date: 2026-08-15

Scope: research/erdos-straus/ and the active theorem arc toward DSC-P / López / ES

1. Effort mass by layer

LayerShare of deposited theorem massNature
Finite certificates (DSC k≤1200/1500, frontiers, selectors)~35%Exact finite theorem-certificates
Structural shields (character, signature, multiplicative two-box, fiber peel)~25%Universal lemmas + envelopes
Ancestry / quotient rigidity~10%Infinite families, casewise proved
C1 / C2 / CN-coprime / lift-room / thinness~20%Local escape theorems
Shared-factor residual (q=3 tight cluster)~8%Reduction + range certificates
López remainder / composite core / ES wall~2%Named open problems

Reading: The program is heavy on finite certificates and local structure, medium on escape theorems, light on the pointwise remainder that would finish López/ES. That is healthy for a depth-spectrum program and insufficient for a claim of ES.

2. Closed vs open (logical weight)

Closed (usable as lemmas)

  • Trap cardinality, shadow relation, ancestry form k=(4s+1)j−s
  • Character-shield completeness; multiplicative coset + two-box geometry
  • Fiber peel + bounded selector on frozen k≤1500 bundle (0 unresolved)
  • Prime-modulus backbone; density-one Type A/B among primes
  • Quotient rigidity for q ∈ {5,9,13,17,21,29}
  • C1 pullback escape
  • CN-coprime for every finite active-core size
  • Lift-room, totient-ratio, C2-thin reduction to q=3
  • 205 → 10 absorption; admissible complementary q=3 has 0 novel covers through k≤1500
  • DSC-P fragment for pairwise-coprime active moduli

Open (blocking universal DSC-P / López / ES)

  1. All complementary q=3 families for every k (only k≤1500 certified)
  2. General q=3 absorption (every q=3 layer an ancestry child of a q=1 anchor)
  3. Shared-factor CN for arbitrary tight clusters beyond lift-room peel
  4. Bound on |N^{act} ∩ tight| for Class-C residuals
  5. López Type A/B for every prime (density one ≠ zero exceptions)
  6. Composite n for full Erdős-Straus

3. Proportion diagnosis

DiagnosisAssessment
Over-investment in finite k-bounds?Mild — k≤1500 is a strong certificate but must not be mistaken for DSC-P
Under-investment in López remainder?Yes — almost no deposited attack on exceptional primes outside Type A/B
Shared-factor focus correct?Yes — after CN-coprime, the only local hole is 3-adic complementary covers
Claim disciplineGood — wall files exist; unrestricted C2-shared was corrected when false

Net: The theorem mass is correctly concentrated on the Class-C / active-core escape route. The next dollar of effort should not expand another finite DSC census; it should remove the k≤1500 cap on q=3 absorption and classify base q=3 layers.

4. Ordered next moves

  1. General strong q=3 absorption (this commit) — prove every layer with an m_i | (m_j/3) trap-reducing ancestor is novel-impossible when q_j=3.
  2. Classify base q=3 layers (no such ancestor) and whether complementary pairs among them appear on admissible candidates for any k.
  3. Peel roomy layers from arbitrary cores; residual = 3/5/7-adic tight.
  4. Census |N^{act}| structure on Class-C residuals (not only median ~1).
  5. López remainder program (separate track).

5. What “solved” would require

GoalMissing
Universal DSC-PShared CN for all geometries + all active-core sizes
López for all primesZero exceptional primes outside Type A/B
Erdős-StrausLópez + composites

None of these are implied by the current closed mass alone.