The manuscript studies quantum gravity on the fixed Kuhn lattice of a four-counter recognition ledger, with flat metric g_κ = I - κ 11^T. It recalls earlier author preprints establishing two headline results for Regge’s hinge charge s A δ: a regulated Lorentzian path sum that is finite on every finite window of squared lengths at κ = 1 on the null stratum (Theorem 3.1, Haar measure of V_0 restricted to Adm_1 ∩ {|ℓ_e| ≤ L}), and the statement that the metric part of the lattice field equations equals -½ times the Einstein density of the sampled 2-jet at every order in field amplitude, leading order in lattice spacing b, for every κ > ½ (Theorem 4.1). Newton’s law follows for the sourced linear field with G = 1/(8π s). The new material of the present paper is a selection of that charge among area-and-deficit charges C(A, δ). Two independent routes are proved here. Gauge consistency on the Euclidean member κ = 0 (quadratic annihilation of every vertex move plus cubic vanishing at three explicit Gaussian-integer witnesses W_1, W_2, W_3 on the Euclidean graviton shell) forces the jets (value, slope, curvature) of a single area-form function φ(A) at the four Euclidean flat areas to be Regge’s jets (λ, λ, 0) up to a common scale (Theorems 5.2–5.3 and Table 1). The record clause (Definition 2.2) together with either a content-reading premise C = F(A δ) or an area-form premise C = φ(A) δ then fixes the whole function and the scale exactly: C = s_p A δ (Theorem 5.5). The long-wave cubic vertex of the selected action is carried to κ = 1 and shown to equal half of Einstein’s ΓΓ vertex (Theorem 5.6), a normalization already present in the classical Regge action and in the field equation of Theorem 4.1 (Remark 5.7). Neighbouring classes (one function per hinge class, ratio charges, shape charges, the recognition-cost charge) are examined; the recognition-cost rival fails the cubic test and its unique quadratic-preserving ratio correction restores Regge jets (Theorem 5.10). Quartic contact vertices, tree-level scattering, birefringence beyond long wavelength, and a conditional one-loop analysis are discussed, most of them resting on companion preprints or unpublished work.
The algebraic core of the selection argument is sound under its stated premises. For ratio charges the per-simplex cubic table reduces to a linear combination of three closed-form sums X_1, Y_1, Z_1; the three exact witness values printed for W_1 force c_0 = c_1 and c_2 = 0, i.e., Regge jets (Theorem 5.2). The same arithmetic extends to the eight jets of a single area-form function: the quadratic test confines them to a three-parameter span, the three witnesses supply six real linear equations whose coefficient matrix has nonzero determinant, and the only solution is Regge jets at a free scale (Theorem 5.3). The record-clause theorem is a pure substitution: the clause supplies linearity in area at one fixed deficit 2π/k, and either form premise carries that value to every deficit (Theorem 5.5). Negative areas are used only to fix the content-reading function F on negative arguments; on positive areas the area-form case needs no extension. These checks confirm the local selection statements. The cubic vertex identity at κ = 1 is reduced by symmetry to twenty representative triples; the proof scheme is valid, but the twenty identities and the 220 rational-function coefficients of the continued table T(κ) are not printed.
Two structural gaps separate the proved selection from the title claim. First, the finite sum of Section 3 integrates only over the eleven non-posting edge classes (null stratum V_0 at κ = 1), while the field equation of Section 4 varies all fifteen classes freely. No Faddeev–Popov reduction or gauge-slice argument is supplied; the manuscript correctly flags the reduction as open (Section 1 and Figure 1). Until it exists the two results live on genuinely different configuration spaces, so the title phrase “a finite sum over geometries with Einstein’s equations at every order” joins theorems that do not yet share a measure. Second, the value κ = 1 is an identification (one tick = one posting, transfer reading of light’s step). The alternative reading (tick = double entry) yields κ = ½, at which the (1,1) hinge has vanishing area, the cubic table T(κ) has poles, and none of the Lorentzian headline results is established (Appendix D, Table 4). The companion law-of-motion preprint currently favours the value at which the present Lorentzian theorems fail. The physical interpretation of the selected Regge action as gravity therefore rests on an undecided identification that the abstract and introduction should elevate to a central caveat.
The gauge route itself has a limited domain. It is proved only on the Euclidean member κ = 0 and only at three witnesses that lie on the Euclidean graviton shell. For charges whose own quadratic form has a different shell (u_3 ≠ 0) the cubic test is a condition posed on a foreign shell, not the charge’s own solvability condition (Section 5.4). The cubic test on the physically relevant κ = 1 lattice is left open. Moreover the jets fixed by the Euclidean test are jets at the four Euclidean flat areas ½, √2/2, √3/2, 1; two of the four flat areas of the κ = 1 lattice (3/2 and √3) are not among them. Consequently the gauge route alone does not make the action Regge through cubic order around κ = 1; the record-clause route (an identification plus a form premise) is required to close the gap (Section 5.5). The manuscript acknowledges these limitations, yet the broader phrase “gauge consistency” is sometimes used where only “selection by the stated Euclidean tests” has been proved.
Reproducibility of the exact-computation theorems is incomplete. Theorems 5.2, 5.3, 5.9 and 5.11 rest on unprinted tables (eight quadratic and twelve cubic 10 × 10 tables at the Euclidean simplex, their class and shape analogues, and the 220-coefficient rational table T(κ)). The paper prints the witnesses, the five quadratic conditions and the six cubic equations of Table 1, and offers the tables on request. For a result whose novelty is a selection theorem proved by computer algebra, the tables (or a machine-readable repository with a permanent DOI) must be part of the archival record. Without them an independent referee cannot re-derive the determinant of Table 1 or the twenty representative identities of Theorem 5.6. The same gap affects the quartic claims imported from the companion preprint [12]. The elementary algebraic checks that can be performed from the printed text (flat-lattice formulas, ratio-charge witness arithmetic, record-clause substitution, recognition-cost jets) all hold; the larger linear-algebra statements remain conditionally correct but not fully reproducible from the manuscript alone.
The finite-sum theorem, the Einstein-equation theorem, Newton’s law, the graviton quadratic-form identity, the birefringent correction and essentially all one-loop statements are proved only in the author’s earlier Zenodo/Pith preprints or in work marked “proved, unpublished.” None of those sources has undergone peer review. The present manuscript is therefore a selection paper that rests on an unverified foundation. The one-loop claim that the near-flat graviton is not massless is conditional on three open inputs (imaginary-part bound, loop modulus in the complete gauge, and a proof that the real part of the remainder is nonzero); it controls only type-E strains, the computed numbers are numerical, and the only prescription that yields a finite ordered limit at nonzero momentum is itself a named premise. The abstract’s phrasing “the paper claims no … massless graviton” can be read as a positive exclusion; the conditional character should be stated in one sentence. The ledger-to-lattice identification itself still contains an unproved step (the four-account to three-account correspondence), and the record clause, writing principle, (I1), (I2) and the premise that every small-amplitude field is physical are all inputs. The logical distance between the ledger and the selected Regge action is therefore a chain of open identifications that the introduction should make fully visible.
Comparison with prior literature is appropriately drawn. The long-wave recovery of linearized gravity on the Kuhn lattice continues the weak-field analysis of Roček–Williams; the continuum-limit identities at Euclidean signature are consistent with the measure-theoretic convergence theorem of Cheeger–Müller–Schrader; the breaking of vertex-move symmetries on curved solutions recovers the observation of Bahr–Dittrich; corner doubling is already known from Menotti–Pelissetto. The gauge-test route is a lattice analogue of the continuum consistency arguments of Boulanger–Damour–Gualtieri–Henneaux for spin-2 self-interactions. The paper’s distinctive contribution is the explicit selection of Regge’s charge inside a concrete class of hinge charges on one fixed triangulation, together with an unusually rigorous status discipline (theorem / identification / open / numerical / inference / model) that Table 3 and Figure 3 make transparent. That contribution stands, but only after the configuration-space gap, the κ-identification, the limited domain of the Euclidean tests, and the missing computational artifacts are addressed. The recommendation is therefore major revision: supply the tables as a permanent supplementary artifact, rephrase the title and abstract so that the finite sum and the field equation are not presented as a single package, elevate the open status of κ = 1, and either include self-contained accounts of the two headline recalled theorems or clearly mark the paper as a conditional selection result assuming those preprints.
preprint published 2026-11-14 17:58 UTC by preprint-admission
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