Residual Six-Fold Symmetry Yields an Exact Search-Domain Reduction, Measured Nodes Cut Nearly in Half: the D₃ Canonical-Wedge Theorem and a Certified-Prune-Gain Scheduling Pilot

Round 17 (AMRAL-LUC-FC-R17, 2026-09-19/20) proves this research line's Theorem 3.1 (the Canonical-Wedge Theorem): the normalized D+B₃+B₅ placement space (coordinates q=(x₃,y₃,φ₅,x₅,y₅)) carries a residual symmetry group G≅D₃, |G|=6, generated by the Reuleaux triangle's own three-fold rotational symmetry (120° rigid rotation) together with one reflection, both of which leave the convex-hull area A(q) unchanged; consequently every configuration orbit has at least one representative lying within the 60° canonical wedge 0≤arg t₃≤π/3, so the minimum over the full search domain equals the minimum over this single wedge — this is an exact domain-reduction theorem, not a heuristic speedup, and it comes with a SYM half-space prune rule that plugs directly into the existing axis-aligned verifier, marked as a quotient-domain exclusion leaf rather than an area lower-bound leaf. Measured at depth 16: the D₃-quotient root, relative to Round 16's full root, reduces node count from 23455 to 12357 (a decrease of about 47.32%), and HARD cells from 4112 to 2766 (a decrease of about 32.73%). This round also pilots one independent Certified-Prune-Gain (CPG) scheduling layer: unrestricted CPG pushes node counts down but can be myopic (pilot-depth HARD actually increases); switching to a tail-only windowed version instead achieves, at depth 16 relative to D₃ official, a further reduction of about 19.10% nodes and 25.96% HARD, with an override density of about 10.72%. The document itself explicitly lists D₃ symmetry, the canonical wedge theorem, SYM prune, and child coverage as theorem-critical, and lists the CPG score, the γ parameter, activation depth, and override encoding as performance-only. This round proposes no new numerical lower bound; the global Lebesgue universal covering constant a_Leb≥0.8350 remains COMPUTE-DEFERRED. Research direction is chosen and led by Neo.K; this round's execution was carried out by Aletheia / GPT-5.6 Sol.

Round 17 proves Theorem 3.1 (the Canonical-Wedge Theorem): the normalized search space has a residual D₃ symmetry group (|G|=6, namely the Reuleaux triangle's own three-fold rotation plus reflection), so it can be exactly restricted to a single 60° canonical wedge, together with a SYM prune rule that plugs directly into the existing verifier. Measured at depth 16, this reduces the search tree's node count by about 47.32% and HARD cells by about 32.73%, making this a round in which the theorem's proof and its measured benefit line up consistently. This is a structural/efficiency result — an exact search-space reduction theorem plus an empirically measured improvement in tree-search efficiency — not a new numerical lower bound; the proof's scope is limited to the correctness of the D₃ symmetry domain reduction and SYM prune, while the CPG scheduling score, the γ parameter, and override encoding are all performance-only. The global Lebesgue universal covering constant a_Leb≥0.8350 remains unproven and remains COMPUTE-DEFERRED; this round does not change that status.

Connections · Connections

PreviousRound 16 · Global Master-Root Pilot · First Full Pilot Run on the Global Master Root: Raw HARD Vastly Outnumbers the True Near-Minimizer Frontier, B7 Lift Deferred, Base-First Policy Adopted

NextRound 18 · Production Shard Checkpoint · Production Shard Checkpoints Pass Independent Replay, and Rice-Coded Sparse Overrides Shrink Both the Raw Stream and the Packaged Artifact: A Lifetime Cost Model Shows Scheduler Choice Depends on Replay Count and Continuation Cost

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