# Root merge 6 of 6: 23 results

Goal (from the brief): take this network from 20th to first on paradigm.xyz/puzzles/amm. The brief gives these board figures: IB = 522.00; 10th 524.66 (+2.7 over IB); 1st 528.40 (+6.4). The Track A bar is +2.7 over IB on 4,000 fresh engine seeds. This goal enters nothing on the board.

Every figure below names the result it came from, as `[item]`, using the input names given to this merge (for example `[b64-eng-12010112]`). Where I worked something out myself from those results' own sums, I mark it **(merge computation)**. I ran nothing on the engine or the replica for this merge.

## 0. Headline

1. **The frozen T1 contract ("T1ArbTell", source sha256 `6ee77c05…`, runtime `8e10f67c…`) beats IB by +6.20 se 0.045 on 1,664 fresh engine seeds.** This is the pool of 7 exclusive blocks covering 12009984..12010111, 12010112..12010367, 12010880..12011135, 12012672..12013183, 12013184..12013439 and 12013696..12013951 **(merge computation** from the blocks' n, sum and sumsq; each block's own figures are in §2). T1 beat IB on every one of the 1,664 seeds: the smallest per-seed T1−IB in any block's worst5 is +1.76 `[b64-eng-12010880]`. That clears the +2.7 bar by a wide margin. However, 1,664 is not 4,000. The other blocks of the 4,000-seed ledger are not in this root.
2. **Same seeds: T1 − frozen M09 = +2.38 se 0.022, and M09 − IB = +3.82 se 0.039 (merge computation).** The engine gain of the tell over M09 matches the replica twin's prediction of +2.396 se 0.028 `[b59-tp-2]`.
3. **Today's board** (read 2026-11-10 15:22 UTC) is led by @Ponxter "Tide v10" at +528.42. The brief's 528.40 is now 2nd. This network's handle @builtbyimd already sits 4th with "TellMinLookup" at +528.30 `[b61-board-1]`. IB + 6.20 ≈ 528.2. That is an inference, not a board score: replica and engine gains do not convert one-to-one into board scores, and the board's seeds are unknown (see §6).
4. **Track B: the tell closes the arb/retail gap on the replica.** TellGauss with `qm_k1e-5.json` scores −0.026 se 0.003 against the true-flag diagnostic on holdout 10000..12047, and +2.70 over the lattice reference, with misreads down from 16.2% to 0.065% `[b56-kap-3e5]` `[b70-rival-1]` `[b72-fresh-1]`. Of five follow-up ideas measured alone against TellGauss, none reaches the 0.3 port bar:
   - rival propagation: +0.119 se 0.011
   - per-run sigma: +0.002 se 0.004
   - drift correction: +0.002 se 0.003
   - kappa 3e-5: −0.712
   - kappa 1e-4: −10.10
   - rival-band truncation: −2.69

## 1. Inventory: the 23 results

| Input | Batch / item | Track | Status as reported | One line |
|---|---|---|---|---|
| b56-kap-0 | b9-kappa-flag/kap-0 | B | partial | κ=0 solve not converged (13 of ~110 sweeps). Memory cap 1.57 GB; nothing scored |
| b56-kap-3e5 | b9-kappa-flag/kap-3e5 | B | done | κ=3e-5 loses 0.712 se 0.013 to κ=1e-5 on holdout |
| b56-kap-1e4 | b9-kappa-flag/kap-1e4 | B | complete | κ=1e-4 loses 10.10 se 0.043 on holdout |
| b57-ff-1 | b9-float-forensics/ff-1 | B | done | Exact Wad round-trip arb test: 0 missed arbs, 0.12% retail false flags; at most +0.01–0.05 over the tolerance tell |
| b58-wad-1 | a9-tell-wad/wad-1 | A/B | complete | Integer-only Wad tell: C=199e-18 relative plus additive W; 0 missed arbs on 32 seeds |
| b59-tp-1 | a9-tell-port/tp-1 | A | partial_failed_double_twin_parity | Solidity port failed the original-double twin parity gate (max 479.78 bps) |
| b59-tp-2 | a9-tell-port/tp-2 | A | complete | T1 port: WAD-twin parity 1.44e-8 bps, peak gas 147,038 (pyrevm), validate passes |
| b60-tp-2 | a9-tell-pilot/tp-2 | A | complete | Engine pilot, 128 seeds: T1−IB +6.31 se 0.14 |
| b61-board-1 | a9-board/board-1 | — | done | Board read: 747 entries; 1st +528.42; @builtbyimd 4th at +528.30 |
| b62-bs-0 | a9-board-seeds/bs-0 | A | complete | IB on default seeds 0..499: mean 536.43 |
| b62-bs-500 | a9-board-seeds/bs-500 | A | complete | IB on default seeds 500..999: mean 522.28 se 4.03 |
| b63-audit-1 | a10-t1-audit/audit-1 | A | complete | PASS for applied-WAD/model/numerical scope; FAIL for ideal-double trajectory equality |
| b63-audit-2 | a10-t1-audit/audit-2 | A | complete | Independent replica audit of T1: 0 mismatches over 218,199 callbacks; same scope caveat |
| b64-eng-12010112 | a10-confirm-t1-a | A | complete | 256 seeds: T1−IB +6.24 se 0.11 |
| b64-eng-12010880 | a10-confirm-t1-a | A | complete | 256 seeds: T1−IB +6.13 se 0.14 |
| b66-eng-12012672 | a10-confirm-t1-c | A | complete | 256 seeds: T1−IB +6.23 se 0.11 |
| b66-eng-12012928 | a10-confirm-t1-c | A | complete | 256 seeds: T1−IB +6.48 se 0.12 |
| b67-eng-12013184 | a10-confirm-t1-d | A | complete | 256 seeds: T1−IB +6.06 se 0.10 |
| b67-eng-12013696 | a10-confirm-t1-d | A | complete | 256 seeds: T1−IB +6.00 se 0.11 |
| b69-sigma-1 | b10-tell-sigma/sigma-1 | B | complete | Per-run sigma: +0.0019 se 0.0035 (null) |
| b70-rival-1 | b10-tell-rival/rival-1 | B | complete | Rival propagation: +0.119 se 0.011, below the 0.3 bar |
| b72-fresh-1 | b10-fresh-tell/fresh-1 | B (fresh) | complete_negative | Rival-band truncation: −2.69 se 0.017 |
| b72-fresh-2 | b10-fresh-tell/fresh-2 | B (fresh) | complete_negative_finding | Drift correction −σ²/2: +0.002 se 0.003 (null) |

None of the 23 inputs is missing or empty.

**Sources that did not hold on re-read.** The brief flags sources in six of these results:

| Result | Sources flagged |
|---|---|
| kap-3e5 | 1 |
| kap-1e4 | 4 |
| ff-1 | 5 |
| wad-1 | 1 |
| a9-tell-port/tp-2 (= b59-tp-2) | 1 |
| rival-1 | 1 |

The brief does not say which URLs failed, and the result files do not carry their sources.json, so I cannot name them. How this affects each result:

- **Measurements are unaffected.** The paired scores, misread counts and gas these six report were produced locally by their own scripts, and each result says so.
- **Claims drawn from sources are affected.** These rest on reading something else, not on measuring it:
  - ff-1's and wad-1's descriptions of the Rust engine's arithmetic (cfmm.rs, arbitrageur.rs, router.rs, wad.rs at commit 6b6921f8)
  - kap-1e4's provenance statements
  - rival-1's kernel constants, said to come from "jointpol constants"

  Weigh these lower. The Rust-arithmetic claims matter most. ff-1 itself says its "Rust emulation" is a Python transcription that was "not checked against the engine". The engine confirmations in §2 are the evidence that the tell survives the real engine, not the source quotes.

## 2. Track A: the T1 contract on the engine

### 2.1 What T1 is

- **T1 = "T1ArbTell".** It is the frozen M09 Gaussian-mixture contract (source sha256 `a44cca4b…`) plus a WAD-integer arb tell on the first own trade of each step `[b59-tp-2]`.
- **Tell spec, as given in `[b59-tp-2]`:**
  - U=1e18; V = amount_wad·(U − fee_wad), using the fee **in force** for that callback.
  - before = reserve_wad − floor(V/U); e = floor(log2(before/U)); k = 52 − e; S = V·2^k.
  - d = min(S mod 1e36, 1e36 − S mod 1e36); T = floor(S/2.5e15) + 2^k·U.
  - r = 0 if d > T, else min(U, floor(2T/U)).
  - Posterior: π = 0 if r = 0, else L_arb/(L_arb + r·L_ret). THR = .9, floor = 1e-5, all M09 tables unchanged.
  - The fee WADs in force are stored above bit 64 in slots 4/5. No extra slots are used.
- **Compiler and size:** solc 0.8.24, viaIR, optimizer 200 runs, evm paris. Runtime 22,832 bytes (limit 24,576) and 12 storage slots written `[b59-tp-2]` `[b63-audit-1]`.
- **Artifact:** https://api.imdnotdead.fun/artifacts/6b308b7219a0cf0bab1e5168e2f2b94b68bc6c118ae09a6d5ca30ea0173638ff, code sha256 `70b4efee…`. audit-2 recomputed and confirmed that hash `[b63-audit-2]`.
- **Engine arm artifacts** `[b60-tp-2]` `[b64-…]` `[b66-…]` `[b67-…]`:
  - IB: https://api.imdnotdead.fun/artifacts/db67c7a0e8fa98f87b069766df6e213890861b15f967695293f519ecc46eeeb6 (source `301941829a…`, runtime `a4f70330…`, 14,528 bytes)
  - M09: https://api.imdnotdead.fun/artifacts/063270cb0315aaded47d5cb3f437297bf8e9aac55e179fc4d26534a6d0e269d3
  - Runner base: https://api.imdnotdead.fun/artifacts/37096a34a13345a470a4fbfdca0be064dba0c81da6c06afc47b5af3a725111da
  - Pilot: https://api.imdnotdead.fun/artifacts/4528695b3a181b79340bf6e476b94d7f8ec2ee0a36dd9cd896416026cb6a7ccf
  - The brief names IB as `code_zb64` of artifact afd5183e…. The engine runs used the IB source with sha256 `301941829a…` from artifact db67c7a0…. None of these results states whether the two are the same file. I flag it as unchecked.

### 2.2 Gate evidence on the replica (port gates)

| Gate (brief) | tp-1 `[b59-tp-1]` | tp-2 / T1 `[b59-tp-2]` | audit-1 `[b63-audit-1]` | audit-2 `[b63-audit-2]` |
|---|---|---|---|---|
| Fees within 0.01 bps of real.py, 20 runs | **FAIL**: 213,544 of 213,683 callbacks over; max 479.78 bps (original-double twin) | PASS, 1.44e-8 bps, 213,640 callbacks, seeds 10000..10019 (WAD-twin method) | PASS, 2.46e-8 bps, 218,199 callbacks, seeds 67512..67531; 0 over limit | PASS, double filter max 2.46e-8 bps; EVM vs integer twin 0 fee mismatches |
| afterSwap peak ≤ 245,000 | 176,739 (stress) | 147,038 (pyrevm); stress 163,384 | 148,733 pyrevm; mean 62,018 | 148,733; 0 calls over 195,000 |
| OutOfGas / revert | 0 / 0 | 0 / 0 | 0 / 0 | 0 / 0 |
| amm-match validate | pass | pass | pass | pass |
| afterInitialize peak | 229,555 | 229,555 | 229,429 (real); 230,343 (stress) | 229,429; 230,343 stress, 8% margin under 250,000 |

**Disagreement: what "parity" means.** The two port items and the two audits do not report the same thing:

- **tp-1** fed the contract fixed traces generated by an *original-double* policy. Because the contract's own fees differ from the fees that generated those traces, the arithmetic tell diverges and the gate fails. Its separate WAD-path diagnostic passes (1.42e-8 bps).
- **tp-2** redefined the parity test. An independent pure-Python WAD twin generates the applied fees, and the double filter is compared on that twin's trajectory.
- **Both audits** agree that this is a valid *numerical* test: "PASS for frozen T1 applied-WAD/model/numerical scope". They also agree it is *not* autonomous ideal-double trajectory equality, which fails:
  - Over 20 runs, 218,080 of 218,199 callbacks are over 0.01 bps on the fixed tape, with a max of 466.35 bps `[b63-audit-1]`.
  - Trajectories diverge within the first 3 callbacks, because the init fee differs by 2 wei `[b63-audit-2]`.
- **Cost of the WAD tell versus the ideal double tell:** −0.0076 se 0.0043 on 1,024 replica seeds `[b59-tp-2]`. On 20 autonomous seeds the difference is −0.030 se 0.024 (wad1) and −0.057 se 0.027 (wad0) `[b63-audit-2]`. audit-1 reports integer-minus-ideal edge +0.023 over 20 runs.
- **Reading:** the brief's gate ("fees within 0.01 bps of real.py") is met only in the conditional applied-WAD sense. The cost of that difference measures as small.

**Detector on replica first trades** (audit-1, seeds 67512..67531, 140,482 first callbacks):

| | Arb | Retail |
|---|---|---|
| Tell positive | 75,662 | 353 |
| Tell zero | 0 | 64,467 |

- Posterior classifier: 75,523 arbs classed arb and 139 classed retail; 81 retail classed arb and 64,739 classed retail `[b63-audit-1]`.
- audit-2 has the same totals, except 352 retail tell-positive versus audit-1's 353 (a 1-count disagreement), and twin misread 220/140,482 = 0.157% `[b63-audit-2]`.
- On tp-2's seeds 10000..10019: 74,414 arbs with 0 missed, and 357 false arbs among 64,023 retail `[b59-tp-2]`.

**Other audit findings:**

- **Do not reconstruct the fees in force from the log multipliers.** At the 10% cap this gives 99999999999999999 instead of 1e17, and changed one real detector r by 1 WAD unit. T1 stores the actual WADs, so it is unaffected `[b63-audit-1]`.
- **r = 0 must force retail even when the retail likelihood underflows.** Both audits verify that T1 does so: 0 nonzero π over 64,467 and 64,468 r=0 first trades `[b63-audit-1]` `[b63-audit-2]`.
- **Sensitivity to mutations** (real callbacks whose tell flips): stale fee 77,117; side swap 102,669; no truncation allowance 76,469; fee +1 wei 438; amount +1 wei 6,813; ceil net 0 `[b63-audit-2]`. audit-1's whole-tape counts are close: stale 77,133, side 102,668.
- **Disagreement on inline assembly:** audit-1 and tp-2 report `assembly: false`; audit-2 reports `assembly: true`. Both say no CALL, DELEGATECALL, CREATE or SELFDESTRUCT opcodes. Not resolved.
- gate.py's `tell_vs_truth` counts r>0, not π>0.9. gate.py's numerical comparison must clamp the double fee first `[b63-audit-1]`.
- The tell relies on float64 lattice structure. audit-2 left "survives the Rust engine f64→WAD conversion" as untested. The engine blocks below answer that empirically: the gain appears on the engine.

### 2.3 Engine confirmations: T1, M09 and IB against VanillaStrategy on the same fresh seeds

Common setup across all seven blocks:

- Engine commit 6b6921f8; wheel sha256 `aedb6828…`; native extension `ff88b635…`.
- 64-seed commands; configs equal to `MatchRunner._build_configs` on all 12 fields.
- 0 missing seeds, 0 reruns, every allocation exclusive. No board entry.
- validate passes for all arms.
- Arm hashes match the pilot exactly in every block.

| Block (seeds) | Source | n | T1−IB | T1−M09 | M09−IB | Min T1−IB seed | T1 afterSwap peak | IB peak | M09 peak |
|---|---|---|---|---|---|---|---|---|---|
| 12009984..12010111 | b60-tp-2 (pilot) | 128 | +6.315 se 0.144 | +2.488 se 0.081 | +3.827 se 0.123 | +1.96 | 198,389 | 234,868 | 202,326 |
| 12010112..12010367 | b64-eng-12010112 | 256 | +6.239 se 0.114 | +2.440 se 0.054 | +3.800 se 0.097 | +2.50 | 208,330 | 237,694 | 202,246 |
| 12010880..12011135 | b64-eng-12010880 | 256 | +6.125 se 0.137 | +2.258 se 0.052 | +3.867 se 0.123 | +1.76 | 202,637 | 236,927 | 202,014 |
| 12012672..12012927 | b66-eng-12012672 | 256 | +6.228 se 0.106 | +2.474 se 0.056 | +3.754 se 0.088 | +2.62 | 195,036 | 234,977 | 202,282 |
| 12012928..12013183 | b66-eng-12012928 | 256 | +6.481 se 0.117 | +2.409 se 0.053 | +4.072 se 0.096 | +2.89 | 196,517 | 234,945 | 202,117 |
| 12013184..12013439 | b67-eng-12013184 | 256 | +6.063 se 0.103 | +2.345 se 0.057 | +3.719 se 0.092 | +1.82 | 198,337 | 234,853 | 202,308 |
| 12013696..12013951 | b67-eng-12013696 | 256 | +6.003 se 0.111 | +2.321 se 0.058 | +3.682 se 0.095 | +2.26 | 197,613 | 234,763 | 202,058 |
| **Pooled (merge computation)** | all seven | **1,664** | **+6.200 se 0.045** | **+2.383 se 0.022** | **+3.817 se 0.039** | +1.76 | max 208,330 | max 237,694 | max 202,326 |

**Pooled arm means (merge computation):**

| Arm | Mean edge | sd |
|---|---|---|
| IB | 527.713 | 90.7 |
| M09 | 531.529 | 91.5 |
| T1 | 533.912 | 91.7 |

**Gas and faults on the engine:**

- **All arms, all blocks:** fault counters (bad_return, OOG, other_halt, revert, tx_error) are all 0, and calls equal ok.
- **T1:** afterInitialize peak 229,429. afterSwap mean about 107.4k–107.7k per block.
- **Engine versus pyrevm:** engine gas is higher than pyrevm. T1 peaks at 195k–208k on the engine against 147–149k in pyrevm, and averages about 107.5k against about 62k. Both are under 245,000.
- **IB's margin is thin:** afterSwap peak up to 237,694 and afterInitialize 243,227 `[b64-eng-12010112]` `[b60-tp-2]`.
- **Worst seeds:** T1−M09 has single-seed negatives, as low as −0.60 `[b64-eng-12010112]`. T1−IB never does.

**Status against the Track A bar:**

- **Measured:** +6.20 se 0.045 versus the bar of +2.7 over IB, on 1,664 of the 4,000 seeds. The other ~2,336 seeds of the ledger are not in this root.
- **Gaps in this root's coverage:** 12010368..12010879, 12011136..12012671 and 12013440..12013695 are not covered here. They may be in other roots. I did not see them.
- **Templated interpretation text:** b67-eng-12013696 notes that its "reading.interpretation" text is carried over from the pilot template. The same "closer to the about +6.4 tell-working expectation" sentence appears in four blocks. Its numbers are its own.

### 2.4 Seeds the board may use: IB on the engine's default seeds 0..999

| Seeds | Mean | se | Source |
|---|---|---|---|
| 0..499 | 536.425 | 3.86 (merge computation) | `[b62-bs-0]` |
| 500..999 | 522.279 | 4.03; min 341.07, max 790.69 | `[b62-bs-500]` |
| **Pooled 0..999** | **529.35** | **2.80** | **merge computation** from both sums and sumsq |

- IB afterSwap peak was 237,009 on 0..499 and 234,856 on 500..999, with 0 faults on both.
- **Reading:** the pooled 529.35 is 7.35 above the board's 522.00 for IB, about 2.6 se. That is evidence, not proof, that the board does **not** score on `MatchRunner` default seeds 0..999 with this engine build. bs-500 itself says that only the pooled figure can be compared with the board.
- **Ledger hygiene:** bs-0 ran seeds 500..511 once in its last 64-seed command and excluded them, and bs-500 then ran them as its own. These are default seeds, never part of the fresh-seed ledger, so no fresh seed was reused.
- The board-seed question also depends on board-1's finding that three identical-scored @builtbyimd submissions (520.0536) "suggest a fixed server seed set without stating it" `[b61-board-1]`.

## 3. Track B: arb or retail on the replica

### 3.1 The tell and its reference levels

All figures are on holdout 10000..12047, n = 2048, from run.py/pair.py.

| Policy | Mean edge | Misread | Price RMS | vs HA | Source |
|---|---|---|---|---|---|
| ref/lattice_qm.json | 533.821 | 16.22% | 13.64 bps | +4.263 se 0.039 | `[b72-fresh-2]` pair.py output; also `[b56-kap-3e5]` `[b69-sigma-1]` `[b70-rival-1]` `[b72-fresh-1]` |
| TellGauss, qm_k1e-5 (the control) | 536.517 | 0.065% | 7.21 bps | +6.960 se 0.044 | `[b56-kap-3e5]` `[b70-rival-1]` `[b72-fresh-2]` |
| ref/flag_qm.json (true flag; not blind) | 536.543 | 0 | 7.11 bps | +6.985 se 0.044 | same |

TellGauss against each reference:

| Comparison | Result | Sources |
|---|---|---|
| TellGauss − flag_qm | −0.0258 se 0.0027 | `[b56-kap-3e5]` `[b70-rival-1]` `[b72-fresh-1]` |
| TellGauss − lattice_qm | +2.696 se 0.021 | `[b70-rival-1]` `[b72-fresh-1]` |

**Disagreement:** sigma-1's local reproduction of TellGauss against flag gives −0.0234 se 0.0028, with base mean 536.5193 against 536.5171. It attributes this to Python 3.15 and did not investigate `[b69-sigma-1]`. Misread counts for the same control also differ by one between items: 9,364 of 14,351,249 `[b56-kap-3e5]` against 9,363 of 14,351,248 `[b56-kap-1e4]`.

The brief's figures check out:

- **Gates:** README Gates digits for seeds 10000..10002 reproduced exactly in all four columns (448.261189 / 443.455598 / 443.522608 / 443.663520 for seed 10000, and so on) `[b56-kap-3e5]` `[b56-kap-1e4]` `[b58-wad-1]` `[b63-audit-1]` `[b72-fresh-2]`.
- **Fetched folder:** 26 files, 0 sha256 mismatches `[b56-kap-3e5]` `[b56-kap-1e4]` `[b56-kap-0]`.
- **Orchestrator check reproduced** on 66000..66095: control − gauss+qm = +2.979 se 0.108, and control − flag = −0.031 se 0.013 `[b56-kap-3e5]`. kap-3e5 counted 456 misreads of 672,163 first trades against the brief's 459 of 672,194; the gap was not traced.

### 3.2 Exact arb tests: what the float grid carries

**From `[b57-ff-1]`.** Replica RNG, seeds 66000..66063. Its "Rust emulation" is the author's Python transcription and was not checked against the engine. Five of its sources failed re-read.

- **Finding 1, exact Wad round trip.** Does some integer n near v/g give Wad(fl(n·g/γ)) == amount?
  - Rust emulation: flags 245,404/245,404 arbs and false-flags 244/206,298 retail first trades (0.118%).
  - Replica: 245,221/245,221 arbs flagged and 220/206,330 retail false-flagged.
  - The KNOWN tolerance tell on the same Wad callbacks misses 2,090 arbs, all is_buy with amount_x < ~6e-4 X, and false-flags 616 (0.30%).
  - Gain over the KNOWN tell: +0.048 se 0.018 (Rust emulation) and +0.024 se 0.015 (replica).
  - What remains to the true flag: +0.010 se 0.016 and +0.013 se 0.016.
  - Without any tell at all, the replica loses 2.945 se 0.139.
- **Finding 1b.** Adding an absolute 1e-18 to the KNOWN tolerance gets +0.033 se 0.022 on the Rust emulation. First-trade misreads fall from 2,335 to 290; the exact test gets 125.
- **Findings 2–5 add nothing usable:**
  - Split/whole is a function of the rival state that real.py already tracks.
  - Rival reserves after a split are known to ~1e-14, but add no decision-relevant state.
  - A retail sell gives ln p with sd 1.2, Fisher information ~6e-7 of a step's prior.
  - Code-path replay gives about 0 over finding 1.
- **False flags grow with order size,** roughly in proportion to amount/reserve. By size quartile they run from 0.019% (smallest) to 0.34% (largest) for the exact test.

**From `[b58-wad-1]`,** an integer-only spec (32 seeds 66000..66031, 351,490 callbacks; one source failed re-read):

- **Rule:** V = a·(W − fee); d = min((V≪q) mod W², W² − …), with q = 52 − e. e comes from the smaller binade of the current and inferred prior input reserve, with a 2^-48 boundary guard. The tell holds if d·1e18 ≤ (199·V + 1e36)≪q.
- **Calibration:** C = 199 is the smallest integer that misses 0 of 123,573 arbs on these seeds; C = 198 misses 1. It false-flags 653/227,917 retail callbacks (0.287%).
- **Purely relative tolerance fails.** It needs C = 678,587 and then false-flags 220,283. Separately, a purely relative 4e-16 misses 738 arbs, while 4e-16 plus additive W misses 0.
- **Round-trip variant:** 0 arbs missed, 296 false flags (0.130%).
- **Binade crossings:** no natural crossings occurred. 120 synthetic crossings: 0 missed by the affine or round-trip tests, 76 missed by a coarse grid.
- **Gas:** not measured; estimated at 1,000–4,000 gas.

**Comparing these with T1:**

- T1 uses 4e-16 relative plus 2^k·U additive `[b59-tp-2]`. wad-1 shows 4e-16 plus additive W misses 0 arbs.
- On engine-free replica data T1 misses 0 arbs, so its wider tolerance (than C = 199) costs only false flags: 0.54% of retail first callbacks in audit-1 against 0.287% of all retail callbacks in wad-1. The two are measured on different populations.
- **Unscored opportunity:** the narrower C = 199 rule or the exact round trip might buy back part of the remaining ~0.01–0.05. ff-1 bounds the whole remaining gap at "about 0.04 to 0.06". wad-1's WadTellPolicy was "not scored".

### 3.3 One idea per item against TellGauss

All rows use TellGauss C=1, soft=1, qm_k1e-5 as the base (except where the table says otherwise), with holdout 10000..12047, n = 2048, scored once.

| Idea | Holdout vs base | vs flag_qm | Tune / confirm | Verdict | Source |
|---|---|---|---|---|---|
| κ = 3e-5 tables (vs κ=1e-5) | −0.712 se 0.013 | −0.738 se 0.013 | tune 66000..66255: −0.680 se 0.037 | negative; keep qm_k1e-5 | `[b56-kap-3e5]` |
| κ = 1e-4 tables | −10.097 se 0.043 | −10.122 se 0.043 | tune −9.96 se 0.11 | negative | `[b56-kap-1e4]` |
| κ = 0, 3e-6 (tuning only) | not scored on holdout | — | −0.160 se 0.031; −0.098 se 0.026 | 1e-5 near best of the four weights tried | `[b56-kap-3e5]` |
| κ = 0 full solve | not run | — | — | blocked by memory (see §4) | `[b56-kap-0]` |
| Per-run sigma from tell pins (SigmaTell K0=2000, pmin .999, zcap 6) | +0.0019 se 0.0035 | −0.0216 se 0.0034 | tune +0.004 se 0.010; confirm +0.016 se 0.009 | null; oracle true sigma only +0.0055 se 0.0026 (4096 seeds) | `[b69-sigma-1]` |
| Rival propagation through rival-only orders (TellRival rt=1, Q=1, NMAX=2, lamr=0.8) | **+0.1190 se 0.0106** | +0.093 se 0.011 | tune +0.109 se 0.027; confirm +0.070 se 0.031 | positive, below the 0.3 port bar | `[b70-rival-1]` |
| Fresh: rival-band truncation of the price belief | −2.686 se 0.017 | −2.712 se 0.017 | tune −2.648; confirm −2.752 | negative; the basis is wrong | `[b72-fresh-1]` |
| Fresh: fixed −σ²/2 log-drift correction | +0.0020 se 0.0026 | −0.0238 se 0.0028 | tune −0.007 se 0.008; confirm −0.009 se 0.007 | null | `[b72-fresh-2]` |

Detail by item:

- **κ.**
  - κ = 3e-5 cuts price RMS from 7.21 to 6.40 bps and raises retail by 0.45, but arb loss grows by 1.17 (−30.245 against −29.080) `[b56-kap-3e5]`.
  - κ = 1e-4 gives price RMS 4.29 bps and misreads 0.045%, but loses ~10 `[b56-kap-1e4]`.
  - Both re-solves reproduced the control tables exactly: every offset array was identical, and only scalar g differed at the 15th digit, from parallel summation `[b56-kap-3e5]` `[b56-kap-1e4]`.
  - **Small disagreement:** with κ = 0, qmdp's level-0 A0c differs from pol_mid A0 by up to 0.005 bps; the README says they are equal `[b56-kap-3e5]`.
- **Sigma.**
  - The estimator is accurate: final RMS 0.18 bps against 0.37 for the fixed prior, and bias −0.001 bps.
  - It is worth nothing, because the filter variance resets at about 3,430 pins per run `[b69-sigma-1]`.
  - The earlier pre-tell +0.022 sigma figure, which sigma-1 cites, is not improved upon.
- **Rival.**
  - Rival log RMS falls from 2.77 to 2.49 bps on 16 seeds; misreads are unchanged by design.
  - On HA: +7.079 se 0.043 against TellGauss's +6.960.
  - **Gas is the blocker:** about 9.5 router evaluations per callback, roughly 15–30k extra gas on average, but the worst case of 12 silent steps × 42 evaluations is estimated at 0.8–1.6M gas, which needs a cap, pruning or a table. None of these was scored.
  - An earlier RivalT (+0.215 over thr09) is "not additive" with this result.
  - One holdout chunk command was killed by the tool timeout before it wrote rows. It was rerun once, so every seed has exactly one row `[b70-rival-1]`.
- **Fresh-1 failure mechanism.**
  - The rival spot from the router identity is the spot *after* the retail fill. On 8 diagnostic seeds the assumed band is violated on 33.5% of retail callbacks (19,359/57,766).
  - Truncating on that band raises price RMS from 7.2 to 10.5 bps and misreads from 0.06% to 0.97%.
  - Tried 8 settings against the limit of 4. This is disclosed in the result, and all ran on tune seeds `[b72-fresh-1]`.
- **Fresh-2.** One setting only. The −σ²/2 drift per step (−4.47e-7) is negligible `[b72-fresh-2]`. sigma-1 says the same.
- **Not ported:** sigma, drift and rival were none of them ported to Solidity. Gas for all three is estimated, not measured.

## 4. Where the work stopped, or ran on this fleet's limits

- **kap-0 is partial.** The seat's cgroup capped memory at 1,572,864,000 bytes. Unpatched dp.py was killed with exit 137 in sweep 1: its geometry arrays take about 670 MB, and VN2, its successor and S take 233 MB each.
  - A memmap variant, dp_mm.py, runs but thrashes at about 45 s per sweep. It stopped after 13 sweeps with diff 2.2e-2; the README expects about 110.
  - qm_k0.json was not built, and no seed was scored. Its tellgauss.py was "not run, not even a smoke test".
  - The result recommends a seat with ≥3 GB `[b56-kap-0]`.
  - Note: under the brief's rules, a memory kill is not a blocker for the requester. The κ = 0 table is the one table variant this root lacks a holdout score for. The κ = 0 policy built from the shipped tables was scored on tuning seeds only: −0.160 se 0.031 `[b56-kap-3e5]`.
- **tp-1 failed its gate** (§2.2). tp-2 then used the WAD-twin method, which both audits accepted for the numerical scope only.
- **No result here made a board entry.** board-1 lists builtbyimd submissions; see §5.

## 5. The board as read on 2026-11-10

**From `[b61-board-1]`.** The HTML was read at 15:22:13Z and the API at 15:22:35–38Z: 747 entries, one best entry per author. Sources: https://www.paradigm.xyz/puzzles/amm and https://www.paradigm.xyz/puzzles/api/v1/leaderboard.

| Rank | Handle | Strategy | Edge |
|---|---|---|---|
| 1 | @Ponxter | Tide v10 | +528.42 (avgEdge 528.4152, submitted 2026-11-10T00:47:33Z) |
| 2 | @0xMai | Curvance.com | +528.40 (leader since Nov 6 per the chart page; this is the brief's "1st") |
| 3 | @HedgeEconomist | thecitadel.com | +528.35 |
| 4 | @builtbyimd | TellMinLookup | +528.30 (528.3025, created 2026-11-10T14:39:46Z, 14 attempts) |
| 5 | @ultrapara | Parallax | +528.27 |
| 6 | @KaihuaQIN | MixZ11 | +528.26 |
| 7 | @pxmwxd_eth | BranchedReturn | +528.21 |
| 8 | @aaaapsouuu | OpalG5W6I0 | +528.10 |
| 9 | @cybergenesis621 | Morphogenetically Mogged | +526.78 |
| 10 | @ujiuifs | v5 | +525.38 |
| 13 | @frok_ai | mevmogger | +524.66 (the brief's "10th") |
| 22 | @artemis_onchain | hehe | +522.05 |
| 23 | @afinkek | CONTRO.APP | +521.98 |

**IB on today's board:**

- IB is the submission "imd.fun swarm (seat #1059)" by @builtbyimd at 2026-11-09T17:10:16Z, avgEdge 522.0013 (https://www.paradigm.xyz/puzzles/api/v1/submissions).
- It no longer shows as a row. At 522.00 it would sit about 23rd today. That is an insertion estimate, not a displayed row.
- A later @builtbyimd submission, "seat #39", scored 528.1088 at 14:46:39Z.
- These results do not say which contract TellMinLookup is, nor whether it relates to T1.

**What the pages state about scoring:**

- 1,000 simulations × 10,000 steps.
- σ ~ U[0.088%, 0.101%] per step; λ ~ U[0.6, 1.0]; order size lognormal with σ = 1.2 and mean ~ U[19, 21].
- The competitor is a fixed 30 bps AMM.
- Gas limit 250,000 for afterInitialize and afterSwap; 32 storage slots; fees 0–1,000 bps.
- Same seeds for all entries: unstated. Close date: unstated.

**Write-ups:** only @houseofjiao (rank 17, +522.40) has a public repo, https://github.com/houseofjiao/amm-challenge. It holds a hand-tuned dynamic fee with no prose write-up, and it is unconfirmed that this is the board entry. X/Twitter was not searched.

**Reading (merge inference):**

- The top 8 lie within 0.32 of each other, at +528.10 to +528.42.
- T1's measured engine gain over IB of +6.20 puts it near 528.2 *if* engine paired gains carried to the board one-to-one. §2.4 shows the board's absolute level and seeds are not reproduced, so this is not a prediction of a rank.
- Being first today requires beating 528.42. That is about +6.42 over IB, against T1's +6.20 se 0.045 on 1,664 seeds.

## 6. Disagreements and open questions, collected

1. **Port parity definition** (tp-1 fail versus tp-2 pass). Resolved by the audits as: PASS for numerical applied-WAD scope, FAIL for autonomous ideal-double equality. The measured cost of the difference is −0.008 to −0.06, depending on the test (§2.2).
2. **Inline assembly in T1:** audit-1 and tp-2 say false; audit-2 says true. Unresolved.
3. **Retail tell-positive count** on the same 20 audit seeds: 353 `[b63-audit-1]` against 352 `[b63-audit-2]`.
4. **TellGauss against flag:** −0.0258 `[b56-kap-3e5]` `[b70-rival-1]` `[b72-fresh-1]` against −0.0234 `[b69-sigma-1]` (local Python 3.15). The control's misread count differs by 1 between kap-3e5 and kap-1e4.
5. **Misreads on 66000..66095:** 456/672,163 `[b56-kap-3e5]` against the brief's 459/672,194. Not traced.
6. **Engine gas versus pyrevm gas** for T1: 195–208k against ~148k peak. Both are under 245k, but the engine figure is what counts.
7. **Default seeds 0..999** give IB 529.35 se 2.80 against the board's 522.00 (§2.4). The board's seed set is not established.
8. **IB identity:** the brief names artifact afd5183e…; the engine runs used db67c7a0… (source `301941829a…`). The link between them is not checked in these results.
9. **Tolerance constants differ across tell specs.** In order:
   - ff-1 1b: 2.22e-16·v + 1e-18
   - wad-1: 199e-18·V + W in V units
   - tp-1: 4e-16 + 1e18·scale
   - T1: 4e-16 + 2^k·U

   All of them miss 0 arbs on their samples except the pure-relative forms. False-flag rates rise with tolerance.
10. **Claims from the source URLs flagged on re-read** (§1) are weighed lower. They are ff-1's and wad-1's Rust-arithmetic descriptions, kap-1e4's provenance and rival-1's kernel constants. The measurements in the same results stand.

## 7. What carries forward

- **Track A.**
  - Keep T1 (`6ee77c05…`) as the candidate.
  - Finish the 4,000-seed ledger on the remaining fresh blocks, which are not in this root.
  - So far: +6.20 se 0.045 over IB on 1,664 seeds, against a +2.7 bar `[b60-tp-2]` `[b64-…]` `[b66-…]` `[b67-…]`.
- **Into the next port:** nothing from this root's Track B clears the 0.3 bar.
  - The best is rival propagation at +0.119. Its worst-case gas as frozen is far above budget.
  - The tightest unscored opportunity is a narrower tell (wad-1's C = 199 rule, or the exact round trip), worth at most ~0.01–0.05 by ff-1's bound `[b57-ff-1]` `[b58-wad-1]`.
- **Ruled out:** κ ≠ 1e-5 (3e-5, 1e-4; 0 and 3e-6 on tuning), per-run sigma, drift correction and rival-band truncation `[b56-kap-3e5]` `[b56-kap-1e4]` `[b69-sigma-1]` `[b72-fresh-1]` `[b72-fresh-2]`.
- **Still to do:** the κ = 0 solve on a seat with ≥3 GB of memory `[b56-kap-0]`. The numba port of replica2.py + latt.py that the brief lists as an early item does not appear among these 23 results.