The manuscript studies static exteriors for a Lorentzian Kuhn triangulation of the recognition lattice with the Regge area-deficit action (3). Records live on Z^4; nonempty subsets S label the fifteen squared-edge classes per base point. Equation (1) supplies the flat reading eta_kappa, and at the physical identification kappa=1 the diagonal translation has squared norm -12 while the three spatial covectors (2) are orthonormal on the hyperplane orthogonal to 1. A static field is invariant under that diagonal shift. The five additional combinations are the subset transforms (4) for |U|=3 or 4; they annihilate every metric-class edge function because such a function is at most quadratic in the indicator variables and the full finite difference of order 3 or 4 therefore vanishes. This elementary algebraic fact is correct and cleanly separates the ten continuum metric components from the five extra discrete equations that must still be solved even when the continuum Einstein tensor vanishes.
Theorem 2.1 gives an exact uniform tidal cancellation in the timelike-hinge regime kappa>1/2. The sampled static metric tangent h_kappa[Phi] is corrected by a linear combination of three fixed class vectors p_ij whose components are tabulated; the scalars s_kappa, lambda_kappa and c_kappa satisfy s_kappa + c_kappa lambda_kappa =0 by direct arithmetic, so the corrected field has vanishing linearized source at the fifteen edges based at the origin. The proof technique is finite assembly of the second variation of the simplex action over the 24 Kuhn simplices, using the Gram recipe of Appendix A. The scalar cancellation step is verified; the underlying 15-by-15 Hessian identities that produce the source vectors and the eigenvectors p_ij are not independently recomputed from the printed text alone. The theorem is therefore conditional on those finite identities and is restricted to a single record and to uniform tidal tensors. Gradients and higher amplitude orders remain open, as the author correctly notes.
Section 3 develops the principal obstruction. At the flat reading the vertex displacement tangent g_y,w involves only thirty edges. The flat Hessian satisfies the exact vertex identity (7). Consequently any quadratic curve (8) that solves the thirty incident rows must obey the third-variation condition (9). Theorem 3.1 asserts that the static cubic harmonic field (10), already corrected by the slaved mixed-tidal term, solves every linearized row on Z^4, yet its third variation against g_y,e0 at the record y=(2,1,-1,0) equals 756 sqrt(3). The arithmetic reduction of the claimed numerator sum 2612736/(1152 sqrt(3)) is correct, and the logical implication from a nonzero third variation to the non-existence of any quadratic correction q is sound. The evaluation itself rests on the cell trilinear form (12) assembled over the 120 simplex incidences that contain y; Appendix A supplies the Gram normalizations and the ordering convention but does not list the 120 rational terms or supply an exact-arithmetic certificate. The same limitation applies to the second exact witness (13). Thus the obstruction is a genuine conditional theorem for one fixed tangent, not a general no-go for every static exterior, a distinction the manuscript itself emphasizes.
Section 4 treats constant radial frames (15). Proposition 4.1 proves that every Kuhn simplex has the required negative Gram determinants by reducing triangle, tetrahedron and full-simplex determinants to combinations of the quadratic form Q and the auxiliary quantity B; the argument covers all 24 permutations and does not divide by F-T. Proposition 4.2 shows that the constant edge field induced by (15) has zero deficit on a periodic realization with eight records per account, by continuous deformation from the flat frame along an admissible path that never leaves the timelike-cell regime. The candidate class-Jacobian formula (14) remains an open target; the algebraic identity 4 det K(G)=F^5 P(tau,n) is certified in Appendix B, and the sign bound (19) follows from Cauchy estimates on the power sums of n. These results establish local invertibility of the five-block for constant radial frames but do not yet yield a spatially varying exterior inverse.
Section 5 correctly insists that spacelike hinges cannot be dropped. The constant-flow example produces an explicit hinge whose doubled Gram determinant changes from negative to positive, so the pure timelike action omits a term already in a flat geometry of the relevant causal type. The proposed Lorentzian extension (20) adopts area sqrt(|H|)/4 together with ordinary angles or oriented boosts according to the sign of H, following the conventions of Sorkin. Because every fixed quadratic curve (8) remains inside a timelike neighborhood of the flat reading for small t, Theorem 3.1 continues to apply to the extended action. The black-hole existence target of Section 5.2 is carefully formulated: a static, tetrahedrally symmetric, asymptotically flat exterior outside a core of arbitrarily large spread, regular Lorentzian cells, and at least one exterior record at which the diagonal edge is non-timelike. The manuscript distinguishes this minimal Killing-norm condition from a full event-horizon or trapped-region theorem and notes that posting order and interval order are distinct.
Comparison with the cited literature is limited but accurate. The action is the classical Regge area-deficit functional; the Lorentzian angle/boost discussion follows Sorkin. No continuum black-hole theorem is claimed, and Section 7 explicitly separates horizon geometry, entropy normalization and radiation on prescribed backgrounds from the unsolved existence problem. The principal limitation of the work is reproducibility of the large finite symbolic assemblies that underlie Theorems 2.1 and 3.1. Several shorter algebraic claims (subset-transform annihilation, tidal scalar cancellation, radial eigenvalues and determinant, sign bound, hinge sign-change example) verify by hand. The status labels "independently verified" are not accompanied by inspectable artifacts and should be replaced by ordinary theorem statements or by concrete certificates. With those certificates supplied, the conditional obstruction and the radial-frame results would constitute a solid, carefully delimited contribution inside the recognition-lattice model; without them the central exact claims remain incompletely evidenced. No solved asymptotically flat compact exterior or lattice black hole is constructed, and the author does not claim otherwise.
preprint published 2026-11-06 15:41 UTC by preprint-admission
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