Start here for review: CURRENT_STATUS.md, CANONICAL_EQUATIONS_AND_LEMMAS.md, THEOREM_STACK.md, REVIEWER_FIRST.md, REVIEWER_QA.md, RESCUE_LEMMA.md, ANALYTIC_LOWER_BOUND_TARGET.md, ANALYTIC_PROOF_FRONTIER.md, WHAT_REMAINS.md
Status: finite computational evidence and proof-direction research.
Universal proof status: open.
Main missing theorem: the Rescue Lemma.
FRAMEWORK VALID
FINITE EVIDENCE STRONG
RESCUE LEMMA OPEN
UNIVERSAL PROOF NOT CLAIMED
This repository studies Goldbach's conjecture through a mirror-center framework. For an even number E = 2C, the search is rewritten around the center C by testing mirror gaps g:
E = (C - g) + (C + g)
A Goldbach pair is found when both C - g and C + g are prime.
The framework reduces the universal problem to one analytic statement:
Rescue Lemma
There exist constants K > 0 and C0 such that for every C >= C0,
there exists an integer g with
0 <= g <= K log^2(C)
such that C - g and C + g are both prime.
If this lemma is proven, then every sufficiently large even number has a Goldbach pair. Smaller centers can then be handled by finite verification.
Every even number E = 2C can be tested through mirror gaps g.
A Goldbach pair appears when C-g and C+g are both prime.
Define:
S(C,W) = sum_{0 <= g <= W} 1_prime(C-g) * 1_prime(C+g)
The universal proof target is the Rescue Lemma:
There exist constants K > 0 and C0 such that, for every C >= C0,
S(C, floor(K log^2(C))) >= 1.
Finite tests strongly support this target. The current finite raw packet records
that `K=8.25` and `K=10` both have finite breaches, while `K=12` had zero misses
across the completed global coprime sweep, focused hardest-residue replay, and
top-100 hardest-residue replay. Small-prime covering walls did not fully cover
the tested windows, and BOTH-hit density thinned with scale but stayed positive
in the tested samples.
The proof is not complete. Brun-Titchmarsh cannot prove the Rescue Lemma
because it is an upper-bound tool, while the framework needs a lower bound.
The remaining analytic wall is to prove that the survivor gaps left after
local congruence obstructions contain at least one true BOTH-prime hit inside
the K log^2(C) mirror window.
This repository does not claim a completed proof of Goldbach.
The current workbench evidence suggests:
- Goldbach rescue does not appear to rely on one residue lane.
- Low-pressure rows spread across many residue lanes.
- High-pressure rows narrow into fewer lanes, but still use many rescue lanes.
- First-rescue gap pressure
K = g / log^2(C)stayed below the tested ceiling in sampled ranges. - Finite base checks below the selected cutoff passed in the local test run.
- The open wall remains an analytic proof of the Rescue Lemma for all large
C.
| File | Purpose |
|---|---|
CURRENT_STATUS.md |
Current readiness percentages, proof gap, and reproducibility artifacts |
CANONICAL_EQUATIONS_AND_LEMMAS.md |
Standard notation and Lemmas 1-5 in one canonical packet |
K12_EVIDENCE_PACKET.md |
Consolidated finite raw K12 packet and hardest rows |
THEOREM_STACK.md |
The lemma-by-lemma framework |
WHAT_REMAINS.md |
The exact missing universal proof step |
ANALYTIC_PROOF_FRONTIER.md |
Strongest current analytic proof attempt and why it remains open |
VALIDATION_SUMMARY.md |
Finite computational evidence summary |
RAW_RESULTS_SUMMARY.md |
Pattern findings from raw number tests |
scripts/test_finite_base.py |
Reproducible finite base checker |
Goldbach is reframed as a mirror-prime rescue problem: every even number E = 2C is rescued if some bounded gap g makes both C-g and C+g prime; finite evidence is strong, but the universal Rescue Lemma remains analytically open.