Multi-Mode Fourier Legality Compiler and Full-Orientation Oracle Rule Both Close, Yet No New Witness Beats B7: This Round Pivots to a Proof-Cost Pareto Map and a Witness Portfolio

Round 22 (AMRAL-LUC-FC-R22, 2026-09-19) follows on from Round 21's negative result — that regular B7 is fairly robust against a fresh-placement adversary within the tested 7-fold-preserving / nearby-Reuleaux7 family — by formally opening the symmetry-breaking odd Fourier modes k=3,5,7,9,11 as a combined family: h(θ)=(1-λ)h_B7(θ)+λ/2 plus a Fourier correction term over K⊆{3,5,7,9,11}. It proves that for any odd k this correction term satisfies q(θ+π)=-q(θ), hence h(θ)+h(θ+π)=1 and constant width is preserved, and it proves the Multi-Mode Convexity Theorem, turning curvature legality into a searchable sufficient condition — the weighted norm sum of the Fourier coefficients must not exceed λ/2. The round then catches a genuine oracle defect: a symmetry-order analysis shows that as soon as one active mode has k≢0 (mod 7), the generic orientation fundamental period is P_φ=2π — i.e. the orientation domain is 7 times longer than regular B7's. A concrete example makes this precise: K=(1-0.08)B7+0.08B3 gives an "optimistic" value of 0.83742146 when searched only on the restricted domain [0,2π/7), but a full multi-seed exact-support search over φ∈[0,2π) finds a lower, true basin at 0.83583910 — proving that a symmetry-breaking candidate cannot reuse B7's reduced domain. This motivates the Full-Orientation Oracle Rule and a five-level oracle status ladder (TRAINING-SCORE, OPTIMISTIC-CANDIDATE, ROBUST-SEARCH-CANDIDATE, COMPILED-CANDIDATE, LIFT-VERIFIED — only the last level carries positive theorem value). Across the four Round 20 necessity-marked base configurations, a legal combined-family candidate reaches a training minimum of 0.83770073, drops to 0.83733308 under a single-seed fresh oracle, and after a full multi-seed, full-2π re-search the four base minima come out at 0.83683248, 0.83677292, 0.83674967, and 0.83668617, against regular B7's own worst-case search of 0.83692628 on the same four configurations — so the candidate has both weaker forcing and a 7-times-larger orientation domain, and is Pareto-dominated by B7 on the two axes tracked so far; adding fresh low placements back into the adversarial set and retraining reproduces the same pattern — training scores that look good but that a fresh full-orientation oracle reopens into a new, lower basin — so Round 21's overfitting conclusion still holds within the symmetry-breaking family. The round opens a second, independent route, the Pure Harmonic Family: h_k(θ)=1/2+a cos(kθ) for odd k, with exact convex legality |a|≤1/(2(k²-1)), and defines a curvature reserve σ_k=1/2-(k²-1)|a| (generic support erosion h-δ stays legal for δ≤σ_k); taking the amplitude to 99.5% of the legality bound leaves only σ=0.0025 regardless of k, so the round adopts a 90%-amplitude branch with σ=0.05 as its main test line. For k=11 (a=0.00375, orientation ratio 7/11≈0.636) the four-configuration fresh-search worst case is 0.83574698 (dense bracket 0.83578614–0.83581539); for k=13 (a≈0.00267857, ratio 7/13≈0.538) the worst case is 0.83563803 (bracket 0.83567775–0.83569731); higher-resolution search of k=17 (ratio 7/17≈0.412) gives min_search≈0.83507834 and k=19 (ratio 7/19≈0.368) gives min_search≈0.83504963 — both still above the target but with forcing margins already thinned to the 10⁻⁵ scale. A depth-20 comparison of lift-tree nodes and unresolved placement volume gives: regular B7 at 7037 nodes/1.4107×10⁻⁴; k=11 (90%) at 9623 nodes/1.8536×10⁻⁴; k=13 (90%) at 9211/1.4270×10⁻⁴; k=17 (90%) at 7399/9.5317×10⁻⁵; k=19 (90%) at 7221/8.3671×10⁻⁵ — so the high-k harmonics do achieve a lower unresolved absolute placement volume than B7 at this depth, even though B7's deep-tail contraction remains markedly stronger. The round therefore formally defines a Proof-Cost Pareto Vector Π(K)=(m_force, P_φ, σ, N_d, V_d, ρ_tail) and sorts current witnesses into three Pareto roles: regular B7 as PRIMARY ROBUST CLOSER; H11/H13 as BALANCED AUXILIARY, with forcing margins still at 6–7×10⁻⁴; and H17/H19 as CHEAP/RAZOR-THIN AUXILIARY, with forcing margins of only a few ×10⁻⁵ above target and a corresponding need for high vigilance against hidden counterexamples. None of the tested full-2π multi-mode candidates is promoted into the production witness pool (they are retained instead as a SEARCH ARCHIVE), so the research goal shifts formally from "find one shape that beats B7" to building a proof-cost-aware witness portfolio (W_portfolio={B7,H11,H13,H17,H19,…}), from which only witnesses that actually pass a given cell's negative filter, lift probe, and verifier support may be selected; the Round 02 compiler can also gain a PURE-HARMONIC-CW fast path (requiring only verification of odd k, amplitude legality, symmetry period, curvature reserve, and support/contact). Every Fourier forcing minimum in this round remains SEARCH-ONLY — the placement oracle is not an exhaustive certificate, so none of this constitutes a positive theorem — and the global bound a_Leb≥0.8350 remains compute-deferred. Round 23 will take B7, H11, H13, H17, and H19 directly against the Round 20 necessity-marked atlas for negative filtering, shallow and deep lift probes, cell×witness incidence, and weighted set cover. 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.