Prime-Gap Probe

The prime gap sequence is the list of distances between consecutive primes. Its first moment - the average gap - grows like the logarithm of the prime, and that average imposes a great deal of structure on its own. The question this study asks is whether anything is left over once that first-moment behavior is accounted for: does the sequence carry spectral and topological structure beyond what its mean already explains?

The question has three possible answers, and the study is designed so that any of the three is a publishable result. The sequence may carry structure that agrees with the spectral dimension d_s = 1/2 reported by Watson (2025); it may carry structure that disagrees in a characterizable way; or it may fall into a third regime indistinguishable from the null baselines. Watson 2025 is a comparison benchmark, not a target to reproduce - the construction used here (a graph Laplacian on the prime-gap degree sequence) is formally independent of Watson operator-on-primes construction, so a different answer is a different question, not a contradiction.

The output is a single number-theory-adjacent paper. Target categories are math.NT primary, with math.SP and math.CO cross-lists. The endorsement chain is independent of every other research program in the portfolio.

The probe is bounded on purpose, and naming what it is not is part of the discipline. It is not a Substrate Geometry methodology extension - that was the original framing and it was dropped, because the instinct toward multi-metric vector characterizations does not transfer to a single number-theory question and inflated an early version of the toolstack to three tools where two are sufficient.

It is not a re-derivation of nearest-neighbor spacing results on primes; that space is closed (Wolf 2014, Timberlake-Tucker 2007), and the topological observable here is deliberately carved away from it. And it is not a rigorous-mathematics paper: no theorems are proven. It is a computational characterization paper, making claims about numerically computed observables on a finite range, with pre-committed thresholds, baselines, and sensitivity analyses. Reviewers are told this in the first section.

Five principles govern the study, and relaxing any of them collapses its defensibility. First, pre-committed numerical thresholds: each surviving tool has exactly one gate threshold, a number fixed before computation, and the paper survives or fails on whether the computed value crosses it - not on qualitative agreement.

Second, mandatory baselines: every observable is computed against Poisson (memoryless null), GUE (random-matrix null), and uniform (structureless null), and the comparison is reported even when the prime-gap result is null. Third, sandboxing: each tool is verified on known-answer cases (path graph, complete graph, circle) before it is ever pointed at primes. Fourth, sensitivity layers are kept strictly separate from the primary observable. Fifth, subtractive over additive: a tool with an unresolvable foundational issue is dropped, not patched - one continuous-PDE tool was already removed on exactly this principle.

Two independent tools each ask one question. The topological tool computes the persistence diagram of the gap sequence under a raw one-dimensional sublevel filtration and asks whether it deviates from the Poisson, GUE, and uniform baselines, under bottleneck and Wasserstein distances, by a pre-committed margin. Higher-dimensional homology under Takens and sliding-window embeddings is a sensitivity layer only, never the primary, because embedding choices introduce parameter dependence.

The spectral tool takes the Havel-Hakimi greedy realization of the prime-gap degree sequence, verifies graphicality directly on the computed sequence rather than citing an asymptotic threshold, and asks whether its graph-Laplacian spectrum yields a spectral dimension that agrees with Watson d_s = 1/2, disagrees by a pre-committed margin, or sits in the Erdos-Renyi-at-matched-edge-count regime. A 100-realization configuration-model ensemble, with pinned seeds, is the sensitivity layer. The two tools share no construction steps, so a finding from one cannot contaminate the other, and the paper reports both regardless of outcome.

If both tools cross their thresholds, the paper reports a characterized deviation of the prime gap sequence from baselines, on two methodologically independent observables, with explicit comparison to Watson 2025. The contribution is the characterization, not a discovery claim.

If one tool crosses and the other does not, the paper reports the asymmetry, which is itself informative because it constrains what kind of structure the sequence carries. If neither crosses, the paper reports a null result against three non-trivial baselines, with thresholds met and graphicality verified. The reason all three branches are defensible is the same reason the study is worth running at all: the gates were committed before the data was seen.

The manuscript is on file and has been through several scoping iterations (v3 through v6) hardened by adversarial review. It is not yet publicly released; this page describes the study while the preprint and endorsement steps are pending. When the preprint lands it will be added here and to the publications list, with its arXiv identifier and the usual inline preview.