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+## 2026-07-16 — E-WH-TWO-SIDES-SIG-CHECKSUM-1 — one Walsh–Hadamard family for both pyramid sides; Hambly–Lyons signature as the replayable-trajectory checksum (probe-gated forward synthesis)
+
+**Status:** CONJECTURE (operator-proposed synthesis; [H] and [S] respectively; two named probes; nothing shipped, no canon changed).
+**What:** two candidate adoptions for the Morton-tile stacked inverse pyramid, harvested from the E-PALETTE-RESIDUAL-LADDER-1 discussion:
+(1) **[H] One WH family, both sides.** OGAR canon already frames the bipolar-phase pyramid's sign side as the Walsh–Hadamard transform of the address tree (sign composition = XOR); bgz-tensor's proven `hadamard_rotate` is the SAME transform family applied to the stored-magnitude side (outlier-flattening preconditioner before cheap uniform quantization). Proposal: WHT₁₆ on a 4×4 tile's 16-cell magnitude vector before nibble quantization — phase keeps address-derived WH signs, magnitude gains the WH preconditioner; the two-algebra rule is untouched (the rotation is coding-side, not a merge mode — I-SUBSTRATE-MARKOV unaffected). **PROBE-WH-MAG:** WHT₁₆+i4/i2 vs direct i4/i2 reconstruction error on real tile magnitudes (bgz-tensor's row-level win does not automatically transfer to 16-cell tiles).
+(2) **[S] Signature as trajectory checksum.** The H.268 "replayable, checksummable" requirement meets Hambly–Lyons directly: the signature is the canonical path digest with a uniqueness theorem, and **tree-like equivalence is the digest's exact null space** — the formal version of "which detours leave no comma" (backtracking excursions vanish; genuine deviations leave non-zero level-2 area). Harnesses exist on both sides (jc Pillar 11 forward/converse; ndarray `hpc/pillar/signature.rs` B7). **PROBE-SIG-CHECKSUM:** depth-2 truncated signature of a temporal-stream trajectory as its digest — replay-identity holds, tree-like edits invisible (correctly), non-tree perturbations caught.
+Neither proposal adds a stored field to the tile (WH is derivable; the signature is a derived digest). Both conditional on the standing §10 ledger (D-MTS-1..3, PHASE-1/PERT-RHO/PYR-1 + J2, WHP-1..4); no kill condition weakened.
+**Cross-ref:** E-PALETTE-RESIDUAL-LADDER-1, E-H268-REPLAYABLE-TILE-1, TD-BGZ-TENSOR-PRE-LANE-REVIEW (the WH-adoption question is evidence FOR axis (b) of that review), ndarray `pr-x12-h268-morton-wgpu-synergies.md` §10 forward-synthesis block (the twin text), OGAR "Bipolar-phase pyramid — Walsh-Hadamard on VSA" (canon unchanged), I-SUBSTRATE-MARKOV / the two-algebra rule (untouched).
+
 ## 2026-07-16 — E-PALETTE-RESIDUAL-LADDER-1 — bgz-tensor's Hadamard-residual ladder = the honourable fourth mode next to the three flavours of 256; Hambly–Lyons signature as the [S] formal anchor
 
 **Status:** HONOURABLE-MENTION RULING (operator-directed) + [S] research anchor. No layout added to the §3 catalogue; no canon changed; the le-contract gains an append-only subsection.