Ball-Lens: Order and Counting on Discrete Substrates

1. What this is

An ongoing research program asks which physical structures emerge from minimal primitives: a set carrying a graph metric and a counting measure, and nothing else. Along the way the program produced results that are self-contained mathematics, independent of whether the physical program succeeds. Those results are published.

The repository is the canonical source. This page exists to say what the program is, where it currently stands, and how to attack it. It deliberately does not summarize the results, because a summary is a second copy that drifts from the first, and a claim that exists in two places will eventually exist in two versions.

2. Scope, stated plainly

What is claimed: the material in the repository's papers directory is proven mathematics, machine verified by the scripts published alongside it, with each claim's evidence strength labeled individually as proof, computational, witness, or open.

What is not claimed: a physical theory. The broader program is ongoing and is not asserted by this release. Any reading of these results as established physics is a misreading, and the repository says so on its front page.

3. Status

The current public state is v2.0.0-rc1, a release candidate dated 3 August 2026. It is deliberately flagged as a pre-release, which means the repository host still shows v1.0.0 as the latest release; the release candidate is the one to read. Final v2.0.0 follows an external review window.

Before the candidate was assembled it was handed to a verification pass with no access to the authoring context and instructions to attack the claims rather than confirm them. That pass returned a verdict of release ready with corrections. It found a geometric misdescription in one theorem, where the counts and formulas were correct but the region was described in an unstated coordinate system. The correction is recorded in the release notes and in the paper, and the review itself is published unedited.

4. The open problems

These are stated as open in the repository rather than deferred to a future version. The first is the one worth an outside mathematician's time.

• Uniform diluted winding rigidity. The published grid result is per-instance exhaustion and is deliberately not claimed as universal. What is wanted is either a checkable hypothesis short of per-substrate exhaustion, or counterexamples showing that none exists. The problem is boxed in the paper, with a complete certifier and a counterexample toolkit provided.
• Symbolic proofs of the whole-boundary forms. Verified computationally over a stated finite range only. A derivation, or a counterexample outside that range, would both be accepted.
• Single-realization concentration. One isotropy result is proved in ensemble mean. Whether it concentrates on a single realization is an open hypothesis supported by measurement, and the qualifier is load bearing.
• Characterization of a measured ordering-fraction invariant. Measured, not explained.

5. How to report an error

Open an issue on the repository, or contact the author directly, with three things: the claim attacked, identified by its manifest entry; the explicit object, meaning the substrate parameters or edge list, the assignment itself, or for a proof defect the exact line and the gap; and a command sequence that exhibits the failure from a clean checkout, if the failure is computational.

The reviewer guide in the repository lists the claims most worth attacking, in descending order of value if broken, together with the author's own list of the weakest joints. Reports that survive verification are recorded with attribution, whether or not they are fatal to the claim attacked. Negative results stay in the record. The objective is not to be right; it is to find out.

6. Priority and provenance

Release tags plus an OpenTimestamps proof of the release commit hash, filed in the repository, constitute the public priority record. It is verifiable against any Bitcoin node, independent of the repository host and independent of the author.

7. Why this is on a business site

Because the method is separable from the subject matter, and the method is the part that transfers. The governance system this program runs on, positions filed before results exist, records that only grow, tests that can be voided, and verification by someone who cannot see the author's reasoning, is described in the companion piece linked below. It is the same system used on client work.