The starting point is a familiar frustration. Dimensionless constants such as the Thomson-limit inverse fine-structure constant, approximately 137.035999, and the proton-electron mass ratio, approximately 1,836.152673, can be inserted into successful gauge theories with perfect consistency, yet those theories do not explain the values themselves. Precision metrology, including atom-recoil measurements of the fine-structure constant and part-per-trillion spectroscopy of molecular ions, pins the numbers down ever more sharply, but precision is not explanation. The new hypothesis addresses this narrower gap: it asks whether the electromagnetic and baryonic sectors might realise two different scalar evaluations of one shared finite-boundary structure, rather than postulating a direct numerical equation between the two constants.

The mathematical heart of the proposal is a rank-three integral carrier reduced modulo three, producing a finite projective incidence geometry. For the structural choice (d, q) = (3, 3), exact projective identities yield 13 ray classes, 52 incident flags, and a restricted incidence relation in which the operator D3 acting on its adjoint gives three times the identity. Quotienting out a single transparent identity direction from the ray-and-covector event module leaves a 25-dimensional punctured event space. These counts are presented as exact finite-geometric results, not retrospective decompositions of measured constants, and the author is careful to distinguish what is proven from what is assumed.

From this common parent, two typed branch maps lead to inequivalent evaluator packages. The electromagnetic branch, called the Frobenius-aperture evaluation, retains the integral rank and the event count, splits the carrier into longitudinal and transverse channels, and evaluates a minimally connected eight-link support with a topology-independent Frobenius metric. A residual correction is modelled by the sinc kernel integrated over its first positive lobe, cut off at its first zero at pi, and averaged uniformly over the 25 event directions. The resulting scalar is 137 plus a small correction involving the sine integral of pi, giving 137.035999171883, which is conditionally identified with the inverse fine-structure constant.

The baryonic branch takes a different route. It retains the full projective ray-covector flag geometry, adds a directed transfer relation under which each flag connects to nine others, and evaluates the data through an event-to-flag leverage ratio, a reservoir capacity, a phase quotient identifying two boundary orientations, and a one-mode Schur complement reduction. Motivated by the order-three centre of the SU(3) colour group, the branch specialises to q = 3, producing an integer backbone of 1,836 and a fractional correction involving a factor of 425 and the orientation-conditional density 1/pi. The output, 1,836.152673425907, is conditionally identified with the proton-electron mass ratio.

What distinguishes the paper from many numerological attempts is its bounded no-retuning audit. Before any comparison with metrology, the author enumerated all 47 non-isomorphic nine-vertex tree topologies and all 256 orientations of each, a total of 12,032 representatives, and tested simple aperture, weighting, phase, reservoir and Schur alternatives one discrete ingredient at a time. No coefficient, kernel width, phase, rank or normalisation was fitted. The audit found that connectedness and primitivity uniquely select the eight-link spanning-tree support, that the Frobenius functional is the only tested elementary positive functional that is both non-trivial and invariant across all representatives, that the first-zero cutoff at pi is isolated by positivity and stationarity, and that uniform averaging over 25 events is the unique unit-total mean weight.

The audit also honestly catalogues what remains unresolved. Three degeneracies survive: a comparably simple sinc-squared aperture kernel gives a slightly different but still plausible value unless an action-level amplitude principle is supplied; the oriented phase circle with density 1/(2 pi) survives unless the physical necessity of the orientation quotient is derived; and an integer reservoir alternative without the inverse-incidence weight remains a live competitor. The author concludes that the ingredients enjoy partial conditional isolation within declared classes, not a global uniqueness theorem, and that the choice (d, q) = (3, 3) itself lacks a dynamical selection theorem.

Crucially, the paper is explicit about what the hypothesis is not. It is not a direct equation linking the two constants, not a unification of the U(1) and SU(3) gauge groups, and not a first-principles calculation from the QED or QCD actions. Both fixed scalars are numerically compatible with the 2022 CODATA recommendations, with signed uncertainty diagnostics of minus 0.244 standard uncertainties for the electromagnetic value and minus 0.0029 for the baryonic value, but these are compatibility diagnostics rather than discovery significances, because the target observables were known during model construction. The word prediction is reserved for prospective comparisons made after the architecture and evaluators have been frozen.

The proposal is nonetheless framed as sharply falsifiable. It can fail mathematically, if an error is found in the incidence counts, quotient or enumeration; through naturality failure, if broader admissible evaluator classes reveal large unresolved degeneracy; through dynamical failure, if no gauge-invariant finite-boundary action produces a branch evaluator; through sewing failure, if the two sectors require incompatible gluing laws; or metrologically, if future recommended values exclude one of the frozen scalars. The immediate programme is to derive the minimal support, Frobenius response and aperture kernel from a finite Abelian boundary action on the electromagnetic side, and the directed flag transfer, reservoir and Schur response from centre-projected baryonic boundary data on the other, with both branches required to descend from a common sewing quotient.

Whether the hypothesis survives that test remains entirely open, and the author does not claim otherwise. But the work offers something rarer than another decimal coincidence: a fully typed logical architecture in which every ingredient is labelled as exact result, representation consequence, discrete hypothesis or conditional identification, together with a reproducible computational audit released as supplementary material. The explanatory claim is deliberately narrower than a theory of everything and stronger than arithmetic luck. The real question it poses is whether finite Abelian and centre-projected gauge dynamics can independently realise the same integral incidence architecture, and if so, whether that shared structure is a hint of something deeper beneath the Standard Model’s unexplained numbers.

Subject of Research: A finite projective-geometry hypothesis deriving the fine-structure constant and proton-electron mass ratio as scalar evaluations of one boundary structure

Article Title: A rank-three integral boundary closure hypothesis for electromagnetic and baryonic scalar evaluations

Article References: McCoss, A. (2026). A rank-three integral boundary closure hypothesis for electromagnetic and baryonic scalar evaluations. Frontiers in Applied Mathematics and Statistics, 12, Article 1965695. https://doi.org/10.3389/fams.2026.1965695

Image Credits: AI Generated

DOI: 10.3389/fams.2026.1965695

Keywords: fine-structure constant, proton-electron mass ratio, finite projective geometry, boundary gauge theory, dimensionless constants, CODATA, QED, QCD, projective incidence, Schur complement, falsifiability, mathematical physics