pillar: bit-exact i128 lattice signature lane + Hambly-Lyons Thm 5/6 certificate

.claude/blackboard.md

@@ -3,7 +3,46 @@
 > **Read this first.** The "Polyglot Notebook" architecture below is a
 > separate/older program, not the current epoch.
 
-## 2026-08-31 (latest) — W1a-#9 masking primitives SHIPPED on every dispatch arm (PR #285)
+## 2026-09-01 (latest) — Pillar-11 lattice lane: BIT-EXACT i128 lattice signature + Hambly–Lyons Thm 5/6 certificate
+
+`src/hpc/pillar/lattice_signature.rs` (feature `pillar`). For unit-step
+lattice walks every level-`k` signature coefficient is a rational with
+denominator `k!`, so the lane stores `k!·S_k` as `i128` and the whole
+computation is bit-exact — identity is `==`, no tolerance. Chen composition
+with a unit step is a binomial convolution against a tensor supported on
+`(a,…,a)` only, so each step costs `O(Σ_k d^k·k)`. Depth policy is INTEGER:
+`theorem2_depth(L) = ⌈47917·L/10000⌉ ≥ ⌊2e·ln(1+√2)·L⌋` (the PUBLISHED
+constant: Annals of Math 171 (2010) Theorem 5, `⌊2e·log(1+√2)·L⌋`; the
+arXiv v2 preprint's Thm 2 states `e` — a version trap CodeRabbit caught on
+lance-graph #1133 and the Annals PDF confirmed; the float never enters the
+kernel), `theorem3_factor(d) = 2⌈log₃(d/2)⌉+3` by integer loop.
+Measured (debug): 52/52 reduced `d=2` words of length ≤ 3 separated at the
+theorem depth (`LATTICE_L_MAX = 3` — the doubled constant makes depth ⌊c·L⌋
+grow to 19 at L=4, and `d^depth` coefficients per level exhaust memory); 64/64 tree-like words EXACTLY the identity at a fixed depth 12
+(`LATTICE_TREELIKE_DEPTH`, since identity holds at every depth and the
+theorem depth for length-6 words is 28); 64
+reduced length-8 words share `S^(2) = 1` with the constant path (the
+paper's §1.6 figure-of-8 class) and every one separates at level 3
+(`3!·S_xxy = 6` for the canonical one); `d = 1` collapses the 64 length-6
+words to exactly 7 tensors (net increment only — the `d ≥ 2` precondition
+is now a pin, not prose). Parity pin against the existing f32 lane
+`signature_d2_deg3` on every lattice word of length ≤ 6 (exact small
+integers). Bit-exactness pin: FNV digest `0x7C9612A734212FC6` over all
+words of length ≤ 3 at theorem depth. `prove_pillar_11_lattice()` reports
+`psd_rate` = separated fraction (1.0), `n_paths` = 52, `n_hops` = 64
+false merges, `lognorm_concentration` = deepest separation level (3).
+**Disambiguation:** `signature.rs` stays the f32 depth-3 kernel-STABILITY
+battery; this lane is the UNIQUENESS half, and it is the ndarray twin of
+lance-graph `jc::hambly_lyons` W6 (PR #1133) with the f64 tolerance
+replaced by integer equality. **SIMD:** scalar integer reference lane on
+purpose; the W1.5 sigker vectorised lane (now unblocked) must reproduce
+these `i128` tensors bit-for-bit. Loose ends: an `i128` lane in
+`ndarray::simd` does not exist; the `d ≥ 3` arm of Theorem 3 is
+implemented (depth formula) but not exercised by a test beyond the factor
+pins; `crates/sigker-parity` should gain a W1b test comparing this lane
+against `sigker::signature_truncated` on lattice words (exact ints vs f64).
+
+## 2026-08-31 — W1a-#9 masking primitives SHIPPED on every dispatch arm (PR #285)
 
 `U64x8`/`U32x16` gained `andnot` (set difference, `self & !other` — argument
 order deliberately differs from the raw Intel intrinsic, same direction on