MATH, 10 PRIMO, 10 - Vol. 14 (v2, corrected): From the Billiard Table to the Elimination Formula - The Complete Documented Procedure: the Full Annals of the Dialogue (Six-Pocket Billiard-Table Problem, Synchronization LCM, Elimination Layers, Collision Congruences k = -r*30^-1 (mod q), the x^2-y^2 Layer with 1008 Surviving y-Trajectories and 93.28 Percent Pre-Elimination for N = 30029, Factor Trajectories, and the Recursive Generator p_{n+1}), Followed by the Formal Paper - Four Laws with Comple

Abstract

Volume 14 (v2, corrected) is both an archive and a paper, as demanded by the author: document the whole procedure exactly as it happened, in English, with the Lean verification attached. CORRIGENDUM RECORDED IN THIS v2 (same day, caught by external point-by-point review of v1 and verified against the sources): the main theorem (Theorem 13.3) was stated 'for every integer n >= 3' without the parity restriction; as stated it is false - every even n = 2p (p an odd prime) meets the right-hand side vacuously or spuriously (counterexamples 4, 8, 10, 14, 16, 20, 22, 26, ...). The correct domain is the odd table itself, n = 2k+1, k >= 1 - the domain of the boxed formula, of Theorems 13.1/13.2, and of the Lean audit, whose theorem sobrevivente_impar already carried the hypothesis n % 2 = 1. The slip was in prose scope only: the boxed formula, the laws' content, the proofs' computations, and every numerical and formal verification are unaffected. v2 restricts the quantifier, fixes the contrapositive reading, tightens the domains of Laws 1-3 (m >= 1; the kill claim quantified over t >= m, matching the filter definition exactly), fixes one broken cross-reference of v1 (sec:sundaram-formal to sec:sundaram), and records everything in a dedicated corrigendum remark (Remark 13.4). REST OF THE VOLUME UNCHANGED. PART I is the complete documentary record: the entire dialogue (35 numbered turns) from a homework problem about a billiard ball on a six-pocket table - equations of trajectories first, the LCM discovered as a synchronization period, never assumed - through elimination layers E2->E3->E5->E7, the wheel classes 30k+r with collision congruences k = -r*30^-1 (mod q), the difference-of-squares layer on N = 30029 (modular sieve of surviving y by CRT: 1008 of 15015 trajectories, 93.28 percent eliminated, only the trivial y = 15014 square), the composite contrast 30031 = 59*509 (y = 225, x = 284), factor trajectories T_a with intersections am = bn, the recursive generator p_{n+1} = min{m > p_n : m not in union T_{p_i}}, the test to 5000 with zero errors, the three publication risks, and the closing verdicts (trial division, set-difference sieve, and x^2-y^2 = Fermat 1643: nothing mathematically new; no internet-security risk). The author's Portuguese is corrected in translation; the verbatim 2,658-line transcript ships with the deposit. PART II is the formal paper: odd table, filters, death rows; the four laws (Position, Infinite row, Activation with d^2 embedded, Redundancy, Boundary, Wheel DNA a_p(r) = r*a_p(1)) with complete proofs; soundness and completeness of the one-line formula P = {2} u {2k+1 : every odd d >= 3 with d^2 <= 2k+1 fails to divide it}; and the exact identity with Sundaram's grid k = i+j+2ij, checked item by item to 1e5 (82,017 = 82,017). PART III re-executes the Lean 4 audit from source in a fresh session (Stage A 1.3 s, B 32.1 s, C 31.3 s): 25 named theorems, 12 strictly zero-axiom (including p_roda_profundo: primoW 10000000141 = true, the deep prime 10,000,000,141 = 30*333,333,338 + 1 = 1 (mod 30), 26,665 wheel candidates walked by the kernel in 96+100 slices of 512), 13 general laws on propext/Quot.sound only, no Classical.choice. PART IV verifies numerically: four independent eliminations (prime filters, all odd filters self-contained, wheel-30 prime columns, wheel-30 all-wheel filters) equal to the reference sieve item by item to 1e8 (pi = 5,761,455), DNA law for every p <= 1e4, product rule 17,076 = 17,076. PART V is the honest novelty ledger under PRIMES-FRONTIER: idea classical (Eratosthenes), odd form classical (Sundaram 1934, exact), wheel classical; the contributions are the closed one-line form, the laws, the certificate chain, the hand protocol, and the dated annals. PART VI delivers the hand protocol (pencil, one page, death-row table p -> (p-1)/2 for p <= 997 in Appendix A) with worked verdicts 25, 91, 841, 899, 907, 30029, 30031. Appendices: full death-row table, series record (Vols. 1-14), reproduction guide, turn index, core code verbatim. Complete sources, fresh Lean outputs, JSON results and SHA256 manifest included. License CC BY 4.0.