probe(EXO-129): a learned 256×256 table does not compose where the codebook is not closed

.claude/board/entries/2026-10-23-exo129-learned-lut-preregistration.md

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+# 2026-10-23 — EXO-129: does a LUT learned from observed transitions compose? (pre-registration)
+
+**Status:** PRE-REGISTERED (written and committed before any probe code)
+**Thread:** TORQUE T-2026-10-23-1, Linear EXO-129. Crumb trail T-CLOSURE (EXO-132).
+**Owner:** CC:session_01G22yT6htkcdyXsihxxXdrg
+
+## Question
+
+P-S2 (#1726) showed that `ComposeTable` does not survive two hops. It snaps
+`entry[a] ⊕ entry[b]` back to the nearest entry, and most bound pairs land
+outside the codebook. This probe asks two things:
+
+1. Does a 256×256 table **learned from observed one-hop transitions**
+   (`PaletteLut`-shaped, cell = modal next state) compose over k hops where
+   `ComposeTable` does not?
+2. **Mechanism M:** does a data-only statistic, the **purity** `P` of the
+   learned table (the share of training observations that equal their cell's
+   modal result), predict how well it composes? This is the testable form
+   of T-CLOSURE's "is the carrier closed under the operation".
+
+## Setup
+
+- **Operation per hop, with key index `b`:** `x ← x ⊕ entry[b]` in Base17
+  (`xor_bind`), as in P-S2.
+- **Truth for a chain** `(x0, b1..bk)`: `assign(x0 ⊕ e_b1 ⊕ … ⊕ e_bk)`, one
+  snap at the end. `x0` is a population row, not a palette entry.
+- **Arms:**
+  - **C:** the `ComposeTable` chain from `s0 = assign(x0)`.
+  - **L:** the learned-table chain from `s0`.
+- **Training:** 20 000 trajectories of 6 hops from population rows (seed A).
+  Each hop records `(assign(x_t), b, assign(x_{t+1}))`. `L[a][b]` is the
+  modal result. Unobserved cells fall back to `ComposeTable`, and the
+  unobserved fraction is reported.
+- **Test:** 2 000 fresh chains (seed B) for each k = 1..6.
+- **Worlds (six settings):**
+  - **W1 clustered** and **W2 uniform:** P-S2's generators and seed,
+    palette from `WeightPalette::build`. These are inherited from P-S2, not
+    built for this probe.
+  - **W3 group-closed:** 256 palette entries built directly as an XOR group.
+    `entry[g]` has `dims[i] = 4096` for each set bit `i < 8` of `g`, so
+    `entry[a] ⊕ entry[b] = entry[a ^ b]` exactly. Population: 4 members per
+    element, each `entry[g] + u`, with `u` non-negative per-dim noise in
+    `[0, n]` (non-negative so that XOR never touches the sign bit).
+    Noise levels: `n ∈ {0, 500, 1500, 3000}`, four settings. Larger noise
+    lets XOR of noise reach bit 12 and break closure gradually.
+
+## Measures
+
+- `agreement_L(k)` and `agreement_C(k)` for k = 1..6: the share of test
+  chains whose arm result equals the truth.
+- `P` (purity of L on its training observations) and the unobserved-cell
+  fraction.
+
+## Decision rules (fixed now)
+
+1. **Harness control:** in W3 with n = 0, both L and C read ≥ 0.99 at every
+   k. Otherwise the harness is broken and nothing below is read.
+2. **EXO-129 torque card:**
+   - **PROMOTE** if L ≥ 0.90 at k = 2 in **both** W1 and W2.
+   - **KILL** if L < 0.90 at k = 2 in both.
+   - Mixed: report, and the row stays TORQUE with the next probe named.
+3. **Mechanism M:** passes if, over the six settings, the Spearman rank
+   correlation between `P` and `agreement_L(2)` is ≥ 0.9. The prediction
+   `agreement_L(k) ≈ P^k` (independent per-hop errors) is reported with its
+   residual, but it does not gate.
+4. **Disables (both must behave, or no claim is made):**
+   - **D1:** shuffle L's result bytes across cells. Agreement at k = 2 must
+     drop to ≤ 0.10 in every setting.
+   - **D2:** set L := C. This must reproduce C's numbers exactly.
+
+## Applying GOLDSTAUB G-2026-10-23-1 to this probe
+
+The world that passes by construction is W3 with n = 0. That is why it is
+only a harness control. The world expected to fail is W2, which has no
+structure for a table to learn. So no conclusion is drawn from a single
+world's pass or fail:
+
+- the **mechanism** verdict comes only from M across the six settings;
+- the **torque** verdict comes from W1/W2, which P-S2 fixed before this
+  probe existed.
+
+## Results (2026-10-23, after the run; nothing above was changed)
+
+`cargo test --release --test exo129_learned_lut_probe -- --nocapture`
+(bgz-tensor, debug info off). Agreement with the truth at k = 1..6:
+
+| setting | purity P | unobserved | L k=1 | L k=2 | C k=1 | C k=2 | L k=6 | C k=6 |
+|---|---|---|---|---|---|---|---|---|
+| W1 clustered | 0.471 | 0.199 | 0.242 | **0.017** | 0.579 | 0.029 | 0.005 | 0.005 |
+| W2 uniform | 0.479 | 0.170 | 0.112 | **0.009** | 0.279 | 0.027 | 0.005 | 0.005 |
+| W3 n=0 | 1.000 | 0.159 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
+| W3 n=500 | 1.000 | 0.159 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
+| W3 n=1500 | 1.000 | 0.159 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
+| W3 n=3000 | 0.544 | 0.340 | 0.263 | 0.145 | 0.231 | 0.260 | 0.041 | 0.248 |
+
+- **Rule 1 (harness control):** holds. W3 with n = 0 reads 1.000 for both
+  arms at every k.
+- **Rule 4 (disables):** both hold.
+  - D1 (shuffled table) reads ≤ 0.005 at k = 2 in every setting.
+  - D2 (L := C) reproduces C exactly.
+- **Rule 2 (torque): KILL.** L reads 0.017 / 0.009 at k = 2 in W1 / W2,
+  against the 0.90 bar. The learned table is worse than `ComposeTable`
+  there, even at k = 1.
+- **Rule 3 (mechanism M): PASS by the registered rule**, Spearman ρ = 0.935,
+  **but the evidence is thin.** Four of the six settings tie at P = 1.0 /
+  agreement = 1.0, so the rank correlation rests on three distinct points.
+  The unregistered magnitude prediction `agreement_L(k) ≈ P^k` fails badly:
+  W1 k = 2 is 0.017 against P² = 0.222.
+- **Pin:** both verdicts are asserted in the test. Disable run: inverting
+  purity (`1 − P`) makes the M assertion fail, so the pin is real.
+
+### What the numbers say (reading, not a ruling)
+
+- **Purity < 1 means the palette state is not Markov for this operation.**
+  The cell does not determine the next cell: members of one cell go to
+  different cells. In W1/W2 the purity is about 0.47. No 256×256 table
+  can compose there, learned or derived, because one byte of state does not
+  carry the information the next hop needs. In Markov-chain terms this is a
+  **lumpability** failure (Kemeny–Snell): the partition is not lumpable for
+  the operation. That is a candidate name for trail T-CLOSURE. It is not
+  adopted here.
+- **Learning made W1/W2 worse at k = 1** (0.242 vs C 0.579). Training cells
+  pool starts from population rows with states reached after several hops,
+  and 5/6 of the observations are reached states. The modal result then
+  favours the reached-state distribution, while the test starts from
+  population rows. This is a real cost of the pre-registered design, and it
+  is itself a non-lumpability symptom: where in the cell a state sits
+  matters.
+- **W3 n=3000 separates two error sources.** C stays flat at about 0.25 from
+  k = 1 to 6. The table is the exact group law, so composition adds no
+  error, and the loss comes from the start and end snaps alone, which
+  depend on the fixed member noise. L decays (0.263 → 0.041), because its
+  cells were filled from mis-snapped observations. A closed carrier
+  therefore composes with its exact law even when snapping is noisy. A
+  table learned from noisy snaps does not.
+
+### Closeout
+
+PR (this branch) | STATUS: measured | OUTCOME: a learned 256×256 table does
+not compose where the codebook is not closed (KILL); purity < 1 marks those
+cases, with thin rank evidence (M PASS on three distinct points) |
+OPEN: whether a purity bar (for example P ≥ 0.99) predicts composition on
+real Moore/DR pair data. That needs labelled data (GOLDSTAUB
+G-2026-10-23-1).

.claude/board/entries/README.md

@@ -25,12 +25,13 @@ index row, (3) no duplicate entry id. Checks 1 and 2 are deliberately
 opposite directions; the stranding this convention prevents shows up in
 exactly one of them, never both.
 
-443 entries, 2026-08-06 .. 2026-10-23.
+444 entries, 2026-08-06 .. 2026-10-23.
 
 | date | entry id | finding | file |
 |---|---|---|---|
 | 2026-10-23 | `ps2-compose-chain-preregistration` |  | [2026-10-23-ps2-compose-chain-preregistration.md](2026-10-23-ps2-compose-chain-preregistration.md) |
 | 2026-10-23 | `ledger-revision-1635-1726` |  | [2026-10-23-ledger-revision-1635-1726.md](2026-10-23-ledger-revision-1635-1726.md) |
+| 2026-10-23 | `exo129-learned-lut-preregistration` |  | [2026-10-23-exo129-learned-lut-preregistration.md](2026-10-23-exo129-learned-lut-preregistration.md) |
 | 2026-10-22 | `probes-pd4-pe2-pd3` |  | [2026-10-22-probes-pd4-pe2-pd3.md](2026-10-22-probes-pd4-pe2-pd3.md) |
 | 2026-10-22 | `pal256-salience-mp3-probe` |  | [2026-10-22-pal256-salience-mp3-probe.md](2026-10-22-pal256-salience-mp3-probe.md) |
 | 2026-10-22 | `pal256-nars-bayes-evidence-contract` |  | [2026-10-22-pal256-nars-bayes-evidence-contract.md](2026-10-22-pal256-nars-bayes-evidence-contract.md) |