Othello as a Dynamic Spectral Lattice System — Research Notebook

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"Can't stop the signal, Mal. Everything goes somewhere, and I go everywhere." — Mr. Universe, Serenity (Joss Whedon, 2005)

Signature epigraph of the spectral-research collection. The body of work — validated results and rigorous falsifications alike — was offered through conventional channels and dismissed as foolery. The math stands independently. The discipline since: ship every result, falsifications included, with full reproducibility and per-row provenance (the Mathematical Provenance Method). A corpus that publishes its own invalidations is harder to dismiss than one that doesn't, and propagates through every channel that ingests open research. The signal is in the world; it goes everywhere now.

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Authors: Steven (mlehaptics Project) & Claude (Anthropic) Date: April 2026 (last update: Phase 1e, PR #58) Status: Active research — §1–§2e computational; §3 computational (faithful-sheaf update in §2e.5); §4 instantiated in §2e.15–§2e.19; §5 resolved in §2e. v0.3.2: encoder pipeline now uses SheafMoveOperator as the replay primitive (§2e.19). Tools: Python 3, NumPy, SciPy; C17 reference encoder (clang); ctypes DLL path.

Living document. Sibling to ../chess-maths/chess_spectral_research_notebook.md (the chess notebook's §10 is the theoretical survey this notebook tests computationally), ../logo-maths/logo_research_notebook.md (the second instance of the split-object-with-fiber-matrix pattern, including an explicit retraction in L7b that this notebook takes as a cautionary template for Phase 2 claims), and ../antikythera-maths/doom_spectral_research_notebook.md — DOOM (1993, id Tech 1) translated into a graph-Laplacian spectral model, where Track 3 (Dynamic Sheaf Laplacian for line-of-sight / raycasting) is mathematically identical to §10.7's ray-flanking mechanic of this notebook. The doom-spectral notebook explicitly cites §10.7 as the reference; sibling-link added here so the bidirectional cross-reference is complete.

Every claim is tagged KNOWN (published, cited), NOVEL (no prior art found), CONFIRMED (computationally verified in this session), FAILED (tested and didn't work), or UNDETERMINED (requires data not available in this session).

Project navigation + state-pointer

ReadTheDocs landing — https://mlehaptics.readthedocs.io/en/latest/ — is the canonical pointer to the current state across all sister notebooks in this project.

This notebook is a snapshot. Future framework additions will not be back-ported into it; the RTD landing tells you whether new sister notebooks or downstream developments are available.

Brief since-summary (as of 2026-05-08): - ephemerides-spectral framework matured through v0.26.0 — per-body action-angle catalogues, Attested Multi-Source Collector framework with MPR v1 normative format, four-tier reproducibility model (T0 / T1 / T2 / T3), schema-gap-driven trigger. PyPI: https://pypi.org/project/ephemerides-spectral/. - The Mathematical Provenance Method (MPM) formalised as the project's discipline (ephemerides notebook §0.0); instrument-first physics critique in ephemerides §20 with three-regime classification (impulse + ring-down / driven sustain / driven with irreversibility). - Inkscape contribution shipped on the spectral-faithful branch — three new SVG filter primitives (feSpectralBilateral, feSpectralDistance, feSpectralNoise) using the same eigenbasis substrate this notebook tests on Othello. - mfo-spectral sister-notebook added (May 2026) — Metric Field Ontology, one candidate foundational-ontology framing hosted in the project (cavity-instrument analogy; fractal metric field; ~11D structure motivated independently of string theory). Not the project's endorsed answer over alternatives; ephemerides §20 cites MFO as a worked example without picking a spatial-structure side. Future MPM target. - Ephemerides §21 — Tool-rejection as MPM-screening failure (symmetric counterpart to §20). Names the evaluator-side screening failure: rejecting work by which tool was used to make it, not by what it claims. Anchored on the historical orbital-mechanics chain DE441 traces back to (Copernicus / Bruno / Galileo / Kepler). §21.3 names the disability-accommodation dimension explicitly: categorical tool bans function as participation barriers for contributors with aphantasia, ADHD, dyslexia, motor disabilities, and many other variations. MPM is tool-agnostic by design.

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0. Framing

Othello shares the 8×8 grid with chess and therefore inherits the board Laplacian eigenbasis (§9b chess), the D₄ symmetry group, and the 8-generator spectral lattice. Everything above the board differs: no movement, no per-piece Laplacians, a single Blume-Capel 3-state fiber at each cell, exact D₄×Z₂ symmetry (Z₂ is approximate in chess because pawns break it — §1b.1 chess), non-local flanking updates, monotone disc-count filling as the global T-breaker.

The verification target is §10 of the chess notebook — a hybrid-framework thesis stitching Blume-Capel spin-1 + D₄×Z₂ + 8-ray decomposition 2 A1 + B1 + B2 + 2 E + Wolff-like flanking under Fraenkel bounded-change CA semantics + Hansen-Ghrist dynamic sheaf + Sagawa-Ueda thermodynamics under a Boltzmann-policy embedding.

The discovery principle: do not commit to a single fiber rank in advance. §10.4 names three candidates — rank 2 (orbit count), rank 6 (irrep count), rank 8 (individual rays) — and this session's job is to probe all three and report what the structure actually yields.

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1. Infrastructure

All numbers below come from the consolidated Phase 1 runner at research/consolidated_tests.py. Status tags per hypothesis match results/phase1_hypotheses.csv.

1.1 Board Laplacian — sanity (H1)

Construction: research/othello_utils.py::grid_laplacian_8x8 builds the P₈ □ P₈ graph Laplacian on 64 vertices. dct_basis_2d constructs the length-8 DCT-II basis and tensors it into the 64-column 2D basis. Comparison handles degenerate eigenvalue subspaces by block- matching ||V V^T − D D^T|| rather than per-vector inner products (critical: some eigenvalue multiplicities reach 4, so individual eigenvectors are only defined up to rotations within their degenerate block).

subspace_gap = 3.390e-14
eig_residual = 7.105e-15

H1 status: PASS (threshold 1e-12). CONFIRMED / KNOWN: identical to chess §2 (5.86e-16 there; the slightly worse number here is the block-subspace metric, not the per-vector cosine). The grid Laplacian eigenbasis transfers verbatim.

1.2 D₄×Z₂ irrep projectors — sanity (H8 substrate)

research/d4_z2.py implements the full 16-element group. Element indexing is linear 2*g + z with g ∈ 0..7 (D₄ spatial) and z ∈ {0, 1} (Z₂ colour flip). Bug note: the initial table had B1 = [1, −1, 1, −1, 1, −1, 1, −1] inherited from chess convention; this broke idempotence by failing to be constant on conjugacy classes {g4, g5} (axis reflections) and {g6, g7} (diagonal reflections) — verified by direct conjugation g1 · g4 · g1^-1 = g5 and g1 · g6 · g1^-1 = g7. The corrected table is B1 = [1, −1, 1, −1, +1, +1, −1, −1], B2 = [1, −1, 1, −1, −1, −1, +1, +1].

Sanity checks all pass at machine precision:

character orthogonality: max|<chi_mu,chi_nu>/|G| - delta| = 0.000e+00
group action identity:   identity_err = 0.000e+00
(g2)^2 = e:              err = 0.000e+00
projection idempotence:  max err = 4.441e-16   (over all 10 irreps)
projection completeness: reconstruction err = 4.441e-16
Z2 parity split:         Z2-odd on '+' irreps = 0.000e+00
                         Z2-even on '-' irreps = 0.000e+00

CONFIRMED: the D₄×Z₂ = D₄ × Z₂ direct-product structure, with 10 irreps labelled {A1±, A2±, B1±, B2±, E±}, projects cleanly.

1.3 Ray Laplacians (H2, H4)

research/ray_laplacians.py builds per- direction graph Laplacians. Two constructions:

• Undirected (standard graph Laplacian, self-adjoint): L_d = L_{-d} because graph edges are unordered, so the 8 ray labels collapse to 4 distinct operators (N=S, E=W, NE=SW, NW=SE).
• Directed (non-symmetric, forward edges only): all 8 operators are distinct.

The 8 rays partition into two D₄ orbits of size 4: orthogonal {N, E, S, W} and diagonal {NE, SE, SW, NW}. Orbit-averaged Laplacians L_ortho and L_diag capture the orbit-level structure.

Verified mean degrees:

ortho rays (N, E, S, W):   mean degree 1.750   (path graph length 8 on each row/col)
diag rays (NE, SE, SW, NW): mean degree 1.531   (diagonal path graphs of varying length)

H2 — 8-ray D₄ decomposition. Applying the character-projection formula to the 8-dim ray-indicator space yields multiplicities

measured: {A1: 2, A2: 0, B1: 1, B2: 1, E: 2}
expected: {A1: 2, A2: 0, B1: 1, B2: 1, E: 2}

H2 status: PASS. CONFIRMED the Gamma_8 = 2 A1 + B1 + B2 + 2 E prediction of §10.4 (NOVEL there; now computationally verified).

H4 — ortho vs diag spectral distinctness.

mean_deg_ortho = 1.7500,  mean_deg_diag = 1.5312  (rel diff 0.125)
bandwidth_ortho = 7.7200, bandwidth_diag = 6.8284 (rel diff 0.116)
lambda2_ortho  = 0.1522,  lambda2_diag  = 0.1522  (rel diff 0.000)

H4 status: PASS (threshold: max rel diff > 0.05). CONFIRMED: ortho and diag Laplacians differ at mean-degree and bandwidth level, though lambda₂ (algebraic connectivity) happens to match — both are path-graph-like structures with the same connectivity gap.

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2. Phase 1 batch — verification & exploration

2.H3 B₁ vs B₂ ray modes are numerically distinct — CONFIRMED

cos(B1, B2) on 8-dim ray indicators = 0.0  (exact)
L_B1 = weighted sum of (L_N..L_NW) with B1 coefficients
L_B2 = weighted sum of (L_N..L_NW) with B2 coefficients

||L_B1||_F = 14.966
||L_B2||_F = 14.966
Frobenius cos(L_B1, L_B2) = 0.000e+00   (exact)
max |L_B1 - L_B2| = 1.000
bandwidth(L_B1) = 6.485, bandwidth(L_B2) = 6.485

Not just distinct — Frobenius-orthogonal. The B₁ mode lives entirely in the orthogonal-ray edge set; the B₂ mode lives entirely in the diagonal-ray edge set; the two subspaces share no non-zero matrix entries, so the lifted 64×64 operators are orthogonal in the matrix-inner-product sense. This is stronger than the originally specified "detectable factor" — B₁ and B₂ are orthogonal modes, not just distinct ones.

The equal Frobenius norms and bandwidths reflect that orthogonal- ray and diagonal-ray path graphs are isomorphic (both are unions of disjoint paths on an 8×8 lattice), just embedded along different axes.

Literature grounding. This realises §10.4's claim that the B₁↔B₂ swap is the group-theoretic signature of the rook/bishop distinction, but derived purely from rays with no reference to piece species.

2.H5 Static ray bundle has nontrivial holonomy — CONFIRMED

Local fiber at site s is the stack of 8 ray-Laplacian rows at s, flattened to R^{512}. Parallel transport by sign alignment at each step. Results on 4 closed loops:

plaquette (4 cells), centred at (3,3):  cos = +1.000  (trivial)
3x3 loop, top-left corner:               cos = +1.000  (trivial)
triangle via (0,0)-(0,7)-(7,0):          cos = +1.000  (trivial, three-step)
rectangle (0,0)-(0,3)-(1,3)-(1,0)-(0,0): cos = −1.000  (Z_2 holonomy)

H5 status: PASS (threshold: at least one loop with |cos − 1| > 0.01). One of the four loops returns a full Z₂ holonomy. This is the direct Othello analog of the chess −0.016 holonomy reported in §8c — here the signal is cleaner (a full sign reversal) because Z₂ is exact in Othello.

Open: the specific loops where holonomy appears are not yet characterised. Expected follow-up: enumerate all minimal plaquettes and record which have trivial vs non-trivial transport, to extract the connection form's curvature structure.

2.H6 Fiber rank candidates — PARTIAL (open exploration by design)

Three stackings evaluated. SVD effective rank at threshold sigma_i / sigma_0 > 1e-6:

undirected stack (8 Laplacians, pairs coincide):          effective rank 4
directed stack (8 non-symmetric operators):               effective rank 8
orbit stack (L_ortho, L_diag):                            effective rank 2
D4xZ2-projected per-site out-degree signature (8 x 640):   effective rank 8

Singular value spectrum:

orbit_sv:       [82.08, 29.64]
undirected_sv:  [41.04, 15.75, 14.82, 14.14, ~0, ~0, ~0, ~0]
directed_sv:    [20.52, 8.54, 8.54, 7.87, 7.41, 7.41, 6.71, 6.71]
proj_sv:        [19.19, 4.55, 4.55, 2.45, 1.66, 1.66, 1.11, 0.80]

Interpretation. Rank-2 is confirmed as the orbit-level structure. Rank-4 is the natural undirected construction (N/S, E/W, NE/SW, NW/SE pairs). Rank-8 is the directed construction. Rank-6 does NOT fall out naturally from any of the four stackings — the 2 A1 + B1 + B2 + 2 E count is an irrep multiplicity, not an operator rank. Following the polarization reframing (§9r chess) — which derives Othello as (theta-class, r=inf, c=0) — rank-2 is the most structurally defensible and most directly motivated candidate.

Recommendation for downstream work: adopt rank-2 by default for the production encoder (D = 768 = (2 + 10) × 64) and keep rank-8 (D = 1152) as an ablation. Rank-6 should be treated as a decomposition of the D₄×Z₂ irrep channels, not a stacked-operator rank.

2.H7 Coprime generators exist — CONFIRMED

From research/coprime_generators.py, exhaustive search over small admissible primes:

rank	D	(row_gen, col_gen)	64 phases unique?
2	768	(7, 11)	yes
6	1024	(3, 11)	yes
8	1152	(7, 11)	yes

H7 status: PASS. CONFIRMED. All three candidate dimensions admit (p, q) generator pairs reproducing the chess §9f structure. Caveat: the first attempt used "smallest primes coprime to D" which produced (3, 7) for D = 1024 — these collide because (r₁−r₂) · 3 + (c₁−c₂) · 7 = 0 has non-trivial (r_diff, c_diff) = (7, −3) within the 8×8 range. The fix is explicit phase-uniqueness checking over all 64 cells, not just coprimality.

2.H8 D₄×Z₂ invariance of the encoder — CONFIRMED

Played 10 random moves from the Othello start to get a non-trivial configuration. Projected the Blume-Capel signal s onto A1−; applied all 16 (g, z) group elements and verified:

max ||A1-(P_g · s) - A1-(s)||_inf over g in D4  =  0.000e+00   (D4 invariance)
max ||A1-((g,1) . s) + A1-(s)||_inf             =  0.000e+00   (Z2 sign flip, as expected for Z2-odd signal)
max ||A1+(P_g · s^2) - A1+(s^2)||_inf           =  0.000e+00   (occupation Z2-even invariant)

H8 status: PASS (threshold 1e-10 on all three). CONFIRMED the D₄×Z₂ projection operates correctly on both Z₂ parities. For the raw magnetisation (Z₂-odd under colour flip), A1− is D₄- invariant and Z₂-odd — its projection picks up the correct sign under (g, 1). For the occupation quadrupole s² (Z₂-even), A1+ is fully D₄×Z₂-invariant at machine precision.

2.H9 A₁ depth-gap transfer — UNDETERMINED

Requires Takizawa 2023 perfect-play data (Zenodo 10.5281/zenodo.10030906) and an Othello engine capable of variable-depth evaluation (edax or equivalent). Neither is available in-session.

Surrogate: A1− energy of 30 random positions (game stages varying 5–40 moves) has mean 3.64, std 7.84 — non-trivial variance, so the protocol is at least sensible. The actual transfer test (Spearman ρ > 0.3, p < 0.01 vs Takizawa depth-gap) is deferred to the sequel.

2.E1 Null tests — CONFIRMED (absence)

sum over directed rays ||A_anti|| / ||A_sym||   = 7.621  (directed case has antisymmetry by construction)
max undirected ray antisymmetric norm           = 0.000e+00  (undirected ops are self-adjoint)

CONFIRMED: undirected ray operators have zero antisymmetric content — the Z₂-breaking "pawn fiber" of chess has no Othello analog, consistent with exact Z₂ colour symmetry. Knight-style DCT orthogonality and rank-5 piece-species fiber have no Othello referents by construction (single disc type, no knight move).

2.E2 Pauli / CP² fermionic analog — PARTIAL

The local Z₂ grading on the 3-state Blume-Capel fiber admits a parity operator diag(−1, +1, −1) in the (+1, 0, −1) basis, decomposing "occupied with sign" vs "empty". This is the local structure of a CP² sigma model restricted to occupied cells plus an empty-state singlet. A full Jordan-Wigner-style fermionic encoding would require global ordering and fermion-sign accounting, which is deferred.

2.E3 Disc density as slow variable — PARTIAL (but suggestive)

Five random games, 300 positions total:

Spearman(rho, A1- energy on magnetisation)   = +0.671  (p = 1.5e-40)
Spearman(rho, D4-only A1 energy on s^2)      = +0.998  (p = 0.000)

The second correlation is near-tautological (occupation integral IS the disc density), so it is a control. The first is a substantive result: A1− magnetisation energy scales with filling density with Spearman +0.67 — not perfectly linear, but strongly monotone. Disc density rho behaves as an effective slow variable gating the magnetisation-sector spectra, consistent with §10.3's Blume-Emery- Griffiths structural analog.

2.E4 Compass ground state — PARTIAL

The 90° compass Hamiltonian H = -sum_rays sum_pairs s_i s_j on s ∈ {±1} is minimised by constant boards. Constant boards ARE reachable Othello terminal states (64-0 sweeps). Sample random-play terminal in this session: 48–16 split (hamming to all-+1 = 16). The compass ground state is therefore reachable as a degenerate extremum, though not by random play.

2.E5 Flank cluster-size distribution — PARTIAL

Sampled all legal-move flip-counts across 20 random-play games:

N = 7216 candidate moves
mean flip count = 2.28
std             = 1.81
max             = 13

Full histogram exported in results/phase1_detail.json. Distribution- fit comparison (power-law vs exponential vs Fortuin-Kasteleyn-Blume- Capel) is deferred to the sequel when WTHOR tournament data is available. Random play produces an exponential-looking tail; it is an open question whether tournament play shifts this toward power-law scaling.

2.E6 Dynamic sheaf — see §3.

2.E7 Disc-count monotone as global T-breaker — PARTIAL

Single random game, A1− energy trajectory. Forward-increment positive fraction = 0.52 (essentially symmetric), correlation of forward and reversed gradients = +0.07 (decorrelated) — no signal at N = 1 game. Aggregate statistics over many games are needed before drawing conclusions; at the level of a single trajectory, the spectral observable does not cleanly distinguish forward from reversed play.

2.E8 Takizawa perfect-play correlations — UNDETERMINED

Blocked on external data, as with H9. Scoped to the sequel.

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2b. Phase 1b — game-trajectory tests on real PGN

Run research/game_trajectory_tests.py on the Barcelona European Grand Prix 2026 transcript (dataset/liveothello_Barcelona_EGP_2026.pgn, 35 games, 2184 position records including auto-inserted passes). Full aggregate in results/phase1b_game_trajectories.json; per-ply CSV in results/phase1b_per_move.csv.

2b.T1 Flip-count distribution on real play

N = 2184 positions
max flip over entire corpus = 12   (single legal move in a single position)
median per-position max flip = 4.0
mean-of-mean flip-per-legal-move over positions = 2.23 +/- 0.70

Comparable in magnitude to the random-play surrogate (E5 mean 2.28, std 1.81, max 13). Strategic play does not dramatically shift the single-move flip-count distribution at this sample size. The §10.10 T1 test — power-law vs exponential — requires larger N and an explicit distribution fit against FK-Blume-Capel and SOC baselines; that is still scoped to the sequel.

2b.T2 B1 vs B2 population asymmetry — confirmed direction

The §10.4 and §10.10 T2 prediction: under rules alone the two orbits are indistinguishable; under tournament strategy the diagonal orbit may register higher because corners are diagonal-reachable first from the centre and edge/corner control is strategically valued asymmetrically.

mean <B1^2> energy over 2184 positions = 3.930
mean <B2^2> energy over 2184 positions = 4.397
ratio <B1^2> / <B2^2> = 0.894
paired diff (B1 - B2) mean = -0.468  (s.d. 4.551)
B2 > B1 in 1351/2184 positions (61.9%)
B1 > B2 in  833/2184 positions (38.1%)

The diagonal orbit (B₂) registers ~12% higher than the orthogonal orbit (B₁) in tournament play. The effect is in the predicted direction. It is not universal — 38% of positions invert the ranking — but the population mean is clearly offset. At this sample size (N = 35 games, ~2200 plies) we cannot yet compare to random-play expectation with statistical power sufficient to rule out finite-sample effects; the random-play baseline is a follow-up measurement.

2b.E3 scale-up — stronger under real play

Spearman(rho, A1- energy) over 2184 tournament positions = +0.772
p-value essentially 0 (Spearman exact limit)

Phase 1 random-play measurement at N = 300: +0.671. Real tournament play scales the correlation UP to +0.772 — structural coupling between disc density and magnetisation-sector spectra is tighter under skilled play, consistent with the Blume-Emery-Griffiths reading of §10.3.

2b.E7 aggregate forward asymmetry — small but consistent

mean forward-positive fraction over 35 games = 0.541 (s.d. 0.038)
fraction of games with forward-positive fraction > 0.5 = 0.857  (30/35)

Small but tight signature — 54.1 ± 3.8% of A₁⁻ ply-to-ply gradients are positive. The monotone disc-filling T-breaker of §10.8 leaves a detectable spectral signature at game level. The tight standard deviation (0.038) suggests this is a real population effect rather than per-game noise.

2b.G9 A₁⁻ peak/drop trajectory — Othello analog of §9h′

Chess §9h' Experiment 2 found the A₁ energy PEAK ply as the decisive crisis predictor across 5 masterpieces, with the ΔA₁ drop ply preceding the peak (the drop detects simplification, the peak detects residual tactical density).

mean peak  ply = 57.9  (92.8% of game)
mean drop  ply = 45.9  (73.7% of game)
corr(peak_ply, drop_ply) across 35 games = -0.298

The ordering is reversed relative to chess. In chess the drop precedes the peak; in Othello the drop (at ~74% of the game) precedes the peak (at ~93%), which makes structural sense — A₁⁻ magnetisation energy in Othello grows monotonically with disc count through most of the game, then plateaus near terminal. The "drop" in Othello is a midgame simplification event, which can happen when a large flanking chain is resolved; the "peak" is the near- terminal maximum of the magnetisation-weighted structure. The chess-style simplification-then-peak reading does not transfer.

The peak/drop correlation across games (ρ = −0.298) is a mildly negative signal — games where the peak occurs especially late tend to have earlier drops. Not sharp enough for significance at N = 35; worth retesting at WTHOR scale.

2b Summary

Phase 1b upgrades five PARTIAL probes to numeric-with-real-games and lands one new prediction (T2) at the corpus-empirical level. T3 (Shannon info per move) and H9 (depth-gap vs edax) remain scoped to further tooling — T3 needs the reversi-scripts opening_book_freq.csv; H9 needs a compiled edax binary from the same repo.

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2c. Phase 1c — reversi-scripts integration

Runs the corpus-level probes that need artefacts from eukaryo/reversi-scripts but NOT the 20 GB figshare perfect-play table. All four sub-phases of the Phase 1c plan (PHASE_1C_PLAN.md) land.

2c.1 OthelloBoard cross-validated — CONFIRMED

Agreement rate against Takizawa's reference bitboard implementation (vendored in research/third_party/reversi_misc.py, GPL v3):

positions compared:    2684
positions in agreement: 2684
agreement rate:         100.000 %

2184 Barcelona corpus positions plus 500 synthetic random-play positions; zero disagreements. OthelloBoard.legal_moves() is independently validated; every probe downstream of legal-move enumeration inherits that confidence.

2c.2 §10.10 T3 Shannon information per move — CONFIRMED

Opening book opening_book_freq.csv.bz2 (24 MB, 2.57M rows, D4-canonical position → WTHOR tournament frequency). For every played ply in the Barcelona corpus compute

I_move = log_2 |M(s)| - log_2 P(chosen | WTHOR empirical,
                                        Laplace alpha=1)

Coverage by game phase (fraction of corpus positions in the book):

empties 60-53 : 100 %
empties 52    : 97 %
empties 51    : 91 %
empties 50    : 86 %
empties 49    : 80 %
empties 30-21 : 14 % down to 3 %
empties <= 20 :  0 %

Overall in-book fraction: 32.2 % of played moves have at least one successor position in the WTHOR book. The ⅔ out-of-book tail is midgame / endgame where Barcelona 2026 has diverged from any 2001-2020 WTHOR precedent.

Headlines (N = 2099 played plies, 35 games):

mean I_move (all plies)        = 5.087 bits  (s.d. 2.025)
mean I_move (in-book only)     = 4.403 bits
mean I_move (out-of-book only) = 5.412 bits

Spearman(I_move, n_legal_moves)      = +0.814, p << 1e-10
Spearman(I_move, A1- energy)         = -0.065, p = 3e-3
Spearman(I_move, A1- energy) in-book = +0.213, p = 2e-8
Spearman(game mean I_move, |disc_diff|) = +0.109, p = 0.53

The dominant correlation is with n_legal_moves as expected (the log_2 |M| term is leading order). The novel finding is the in-book-only positive correlation between I_move and A1- energy: ρ = +0.213, p = 2 × 10⁻⁸ (N ≈ 676 in-book plies). In book- covered positions — where we have a real empirical-policy reference — the spectral A1- observable tracks how much a chosen move diverges from the most-common tournament line. This is the first direct connection between the D₄×Z₂ spectral decomposition and the §10.10 information-theoretic bookkeeping.

The null on game mean I_move vs |disc_diff| (ρ = +0.109, p = 0.53) says that at the GAME level, information-rich games are not necessarily decisive games. Consistent with tournament play containing both forced-sequence blowouts and balanced fights.

2c.3 Edax 50-empty knowledge anchor — PARTIAL / striking but underpowered

Cross-reference against empty50_tasklist_edax_knowledge.csv (2587 D4-canonical 50-empty positions with edax's predicted score). Match count:

50-empty positions in Barcelona corpus:    35  (one per game)
canonical matches against 2587-row list:   15  (42.9 %)

Below the --min-matches = 20 threshold from the Phase 1c plan. The default runner therefore reports "deferred". Peek at --min-matches = 10:

Spearman(A1- energy, edax_score) at matching positions
    = +0.820, p = 1.8 × 10⁻⁴  (N = 15)

Large effect size with tight p-value — but N = 15 is below conventional power thresholds; this is a preliminary anchor, not a confirmed result. The direction is intuitive: edax's score is signed with "good for side-to-move"; at 50-empties only 14 discs are down, and large A1- magnetisation at that stage typically reflects one player having flipped more stones overall, which edax rightly evaluates as an advantage. Worth retesting at WTHOR scale — a 2000-match corpus would either replicate the effect or surface selection bias.

2c.4 H9 surrogate (A1- energy vs edax d=1 / d=20 gap) — CONFIRMED (disc-count-mediated)

Run against the Takizawa-delivered edax 4.5.5 binary (wEdax-x86-64.exe) at depth 1 and depth 20 across the full Barcelona corpus.

n_positions_evaluated: 2180 / 2184  (99.8% — 4 parse failures
                                     on rare edax output edge cases)
Spearman(A1- energy, |d1 - d20|)              = +0.151, p = 1.6 x 10^-12
Partial Spearman controlling for |disc_diff|  = +0.058, p = 0.007

Direction matches chess §9h' (which reported +0.452 at N = 55 Stockfish d=1 vs d=20). Effect size is roughly a third of chess; p-value is much tighter because N is ~40x larger. The chess §9h' partial-with-piece-count was essentially identical to the raw Spearman (+0.456 vs +0.452), meaning piece count did NOT mediate the effect. In Othello, the partial collapses from +0.151 to +0.058 — the signal is mostly a disc-count artifact, not a pure spectral signal.

Interpretation. Othello's A₁⁻ channel is the Z₂-odd magnetisation projection; it is strongly coupled to disc density (§10.12 E3 scale-up: Spearman(ρ, A₁⁻) = +0.772). Chess's A₁ channel is the full-D₄-invariant component including magnitudes, which does not reduce to a piece-count proxy. So the natural chess/Othello analog is not identical — the Othello A₁⁻ is closer to the chess signed-sum A₁ than to the chess energy A₁, and chess §9h' itself shows that A₁ signed sum collapses to a material-counting proxy (§9h' partial controlling material = −0.057 vs raw +0.527). Our Othello H9 result is the direct analog of the chess A₁-signed-sum observation, not the A₁-energy observation, and the +0.058 partial is the spectral-beyond-counting residual.

A cleaner chess→Othello transfer would use the Z₂-even D4-only A1 projection of s² (the occupation magnitude, which does not reduce to disc count because it sums a Z₂-invariant quantity). That probe is scoped as a Phase 1d item.

H9 status: CONFIRMED as a disc-count-mediated effect with a small residual spectral signal. Re-reading §9h' chess against this Othello finding, the distinction between the A₁-energy and A₁-signed-sum channels may be load-bearing for the cross-game transfer of the complexity-vs-depth-gap correlation. The single-number headline (chess +0.452 vs Othello +0.151) is misleading without this breakdown.

2c Summary

All four Phase 1c sub-phases land numeric results. The headlines:

1. 1c.1 CONFIRMED — OthelloBoard 100% agrees with Takizawa's reference bitboard engine across 2684 positions.
2. 1c.2 CONFIRMED (conditional) — Shannon I_move vs A₁⁻ energy is null globally (ρ = −0.065) but significantly positive on the in-book subset (ρ = +0.213, p = 2 × 10⁻⁸, N ≈ 676). Novel: first direct connection between §10.10 information-theoretic bookkeeping and §10.4 spectral decomposition, no direct chess analog.
3. 1c.3 PARTIAL (flagged) — edax 50-empty anchor at N = 15 matches shows ρ = +0.820, p = 1.8 × 10⁻⁴. Large effect, small N; worth retesting at WTHOR scale.
4. 1c.4 CONFIRMED (disc-count-mediated) — A₁⁻ energy vs |edax d=1 − d=20| Spearman = +0.151 (p = 1.6 × 10⁻¹²) on N = 2180. Partial controlling for |disc_diff| collapses to +0.058 (p = 0.007). Direction matches chess §9h' but the signal is mostly a disc-count artifact; chess §9h' partial did NOT collapse (chess A₁ energy is not a piece-count proxy, Othello A₁⁻ magnetisation largely is). The cleaner chess-transfer analog uses the D₄-only A₁ projection of s² (occupation) — scoped as Phase 1d.

Remaining §10.10 tests (T4 T_eff / D_eff trajectory, T5 FK-BC cluster fit) stay scoped to the sequel / external-data-dependent work. Strict H9 / E8 are now runnable following the arrival of Takizawa's figshare knowledge archive — see §2d.

Framework lesson carried forward. The "state richness vs strategic structure" axis split (§10.13 in the chess notebook) is the most reusable observation: Shannon I_move and sheaf λ₂ both track n_legal_moves at ~+0.77 to +0.81; A₁⁻ energy tracks ρ_disc at +0.77. Two independent axes of position description, each with its own natural observable. Chess counterpart experiment has not been run; would benefit from a Phase 1d pass.

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2d. Phase 1d — strict H9 / E8 against Takizawa perfect-play archive

The Takizawa figshare download (17 GB compressed → 171 GB decompressed at a local drive path, outside repo) provides per-50-empty-position knowledge files. Each of 2587 canonical 50-empty positions has an associated knowledge_<OBF>.csv containing every 36-empty sub-problem reached from it under optimal play, with exact game-theoretic bound pairs (score_lb, score_ub) and accuracy codes (100 = exact, 99 = single-point acceptable).

The 2587-row small CSV empty50_tasklist_edax_knowledge.csv (already committed under dataset/reversi_scripts/ since Phase 1c) has the pre-proof edax score prediction for each 50-empty position plus its WTHOR frequency count.

2d.a Tasklist-scale correlation (N = 2587) — CONFIRMED the predicted transfer channel

Run: python research/h9_strict_runner.py. Computes the D4-only A₁ projection of s² (the Z₂-invariant occupation observable that Phase 1c.4 identified as the "cleaner chess-transfer analog" of A₁ energy), plus A₁⁻ magnetisation and the B₁/B₂/E channels, for every 2587 tasklist position. Correlates with edax's predicted score. Full per-position CSV in results/phase1d_spectral_vs_perfectplay.csv; aggregate JSON at results/phase1d_correlations.json.

Headline correlations (N = 2587):

raw:
  D4-only A1(s^2)     vs edax_score      = -0.349  p = 7.4e-75
  A1-                 vs edax_score      = +0.181  p = 1.5e-20
  B1, B2, E           vs edax_score      |rho| < 0.07, mixed significance
  D4-only A1(s^2)     vs |edax_score|    = -0.320  p = 1.7e-62

partial (controlling |disc_diff|):
  D4-only A1(s^2)     vs edax_score      = -0.279  p = 2.3e-47   SURVIVES
  A1-                 vs edax_score      = -0.054  p = 6e-3      collapses
  E                   vs edax_score      = +0.075  p = 1.3e-4

The predicted D₄×Z₂ Z₂-invariant channel carries robust spectral- beyond-counting signal (ρ = −0.279 partial, p = 2 × 10⁻⁴⁷, N = 2587). This confirms the §2c.4 conjecture that the cleaner chess- transfer analog uses the occupation projection rather than the magnetisation projection. The A₁⁻ magnetisation channel, consistent with the Phase 1c.4 finding, collapses under the partial (ρ = −0.054) — its raw correlation is a disc-count artifact.

Direction. Negative ρ means: positions with higher D4-symmetric occupation structure tend to have lower edax predicted score (worse for side-to-move). At 50-empties (14 discs placed), this suggests that tournament strategy prefers asymmetric configurations over D4-symmetric ones, and edax's evaluator encodes that preference.

2d.b Strict augmentation via archive bounds — CONFIRMED, stronger than 1d.a

Archive scan completed: 2587 / 2587 knowledge files, ~80-minute walltime on the N:\ mount, zero parse errors. Per-parent aggregates (mean / median / min / max of each file's 36-empty child score_lb and score_ub) joined with the per-position spectral observables; full summary in results/phase1d_archive_summary.csv. Each 50-empty parent's file contained between ~300 k and ~600 k 36-empty children under Takizawa's proof tree — proved values aggregated across that many positions per parent.

Headline: the D₄-only A₁(s²) channel is a STRONGER predictor of Takizawa's proved game-theoretic bound than of edax's pre-proof predicted score, and the effect survives the disc-count partial control MORE robustly than the prediction version.

Raw Spearman correlations (N = 2587):

D4-A1(s^2)  vs  edax_score            = -0.349   p = 7.4e-75
D4-A1(s^2)  vs  archive_mean_lb       = -0.498   p = 1.7e-162   <== strongest
D4-A1(s^2)  vs  archive_mean_ub       = -0.064   p = 1.1e-3
D4-A1(s^2)  vs  archive_median_lb     = -0.405   p ~ 5e-100
D4-A1(s^2)  vs  archive_median_ub     = -0.318   p ~ 1e-61

Partial Spearman controlling for |disc_diff|:

D4-A1(s^2)  vs  edax_score            = -0.279   p = 2.3e-47
D4-A1(s^2)  vs  archive_mean_lb       = -0.319   p = 4.4e-62    <== survives, stronger
D4-A1(s^2)  vs  archive_mean_ub       = -0.163   p = 6.2e-17
D4-A1(s^2)  vs  archive_median_lb     = -0.268   p = 6.3e-44

Interpretation — the 43 % gain, two readings. The raw Spearman magnitude goes from |ρ| = 0.349 (spectral vs edax's pre-proof heuristic score) to |ρ| = 0.498 (spectral vs Takizawa's proof- grounded archive_mean_lb), a ~43 % relative gain (0.498 / 0.349 ≈ 1.43). The partial version goes from 0.279 → 0.319 (14 % gain), confirming the signal is not a disc-count artefact. Two possible mechanisms contribute to the gain; they are NOT mutually exclusive and a follow-up experiment is needed to disentangle them:

• Reading A — aggregation / noise-reduction. archive_mean_lb averages over 300 k – 600 k proof-grounded 36-empty child bounds per 50-empty parent. Heavy averaging produces a smoother target than edax_score, which is a single pre-proof heuristic integer. Spearman against a smoother target is naturally tighter. Some fraction of the 43 % gain is a noise-floor effect on the y-variable, not a substantive property of the spectral observable.

• Reading B — alignment / substantive. The D₄-A₁(s²) projection is a simple linear functional of the occupation pattern with no search or heuristic content. If it correlates with the proved game-theoretic value more tightly than with a strong engine's pre-proof prediction of that value, the spectral observable is genuinely picking up structural content aligned with ground truth that the engine's heuristic does not fully capture. That would be a substantive statement about the D₄-A₁(s²) channel's informational content.

How to separate the two readings (not yet run). Run edax at matched deep search depth (e.g. d = 20, roughly comparable to the effort of Takizawa's individual sub-problem solves) on the same 2587 positions and take that as a third target, edax_d20_score.

• If ρ(spectral, edax_d20) ≈ ρ(spectral, archive_mean_lb), the 43 % gain is mostly Reading A (aggregation / smoother target). The spectral observable agrees with ANY strong evaluator, pre- or post-proof, once noise is averaged down.

• If ρ(spectral, edax_d20) is noticeably weaker than ρ(spectral, archive_mean_lb), Reading B is load-bearing: the spectral channel carries ground-truth-aligned content that even a strong deep-search engine doesn't fully capture.

At the time of this writing, only the pre-proof edax_score and the proof-grounded archive_mean_lb are computed. The deep-search comparison is scoped as Phase 1e.1 (edax wrapper + runner are already in place from Phase 1c.4 — a1_depth_gap_runner.py).

The earlier phrasing in an interim commit that the spectral channel "knows more about the TRUE game-theoretic value than edax's heuristic eval" was too strong — it conflated Reading B with the narrower Reading-A-or-B ambiguity the data actually supports. The two-reading framing is the accurate one until the deep-search edax comparison lands.

The asymmetric behavior between archive_mean_lb (very strong) and archive_mean_ub (much weaker) is structurally interpretable: the lower bound aggregate represents the "worst case under optimal play" for the side-to-move — a position's defensive floor — while the upper bound aggregate is the "best case if opponent mistakes" (often saturated at 64 for any reachable chain). The spectral observable tracks the defensive floor much better, suggesting it encodes something like "how much optimal-play value this position contains" rather than "how much tactical upside exists."

A₁⁻ magnetisation also shows modest negative partial correlations with archive bounds (ρ ≈ −0.07 to −0.08), ~4× weaker than the D₄-A₁(s²) signal. B₁, B₂, E channels: near-zero, consistent with the 1d.a finding.

This is the strict H9 / E8 result that §2c.4's Phase 1c.4 caveat predicted: an occupation-based (Z₂-invariant) observable needed to find the chess-style A₁-energy-vs-depth-gap analog in Othello. It lands at N = 2587 and p ~ 10⁻¹⁶² — the largest-N and lowest-p spectral-to-ground-truth correlation in this notebook or any of its siblings.

Open items for a Phase 1e or sequel:

• Does the effect generalise to other child-aggregate functions (variance, per-child correlation)? Preliminary: mean is strongest, median survives the partial, min/max saturate at ±64 and are uninformative.
• Does the correlation transfer to earlier game phases (55- or 60-empty positions) when larger corpora become available? The Takizawa archive only covers 50-empty and 36-empty levels.
• Is there a cleaner chess-side analog experiment? Chess lacks a weak-solution table, but the §9h′ fishtest corpus could be re-run with a Z₂-invariant D₄-only A₁(|signal|) projection paired with the Stockfish depth-20-vs-depth-12 gap; prediction is that the chess partial ρ against Stockfish would rise above the +0.456 A₁-energy baseline if the Othello structural finding transfers.
• Does the result hold for Takizawa's own accuracy-weighted scoring (weighting children by accuracy = 100 vs 99)? The current aggregate includes both equally; an exact-only restriction would reduce N per parent but tighten the signal.

2d.c Deep channel analysis — the full D₄ battery on s² (post PATCH 6 audit)

The 1d.a and 1d.b passes reported A₁(s²) as the headline and mentioned that B₁/B₂ on magnetisation came back null. A follow-up pass extended the observable battery to all FIVE D₄ irreps projected onto the occupation s² signal (Z₂-invariant probes), plus all five D₄×Z₂ '−' irreps on the magnetisation s signal (Z₂-odd probes). Motivated explicitly by the question "after the PATCH 6 B₁/B₂ character-table fix, did we re-evaluate B₁/B₂ or just assume they stayed null?" Answer: on magnetisation they are null, but on occupation the full D₄ battery contains a second strong signal we had not reported.

Complete post-fix channel table, raw Spearman vs archive_mean_lb (N = 2587):

Channel	raw ρ	raw p	partial ρ	partial p
Magnetisation (Z₂-odd)
A₁⁻	+0.187	8.6×10⁻²²	−0.069	4.4×10⁻⁴
A₂⁻	−0.006	0.76	+0.000	1.00
B₁⁻	−0.015	0.45	+0.004	0.86
B₂⁻	−0.041	0.04	+0.009	0.65
E⁻	+0.073	2.1×10⁻⁴	+0.102	1.8×10⁻⁷
Occupation (Z₂-even)
D₄-A₁(s²)	−0.498	1.7×10⁻¹⁶²	−0.319	4.4×10⁻⁶²
D₄-A₂(s²)	+0.151	1.4×10⁻¹⁴	+0.133	1.1×10⁻¹¹
D₄-B₁(s²)	+0.148	3.5×10⁻¹⁴	+0.074	1.6×10⁻⁴
D₄-B₂(s²)	+0.034	0.08	−0.007	0.73
D₄-E(s²)	+0.484	5.9×10⁻¹⁵²	+0.310	9.3×10⁻⁵⁹

Two headline findings, not one. The raw-correlation magnitudes sort into a striking pattern:

Strong (|rho| > 0.3):   A1(s^2)  -0.498,   E(s^2)  +0.484
Moderate (0.1-0.2):     A2(s^2)  +0.151,   B1(s^2) +0.148
Null (|rho| < 0.05):    B2(s^2)  +0.034

Under the disc-count partial:

Strong:     A1(s^2) -0.319,  E(s^2) +0.310
Moderate:   A2(s^2) +0.133
Weak:       B1(s^2) +0.074
Null:       B2(s^2) -0.007

D₄-E(s²) is the near-mirror of D₄-A₁(s²). Opposite sign, magnitudes within 3 % of each other raw and within 3 % partial. These are the two channels that carry essentially all of the "spectral-beyond-counting" signal in the D₄ decomposition of the occupation pattern, and they pull in opposite directions.

Interpretation. The A₁ projection picks out the D₄-invariant (fully symmetric) component of the occupation pattern — "how evenly distributed the 14 discs are under all 8 spatial symmetries." The E projection picks out the D₄-covariant 2-dim (x, y)-transforming component — "how much oriented anisotropy the occupation carries." The mirror relationship is intuitive: positions with LESS uniform fill (smaller A₁, worse for side-to-move) tend to have MORE oriented fill (larger E, also worse for side-to-move), and vice versa. Because D₄-A₁ + D₄-A₂ + D₄-B₁ + D₄-B₂ + D₄-E sums to the full occupation norm (Plancherel), a decrease in one channel must be compensated by an increase elsewhere; the A₁/E pair soaks up the dominant trade-off.

Multivariate implication. If A₁(s²) and E(s²) partially cancel each other, their DIFFERENCE D4-A1 - D4-E (or any rotation in the 2-D span) might be a sharper univariate predictor. Both channels individually explain ~10 % of the variance in archive_mean_lb (partial R² ≈ 0.32² ≈ 0.10 each); combined they might explain 25–30 %. A multivariate regression on these two alone is scoped as a Phase 1e item.

B₁(s²) / B₂(s²) asymmetry revisited. In chess, the PATCH 6 reprocess of archive/chess_a1_followup.py on 2026-04-23 (55-position hand-picked depth-gap corpus, Stockfish d=1 vs d=20) gave B₂ partial ρ = +0.490 against |depth_gap| — the single strongest complexity predictor in THAT sample, beating A₁ at +0.456 (chess §9h′ Table 1). In Othello B₁/B₂ on either signal (magnetisation or occupation) at N = 2587 vs Takizawa proved bounds are essentially null.

Scope caveat. The chess "B₂ is the best predictor" claim is based on a single 55-position corpus, not on a broader Stockfish- matched post-PATCH-6 corpus. A chess-side batch is in flight (see docs/chess-maths/results/sweep_chain_lichess_ashchess_2026-04-21_N50/) re-encoding a 50-game lichess sample on post-fix characters; the corpus-level aggregates (corpus_index.csv) and per-ply Stockfish correlations have not been regenerated from the re-encoded spectralz at the time of this notebook edit. A structural sanity scan on the 20-game post-fix spectralz subset did confirm:

• 1953 plies scanned, 99.7 % show B₁ ≠ B₂ (the correct post-fix signature; bug would pin the ratio near 100 % equal).
• Mean channel energies in that subset: A₁ = 18.94, B₁ = 12.65, B₂ = 32.66 (B₂ is the largest channel by mean), consistent with B₂ being an active complexity channel at chess-corpus scale — but not yet confirmed as the strongest Stockfish predictor at that scale.

So the chess vs Othello structural-divergence claim ("chess B₂ carries piece-species content; Othello has one disc type so the B₁↔B₂ swap has no modulation") is defensible on theory plus the 55-position empirical anchor; confirmation at broader corpus scale is pending the chess-side re-correlation batch.

Structural interpretation of the divergence (theory-side). The chess B₂ signal captures diagonal-sliding vs orthogonal-sliding structure across piece species — rook-like adjacency in the ortho orbit vs bishop-like in the diag orbit, with the "which-orbit" distinction carrying strategic content about pawn structure and piece mobility. Othello has a single disc type, so the B₁↔B₂ swap has no piece-species content to modulate. The orthogonal-vs- diagonal RAY structure still exists (it sits in the H2 8-ray decomposition 2 A₁ + B₁ + B₂ + 2 E), but a single 50-empty configuration of 14 indistinguishable discs projects into B₁(s²) and B₂(s²) as close to random-signal noise once disc count is controlled — no structural lever amplifies the anisotropy.

This is the structural interpretation of the §10.4 rank-6 irrep count not showing up as an operator rank: B₁ and B₂ are real geometric modes of the 8×8 grid, but they need piece-species or directional-movement content to carry strategic signal. Othello provides neither.

B₁(s²) moderate partial (ρ = +0.074) as a ghost signal. It is not null — raw +0.148, partial +0.074 with p = 1.6×10⁻⁴ — but it is ~4× weaker than A₁(s²) and E(s²). Most likely explanation: B₁ picks up edge/corner vs centre occupation asymmetry on the 8×8 grid (transforms as x² − y²), and tournament play at 50-empties has a mild systematic bias toward centre-filled configurations. Not a headline, but worth noting as a weak corroborating signal alongside A₁ and E.

2d.d A₂(s²) — a third meaningful channel (weaker than A₁/E)

The D₄-A₂(s²) channel transforms as R_z (rotation about the board normal) and is invariant under rotations but sign-flips under reflections. Partial ρ = +0.133, p = 1.1×10⁻¹¹ at N = 2587. Not as strong as A₁ or E but clearly above noise and with a tight p-value. Combined with A₁/E, the 2d battery on occupation has three channels with meaningful signal. The full channel-energy vector (A₁, A₂, B₁, B₂, E) on s² could be a 5-D encoding of "structural content of occupation" worth testing in a downstream classifier.

2d.e Chess vs Othello cross-game comparison (post-PATCH-6, both sides)

Cross-game comparison is now meaningful — PR #57 landed chess §9c′ (corpus-level post-fix B₁/B₂ analysis, 23 games across 4 lichess corpora) on main on 2026-04-23, with the chess manifest.json / corpus_index.csv / corpus_summary.md all regenerated from post- fix spectralz. The matched-Stockfish per-ply chess correlation is still a pending batch item (referenced in chess §9c′ but not yet computed against the re-encoded spectralz), but the structural channel-level numbers are available on both sides. Static structural comparison only; game-theoretic-value correlation comparison deferred until chess Stockfish re-correlation lands.

Similarities — universal grid-topology properties

Both chess (static §9c′ 23-game sample) and Othello (N = 2587 Takizawa 50-empty positions) show:

1. B₂ > B₁ in mean energy, at a consistent ratio:
2. Chess B₂/B₁ ≈ 1.4–1.7× across all four corpora (nykt 1.69, ash 1.68, hf 1.52, single 1.35; chess §9c′.2).
3. Othello B₂⁻/B₁⁻ = 2.01 (magnetisation), D₄-B₂(s²)/D₄-B₁(s²) = 1.82 (occupation).

4. Structural interpretation: the diagonal-orbit vs orthogonal- orbit asymmetry of the 8×8 grid under standard occupation patterns (initial position, tournament fills) favours B₂. Independent of piece-species / single-species structure.

5. B₁ and B₂ projections are exactly orthogonal per-ply. Chess §9c′.1 measures Frobenius cos(B₁, B₂) = 0.0000 over 2112 plies. Othello H3 shows the same via direct grid-topology argument (E_ortho ∩ E_diag = ∅). Identical structural fact on both games.

6. E is the largest D₄ channel by mean energy. Chess E mean 11–14 across corpora vs A₁ mean 3.9–4.9 and B₂ mean 4.9–5.9. Othello E⁻ mean 7.2 vs A₁⁻ mean 2.7 vs B₂⁻ mean 2.3. E/A₁ ≈ 2.7× and E/B₂ ≈ 2–3× on both sides.

7. A₁ and E anti-correlate in both games, but magnitude differs:

8. Chess corr(A₁, E) = −0.293 (moderate; §9c′.4).
9. Othello corr(A₁⁻, E⁻) = −0.532 (on magnetisation).
10. Othello corr(A₁(s²), E(s²)) = −0.834 (on occupation — the near-mirror we flagged in §2d.c).
11. Interpretation: A₁ and E decompose the full D₄ occupation energy into "fully symmetric" vs "2D (x, y)-oriented" components and trade off under Plancherel. Chess's signal carries additional content from the fiber channels (FS1, FS2, FS3, FA, FD) that dilutes the A₁/E anticorrelation; Othello's single-disc-type signal concentrates the trade-off into exactly two D₄ channels, producing the tighter −0.83 mirror.

Divergences — piece-species content as the amplifier

Chess B₁/B₂ and Othello B₁/B₂ behave very differently once we look beyond mean energy into cross-channel structure and strategic content:

1. Cross-channel correlation structure is opposite-signed on (A₁, B₁).

Pair	Chess ρ (§9c′.4, raw signal)	Othello ρ (D₄/s² channels)
(A₁, B₁)	+0.575	−0.299
(A₁, B₂)	+0.221	−0.230
(A₂, B₂)	+0.358	−0.003
(A₁, E)	−0.293	−0.834

Chess B₁ co-moves with A₁ ("symmetric-fiber-aligned" per §9c′.4), while Othello B₁(s²) anti-correlates with A₁(s²). The sign flip on the (A₁, B₁) pair is the sharpest structural divergence. It reflects the Plancherel accounting: in chess the signal has many channels (5 D₄ irreps + 5 fiber channels = 10 total, 640 dims) and A₁/B₁ can co-move without exhausting the norm budget; in Othello the 5 D₄ occupation channels alone account for most of the norm, so A₁(s²) and B₁(s²) compete for the same budget and anti-correlate.

1. Strategic content on B₂ is chess-specific. Chess §9h′ Table 1 (55-position hand-picked depth-gap corpus, post-PATCH-6) puts B₂ partial ρ = +0.490 against |depth_gap|, the strongest single-channel complexity predictor there. Chess §9c′.5 decomposes this by move type: rooks and kings excite B₂ preferentially (×1.28 and ×1.53), castling produces a ×2.05 B₂ spike — castling is the archetype of σᵥ-symmetry breaking and B₂ is the σᵥ-antisymmetric channel. Othello has no castling, no piece species, no rook/king asymmetry. Its D₄-B₂(s²) partial ρ = −0.007 (p = 0.73) against Takizawa's archive_mean_lb is genuinely null. The structural ray geometry is shared, but the strategic amplification requires piece-species or directional-movement content to show up.

2. The "useful" headline spectral channel differs.

3. Chess (narrow §9h′ sample): B₂ on raw piece-value signal; partial ρ = +0.490 vs |depth_gap|.
4. Othello (Takizawa-strict N = 2587): D₄-A₁(s²) and D₄-E(s²) on occupation; partial ρ = −0.319 and +0.310 vs archive_mean_lb.
5. Both findings are "spectral beyond counting" — they survive the disc-count / piece-count partial control. They just live on different (signal, irrep) coordinates.

Framework reading

The cross-game picture is:

• Static D₄ grid geometry is universal: B₂ > B₁, A₁/E anti-correlate, B₁ ⊥ B₂. Both games inherit this from the 8×8 grid topology, no piece-species amplification needed.
• Strategic signal on the B-irreps is piece-species-dependent: chess B₂ carries rook-vs-king / castling-vs-plain content that Othello's single-disc-type play has no analog for, so chess B₂ becomes the complexity headline while Othello B₂ is inert.
• Strategic signal on the A₁/E axis transfers: chess A₁ energy (§9h′ +0.452 partial vs depth_gap) and Othello A₁(s²) energy (−0.319 partial vs archive_mean_lb) are both "fully-symmetric" invariants of their respective occupation signals, and both predict ground-truth-aligned quantities.
• The E channel's partial ρ matches A₁'s in magnitude with opposite sign on the Othello side (+0.310 vs −0.319). Chess §9c′ does not yet have an E/depth-gap partial; the corpus-scale Stockfish re-correlation batch (in flight) may find a symmetric A₁/E twin result if the Othello framing transfers.

Pending confirmation items (each requires the chess Stockfish re-correlation batch to land):

• Chess post-fix B₂ partial ρ at corpus scale vs |depth_gap| / |eval_change| — does the §9h′ +0.490 hold at ~2000 plies, or was that a hand-picked-corpus artefact?
• Chess post-fix E partial ρ vs |depth_gap| — does the Othello A₁/E mirror have a chess analog?
• Chess post-fix A₁/E cross-channel Pearson at corpus scale — does the Othello −0.83 have a chess analog, or is the tight mirror an Othello-specific consequence of single-species signal?

None of these are blockers for the §2d.c A₁/E-twin claim; they add a chess-side symmetry check if the framework transfers.

2d summary (updated after deep channel analysis)

1. D₄-A₁(s²) and D₄-E(s²) are the twin headline channels, opposite signs, ~0.32 partial ρ magnitude each against Takizawa proved bounds, p values of 10⁻⁶² to 10⁻⁵⁹. Together they carry the bulk of the Z₂-invariant spectral-beyond-counting signal.
2. D₄-A₂(s²) is a supporting third channel at partial ρ = +0.13.
3. D₄-B₁(s²) is a weak fourth channel at partial ρ = +0.07.
4. D₄-B₂(s²) is genuinely null in Othello, in contrast to chess B₂ on the 55-position §9h′ hand-picked depth-gap corpus where partial ρ = +0.49 against |d1−d20| beats A₁ (+0.456). The chess claim is narrow to that corpus; broader post-fix chess-side re-correlation (50-game lichess sweep) is in flight and its corpus_index / Stockfish tables had not been regenerated from post-fix spectralz at the time of this edit. Structural reading: chess B₂ needs piece-species content to carry signal, Othello's single disc type kills that modulation.
5. All magnetisation-side projections collapse under the disc-count partial control, consistent with the Phase 1c.4 finding that Othello A₁⁻ is a disc-count proxy.

Revised framework statement: the useful spectral observables for Othello perfect-play correlation live in the D₄-only projection of the Z₂-invariant occupation signal s², not in the D₄×Z₂ projection of the magnetisation signal s. This cleanly separates Othello from chess, where magnetisation-based A₁ energy remains the complexity headline. The chess-style "A₁ energy" and Othello-style "D₄-A₁(s²)" are structurally analogous — both are the fully- symmetric irrep on the Z₂-invariant aggregate — but they live on different Z₂ parities of the underlying signal because Othello's piece content is Z₂-invariant (single particle type) while chess's is not (piece-species polarizes the signal).

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2e. Phase 1e — follow-ups, encoder, C-parity, sheaf-as-reference

Phase 1e extends 1d in five parallel tracks. Full per-item detail in the running doc PHASE_1E_FINDINGS.md; this section is the notebook-level summary with status tags.

2e.1 Multivariate A₁(s²) + E(s²) — CONFIRMED the Plancherel pair

phase1e_multivariate.py. Joint OLS at N=2587: (A₁, E) gives R² partial 0.100; full D₄ 5-channel adds only +0.012. The A₁ − E rotated direction at 135° lands partial ρ = +0.337, marginally stronger than A₁ alone. The A₁ + E (45°) direction is near-null (−0.038), confirming the §2d.c mirror.

2e.2 edax d=20 on the 2587 tasklist — Reading B CONFIRMED

phase1e_edax_d20_tasklist.py, phase1e_edax_d20_correlations.py. Spearman(D₄-A₁(s²), edax_d20) = −0.342 — essentially identical to pre-proof edax_score (−0.349, gap of 0.007). Both are ~35 % short of the archive_mean_lb correlation (−0.498). The 43 % gain in §2d.b is not a search-depth artifact — deep-search edax doesn't bridge the gap. Reading B (the spectral channel sees structural content beyond deep-search heuristics) is load-bearing.

2e.3 Shannon info × D₄-A₁(s²) — retraction via Simpson

phase1e_shannon_observables.py, phase1e_signflip_decomposition.py. Initial §2c.2 headline Spearman(I_move, A₁⁻) = +0.213 doubles to +0.465 when the D₄-A₁(s²) observable replaces A₁⁻. §1e.6 decomposition by n_empties bucket then shows both aggregates are Simpson's paradox artifacts of game-phase structure — within any single phase bucket, A₁⁻ correlates NEGATIVELY with I_move (opening in-book −0.274, late-midgame in-book −0.419). The "A₁⁻ tracks strategic divergence" framing in §2c.2 is retracted.

2e.4 Predictive sheaf (§3 L7b caveat) — target-specific trajectory memory

phase1e_predictive_sheaf.py, phase1e_predictive_sheaf_pgn.py, phase1e_predictive_sheaf_faithful.py, phase1e_faithful_gain_investigation.py.

The sheaf spectrum carries FORWARD-PREDICTIVE content — contra logo §L7b.5. On 30-seed random-play trajectories, sheaf λ₂ at time t predicts n_legal_moves(t+10) with R² = 0.56, matching sheaf λ₂ at t+10. Gain vs persistence baseline grows from +0.24 (Δ=1) to +0.49 (Δ=10). On the 2178-game 2025 tournament corpus the pattern replicates, but at scale the picture sharpens: trajectory memory is target-specific.

target Δ=10 gain_vs_snap (2025) d4_e_occupation_energy +0.159 d4_a1_occupation_energy +0.070 (grows to +0.150 @ Δ=20) rho +0.079 (grows to +0.154 @ Δ=20) n_legal_moves +0.012 (near-zero) a1_minus_energy −0.041 (snapshot wins)

Train/test split (1089 train, 1089 test) confirms the signature is not overfit (OOS gain LARGER than in-sample on most targets). Per-feature ablation shows the effect is joint feature decorrelation: at time t the 8 sheaf features (λ₂/entropy/ kerdim/λmax × owner) encode diverse trajectory-relevant info; at t+Δ they collapse to redundant current-state summaries.

2e.5 Faithful sheaf restriction maps — bracket-aware

faithful_sheaf.py. Replaces §3's endpoint-only crude restriction with an R1/R2/R3/R4 bracket classifier: edges exist only across active flanks (R2 stable, R3 pending). Breaks §3.4's constant-128 kernel artifact — faithful kernel dim is position-dependent (141–188 on seed=123). λ₂ differs from crude by mean +0.29. The per-cell R3_PENDING count, projected through D₄, becomes the headline observable in 2e.6.

2e.6 Sheaf-derived D₄ channels beat occupation by 40 % — NEW HEADLINE

phase1e_sheaf_vs_archive.py at N=2587. Against Takizawa's archive_mean_lb:

channel raw ρ partial ρ D₄-A₁(s²) §2d.b baseline −0.498 −0.319 D₄-E(s²) §2d.b baseline +0.484 +0.310 E_pending_white_A1 sheaf-derived −0.621 −0.447 E_degree_white_E sheaf-derived +0.638 +0.429 E_pending_black_A1 sheaf-derived +0.224 +0.370 E_pending_black_E sheaf-derived +0.565 +0.362

The per-cell pending white bracket count, projected through D₄-A₁, lifts the partial ρ magnitude from 0.319 to 0.447 — a 40 % improvement. Sign interpretation: many white-side pending flanks ⇒ bad news for side-to-move (BLACK on the tasklist) ⇒ lower archive_mean_lb.

Consequence: sheaf-derived channels are promoted to the encoder's reference fiber layout (v0.3.0). Channels 10-11 of the 768-dim encoding are now sheaf_pending_black and sheaf_pending_white instead of L_ortho @ s / L_diag @ s. The legacy rank-2 (orbit-Laplacian) fiber remains available via encode_768(state, fiber_mode="rank2") — and is what the C reference encoder supports until the bracket walker is ported (2e.7).

2e.7 C encoder + bit-identical engine dispatch

othello_spectral/c_encoder/, othello_spectral/codegen/emit_c_tables.py, othello_spectral/runtime.py.

ANSI C17 reference encoder with clang -Wall clean build. Python-side ctypes path (v0.3.0): loads othello_spectral.dll, direct-calls othello_spectral_encode_768 — no subprocess overhead. Benchmarks (APR 2026, 25 447 states):

path encode wall python 3.0 s 18.4 s c (ctypes DLL) 3.1 s 17.8 s c (subprocess exe) 7.6 s 21.7 s

All three paths are bit-identical on rank-2 fiber. Parity test tests/test_c_py_parity.py compares 9 states at float32 precision (subprocess) and float64 (ctypes). Sheaf-fiber C port is a future item (the bracket walker in faithful_sheaf.py is Python-only).

Engine dispatch: --engine {py, c, auto}, env vars OTHELLO_SPECTRAL_BIN / OTHELLO_SPECTRAL_DLL / OTHELLO_SPECTRAL_ENGINE; auto mode verifies the C binary against Python (version + smoke test) before selecting, silent fallback on any failure. When --fiber sheaf is active the C path is skipped (rank2-only).

2e.8 T1 flip-count + T5 chain-size — exponential at corpus scale

phase1e_flipcount_distribution.py, phase1e_t5_cluster_distribution.py.

Both §10.10 T1 (per-move flip count) and T5 (per-chain same-colour run length) reject the power-law prediction across all four committed PGN corpora (WC 2005, Barcelona 2026, APR 2026, 2025 all — 23 to 2178 games each).

T1 (flip-count): all 4 corpora prefer exponential by ΔAIC 3 k to 346 k; λ ≈ 0.548; mean 2.37 T5 (chain length): all 4 corpora prefer exponential by ΔAIC 3 k to 221 k; τ ≈ 2.68 (close to FK-BC critical τ ≈ 2.05 but exp wins); mean 2.94, max 8

Remarkably stable across a 20-year span and 3 orders of magnitude of N. Strategic play does not shift these distributions from random-play expectation (§2.E5 mean 2.28).

2e.9 T4 thermodynamic trajectory — negative T_eff robust

phase1e_t4_thermodynamic_trajectory.py, phase1e_t4_robust.py, phase1e_t4_logtransform.py.

Per-ply T_eff = dE_tot / dS_eff heavy-tailed (median −11 k, mean +70 k on Barcelona). Windowed OLS on high-R² windows: median −22 k with IQR [−35 k, −14 k]. Log-transform log(E_tot) vs S_eff: median slope −6.04 with IQR [−8.48, −4.28] — robust negative signal, interpretable as "1 unit increase in Shannon entropy corresponds to factor exp(−6.04) ≈ 0.002 change in E_tot."

Structurally consistent with §2d.c's A₁ / E mirror: as spectral energy grows with disc count, the channel-energy distribution concentrates (Shannon entropy shrinks).

2e.10 Holonomy structural map — Z₂ holonomy exclusive to lj_rect_W×1, W≥3

phase1e_holonomy_plaquettes.py. Enumerated 1192 loops across 7 shape families on the 8×8 static fiber bundle. Exactly 105 / 1192 loops carry Z₂ holonomy (cos = −1), and they are ALL of shape lj_rect_Wx1 with W ≥ 3 (long-jump rectangles of height 1). All other 1087 loops are trivial. The effect is path-orientation-dependent, not homotopy invariant: lj_rect_1×H (transpose) is trivial.

Generalises §2.H5's single-loop finding (the 0,0→0,3→1,3→1,0 rectangle, one of the 35 lj_rect_3×1 instances) to the full curvature map.

2e.11 Chess vs Othello A₁ / E — Simpson's paradox mechanism

phase1e_chess_a1_e_pair.py, phase1e_chess_a1_e_bootstrap.py, phase1e_simpson_mechanism.py, phase1e_othello_a1_e_per_game.py.

§2d.e deferred the chess A₁/E correlation check pending Stockfish re-correlation. Completed here via direct chess- spectral spectralz extraction at N=10 729 plies.

species pooled within between chess/ashchess +0.011 −0.117 +0.667 chess/drnyk −0.201 −0.315 +0.605 chess/hf +0.116 +0.022 +0.663 othello/2025 magn +0.330 +0.385 −0.479 othello/2025 occ −0.072 −0.051 −0.592

Between-game Pearson of game-means is the dominant contribution and has OPPOSITE sign between the two species:

• Chess: between +0.60 to +0.67 — games with high mean A1 also have high mean E. Pooled ≈ 0 because positive between and slight negative within partially cancel.

• Othello magnetisation: between −0.48 — games with high mean A1⁻ have LOW mean E⁻. Pooled positive because within-game co-correlation (+0.39) outweighs.

• Othello occupation: between −0.59, within −0.05. Driven by between-game baseline structure.

The Othello §2d.c 50-empty static Pearson of −0.834 is a phase-specific 14-disc configuration property, not a general trajectory signature. Per-game trajectory-level A₁/E is game-species-specific. The Plancherel-budget reading (fewer channels in Othello ⇒ tighter mirror) only holds at the static 50-empty cross-section.

2e.12 Sheaf-phased encoder v0.3.0 benchmark — sheaf is new reference

Pulls together 2e.5 + 2e.6 + 2e.7: the faithful-sheaf pending- bracket channels replace the v0.2.0 orbit-Laplacian fiber. Default encoder now v0.3.0 sheaf-default; rank-2 mode survives as fiber_mode="rank2" for C parity.

fiber mode partial ρ vs archive_mean_lb (A1 channel) rank2 (v0.2) D4-A1(s^2) −0.319 sheaf (v0.3) D4-A1(pending_white) −0.447 (40 % gain)

2e.13 C sheaf port — bit-identical v0.3.0/0.3.1 parity + 25× speedup

othello_spectral/c_encoder/src/othello_spectral.c gains classify_ray + sheaf_pending_counts (~80 lines, direct port of faithful_sheaf.classify_ray). New public API othello_spectral_encode_768_v2(state, fiber_mode, out) with fiber_mode ∈ {FIBER_RANK2, FIBER_SHEAF}; legacy encode_768 retained as alias for rank-2. CLI + ctypes wrappers thread --fiber {rank2, sheaf} through. Tests at v0.3.1 verify 36 bit-identical py/c comparisons (2 fiber modes × 2 C paths × 9 states) at float32 (subprocess) and float64 (ctypes).

End-to-end on 35-game Barcelona corpus:

path encode wall py (fiber=sheaf) 5.0 s 7.0 s c ctypes DLL (fiber=sheaf) 0.2 s 2.4 s 25× encode speedup

Byte-identical .spectralz SHA256 = 461bd1219357041a.

2e.14 s-vs-s² asymmetry mechanism — three-regime structural explanation

phase1e_s_vs_s2_asymmetry.py runs four per-game diagnostics on 2178 games of 2025 corpus: volatility, lag-1 autocorrelation, Pearson with disc-count, monotone fraction.

target volatility autocorr1 r_disc mono_frac a1_minus_energy +0.236 +0.842 +0.755 0.550 d4_a1_occupation_energy +0.051 +0.9996 +0.991 1.000 d4_e_occupation_energy +0.277 +0.961 +0.083 0.500 rho +0.056 +1.0000 +1.000 1.000 empty_count +0.056 +1.0000 −1.000 1.000

Three regimes fully account for the §1e.7.4b trajectory-memory gain pattern:

1. Game-clock monotone (ρ, empty_count, d4_a1_occ) — trivially predictable. The sheaf "predictive gain" for these is a disc-count correlation artefact.

2. Non-monotone structural (d4_e_occ, r_disc = +0.083). Genuine trajectory memory. The sheaf's pending-bracket count predicts how oriented occupation anisotropy evolves because active flanks drive directional shifts.

3. Turn-order-coupled (a1_minus). The sheaf captures WHAT happens structurally (brackets) but not WHO places/ flips the disc. Sign of A1⁻ depends on turn order; the bracket classifier doesn't encode that → snapshot wins.

Z₂ representation theory doesn't explain it; the distinction is target's coupling to (disc count, turn order), not Z₂ parity.

2e.15 Sheaf-phased operator — 12 % additional gain over §2e.6

othello_spectral/phase_operator.py. The first concrete instantiation of a §4-style phase operator for Othello. Applies a state-dependent diagonal scaling D(f(state)) to the encoding vector, where f maps sheaf spectrum features (λ₂, λ_max, entropy, kerdim per owner) to channel weights.

Four modes provided (v0.3.1):

• identity no-op baseline
• scale_fiber_by_owner_lambda2 weight fiber channels by 1/(1+λ₂^{owner})
• scale_all_by_kernel_dim weight all channels by ⟨kerdim⟩/192
• rotate_E_by_entropy cos(entropy_B) damping of E channels

Benchmark on 2587 tasklist against archive_mean_lb:

mode fiber_pw through D4-A1 partial ρ identity (baseline Exp 1) −0.447 scale_fiber_by_owner_lambda2 −0.503 (+12 % gain) scale_all_by_kernel_dim −0.449 rotate_E_by_entropy −0.447

scale_fiber_by_owner_lambda2 lifts the partial ρ magnitude from 0.447 to 0.503 — another 12 % on top of the 40 % gain sheaf fiber gave over occupation in §2e.6. Total improvement over the §1d.b baseline D₄-A₁(s²) (ρ = −0.319): 0.503 / 0.319 ≈ 58 % cumulative gain.

Interpretation: λ₂ of each owner's faithful sheaf Laplacian measures the algebraic connectivity of that owner's bracket graph. Dividing the per-cell bracket counts by 1 + λ₂ normalises for "how busy" the bracket network is, letting per-cell structure drive the correlation rather than total activity.

Framework is minimal — diagonal scaling only. Phase operators with off-diagonal coupling (SO(2) rotations within the E subspace, state-dependent mixing between channels) are future work. The current module is pure Python; C port would need sheaf eigendecomposition (scipy eigh; not bit-identical across LAPACK versions, so the operator is platform-reproducible but not bit-portable at v0.3.1).

Open items (updated after this pass): - Port faithful_sheaf.classify_ray to C — CLOSED (2e.13) - Z₂-cancellation Exp 2 on Barcelona — CLOSED (2e.14) - Sheaf as phase-operator coupling — CLOSED (2e.15)

2e.16 Learned coupling matrix — +114 % cumulative gain

phase1e_learned_coupling.py. Fits the scale_by_feature_coupling mode's coefficient row M[11] (the one that determines the weight for pending_white fiber) by Nelder-Mead minimisation of −|Pearson(E_pw_A1, archive_mean_lb)| on a 75 % training split (1941 positions). Held out 646 positions for validation. 6 restarts to escape local minima.

stage partial ρ (Spearman) §1d.b D4-A1(s^2) baseline −0.319 §2e.6 raw pending_white_A1 −0.447 §2e.15 scale_fiber_by_owner_lambda2 −0.503 §2e.16 learned coupling (val) −0.683 §2e.16 learned coupling (train) −0.636 §2e.16 learned coupling (overall) −0.648

Val partial ρ is slightly BETTER than train (no overfitting signal); generalisation is clean. Cumulative improvement over §1d.b's occupation A₁(s²) baseline: 0.648 / 0.319 ≈ 2.03×.

Learned coefficients (for the 8-d sheaf feature vector):

feature coefficient lambda2_B -0.016 lambda2_W +0.001 lambdamax_B +0.000 lambdamax_W -0.009 entropy_B +0.256 entropy_W -0.283 kerdim_B +0.027 kerdim_W -0.026

The dominant structure: weight ≈ 0.256 · entropy_B − 0.283 · entropy_W — almost exactly the difference in sheaf entropies between owners.

Interpretation: spectral entropy of the faithful sheaf Laplacian measures how DIVERSE the eigenvalues are (high entropy = spread distribution = rich mode structure; low entropy = concentrated distribution = simple bracket topology). The owner-entropy difference encodes which player's flanking network is more structured. When white's bracket network is MORE structured than black's (i.e. W_entropy > B_entropy), weight is negative, which inverts the pending_white fiber sign, making it CORRELATE POSITIVELY with archive_mean_lb (it was negatively correlated raw).

The operator encodes a structural "who has more bracket complexity" coupling that the hand-tuned λ₂ modes couldn't capture. Worth noting: this is NOT a simple algebraic coupling (not a weight-function of a single spectral quantity). It requires comparing two owners' spectra — a feature of the bracket structure that the sheaf uniquely exposes.

2e.17 Off-diagonal SO(2) on the E irrep subspace

othello_spectral/phase_operator.py adds rotate_E_so2_by_entropy and rotate_E_so2_by_kerdim modes that apply a D₄-equivariant SO(2) rotation within the 2-dim subspace spanned by the lowest-order E-irrep basis:

phi_x(r, c) = (c - 3.5) / ||·||
phi_y(r, c) = (r - 3.5) / ||·||

on the magnetisation-E and occupation-E channels. Because the rotation is orthogonal, it preserves channel ENERGY exactly; it only changes the (proj_x, proj_y) PHASE within the E-isotypic pair. The benchmark against archive_mean_lb through channel ENERGIES (our standard metric) is therefore insensitive to the rotation — confirmed empirically (partial ρ unchanged at −0.4467 across the two new modes).

SO(2) E-rotation is a scaffolded capability: downstream models that use direction-sensitive E-channel summaries (e.g., a phase difference between E-magn and E-occ) can benefit. For now it sits in the API alongside the diagonal-scaling modes as part of the §4 phase-operator toolbox.

2e.18 Move-type phase operators

othello_spectral/move_operator.py. Chess-style per-cell placement operators P_place_i for i in 0..63, using the faithful-sheaf bracket classifier to determine which opponent discs flip. Public API:

SheafMoveOperator.is_legal(state, move, side) -> bool SheafMoveOperator.flip_count(state, move, side) -> int SheafMoveOperator.flipped_cells(state, move, side) -> list[int] SheafMoveOperator.apply(state, move, side) -> post_state SheafMoveOperator.encode_post_move(state, move, side) -> post_enc SheafMoveOperator.legal_moves(state, side) -> list[int] SheafMoveOperator.flip_count_vector(state, side) -> np.ndarray[64]

Parity tests verify byte-for-byte agreement with OthelloBoard.play() across 20 random-play games. 6/6 tests in tests/test_move_operator.py pass. No mutation of the input state.

Operational usefulness: - flip_count_vector is a 64-dim state descriptor that ranks cells by "impact if played" — candidate extra channel for a future v0.4.0 encoder. - encode_post_move allows look-ahead search algorithms to get the encoding of any reachable state without re-replaying through OthelloBoard, skipping the replay overhead (~3× on the sheaf-fiber encoder which does its own bracket walk). - Composition of multiple P_place_i gives multi-ply look-ahead encoding chains, the analog of chess's P_rook · P_bishop sequences. Sheaf spectrum of intermediate states is computable via the same runtime.

2e.19 Encoder replay path now uses SheafMoveOperator — v0.3.2

The encode-pgn CLI previously delegated per-ply legality to OthelloBoard.legal_moves() via othello_pgn_loader.replay() — a 64-cell scan on every ply even though only one cell moves. v0.3.2 swaps that for pgn_loader.replay_sheaf(), which uses SheafMoveOperator.flipped_cells(state, move, side) to walk only the 8 rays from the target cell. Slow path (forced-pass detection on genuinely illegal-looking plies) still defers to op.legal_moves so semantics match exactly.

End-to-end speedup on liveothello-2026-APR (414 games, 25,447 plies):

stage	OthelloBoard	SheafMoveOp	speedup
replay	7.02 s	1.42 s	4.9×
encode (C ctypes)	1.9 s	2.2 s	~1× (unchanged)
total	8.92 s	3.62 s	2.5×

SHA-256 of the resulting .spectralz is byte-identical across replay engines (78,275,228 bytes, hash 3c10951d36e926694e94f6539476d96a53d5a8d8b1f5d3c27862a59a44235ba3) — the speedup is pure. Byte parity is locked in by tests/test_pgn_loader_replay_parity.py: 3 tests covering the legal control game, both illegal fixtures, and 10 random-play games.

The legacy path remains available via encode-pgn --replay-engine othelloboard for audit / debugging. An illegal-move corpus fixture tests/fixtures/illegal_move_corpus.pgn (2 bad games + 1 legal control) plus 5 rejection tests in tests/test_encoder_rejects_illegal_pgn.py verify that both replay engines raise ValueError("... not legal ...") on the same inputs.

This closes the circuit: the SheafMoveOperator instantiated in §2e.18 as a chess-style P_place_i toolkit is now also the production replay primitive that drives every published encoding run, unifying validation, mutation, and lookahead-encoding under the faithful sheaf's bracket walker.

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3. Dynamic fiber — sheaf Laplacian instantiation

research/dynamic_sheaf.py builds a minimal Hansen-Ghrist sheaf Laplacian whose vertex stalks are the 3-state Blume-Capel fiber F(v) = R^3 carrying (empty, black, white) one-hot components. Edge stalks are R^3; restriction maps depend on the cell state at the edge's endpoints (empty -> project onto empty coordinate; friendly -> identity on that player's component; opponent -> half weight, modelling an in-progress bracket).

The map is a crude surrogate — a more faithful construction would make restrictions depend on the full bracket validity of the segment, not just its endpoints. We proceed with the crude version to get a working spectral trajectory and leave faithful implementation as sequel work.

3.1 Sample game trajectory

Random play from the starting position; 60 moves to terminal. Per-move spectra:

trajectory length: 60 moves
rho range:         0.062 to 0.984  (monotonically increasing, as expected)
sheaf lambda_2:    min = 0.223, max = 0.934, mean = 0.571
sheaf entropy:     min = 3.881, max = 4.029
kernel dim:        constant at 128 (2/3 of total dimension 192)
legal-move count:  min = 1, max = 18

3.2 Correlations

Spearman(rho, lambda_2)           = -0.008  (p = 0.95, uncorrelated)
Spearman(legal_moves, lambda_2)   = +0.765  (p = 1.1e-12, strongly positive)

The headline: the sheaf spectral gap tracks legal-move count, not disc density. This is consistent with λ₂ measuring structural connectivity of the flank graph — positions with many legal placements have more "connected" flank structure, producing a wider spectral gap.

L7b caveat. The logo notebook's L7b retraction established that a snapshot fiber does not necessarily extrapolate forward in time. The sheaf spectrum reported here is a per-move snapshot; we have NOT tested whether it predicts the spectrum 5 or 10 moves into the future. The correlation with legal-move count tells us the spectrum varies with game state in a structured way; it does NOT tell us the spectrum is a predictive summary of future state. Any sequel that uses the sheaf spectrum as a game-state summary must test predictive power explicitly, following the L7b template.

3.3 What the sheaf-spectrum observable is good for

Based on the 60-move trajectory:

• Positive: discriminates "many-option" (midgame) from "few-option" (opening / endgame) positions at Spearman +0.77.
• Null: does not track disc-count filling independently of legal-move count.
• Unknown: whether the spectrum is a USEFUL observable depends on the sequel's validation tests. If spec(L_F(t)) is flat across legal moves at a given position, the sheaf adds no discriminative signal over the static fiber; if it varies meaningfully across positions with the same disc count, it does.

Preliminary indication: the spectrum varies more than flatly within a 60-move trajectory (range [0.22, 0.93]), so the observable is not trivially degenerate.

3.4 Kernel dimension

The crude sheaf's kernel dim is constant at 128 across the trajectory. This is an artefact of the simplified restriction maps: empty cells project onto a single coordinate, leaving two kernel directions per cell.

CLOSED in §2e.5 — the faithful (bracket-aware) sheaf (faithful_sheaf.py) breaks the constant kernel: on the seed=123 trajectory the kernel dim ranges 141–188, varying with how many cells participate in active brackets. The faithful restriction maps depend on R2_COMPLETE / R3_PENDING classification of each ray segment, not just cell-endpoint state. λ₂ differs from crude by mean +0.29. All downstream spectrum-based analyses (§2e.4, §2e.15, §2e.16) use the faithful sheaf.

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4. Phase-operator preflight — instantiated in §2e.15–§2e.18

See OTHELLO_PHASE_OP_PREFLIGHT.md for the original handoff document. Phase 1e replaced "scaffold only" with a working toolbox:

• Encoder (D = 768 at v0.3.1): default fiber_mode="sheaf" uses per-cell R3_PENDING bracket counts as channels 10/11 (§2e.6). Legacy fiber_mode="rank2" — the §9r polarization- collapse default — kept for C-parity and ablations.
• C reference encoder at research/othello_spectral/c_encoder/, bit-identical to Python on both fiber modes (§2e.7, §2e.13). 25× speedup on sheaf encoding via ctypes.
• Sheaf-phased operator (research/othello_spectral/phase_operator.py) with 6 modes: identity, scale_fiber_by_owner_lambda2, scale_all_by_kernel_dim, rotate_E_by_entropy, rotate_E_so2_by_entropy, scale_by_feature_coupling.
• Learned coupling (§2e.16): Nelder-Mead fit produces partial ρ = −0.683 against archive_mean_lb — 2.03× the §1d.b D₄-A₁(s²) baseline. Dominant coefficient pattern +0.256 entropy_B − 0.283 entropy_W: cross-owner sheaf- entropy difference as the structural coupling.
• Move-type operators (research/othello_spectral/move_operator.py), chess-style per-cell P_place_i using the bracket classifier to compute flipped cells. 6/6 parity tests against OthelloBoard.play() on 20 random games.
• Encoder replay primitive (§2e.19, v0.3.2): encode-pgn now uses pgn_loader.replay_sheaf() — wrapping the same SheafMoveOperator — as its replay backend. 2.5× end-to-end speedup on APR 2026 (8.92 s → 3.62 s), byte-identical output. The faithful sheaf's bracket walker is now the single source of truth for validation, mutation, and lookahead-encoding.

Residual preflight items: off-diagonal phase operators beyond SO(2)-on-E (state-dependent channel mixing among non-E subspaces), phase-operator composition for multi-ply look-ahead, and a C port of the phase operator itself.

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5. WTHOR empirical tests — resolved in §2e

§10.10 tests 1–5 plus the A1 depth-gap transfer. Phase 1c–1e answered every item; original section preserved below for historical context. Current status:

• T1 Flip-count distribution — FAILED power-law hypothesis, PASSED exponential with λ ≈ 0.548 across all 4 tournament corpora (§2e.8).
• T2 B1 vs B2 population asymmetry — CONFIRMED (§2b.T2); B2 > B1 on the static 50-empty tasklist (2.01×) and on trajectory averages (1.07–1.25×, §1e.7-era commits).
• T3 Shannon information per move — CONFIRMED via opening_book_freq.csv.bz2 (§2c.2) with Simpson's-paradox retraction of the aggregate reading (§2e.3 / §1e.6).
• T4 Trajectory in (T_eff, D_eff) plane — PARTIAL; median T_eff = −22 k robust across corpora via windowed OLS, or −6.04 log-E slope; heavy-tailed at short windows (§2e.9).
• T5 Flank-cluster size distribution vs FK-BC — FAILED power-law, PASSED exponential (τ ≈ 2.68 across all 4 corpora; §2e.8).
• H9 strict perfect-play depth-gap — CONFIRMED (§2d.b at N = 2587) with Reading B separation confirmed by edax d=20 (§2e.2).

research/wthor_loader.py scope became unnecessary: the 2178-game liveothello_2025_all.pgn + Takizawa tasklist + archive summary together provide the WTHOR-scale corpus.

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6. Vocabulary collisions specific to Othello

• "Flip" — sign change on an opponent disc (s -> -s). Distinct from "flank" (the geometric structure of the bracket) and "bracket" (the ray segment same — opposite^+ — same).
• "Ray" (direction, one of 8) vs "line" (a segment, possibly long, possibly empty). The Laplacians in §1 are per-ray; the sheaf edges in §3 are per-segment.
• "Fiber" — adopted from chess §7; here means the non-spatial rule-coupling content of the 8-ray structure projected into the grid eigenbasis. Unlike chess, the fiber is dynamic: its restriction maps evolve with the board state.
• "Polarization" — adopted from chess §9r. In Othello the polarization reduces to (theta-class, r = infinity, c = 0) — a single excitation with 8 angle classes, infinite range (flip propagates until bracket close), zero chirality (Z₂ exact).

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7. Appendix: Environment and reproducibility

• Python 3.12 (tested), NumPy, SciPy.
• Deterministic: all random-draw tests use np.random.default_rng with explicit seeds. See individual functions in research/consolidated_tests.py, research/dynamic_sheaf.py.
• Wall time (this session, all tests run back to back on laptop): consolidated_tests.py ~3 s, dynamic_sheaf.py ~15 s.

• Reproduce Phase 0–2 with:

  cd docs/othello-maths/research
  python othello_utils.py
  python d4_z2.py
  python ray_laplacians.py
  python consolidated_tests.py
  python dynamic_sheaf.py
  

• Results land in docs/othello-maths/results/. Notebook + preflight

• instructions live at docs/othello-maths/.

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How to cite this notebook

BibTeX:

@misc{kirkland_othello_spectral_2026,
  author       = {Kirkland, Steven},
  title        = {Othello as a Dynamic Spectral Lattice System --- Research Notebook},
  year         = 2026,
  howpublished = {\url{https://github.com/lemonforest/mlehaptics/blob/main/docs/othello-maths/othello_spectral_research_notebook.md}},
  note         = {Part of \emph{mlehaptics: Spectral-Research Portfolio}; Reversi piece-flip dynamics + sheaf-Laplacian instantiation. Project-level citation metadata at \url{https://github.com/lemonforest/mlehaptics/blob/main/CITATION.cff}. Co-authored with Claude Opus 4.7 (Anthropic, 1M-context configuration) per project memory \texttt{feedback\_orchestration\_metaphor}. Framing is one candidate within the project's research portfolio per \texttt{feedback\_no\_lineage\_claims\_in\_notebook}.}
}

Plain text: Kirkland, S. (2026). Othello as a Dynamic Spectral Lattice System — Research Notebook. mlehaptics Spectral-Research Portfolio. https://github.com/lemonforest/mlehaptics/blob/main/docs/othello-maths/othello_spectral_research_notebook.md

Per-result citation discipline. Specific technical claims cite their canonical sources directly (WTHOR archive, sheaf-cohomology textbooks, etc., PDF-verified per [[feedback_pdf_extraction_citation_discipline]]). When citing a specific result, prefer citing both this notebook AND the underlying canonical source. Framings presented here are candidate methodological readings per [[feedback_no_lineage_claims_in_notebook]], not endorsed over alternatives without explicit empirical convergence.

Project-level citation. See CITATION.cff at the repo root for the project-as-a-whole citation form.