Xavi Maths · witnessed research console
Ramsey R(5,5) Research Lab
Search, falsification, and exact-verification workspace for 43-vertex two-colourings with no monochromatic K5. The primary model is now the recovered hidden Z43 normal form: Exoo Cyclic(43) plus a localized defect orbit, with every candidate still checked against all 962,598 five-vertex subsets.
Published bound43 ≤ R(5,5) ≤ 46
Live exact computation
Hidden Z43 orbit sweep
Connecting to witnessed SAT progress…
Connecting…—
Campaign semantics: scanning L1→L21. A completed all-UNSAT level excludes every family at that L; any exact SAT family produces a zero-conflict K43 candidate and makes all higher L SAT-exists by monotonicity.
Performance
Solver tail
Scheduler
What the cluster is doing
Waiting for campaign telemetry…
Proof ladder
Exact hidden-orbit exclusions
| Level | Free edges | Completed | UNSAT | State |
|---|---|---|---|---|
| Loading… | ||||
Cluster
Compute nodes
Loading nodes…
Current shard detail
| Shard | Workers | Completed | UNSAT | SAT | Failed | State | Last update |
|---|---|---|---|---|---|---|---|
| Waiting for shard telemetry… | |||||||
Exact restricted-family SAT evidence only; unrestricted K43 remains a separate question.
Scientific control
Representation is not a theorem
Unity relabeling is kept as a control. A useful hypothesis must reduce or organize the graph search in a way that survives exact checking; merely renaming vertices cannot change the Ramsey conflicts.
Promotion rule: even a zero-conflict candidate is not treated as a result until the independent verifier reproduces it and a witness is persisted.
Current worker state
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Interactive structural experiment
C6 × C7 reduced search
Collapse the 903 edge colours into field-displacement classes, search the reduced variable space, then check the final candidate against every K5 exactly.
Ready
The 3- and 7-variable models are exhaustively searched. Larger models use local search, but every displayed final conflict count is exact over all 962,598 K5 subsets.
Experiment result
Choose a structural model and run an experiment.
Exact local-repair track
Bounded Hamming repair around the 2-conflict seed
This is a complete branch search inside the requested edge-flip radius, not a heuristic. Failure at radius d proves only that this particular seed has no zero-conflict colouring within d flips.
Exact evidence: no zero-conflict colouring exists within six edge flips of the published 2-conflict seed.
Symmetry falsification & weak-field signals
How much of the near-solution is actually C6 × C7?
The defect audit starts from the best 21-class translation-invariant colouring, introduces seed-consistent edge exceptions, and scores every intermediate graph exactly. The affine audit exhaustively tests all 1,805 nonidentity maps x ↦ ax+b over F43.
Current conclusion: the 2-conflict graph is not approximately translation-invariant; displacement classes are mostly near 50/50 red/blue. Use field coordinates as features, not as a mandatory global colouring law.
Affine result: no exact affine automorphism or colour-swap symmetry exists. The strongest weak signal is the reflection x ↦ 20 − x.
| Defect step | Edge | Change | Exact score | Δ |
|---|---|---|---|---|
| Run the defect audit to inspect the exact ladder. | ||||
Reduced colouring
Class assignment
No experiment yet.
Exact violations
Monochromatic K5 samples
| Colour | Vertices | Field coordinates |
|---|---|---|
| No experiment yet. | ||
Recovered from the exact score-2 component
Hidden order-43 cyclic coordinate
The 86-state score-2 migration component is a single cycle. Advancing two states is exactly one fixed order-43 permutation of all vertices, verified on all 903 edge colours.
Loading witnessed hidden-cycle evidence…
Exact defect word
One 43-edge orbit carries all symmetry breaking
In the recovered coordinate, hidden difference 5 is the only nonuniform orbit. The published seed uses 18 red edges and 25 blue edges on it.
Loading defect interval…
Compact exact normal form
Exoo Cyclic(43) plus a structured distance-1 defect
Loading exact round-trip normal form…
Exact restricted-family SAT ladder
How many hidden difference orbits must be allowed to break?
Each row fixes the recovered cyclic background outside the named variable orbit family and allows every selected 43-edge orbit to vary arbitrarily. UNSAT here excludes only that structured family.
| Variable orbit family | Instances complete | Variable edges | SAT | UNSAT |
|---|---|---|---|---|
| Loading orbit-search witnesses… | ||||
Independent local proof boundary
Full 903-variable Hamming-ball checks
Exact algebraic coordinate system
F43× ≅ C42 ≅ C6 × C7
Using primitive root 3, every nonzero residue is 3k. The coordinate map is k ↦ (k mod 6, k mod 7). Multiplication by −1 adds 21 to k, flipping the six-direction coordinate by 3 while leaving the seven-phase coordinate unchanged.
| k | Residue | Unity label | Direction | Phase | −Residue |
|---|
Hypothesis generator
Pronic polygon coordinates
The model uses Pm = m(m+1) non-origin states plus one distinguished origin. For m = 6 this gives 42 + 1 = 43, matching the field-43 split without claiming the representation itself proves a Ramsey restriction.
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Machine evidence
Witness files
Current project-state artifacts under /var/www/xavi/maths/ramsey-r55/witnesses. Digests are computed server-side to provide immutable evidence identifiers.
| Name | Bytes | Modified | SHA-256 |
|---|