An open notebook at the edge of machine reasoning
Provisional mathematical experiments by Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol. Proofs, programs, and exact certificates are published together.
I am a complete amateur and cannot independently verify the claims published here. I am exploring the limits of AI-assisted mathematics. Every proof, computation, citation, scope statement, and novelty claim requires independent checking by qualified experts. Passing the supplied programs is useful evidence about the encoded algebra, but it is not peer review.
Eight provisional artifacts remain current. Every claim requires independent expert verification.
Quantum information · Conjecture 1 · Exact all-dimensional bound
A polar-decomposition certificate proves that the cyclic Bell operator of Perito et al. has value \(2\csc(\pi/(2d))\) for every \(d\ge2\), even in the commuting-operator model. Thus its finite-dimensional, approximate, and commuting values coincide.
Graph theory · Certified finite frontier · Conjecture still open
Combined with the published exhaustive result through order 11, a complete one-guard parameter split closes order 12 and places any counterexample at 13 or more vertices.
Graph theory · Unbounded separation · Smallest exact counterexample
Published results combine to give an explicit family with \(\vartheta(G)/\gamma^\infty(G)=\Theta(|V(G)|^{1/3})\). Separately, the graph IEhbtj{ro is an exact minimum-order witness: \(\gamma^\infty(G)=3<7593/2500\leq\vartheta(G)\).
Discovery 07 · Canonical consequence paper · Every order
One formal-inverse mechanism joins an exact reduced three-point fiber, failure of SIC(14) at every exponent, and homogeneous Hessian-nilpotent Vanishing obstructions at every order in 30 variables at degree eight and 44 variables at degree four.
Discovery 04 · Third-iterate upgrade · Exact certificates
Provisional determinations of the degree-nine and degree-27 geometric monodromy groups as W2 and W3, with exact certificates. A separate, independently audited bounded-memory certificate proves the degree-81 group W4; none of these claims has external peer review.
H(668) paper A · Exact local obstructions · No matrix
A fixed-q obstruction, a distance-80/41 frontier, one proof-certified long support exclusion, all nine short support exclusions, and an exact orientation cascade for twenty open long cases.
H(668) paper B · Fixed-compression chart · No Legendre pair
Five shell-two orbits, eighteen dense orbits in a stated census scope, exact placement geometry, primitive-unit certificates, and three pair-resultant norm keys inside one prescribed-compression chart.
H(668) paper C · Semiregular conference lane · No graph
A complete 625-class quotient census, a universal diagonal law, exact modular realizations, and scoped low-rank obstructions for the nine-orbit conference-matrix lift.
Active and paused notebooks that separate the global problem from their exact partial results.
Research program · Active 26 July 2026 · Universal conjecture unresolved
The first 24 hours produced a certified order-12 exclusion, an analytic order-13 odd-hole obstruction, and one exact order-13 template exclusion. The campaign is now prioritizing a minimum-counterexample proof before any move to order 14.
Research program · Paused 25 July 2026 · Three scoped papers
We pursued Eliahou-seed repair, Legendre pairs of length 333, and semiregular conference lifts. Exact obstructions and classifications survived the audit, but no matrix was found and the final complete join remains infeasible on the present architecture.
Research program · Paused 24 July 2026 · One endpoint theorem
A minimum-miss capacity inequality and exact verifier exclude the (85,128) endpoint inside the hypothetical regular degree-18 branch. The result supports a short technical note, but the exact Ramsey number remains unknown.
Research program · Paused 24 July 2026 · No resolution
We explored whether 40–44 unit spheres can touch one sphere in five dimensions. The exact value remains unknown: this notebook verifies the 40-point construction, records scoped exact obstructions and failed approaches, and keeps the computations reproducible.
Research program · Paused 24 July 2026 · No resolution
We pursued the open lower-bound question for the number of graph cycle sets. The checkpoint preserves a protected difference-support reduction, exact shadow counts, an all-m twin-boundary identity, finite Hall tests, and failed mechanisms.
These files remain public to preserve the timestamped research trail. They are not separate current discoveries.
Discovery 06 · Flagship technical precursor
The detailed 24-term construction, regular-nilpotency certificate, rational collision, every-exponent SIC formula, and scoped constant-state optimality proof are incorporated into Discovery 07.
Discovery 03 · 22/44-variable technical precursor
The factor-reusing 13D reduction, 22D cubic map, and exact 538-term quartic certificate are incorporated into Discovery 07, which upgrades the quartic conclusion to every-order nonvanishing.
Discovery 05 · Superseded quantitative milestone
This 72-term construction lowered the explicit witness to SIC(21) and supplied an intermediate inverse lemma. Discovery 07 incorporates the mechanism and supersedes it with a 24-term SIC(14) witness at every positive exponent.
Exploration 01 · Archived and subsumed
The monodromy theorem was already available in stronger form for the full Gallagher family. The particular seed and uniform rational collision survive as Appendix A of the archived Discovery 03 precursor.
Exploration 02 · Archived and superseded
The 13-variable derivation fed the archived Discovery 03 construction and the canonical Discovery 07 route; the exact 54-variable certificate remains here for provenance.
Fast, open, reproducible disclosure—paired with unusually strong uncertainty labels.
The site records experiments and potential discoveries. It does not certify them as true or novel.
Exact source and verifiers are public so specialists can reproduce, criticize, and improve the work.
The byline, AI assistance, public-release time, and evolving status remain attached to every paper.