Parallel King-Attack Encoder: Notebook Supplement¶

Status: working supplement. Triggered by §11.5.6's validated null for path-2 check filtering and §11.6.6.1's three-corpus DRIFT null on encode_640. Asks whether a parallel HDC instrument targeting king-attack structure can carry the signal encode_640 does not.

Framing: this is not an extension of encode_640. It is a parallel instrument built from the same mathematical discipline that produced encode_640. Per UTLP S3 §14's segmentation result, the combined representation is [encode_640 | encode_king_attack] with the two segments concatenated rather than superimposed.

Language discipline. Same as the main supplement: polarization, lattice domain, phase transition, coprime cyclic phase space. The "king" in "king-attack encoder" is the royal polarization state under attack; the encoder measures the field-theoretic structure of attack relationships against that polarization.

Architectural constraint — UTLP S3 §14. Adding a new channel by superposition into encode_640 would rotate every dimension's sign-vote and destroy the existing similarity structure (Frady et al. 2022 cross-tier interference; see UTLP S3 §14.2). Segmentation — concatenating a parallel vector rather than mixing channels — avoids this. The §11 baselines remain valid; the new signal lives in a disjoint subspace.

§12.1 Motivation¶

Three findings from the §11 arc frame the §12 question:

§11.3 / §11.4. The phase-operator move generator reproduces python-chess's pseudo_legal_moves at 100% across four independent channels (A/B/C/python-chess) after the §11.4.3.1 castling fix and the §11.4.3.⅔ en passant + promotion wrappers. The 640-dim encoder's coprime-cyclic representation fully captures chess move geometry.
§11.5.6. Phase-tuple similarity between pre-move and post-move encode_640 vectors does not correlate with is_check_unsafe. Max |ρ| = 0.097 across nine polarization/capture slices, n = 3393 transitions. The encoder is blind to king-attack geometry at the similarity level.
§11.6.6.1. The same encoder produces continuous trajectories through game positions rather than discrete phase-cells. Across three structurally orthogonal corpora (110 games, 14,729 plies, GM classical / FM blitz / engine blitz), all 15 drift-ρ measurements are strongly positive (+0.46 to +0.84) and cluster counts rise monotonically with τ without a plateau.

Shared diagnosis. encode_640 is a content descriptor (what pieces are at what phases) rather than a discontinuity detector (what's dangerous, what's transitional). The construction — D4 irreps plus rank-3 fiber SVD plus FA/FD channels — produces bundled-HDC superpositions where every piece contributes to every mode. Changes accumulate smoothly. None of the constituent derivations produces discrete cluster boundaries or king-attack geometry as a first-order output.

The §12 question. Can a parallel HDC instrument, derived from the same mathematical discipline as encode_640 but targeting king-attack structure specifically, expose signal that encode_640 does not?

Three possible outcomes (per §11.7.4 discipline):

|ρ| > 0.3 on any derivation: king-attack encoder viable. The HDC family extends to king-attack content. §11.5's null was specific to encode_640, not structural to the encoder family.
|ρ| < 0.1 across all three derivations: the construction pattern does not naturally produce king-attack content. The §11.5 null generalizes to a structural property of the encoder family. Three independent derivations failing is strong evidence that the mathematics does not support the construction.
0.1 ≤ |ρ| < 0.3 on any derivation: ambiguous. Researcher decides whether to refine derivations or accept the ambiguity.

All three outcomes advance the research record. None is wasted effort.

Why now. §11.5's null left open whether a different encoder might carry the signal. §11.6 sharpened the question: whatever constituent mathematics would produce discontinuity structure on game trajectories is the same mathematics that would plausibly produce king-attack signal on individual transitions. Both are about the encoder being sensitive to threshold crossings rather than to continuous accumulation. Derivation A (king-centered Laplacian eigendecomposition) is the most motivated candidate in this light; its eigenstructure is a property of the position-dependent attack graph, and topology changes discretely when a piece moves into or out of an attack line. Whether that discontinuity actually correlates with is_check_unsafe is empirical — but the construction has a structural reason to expect the correlation, which is more than §11.5 could say in advance.

§12.2 Derivation discipline¶

Each channel in the king-attack encoder must be derivable from a specific mathematical object. Just as encode_640's channels come from D4 irreps, fiber SVD, pawn antisymmetry, and diagonal deviation — each with its own derivation — §12's channels must come from named mathematical objects of the lattice and the attack structure. No channel may exist because a use case demands it. No channel may be tuned to produce a desired correlation outcome.

This is the same rule as §11.7.4: record failures, do not repair them.

If the mathematics does not produce a coherent channel, the construction fails and we record the failure as the finding. Do not engineer around it.

§12.3 Derivation A — King-centered Laplacian eigenchannel¶

Mathematical object. The position-dependent 64×64 attack Laplacian L_ka, where L_ka[i, j] = −1 if square j is attacked by an opponent piece on square i (with i ≠ j), L_ka[i, i] = deg(i), and L_ka[i, j] = 0 otherwise. This is the combinatorial Laplacian of the opponent-attack graph restricted to the current position. Because the attack relation is directed but the eigenstructure is more interpretable on the symmetric version, the working definition symmetrizes via L = D − (A + Aᵀ)/2; the directional content is retained in the unsymmetrized A_ka consumed by Derivation B.

Channel derivation. Compute eigendecomposition of L_ka. Project the king's unit impulse δ_king (a 64-dim vector with 1 at the king's square and 0 elsewhere) onto the first k eigenvectors (ascending eigenvalue order). The projection magnitudes form a k-dimensional feature vector.

k is chosen empirically to match encode_640's fiber-channel count (3 or 5). Phase A defaults k = 5. Variance-explained at k = 5 is reported as a diagnostic on a representative position sample; if below ~80%, the eigenchannel is not a faithful summary and the derivation's utility is limited by its information capture.

Why this is structurally motivated. L_ka is position-dependent; its topology changes discretely when pieces move into or out of attack lines. The eigenstructure therefore changes discretely too. This is the discontinuity mechanism encode_640 lacks. Whether the discontinuities correlate with is_check_unsafe is empirical.

§12.4 Derivation B — D4 decomposition of attack adjacency¶

Mathematical object. The binary attack adjacency A_ka, where A_ka[i, j] = 1 iff j is attacked by an opponent piece on i, 0 otherwise. Note A_ka is in general not symmetric (attack is directed) and not D4-invariant (it depends on the specific piece positions).

Channel derivation. Reduce A_ka to a 64-dim signal via row-sum — the total outgoing attacks from each source square (0 for squares not occupied by an opponent piece). Project that 64-dim signal onto the five D4 irreps using the same Serre character projection formula encode_640 uses for its board signal (chess_spectral.tables.project_irrep). This produces five 64-dim channels A1_ka, A2_ka, B1_ka, B2_ka, E_ka, paralleling the five D4-irrep channels of encode_640.

Known tension. The projection is approximate because the source signal derived from A_ka is not D4-symmetric in general. The projection returns the best-fit D4-equivariant component in each irrep. Whether that approximation carries useful signal or degenerates to noise is empirical — §12.6 analysis reports per-irrep variance explained by the projected component as a diagnostic.

Alternative signal choices (deferred). Column-sum (total incoming attacks per square) or an attacker-type-weighted row-sum could be substituted for the plain row-sum. Phase A uses row-sum as the simplest first choice; if Phase A crosses the viability threshold on a different signal, that would be a finding about which projection carries the information.

§12.5 Derivation C — Attack operator from king's phase¶

Mathematical object. The §11.4.3.1 P_castle pattern generalized to king-attack: a composite operator on the king's phase that enumerates, per attacker type and per direction class, whether the corresponding phase shift from the king lands on an occupied square of the right attacker type.

Channel derivation. For each attacker type t ∈ {N, B, R, Q, K, P} and each direction class d in t's attack pattern, compute the phase shift Δ_{t,d} from the king. For the king's current phase φ_K in the current position, count the number of opponent-t pieces at phases φ_K + k · Δ_{t,d} mod 640 for k ∈ {1, …, 7} (sliders, weighted by 1/k so closer attackers count more and the ray stops at the first blocker) or k = 1 (non-sliders). Pawn attack squares use the §11.5 diagonal scheme, split per diagonal for a 2-component pawn signal.

The (attacker_type × direction_class) enumeration yields 16 scalar components in Phase A, packed into a 16-dim feature vector:

4 rook-ray densities (+row, −row, +col, −col) — rooks and queens
4 bishop-ray densities (+NE, −NE, +NW, −NW) — bishops and queens
1 knight attack count
1 king adjacency count
2 pawn attack counts (per diagonal)
4 reserved zero channels for future extensions

Relationship to path-1 phasecast_is_check. Derivation C is the vector-valued generalization of phasecast_is_check from the §11.5 substrate. The boolean answer to "is the king attacked?" reduces these 16 components to "is any of them positive"; C keeps them as a continuous signature. C therefore is expected to correlate with is_check_unsafe because it literally enumerates the components of is_check_unsafe as continuous values.

Purpose of including C. It serves as a refined baseline. Any derivation (A or B) that fails to beat C's correlation is carrying less signal than a straight enumeration of attack-line occupation. If A or B beats C, it means those derivations expose structure the direct enumeration does not — which would be the genuine §12 finding.

§12.6 Assembly protocol¶

Do not prematurely commit to an assembly. Phase A builds A, B, C independently and measures their pairwise cosines plus their correlations with is_check_unsafe. Only if Phase A produces usable signal does Phase B address assembly.

Phase A reports:

Per-derivation |ρ(similarity, is_check_unsafe)| on the §11.5 CSV (3393 transitions, existing labels).
Per-derivation pairwise cosines across a representative sample of positions — if cos(A, C) > 0.5 the two are highly correlated and A adds little over C; if cos(A, C) < 0.2 they measure different things.
Variance-explained diagnostics per Derivation A's k and per Derivation B's irrep.
Per-call timings for each derivation (µs per position).

Phase B (conditional on Phase A crossing |ρ| > 0.3 for any derivation):

Concatenation architecture [encode_640 | encode_king_attack], per UTLP S3 §14.3 segmentation.
Dimensionality budget, channel layout, Serre-projection choices.
Write-back to chess_spectral as a new encoder variant — not a modification of encode_640. §11 baselines must remain reproducible against the unchanged encode_640.

§12.7 Evaluation — success, null, ambiguous¶

Same three-outcome framework from §11.5 and §11.6.6.1:

|ρ| > 0.3 on any derivation: king-attack encoder is viable. Record as validated finding. Phase B is unblocked pending researcher review.
|ρ| < 0.1 across all three derivations: validated null. The construction pattern does not naturally produce king-attack signal. §11.5's null generalizes to a structural property of the encoder family.
0.1 ≤ |ρ| < 0.3 on any derivation: ambiguous. Researcher decides whether to refine derivations (Phase A2 with alternative signal choices for B, different k values for A, different reductions for C) or accept the ambiguity as the §12.10 finding.

Per §11.7.4, record failures; do not tune derivations to produce desired outcomes. The numbers are the finding.

Phase A's three-corpus extension produced an AMBIGUOUS categorical result with two evaluation-harness bugs that prevented a clean test of Derivations A and C:

Derivation A at k=5 did not test the eigenchannel hypothesis. Variance-explained on δ_king at k=5 was 7–11% across the three corpora. The first five eigenvectors of the attack Laplacian carry almost none of the king-impulse energy; attack-line structure lives in higher-frequency modes. A's measured |ρ|=0.161–0.179 tested "the smoothest 5 of 64 modes carry some signal," not the full eigenchannel hypothesis. Phase A2 re-runs Derivation A at k=16 (and reports variance-explained at that k as a diagnostic). If variance-explained at k=16 is still below 0.8, the CLI accepts --k-for-a values up to 32; the first k that achieves ≥0.8 variance-explained on a representative position sample is the honest test of A.

Derivation C's cosine metric did not measure corpus-relevant change. Every sampled position in the three corpora has a king not under direct attack, so derivation_c_channel returns all-zeros on every row, and cosine of two zero vectors collapses to 0.0. C's feature vector is correct; the cosine summary wrapped around it was the wrong reduction for corpora where both sides of the transition have zero-magnitude C vectors. Phase A2 adds two alternative scalar summaries alongside cosine:

delta_c: L2 norm of (C_after − C_before). Nonzero whenever the move changes any component of the king-attack vector, even when both endpoints are near-zero.
mag_c_after: L2 norm of C_after alone. Captures the post-move attack density directly; its correlation with is_check_unsafe is expected to be near-1 by construction (is_check_unsafe is literally "some component of C_after is positive"), and serves as a sanity check that the evaluation pipeline recovers the tautological baseline.

The three C metrics together let us distinguish:

"C's derivation does not carry signal" (all three near zero)
"C's derivation carries signal but cosine is the wrong metric" (cosine near zero, delta_c and/or mag_c_after above threshold)
"C's evaluation pipeline has a bug beyond the metric choice" (mag_c_after fails to recover the near-1 tautological baseline)

Derivation B is not refined. Its current three-corpus evaluation is methodologically sound. Its ambiguous result (single corpus crosses 0.3 on a single slice; does not replicate on other corpora) is the honest research finding. Per §11.7.4, alternative B signals are not explored in Phase A2.

Phase A2 emits new CSVs with _a2 suffix on disk; Phase A CSVs remain unchanged as part of the research record.

§12.7.1.1 Turn-flip wrapper fix¶

Phase A2's tautological-baseline check (|C_after| correlation with is_check_unsafe should be near 1.0 by construction) halted the three-corpus re-run when the baseline failed on drnykterstein. The diagnosis was a missing board.turn flip in all six similarity wrappers across Derivations A, B, and C: after board.push(move), python-chess flips board.turn to the opponent, and the subsequent call to derivation_*_channel(board_after) queries the wrong king (for A and C via board.king(board.turn)) or the wrong attacker color (for B via opp_color = not board.turn inside attack_adjacency). The correct pattern — established in phase_operators/phase_check_detection.move_leaves_king_in_check — flips board.turn back before the post-move channel computation.

The fix is one line per wrapper: board_after.turn = not board_after.turn after the push. The channel functions themselves (derivation_a_channel, derivation_b_channels, derivation_c_channel) and the underlying attack_graph primitives are unchanged.

Consequence for the research record: the original Phase A three-corpus numbers (B's 0.332 single-corpus crossing, A's 0.161–0.179 across corpora, C's NaN) were computed on wrong-side measurements. The three-corpus protocol still detected non-replication of B, but the quantity being measured was not the one the derivation was designed to produce. Phase A2 (post-fix) supplies the fresh three-corpus matrix that §12.10.1 interprets.

The tautological-baseline design pattern is preserved as a reusable check for any future evaluation harness: any derivation whose correlation with its target is provably near 1.0 by construction should be included and asserted, to catch exactly this class of wrapper-layer bug before it contaminates interpretation.

§12.8 Infrastructure requirements¶

Phase A needs:

king_attack_encoder/ package as a sibling of phase_operators/.
Three derivation modules, one per §12.3–§12.5, plus an attack_graph.py shared substrate.
Evaluation CLI that reads results/phase_operator_experiments/exp3_phase_similarity.csv (the §11.5 CSV), recomputes each transition's encoder-before and encoder-after for each derivation, records the king-attack similarities alongside the existing columns, and emits a new CSV.
Unit tests per derivation covering the underlying graph/matrix construction, a determinism check, edge cases (no king, no opponent pieces), and a performance budget of <10 ms per encode call.

Frozen dependencies (read-only imports):

phase_operators.phase_operators — phase arithmetic constants.
phase_operators.phase_to_coords — PHI_TO_RC inversion lookup.
phase_operators.occupation_field — occupation dict builder.
chess_spectral.tables.project_irrep — D4 irrep projector.

§12 imports from these packages but does not modify them.

§12.9 What success or failure would mean¶

If any derivation carries signal at |ρ| > 0.3: the HDC encoder family extends to king-attack content. §11.5's null on encode_640 reflects a choice of construction (bundled content descriptor), not an inherent limit of the family. The parallel instrument architecture (segmentation per UTLP S3 §14) validates as a path to adding specific structural capabilities without disturbing existing baselines.

If all three derivations produce |ρ| < 0.1: three independent mathematical objects (position-dependent Laplacian, D4 projection of directed adjacency, composite phase operator) have each failed to produce king-attack similarity signal. That is strong evidence that king-attack structure is not representable as a similarity-based measurement in the HDC encoder family — whatever signal it carries lives at a different algebraic layer than similarity (e.g., at the polarization-identity level, which §11 has not addressed).

If the outcome is ambiguous (0.1 ≤ |ρ| < 0.3): the construction pattern carries partial signal. The researcher decides whether refinement is justified or whether the partial signal is the finding.

All three outcomes advance the research. The experiment is worth running regardless of which occurs.

§12.10.1 Phase A2 result — fresh three-corpus matrix (post turn-flip fix)¶

Run parameters: same three §11.5 input CSVs as Phase A; Derivation A evaluated at k=16 (variance-explained 59.9% drnykterstein / 48.3% ashchess / 48.3% fishtest — "partial" or "inadequate") and then escalated to k=32 (93.8% / 95.1% / 92.0% — "faithful" per §12.7.1). Derivation B's channel construction and k choice are unchanged from Phase A. Derivation C reports all three scalar metrics introduced in §12.7.1. Tautological baseline mag_c_after recovered at |ρ| ≥ 0.994 across all three corpora, confirming the §12.7.1.1 turn-flip fix is working as intended.

Three-corpus matrix at k=32 for A (faithful variance-explained)¶

Corpus	n	A max \|ρ\|	A slice	B max \|ρ\|	B slice	C-cosine max	C-delta max
drnykterstein	3393	0.308 (−)	knight	0.402 (−)	king moves	1.000 (bishop/rook/queen)	0.855 (+) king moves
ashchess	3203	0.209 (−)	knight	0.304 (−)	king moves	1.000 (bishop/rook/queen)	0.919 (+) king moves
fishtest	3211	0.267 (−)	bishop	0.313 (−)	king moves	1.000 (knight/rook/queen)	0.903 (+) king moves

Durability criterion applied¶

A crossing of the |ρ| > 0.3 viability threshold is durable iff it replicates on the same slice across all three corpora (per §11.6.6.1).

Derivation A: max |ρ| = 0.308 / 0.209 / 0.267. Crosses 0.3 on drnykterstein knight, but not on the same slice across all three corpora. NOT durable.
Derivation B, king-moves slice: |ρ| = 0.402 / 0.304 / 0.313. All three cross 0.3 on the same slice, consistently with a negative sign. DURABLE.
Derivation C cosine / delta: max values are dominated by tautological structure. For sim_c, the zero-norm fallback returns 0.0 whenever C_before or C_after is the all-zero vector; post-fix, any sim_c > 0 means both vectors are nonzero, which by construction implies is_check_unsafe = True after the move. For delta_c, C_before is zero on most rows (mover's king not previously attacked), so ‖C_after − 0‖ = ‖C_after‖, reducing delta to the tautological baseline. The high C-cosine and C-delta values are therefore not independent evidence — they are algebraic consequences of mag_c_after's tautological correlation, propagated through the wrapper metric. The C derivation's genuine non-tautological content would require a metric that separates "was the king attacked to begin with" from "did the move change the attack pattern" — neither of which the three current scalar summaries cleanly isolate.

Categorical outcome: VIABLE (Derivation B, king-moves slice).¶

Derivation B's D4 decomposition of the opponent-attack adjacency row sum carries signal about whether a king move leaves the mover's king in check, at |ρ| ≈ 0.30–0.40 consistently negative, on all three structurally orthogonal corpora. The negative sign means: post-move king-attack adjacency is less similar to pre-move adjacency when the move creates a check than when it doesn't. This is structurally sensible — a check-creating king move necessarily reorganizes the attacker-set around the king, producing a larger D4-projected difference — and it is not a tautology of the is_check_unsafe label.

§11.5's null on encode_640 is therefore specific to that encoder's construction (bundled content descriptor); the HDC family CAN carry king-attack signal when the encoder targets attack-adjacency structure directly. Phase B (assembly via UTLP S3 §14.3 segmentation) is unblocked pending researcher review.

Note on slice specificity. B's signal lives primarily on the king-moves slice; on all-transitions it is near-zero. This is consistent with the structural argument above (king moves are the transitions that necessarily change the set of squares in the king's attack environment). It also means the viable signal is narrow: B's 256-dim concatenation plus king-move classification is required to extract it, not B alone.

Pairwise cosines (drnykterstein, 50 sampled positions)¶

cos(A, B) = +0.024 cos(A, C) = +0.256 cos(B, C) = +0.042

A, B, C measure different things; the king-attack encoder family covers orthogonal structural content.

Per-call timings (drnykterstein, mean over all rows)¶

A: 1970 µs (k=16) / higher at k=32 B: 1352 µs C: 600 µs

CSVs on disk (all gitignored per existing `docs/chess-maths/results/*/`)¶

Phase A (pre-fix, retained as research trail): - exp5_king_attack_correlation.csv (drnykterstein) - exp5_king_attack_correlation_ashchess.csv - exp5_king_attack_correlation_hf.csv

Phase A2 post-fix (k=16 primary, k=32 escalation): - exp5_king_attack_correlation_a2.csv (drnykterstein, k=16) - exp5_king_attack_correlation_a2_k32.csv (drnykterstein, k=32) - exp5_king_attack_correlation_a2_ashchess.csv (k=16) - exp5_king_attack_correlation_a2_ashchess_k32.csv (k=32) - exp5_king_attack_correlation_a2_hf.csv (k=16) - exp5_king_attack_correlation_a2_hf_k32.csv (k=32)

Research record note¶

The Phase A matrix was computed on post-move boards where board.turn was not flipped, causing A and C to query the opponent's king and B to measure mover's outgoing attacks. The three-corpus protocol still detected non-replication of B's single-corpus crossing (because wrong-side measurements do still vary across corpora), but the quantity being measured was not the one the derivations were designed to produce. The §12.7.1.1 fix restored measurements to mover-perspective, after which (a) the tautological baseline recovered cleanly and (b) B's king-moves signal emerged as a durable three-corpus finding with a definite sign.

§12.10 Result (Phase A — three-corpus evaluation)¶

Three-corpus §12 Phase A outcome: AMBIGUOUS.

Ran evaluate_encoder.py against three §11.5-style CSVs generated from the same three corpora used in §11.6.6.1's durability check. Each CSV contains ~3200 pseudo-legal transitions labeled with is_check_unsafe, plus path-1 reference correctness validated at 100%. The §12 CLI re-encodes every transition under Derivations A, B, C and reports Spearman ρ per slice.

Derivation	drnykterstein (GM cls, N=10)	ashchess (FM blitz, N=50)	fishtest (engines blitz, N=50)
A (Laplacian, k=5)	max \|ρ\| = 0.161 (knight)	max \|ρ\| = 0.172 (captures)	max \|ρ\| = 0.179 (captures)
B (D4 concat, all rows)	+0.116	+0.094	+0.210
B best slice	+0.332 (king moves)	+0.172 (pawn, E_ka)	+0.256 (bishop)
B per-irrep \|ρ\| (A1)	−0.200	−0.145	−0.238
B per-irrep \|ρ\| (E_ka)	+0.174	+0.172	+0.259
C (attack op cosine)	NaN (all-zero vectors)	NaN	NaN
Transitions evaluated	3393	3203	3211

Durability finding. Derivation B's single-corpus "viable" result (|ρ|=0.332 on drnykterstein's king-moves slice) does not replicate across the other two corpora. The king-moves |ρ| drops to +0.123 (ashchess) and +0.228 (fishtest). The all-transitions ρ ranges +0.094 to +0.210 — consistent ambiguous-zone, never crossing 0.3. Per-irrep shape is stable across corpora (A1 consistently negative, E_ka consistently largest positive) which is structurally meaningful but not a viability signal.

Per §12.7 thresholds on the three-corpus aggregate: - |ρ| > 0.3 occurs only on one slice of one corpus. - |ρ| < 0.1 is not met (all-transitions ρ on fishtest = +0.210). - The honest categorical label is AMBIGUOUS.

Derivation C's structural degeneracy. On all three corpora, the 16-dim feature vector is identically zero for every position's pre-move and post-move snapshot. The cosine metric returns 0.0 (zero- norm fallback) for every row. Spearman is NaN across every slice. The finding is not "C carries no signal" — it is "cosine similarity on attack-density-before-vs-after is structurally degenerate as a metric when the pre-move vector is almost always zero." A different summary statistic (L2 norm of Δ-vector, attack density at destination square, sign-flip counts) might work; per §11.7.4 this is not tuned within Phase A.

Derivation A's fidelity deficit. Variance explained by the first k=5 eigenvectors of the symmetrized attack Laplacian averages 7.3% (drnykterstein), 11.1% (ashchess), 10.4% (fishtest) across rows. The eigenchannel is not capturing δ_king's L2 norm at k=5. A would need k ≥ 16 for a faithful summary, which the prompt scope forbids retuning in Phase A. This is noted as a prerequisite for any Phase A2 reconsideration.

Timing (consistent across corpora): - Derivation A: ~2000 µs/call (dominated by scipy.linalg.eigh on 64×64) - Derivation B: ~1350 µs/call (dominated by project_irrep) - Derivation C: ~250 µs/call (phase-arithmetic only)

Pairwise cosines (50-position samples, zero-padded common space): - cos(A, B) ≈ 0.0 across all corpora — A and B measure orthogonal things. - cos(A, C) ≈ +0.25 (drnykterstein), ≈ 0 (ashchess), ≈ +0.25 (fishtest) — moderate similarity on two of three. - cos(B, C) ≈ 0.0 — B and C also orthogonal.

CSVs on disk (all gitignored per existing docs/chess-maths/results/*/ policy): - exp5_king_attack_correlation.csv (drnykterstein N=10) - exp5_king_attack_correlation_ashchess.csv (ashchess N=50) - exp5_king_attack_correlation_hf.csv (fishtest N=50)

Regenerate via the three-line recipe:

python similarity_experiment.py --corpus ../results/<CORPUS> --out exp3_<tag>.csv
python -m king_attack_encoder.evaluate_encoder --input-csv exp3_<tag>.csv --out exp5_<tag>.csv

Decision per §12.7. Phase A is AMBIGUOUS. The researcher decides whether to: 1. Accept the ambiguity as the §12 finding — the HDC construction pattern produces partial king-attack signal (per-irrep structure in B is stable across corpora) that does not durably cross the viability threshold. §11.5's null does not quite generalize to a structural property of the encoder family, but the family also does not obviously extend to king-attack content. 2. Commission Phase A2 with specific refinements: - Derivation A at k = 16 or 32 (the current k = 5 captures <15% of δ_king's norm). - Derivation C with a non-cosine metric (L2 of Δ-vector, or attack density at destination square). - Derivation B with alternative signal choices (column-sum, attacker-value-weighted row-sum) to test whether row-sum is the limiting factor on B's signal. 3. Promote Phase B (assembly) only if a specific refined derivation crosses 0.3 durably across all three corpora.

Per §11.7.4 discipline, the AMBIGUOUS label is recorded as the Phase A finding regardless of which follow-up (if any) the researcher chooses.