The Law of Motion on the Recognition Ledger

The manuscript develops a four-account double-entry ledger whose postings are vectors in R^4. A debit on account a is the unit vector e_a; a credit is the reflected vector (1/2)1 - e_a. Definition 1 and Proposition 2 establish the associated legs s_a = e_a - (1/4)1, which form a regular tetrahedron with |s_a|^2 = 3/4 and s_a · s_b = -1/4. The three pair sums E_i = s_0 + s_i are orthogonal unit vectors and define the record cube of edge length 1. These geometric identities are internally consistent under the Euclidean J-length |y|^2 = sum y_a^2 and were re-checked from the page images; they fix the four natural spheres of Table 1 and Figure 2, with the midsphere corresponding to kappa = 1/2 (c_J = 1/2 cube edge per tick). The event lattice Lambda generated by the eight posting vectors is Z^4 union (Z^4 + (1/2)1). Proposition 5 correctly enumerates its twenty-four shortest vectors (eight debits, eight credits, the unit pair, six cube edges) and proves the moment identities sum_b (K · b)^2 = 6|K|^2 and sum_b (K · b)^4 = 3|K|^4. In the (u; x, y, z) coordinates these vectors are the vertices of the 24-cell (Hurwitz units), a spherical 5-design. The metric premise of Section 2.5 (that the cost measures a step in time exactly as it measures a step in an account) is explicitly marked Working; every subsequent equality of a dynamical cone with the midsphere rests on it, while the pure reach bound of Section 3 does not.

Theorem 6 is a clean, correctly proved CFL-type statement: any stable, strictly local linear update whose one-tick displacements lie in a fixed finite set P has isotropic light speed at most the in-radius of conv P. The proof uses the support-function bound on complex wave vectors together with a polynomial kernel estimate; Remark 7 recovers the classical leapfrog threshold r = 1/3 as a counter-example when stability is dropped. Corollary 9 then places one-posting laws on the cube (in-radius 1/2) and one-recognition laws on the cuboctahedron (in-radius 1 per two ticks), both giving the midsphere. The theorem applies only to explicit forward updates; the later least-cost and loop laws lie outside its hypotheses, a limitation the body text notes but the abstract does not always repeat. Proposition 10 excludes a wide tick under the additional identification that light's steps are ledger histories. The one-column paradox of Section 3.2 is resolved correctly: the signed total sigma = sum x_a is conserved by every recognition and is identified with charge, while the clock is the posting count tau = sum (d_a + c_a).

Section 4 treats one recognition as a debit followed by its reflection. Lemma 11 gives the closed form F(k) = c_x c_y c_z + i s_x s_y s_z for the leg sum. Proposition 13 shows that the equal-weight recognition move T = |F|^2 has rest probability exactly 1/4 and that this quarter is the positivity edge of any transfer that mixes rests with the twelve face-diagonal transfers. Identification 14 (T = D^dagger D in every sector) and the real-time reading Theorem 15 are central. The single-mode functional equation is sound under the three stated premises, but the whole-operator, infinite-lattice and gapped-zero extensions are deferred to the unresolved companion [20]. The unit c = 1 that converts decay rates into physical frequencies is adopted (Working). Consequently the claim that the adopted laws sit on the midsphere is conditional on both the metric premise and this unit.

The least-cost law between records is developed rigorously. Theorem 16 proves existence, uniqueness and the maximum principle for the convex cost C[phi] = sum w_ij (cosh(phi_i - phi_j) - 1). Theorem 19 shows that well-posedness between every pair of records forces the least-cost (c(k) >= 1) two-tick rule, at the price that the same rule cannot be a stable forward tick (Proposition 20). On the twenty-four bonds the small-strain law (6) yields the exact shell of Proposition 27. Because the second and fourth moments are isotropic, the phase speed has no relative O(|k|) or O(|k|^2) correction; the first departure is the sixth-moment term of relative order |k|^4, vanishing on the cube axes. The coefficients -1/960 (face diagonal) and -1/810 (body diagonal) and the group-speed factor 5 were re-derived from the printed sixth-moment formula and match. Equal bond weights are forced inside the one-parameter family that keeps a cost and kills the second-order term; the resulting cone is again the midsphere only when the unit bond is weighed equally with a cube edge (the metric premise).

Section 6 embeds the D'Ariano-Perinotti Weyl walks on the same lattice. Proposition 29 shows that the half-trace is precisely Re[(1 + i eta) F(k)], so the walks are the two hands of the ledger's own leg sum. Long-wave expansion (Proposition 30) recovers speed 1/2 together with the first-order odd term eta |k| n_x n_y n_z. Theorem 31 claims that every one-tick, two-state, no-rest unitary walk on the eight half-diagonals with isotropic long-wave speed is forced to this half-trace. The moment argument that reduces the half-trace to c_x c_y c_z + lambda s_x s_y s_z is correct, but the finite evaluation that forces lambda^2 = 1 is explicitly omitted while the theorem is marked Proved. Because this theorem underwrites the exclusion of the whole class as matter near the Planck scale (Section 6.7 and Table 3), the omitted computation is a consequential presentation defect. The published classification [3,4] cannot be invoked directly, since Theorem 31 assumes less covariance. Proposition 33 correctly shows that no two-state unitary walk exists on the four debits alone, so credits are forced into any such tick.

Light is identified with the loop cost on the triangles of the twenty-four bonds (Identification 41, Working). Proposition 42 mixes Proved gauge and symmetry statements with Hand-derived and Reported dispersion coefficients; the first nonzero light correction and the off-axis transfer remain unchecked in the manuscript. The averaging law of Section 7.3 is the record's own move and is therefore not light. No lattice action with the full 384-element account symmetry and a first-order angle term has been constructed. Section 9's four-state matter law and strain-based gravity are almost entirely Working/Reported; the weights (1, 1/2, 1) are chosen, not forced, and the mass remains free.

Observational comparisons in Section 10 correctly quote the GW170817 cone-equality bound and convert the walk's first-order coefficient into E_QG, xi and xi_e. The use of Planck N_eff = 2.99 +/- 0.17 to assert no room for an extra light species overstates the cosmological constraint; N_eff bounds relativistic energy density only after a thermal history is specified, while the manuscript's two-polarization count already follows from gauge-mode counting (Propositions 21 and 36). A possible order-of-magnitude mis-statement of the LHAASO quadratic bound for GRB 221009A appears in the printed E_QG,2 figure and propagates into Table 3; it does not affect the structural conclusions. Against the paper's own requirement list (Section 1.2), the forced-law criterion is unmet: the step, the metric, the real-time unit, the reach and the loop weight remain adopted. The verified mathematics (reach bound, 24-cell moments, least-cost well-posedness, Weyl-walk embedding, scalar sixth-moment shell) is sound under its stated hypotheses and constitutes a conditional lattice framework of independent interest. The physical law of motion, the action of gravity and any positive testable prediction remain open, exactly as Sections 11.3-11.5 state.

If the named premises hold, the ledger geometry supplies a single lattice on which light, matter candidates and a shared long-wave cone can sit at kappa = 1/2, with dispersion corrections that begin only at high order and with a clean link to known Weyl quantum walks. Several conditional theorems (reach bound, 24-bond moments, record rest probability, least-cost well-posedness) are independently checkable and of independent mathematical interest. The physical significance remains limited: the central step, metric, time unit, light identification, matter weights and gravity action are adopted rather than derived, the tick size is unfixed, and the paper itself states that nothing is yet positively testable. The work is therefore a transparent conditional map and status report, not a completed derivation of the speed of light or of Maxwell/Einstein dynamics.

1. Status taxonomy ([Proved], [Hand derivation], [Published], [Reported], [Working], [Open], [Identification]) and Table 4 make premises and gaps reviewable.
2. Proposition 2 posting geometry and record-cube coordinates check under the Euclidean J-length.
3. Proposition 5: event lattice Lambda has exactly twenty-four length-1 bonds with second moment 6|K|^2 and fourth moment 3|K|^4 (24-cell / Hurwitz units).
4. Theorem 6 / Remark 7 / Corollary 9: stable explicit local linear updates have isotropic light speed at most the in-radius of the reach; leapfrog recovers the classical CFL threshold r=1/3; one posting and one recognition both give the midsphere kappa=1/2.
5. Proposition 13: equal-weight recognition move T=|F|^2; rest probability exactly 1/4; positivity edge s>=1/4.
6. Theorems 16 and 19: least-cost histories between records exist, are unique, obey a maximum principle, and are forced by well-posedness of linear two-tick rules with c(k)>=1.
7. Proposition 27: equal-weight 24-bond least-cost shell has no O(|k|) or O(|k|^2) anisotropy; sixth-moment |k|^4 coefficients match independent expansion.
8. Proposition 29(a) and Proposition 33: Weyl-walk half-trace equals Re[(1+i eta)F]; no two-state unitary walk on four debits alone.
9. Theorem 37: within h=I+beta 11^T, unique stable massless form is beta=-1/2.
10. Proposition 39: averaging-law group speed <=1/2 per tick with equality on cube axes.
11. Proposition 47 correctly diagnoses the gravity slope gap: J has zero slope at balance.
12. Honest inventory of owed items (reflection, reach, tick size, gravity action, matter weights) and of negative results (one-column paradox, wide tick, first-order Weyl term).
13. GW170817 cone-equality bound is quoted and interpreted correctly as constraining equality, not the absolute sphere.

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