Finite-Witness Completeness Proved, the Saturation Gate Formally Established: Finite Witness Attainment Remains an Open Conjecture

Round 05 (AMRAL-LUC-FC-R05, v0.1, 2026-09-18), after Rounds 01–04 completed the three-layer target→placement→candidate cover finite-resolution compiler, rewrites the question from "can it be made finite" to "when can a finite hierarchy actually stop at a finite level," formally opening the Saturation phase of the LUC-FC research line. This round establishes the finite-witness lower-bound functional Λ(F) (taking the min over a finite witness family F, and proving that this minimum is attained rather than merely an inf), and proves the Finite-Witness Completeness theorem: a_Leb = sup over all finite F of Λ(F) — that is, the true value can indeed be approximated by finite families in the limiting sense. On top of this it defines the cardinality ladder λ_m (the best finite lower bound obtainable from at most m witnesses), proves λ_m↑a_Leb with every level λ_m attaining its maximum, and proves the No False Permanent Saturation theorem: as long as λ_m is still strictly less than a_Leb, there must exist a larger m′ such that λ_m′ is strictly larger — several consecutive levels holding the same numerical value is not by itself sufficient to prove a permanent plateau. This round additionally establishes, for a future global candidate-cell branch-and-bound, cell-local area lower bounds (LB_outer, LB_env, LB_tip) and two safe pruning rules (outer non-universality prune, area prune), and proves that candidates which are strictly non-universal or strictly suboptimal will eventually be pruned under infinite refinement. The single most critical theorem in the whole document, the one carrying its global safety, is Conclusion A: "finite-resolution convergence does not imply finite exact closure," proved by an exact analogy with irrational-number approximation — a sequence of rational numbers can converge to an irrational number without any single term ever equaling that irrational number; the way finite witness families approximate a_Leb is isomorphic to this. This round therefore explicitly does not claim that exact finite saturation has already been achieved, but instead lists it as an independent, still-open Finite Witness Attainment Conjecture (whether there exists a finite m⋆ such that λ_{m⋆}=a_Leb), and states clearly that this conjecture may be true or may be false, and should not be prematurely declared proven merely because the numbers have stabilized. The document also re-verifies the known result of Mishra 2026: its three-witness family provides the strict lower-bound certificate λ_3 ≥ 0.8344, not the exact value λ_3 = 0.8344; together with Gibbs's upper bound of 0.8440935944, the publicly known squeeze interval has a width of about 0.0096935944, and this round does not change that interval. Research direction and methodology source: Neo.K. This round's execution was carried out by Aletheia / ChatGPT, GPT-5.6 Sol.