B7's Deep Tail Becomes a Two-Lobe Phase-Locked Atlas, and the Regular Heptagon Still Wins Out Under a Counterexample-Guided Cutting-Plane Loop: The Finite-Tail Optimism Theorem Proves Why Tail-Fitting Is Always Overoptimistic

Round 21 (AMRAL-LUC-FC-R21, 2026-09-19) takes up the question Round 20 left open: B₇'s deep lift along the tracked near-minimum branch still leaves a very small unresolved tail, and this round for the first time turns those unresolved leaves into structured adversarial geometry and uses them, in reverse, to drive witness synthesis. The B₇ lift-tail pending volume is 1046/2²²≈2.494×10⁻⁴ at depth 22, 1397/2²⁴≈8.327×10⁻⁵ at depth 24, and 2143/2²⁶≈3.193×10⁻⁵ at depth 26; measured by δ_lock=wrap(arg t₇−7φ₇), the median |δ_lock| contracts from 0.1886 to 0.1372 to 0.1081 across depths 22→24→26 (the 95th percentile from 0.4785 to 0.2446), showing that arg t₇≈7φ₇ becomes more pronounced as the certificate tail deepens; the 2143 tail boxes at depth 26 collapse almost entirely into two narrow placement lobes with circular concentration near 0.98–0.99 (Lobe A, about 1000 boxes, 7φ≈2.5255; Lobe B, about 1143 boxes, 7φ≈5.8819), and a full area-minimizing ridge computed by fixing φ₇ and minimizing over (x₇,y₇) falls in the same two orientation channels, supporting the reading that the deep certificate tail is essentially a thickened near-minimizing placement ridge. The round also separates this from Round 20's B₂₃ counterexample as a distinct failure geometry: B₇'s deep tail, with |t₇|~0.012, is a finite-translation phase-locked channel, while B₂₃'s |t₂₃|≈3.19×10⁻⁴ is a near-disk hiding channel that sits close to an existing disk. Witness design is formalized as the shape-player-versus-placement-adversary minimax V*=max_K min_p F(K,p), pursued through a cutting-plane protocol — a shape step K_m∈argmax_K min_(p∈P_m) F(K,p) followed by a fresh oracle p_(m+1)∈argmin_p F(K_m,p) that grows the adversarial set P_m — and the round proves the Finite-Tail Optimism Theorem, v_(m+1)≤v_m, which formally explains why fitting only to a known tail without a fresh oracle is necessarily overoptimistic: a tail-targeted training score can reach as high as 0.8377708, only for a fresh oracle to immediately uncover a new low of 0.8369560, with three further cutting-plane iterations (0.8370455→0.8371292→0.8371286) converging back to near the regular-B₇ baseline of 0.83712806. This round also proves a sufficient legality condition, |a|≤λ/96, for the 7-fold-preserving deformation family h_(λ,a,ψ)=(1−λ)h_(B₇)+λ/2+a cos(7θ+ψ). Against that B₇ baseline, none of the tested alternatives does better: a smooth 7-fold wave family (a=0.0103, already close to the convexity boundary) bottoms out at 0.8369528 under fresh search; the best robust value among 30 random nearby irregular Reuleaux7 candidates is about 0.83697744; and an early k=3 symmetry-breaking pilot tops out at about 0.83628622. The status ledger closes both the B₇ lift-tail atlas and the counterexample-guided synthesis loop, and records NEW SUPERIOR WITNESS as NOT FOUND; Round 22 is scoped to open the multi-tail symmetry-breaking Fourier modes k=3,5,7,9,11 under a certificate-cost penalty J(K)=V(K)−λ_cert·Cost_lift(K) that weighs forcing against proof cost. The global bound a_Leb≥0.8350 remains STILL COMPUTE-DEFERRED, unchanged by this round. Research direction and methodology are due to Neo.K; this round's AI collaborating researcher and primary executor was Aletheia / ChatGPT, GPT-5.6 Sol.