The manuscript presents a conditional numerical construction for the inverse fine-structure constant. At a cell of edge λ = 2√(ln 2) ℓ_P ≈ 1.665109 ℓ_P (Section 2.1, fixed by a one-bit-per-area relation for Gℏ imported from the companion note [19]), the electromagnetic inverse coupling is identified as 1/e^{2}|_cell = 8K and the strong inverse coupling as 1/g_{3}^{2}|_cell = 4K. The dressing K is obtained from a scalar carrier field θ on the D_{4}* lattice of events (integer points union half-integer-shifted points, 24 shortest bonds of equal length, 12 bonds per event). The carrier weight is P[θ] ∝ exp(-∑_b (cosh ℓ_b - 1)) with unit coefficient and cost function taken from [20]. An integration-by-parts identity on the shift-invariant measure yields ⟨ℓ_b sinh ℓ_b⟩ o 1/12. The mean loop weight is computed as K_m = ⟨cosh ℓ_b⟩ = 1.040851 ± 0.000004. Light is defined as a link field whose energy is ∑_T w_T F_T^{2} with triangle weights equal to the three-link mean of cosh ℓ; its long-wavelength stiffness is the minimized energy of a uniform field strength, giving the second-order result 1.040378 on 4^{4}/6^{4}/8^{4} volumes and the all-orders value K = 1.040399 ± 0.000005 from a single unreproduced computation (Section 2.3). The continuum matching is pure logarithms with zero constant in a dimensional-reduction scheme at precisely μ = ℏc/λ (Section 2.4). Standard-Model three-loop running (Alam–Martin 200 GeV state [22] plus the beta functions of Mihaila et al., Chetyrkin–Zoller and Bednyakov et al. [23–25]) then carries the cell values down, producing the headline α^{-1} = 137.0281 ± 0.0102 (0.77 of the carry error relative to CODATA 2022) together with colour at -0.40 and a weak-share formula at -0.26 (Sections 3–5). A falsifiable target Δα_had^{(5)}(M_Z^{2}) = 0.027878 ± 0.000056 is filed in Section 7.
All internal arithmetic that can be checked from the printed numbers is consistent. The cell energy E_P/1.665109 = 7.3322 imes 10^{18} GeV follows at once. The Gaussian benchmark s e^{s/2} = 1/12 yields s = 0.0800633 and ⟨cosh ℓ⟩ = 1.040844. The second-order gap subtraction 1.040851 - 0.000473 = 1.040378 matches the table in Section 4. The five-component light error budget (hadronic 0.000534, 2022-vs-2019 state 0.000573, α_s 0.000195, top 0.000058, truncation 0.00005) combines in quadrature to the tabulated 0.0008109. With K = 1.040399 one obtains 8K = 8.323192, residual -0.0006258 against the asked 8.3238178, pull -0.77; division by the stated slope d(1/e^{2})_cell/d(1/α) = 10.888/137.036 = 0.0794553 recovers α^{-1} = 137.0281 ± 0.0102 exactly. Colour 4K = 4.161596 pulls at -0.40 and back-solves to α_s(M_Z) = 0.11835. The weak-share algebra with D = 4K_m/(K_m + 3) produces (8 - D)/(18 - 3D) = 0.4674804 and pull -0.26. The filed hadronic target and its three-sigma decision thresholds are recovered from the same residual and slope. Figure 1 and the Section 6 table of Δα_had determinations are likewise internally coherent. These checks confirm that the numerical pipeline, once the discrete package is fixed, is free of arithmetic error.
The decisive limitation is that the package itself is not derived. Section 2.2 states that the integers 8 and 4 (and the doubling that converts a geometric link count of 4 into light’s 8) “were identified with the measured couplings in view (the measured couplings over K are 8.0006 and 4.0019) and are adopted … together with the doubling for light, which is not derived.” Section 2.3 adopts the long-wave stiffness among three computed carrier numbers (mean, own stiffness, long-wave) because only it lands. Section 2.4 adopts zero-constant dimensional-reduction matching at exactly ℏc/λ; the author notes that a typical lattice matching constant of O(0.1–1) would move light by hundreds of its errors and that this choice is “the largest of the readings adopted here.” Section 5 adopts the algebraic form of the weak share after inspection of the measured ratio. Section 4.1 enumerates eighteen recorded combinations of read, convention and share form and records that only one lands all three couplings inside one error; it further states that unrecorded alternatives (matching constant, scale multiple, triangle weighting, weight coefficient) each move light by several to hundreds of errors, so “the number of alternatives behind the landing cannot be bounded from what is written here.” Section 8 lists the same items as still owed. Consequently the claim “computed … with no continuous number fitted to light” is narrowly true of continuous parameters yet false of the discrete structural freedom that actually controls the result. The -0.77 landing is the success of one selected package, not a unique consequence of the ledger axioms.
Two further technical gaps prevent independent verification. First, the central number K = 1.040399 ± 0.000005 is obtained from “a single computation that has not yet been repeated independently” (Section 2.3). No ensemble size, update algorithm, autocorrelation analysis, covariance construction, finite-volume extrapolation, or higher-order correction method is supplied; the all-orders value lies 0.000021 (four times the quoted error) above the second-order result, with no explanation of the shift. Until a reproducibility package exists, the uncertainty on K is an internal statistical claim only. Second, the renormalization-group carry assumes that the Standard Model spectrum is complete up to ~7.3 imes 10^{18} GeV (three reduced Planck masses) and that gravitational corrections to the beta functions remain smaller than the 10^{-4} target. Neither assumption is bounded (Section 3). An O(1) Planck-threshold correction would invalidate the entire numerical comparison, yet the quoted ±0.0102 error on α^{-1} and the ±0.00003 claim on α_s(M_Z) exclude it.
The auxiliary checks do not restore independence. Colour shares the same K and matching convention and has a loose window (±0.0012). The weak-share form was chosen with the data in view; nearby simple fractions also land, and the same form sides the opposite way under minimal subtraction. The quark-mass factor-of-three comparison rests on an unpublished mass table whose six free terms were already fitted to the observed masses; the asked dressing 1.040243 ± 0.000241 is an interpolation, not a direct run, and supplies no independent constraint on K. Closure itself holds inside one error only on lattice determinations of Δα_had^{(5)}(M_Z^{2}) that already lie near the required 0.027878; pure e^{+}e^{-} evaluations lie 2–3 σ lower (Section 6 and Figure 2). Filing a test value above every published determination converts the tension into a future prediction, but does not constitute present agreement.
Relative to the historical literature the paper correctly recalls that α remains an unexplained input of the Standard Model (Sommerfeld, Dirac, Eddington, Feynman). The construction is not a conventional lattice-gauge or GUT calculation; it is a discrete statistical-mechanical model whose continuum identification is postulated. The fluctuation identity and the second-order relaxation formula are standard, but the physical readings that convert them into gauge couplings are not. The work therefore functions as a precisely documented consistency check inside an unfinished framework, together with a sharp, falsifiable conditional prediction for the hadronic vacuum polarization. It does not yet constitute a first-principles computation of the fine-structure constant. The abstract and Section 1 statements that the constant “is computed” should be aligned with the candid conditionality already present in Sections 2.2, 2.4, 4.1 and 8; the central K computation must be made reproducible; and the error budget must display the unbounded Planck-scale systematic. Until those steps are taken the numerical agreement, while internally consistent, carries no calibrated statistical weight beyond the success of one selected package among an unbounded discrete menu.
preprint published 2026-11-14 19:34 UTC by preprint-admission
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