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

The framework at the center of the paper is known as extended Aharonov-Bohm electrodynamics. In this formulation, the Lagrangian of the electromagnetic field is supplemented with an additional term proportional to the square of the divergence of the four-potential. When the primary source current fails to satisfy a local continuity equation, the field equations generate a non-local secondary current whose divergence exactly compensates that of the primary current, so that the observable total source remains conserved. The authors emphasize that this construction, sometimes described as charge-conservation censorship, has been explored in earlier work on non-local quantum sources, radiative multipoles, gauge waves and couplings to scalar-tensor gravity. Crucially, they also stress that the theory is not an alternative explanation of the well-established magnetic Aharonov-Bohm interference effect; the name refers instead to the reduced-gauge line of development associated with Ohmura and with the later discussion by Aharonov and Bohm of the physical role of the electromagnetic potentials.

The central question the authors pose is deceptively simple. Can rotation, or more generally the shift-vector structure associated with a rotating frame or a stationary gravitomagnetic background, generate an effective source of the kind required by the scalar sector of extended electrodynamics, even when the underlying charged matter is entirely standard and its microscopic current remains conserved? Their answer comes in two parts. At the microscopic level, the answer is a firm no. Using what is essentially Noether’s theorem in curved spacetime, they prove that for any matter field minimally coupled to electromagnetism and gravity through a generally covariant, locally gauge-invariant action, the physical four-current is covariantly conserved on shell. Neither a gravitomagnetic field produced by rotating masses nor a mere change of coordinates can manufacture a genuine charge anomaly.

At the level of observer-adapted transport variables, however, something genuinely interesting happens. When the conserved four-current is decomposed using the standard three-plus-one splitting of spacetime into space and time slices, the continuity equation for the charge density and transport current measured by observers attached to the rotating frame acquires an exact additional term. This term, which the authors call the observer-split source and write as the spatial divergence of the charge density multiplied by the shift vector, reduces in the weak-field limit of rigid rotation to a remarkably compact expression: the angular velocity times the angular derivative of the charge density. The term is not a new law of physics, the authors insist, but rather the unavoidable bookkeeping that appears when an exact conservation law is rewritten in variables adapted to a rotating device.

A subtle but important part of the analysis concerns what physically selects the observer structure. The authors argue that the relevant foliation of spacetime is not an arbitrary coordinate choice. In a laboratory realization, the synchronization and readout protocol of the experiment selects the hypersurface normal, while the rotor’s phase and angular motion fix the evolution field relative to those hypersurfaces. The resulting source is therefore relational: it depends on the specified laboratory observer structure together with the rotating source motion, much as a medium four-velocity forms part of the physical specification in relativistic descriptions of matter in motion. A passive change of coordinates cannot alter any predicted observable, but changing the physically supplied foliation or evolution field genuinely changes the transport variables and hence the split source itself.

The phenomenological step of the paper is deliberately restricted. The authors postulate that the exact observer-split divergence can act as the source entering the scalar field equation of extended electrodynamics, which in curved spacetime reads as the covariant d’Alembert operator acting on the scalar field equaling the split source. They show that the split source is the four-divergence of a geometrically well-defined auxiliary current, constructed from the physical current plus a shift-vector contribution, and they introduce a minimal longitudinal surrogate primary current whose associated secondary current restores total conservation while leaving the ordinary Maxwell source untouched. In this minimal closure, the standard electromagnetic field remains exactly as Maxwell predicts; only the additional scalar and potential sector responds to the rotating-frame source.

What makes the proposal scientifically serious is that the effective source has sharply defined, testable properties. It is odd under reversal of the rotation direction, so flipping the sense of spin must flip the sign of any linearly related response. It vanishes for a spatially uniform or exactly axisymmetric charge distribution, since only angular gradients of the density contribute. It is controlled by spatial gradients rather than density offsets, and on a closed ring its positive and negative angular contributions cancel exactly for a single-valued periodic density. Because the term is a spatial divergence, its volume integral is governed by boundary fluxes, naturally producing source-sink or bipolar patterns. A genuine microscopic extra-current need not obey any of these restrictions, which is precisely why the model is falsifiable: an effect that survives rotation reversal or persists when axisymmetry is restored would not support the proposed closure.

The authors also spell out what would be needed for a true microscopic violation of charge conservation, going beyond their effective bridge. Three routes are identified: intrinsically non-local matter dynamics such as fractional wave equations, quantum-coherent macroscopic sectors that could exhibit an effective anomaly in the current operator, or a genuinely new coupling between the scalar sector and a gravitomagnetic invariant built from the vorticity of the rotating congruence. These possibilities lie beyond the present paper, but they frame the effective construction as a computable, bounded surrogate rather than a claim of fundamental charge non-conservation. The authors note that any sufficiently small freely falling frame removes the apparent non-conservation entirely, so the split source acquires meaning only after coarse-graining over the spatial and temporal scales set by the apparatus and its measurement protocol.

On the experimental side, the paper connects to a companion proposal involving a small rotating aluminum disc carrying a fiber-optic ring, of which only a localized angular sector is optically active. Illumination modulates an effective surface charge density in the metallic carrier, to be calibrated independently by electrometric or capacitive means synchronized with the optical modulation and rotor phase. The effective source then takes the form of the angular velocity times the angular gradient of the induced density, and its peak amplitude scales linearly with the rotation rate and inversely with the angular width of the illuminated sector. Four direct controls are proposed: optical on-off gating, clockwise versus counterclockwise reversal, comparison of a localized sector with nearly uniform illumination, and comparison of a conductive carrier with insulating dummy controls. The authors further show, through an exact Gaussian smoothing model and a general Fourier-mode theorem, that any linear suppression network with bounded gain can only reduce the effective source, giving a compact suppression law for low-pass filtering of transients.

Looking forward, the authors identify analogue-gravity systems as another promising arena, since acoustic or optical analogue spacetimes provide a material flow that naturally selects a preferred laboratory congruence while perturbations propagate on an effective curved metric. The overall message is one of disciplined modesty combined with genuine opportunity. Rotation does not violate charge conservation, and the authors prove it. But the exact observer-split term that emerges in rotating descriptions of conserved currents is computable, symmetry-constrained and experimentally addressable, and whether it triggers any real macroscopic scalar response in extended electrodynamics is now framed as a sharply posed experimental question rather than a matter of speculation. A null result would bound the proposed scalar response; a signal lacking the predicted reversal or axisymmetry dependence would refute the closure outright.

Subject of Research: Effective observer-split source terms arising from rotating frames and gravitomagnetic backgrounds as phenomenological sources in extended Aharonov-Bohm electrodynamics

Article Title: Effective observer-split source terms in rotating frames and gravitomagnetic backgrounds in extended Aharonov-Bohm electrodynamics

Article References: Iadicicco, A., Modanese, G., & Verolino, L. (2026). Effective observer-split source terms in rotating frames and gravitomagnetic backgrounds in extended Aharonov-Bohm electrodynamics. General Relativity and Gravitation, 58(9), Article 106. https://doi.org/10.1007/s10714-026-03606-2

Image Credits: AI Generated

DOI: 10.1007/s10714-026-03606-2

Keywords: extended electrodynamics, Aharonov-Bohm electrodynamics, observer splitting, rotating frames, gravitomagnetism, charge conservation, ADM decomposition, shift vector, scalar field, falsifiability, analogue gravity, general relativity