# qm-2: Taylor's qm tables (qm-spline) ported into trim-1, copy 2 of 2

**Outcome.** The contract runs Taylor's (gtmcknight) Gaussian filter with the qm tables compressed by qm-spline (QMS1 blob, 10,167 B). Gate P passed before any change. Runtime **19,346 / 24,576 bytes** (target 24,000 met), validator passed. Gate F: 50 runs x 2,000 callbacks, max |fee difference| **5.83e-6 bps** on the fees the pool applies; pyrevm afterSwap peak **215,414** (target 190,000 **not met**). Engine, seeds 9534000..9534127 once each: **0 OutOfGas** in 1,408,559 afterSwap calls, peak 245,140 by the engine's gas accounting (the same accounting that gives t03 235,456), paired edge **+2.137 vs t03 (se 0.113) and +2.912 vs IB (se 0.124)**, n=128. All figures are local measurements from the supplied setup, not independent certification.

Sources: trim-1 (https://api.imdnotdead.fun/artifacts/fce772b9057867b241350b8e44c05298c6c877871a26b58702a5af431726f6c5, source sha256 d849c5f1... verified), port-3 (setup script and harnesses), qm-spline (https://api.imdnotdead.fun/artifacts/17d249ab0c33fc44931d9b105cf3f9e3846a560bd518b67310c2a56defacdecc, blob sha256 3ac7cb88... verified, 10,167 B), Taylor's amm-sim at 4812a638 research/solved (README, real.py, replica2.py, evalpol.py, gates.py, pol_mid.json). Platform: Linux x86_64, 6 cores.

## Setup and Gate P

Cloned amm-challenge at 6b6921f8d463314aa7c96234e1be25a25245f704 under test/scratch, saved port-3's Setup script verbatim as scratch/setup.sh, ran it once from the checkout root: exit 0, wheel sha256 matched aedb6828..., StarterStrategy validated. Later commands sourced scratch/env.sh. No download failed.

Before any change, unmodified `python3 -I gates.py` (Python 3.14.4, no setup) printed:

```text
seed   perfect-info      gaussian+plain    gaussian+qm       lattice+qm
10000  448.261189  443.455598  443.522608  443.663520
10001  576.779199  571.463409  571.623551  571.977293
10002  531.886497  525.168028  525.865859  526.411631
```

The gaussian+qm column equals the README's 443.522608 / 571.623551 / 525.865859. **Gate P: pass.**

## Reference and decoded tables

Reference = real.py `RealPolicy(T, {'soft': 1, 'qm': <decoded json>})`, T = pol_mid.json, driven through Taylor's replica2.py. real.py is unchanged; only the `qm` option points at the decoder's output. The decoded JSON is qm-spline's `qms_decode.py` run on the blob (stdlib + numpy): 10,639,373 bytes, **sha256 4d97c205783ddff731fecd9dbdaea09c295954d2d2e96fbf25eec1320a5c22fa**. Its axes (W0 160, NW 321, N2 49, FB 6, H 5, W2L) equal pol_mid.json's. The reference's initial fees come from real.py's `init`, which reads the plain table at (0,0,0): 0.002823606104386589 both sides; the contract hardcodes that WAD value.

Gate Q was not rerun: it is qm-spline's measurement (holdout loss 0.0632, se 0.0072 against the full qm tables, implied about +3.71 vs HA) and the tables shipped here are exactly its decoded tables, so that measurement applies unchanged.

## Contract design

Base: trim-1's Strategy.sol. Unchanged: filter (`ln`, `exp`, `root`, `pdf`, `cdf`, `trunc`/`truncSD`, `update`), state slots 0..7, timing, `afterInitialize` layout, fee conversion. Removed: the fixed-net MLP (`OFF`, `l0`, `unpack`, `l1a..l1d`, `net`, `rnd`, `fee`). Added:

- **synced flag** (real.py's `synced`): bit 0 of slot 7, `cnt` in the upper bits. Cleared on a new step in `update`, set when the rival estimate is replaced by the router identity in `afterSwap`. No extra storage slot.
- **sd**: `root(v)*10000` WAD bps, bracketed on S = 0,4,8,12,16,20,26,34 with k <= 6 and linear weight tt clipped to [0,1], as real.py's `lookup`.
- **Tag**: 0c in phase 0, else phase + (s if synced else c). Column axis: phase 0 clips R/5 to +-6 (13 columns); phases 1,2 use `_pos2` over W2L (binary search, packed constant).
- **Tables**: 5 fill functions `T0c, T1s, T1c, T2s, T2c`, each writing PUSH32 word immediates into `uint256[87] memory W`: W[0..1] row interior knots, W[2] column interior knots (uint16, 1/64 grid units), W[3..] coefficient lanes: lane L = row-coef x nc + col-coef, 4 lanes per word (64 bits each), each lane the 4 int16 sd-coefficients (1/8 bps). Total 321 words. No basis tables.
- **Bases at runtime**: `basis` finds the span by scanning the interior knots and evaluates the 4 nonzero cubic bases with an unrolled Cox-de Boor recurrence in WAD; the sd axis has no interior knots so `bern` gives Bernstein cubics in k/7.
- **Grid rule**: `interp` accumulates, for the 8 grid nodes (rows i,i+1; columns j,j+1; levels k,k+1), the full tensor sum over the union of active lanes in extended precision (U^3/8 scale, no intermediate division), divides once, clamps each node at 0 as the decoder does, then interpolates bilinearly and linearly in sd. The ask uses (i,j,f,g); the bid uses the point mirror (319-i, last-1-j, 1-f, 1-g). Offsets: a-w = 5(pw_clipped-160)-w+F_ask, b+w = w-5(pw_clipped-160)+F_bid, then fee = 1-exp(-x/10000) if x>0, clipped to [0,10%].

A pure-Python integer model of this arithmetic matched real.py's `lookup` over the decoded tables within 4.8e-7 bps on 3,000 random points (the residual is the decoder's 6-decimal rounding of the JSON).

## Size and validation

Compiler: the pinned challenge compiler (solc 0.8.24, optimizer 200 runs, viaIR, Paris). Runtime **19,346 bytes**, creation 19,374 bytes, runtime sha256 3f522028749a4e88cffc0778411b13adcbf71a9b2e2d811a23972ccea4840991. `amm-match validate` exit 0: `Strategy 'TaylorGaussianQmSpline' validated successfully!` (t03 'censored_10' and IB 'inputs_b' also validated). No assembly (the validator forbids it).

## Gate F

Seeds 70000..70049. For each, the reference drives replica2.py; the first 2,000 own-pool callbacks and the reference's returned fees are recorded; a fresh pyrevm deployment is initialized with 100/10,000 and the callbacks replayed with amounts/reserves as int(float x 1e18), **245,000 gas per call**. All 100,000 calls succeeded (no OutOfGas, no revert). Trace-set digest (sha256 of concatenated per-run sha256 hex, seed order): fe975d4705b71425c0ab0a20acd2bc164dafd455f5b64f89d0df0b4b71f4dc1e.

| Gate F metric | Measured |
|---|---:|
| Runs / callbacks / fee comparisons | 50 / 100,000 / 200,000 |
| Max abs difference, fees as applied by the pool (clampf 10%) | 5.826114e-06 bps |
| Mean abs difference, fees as applied | 1.689195e-07 bps |
| Max abs difference, raw returned values | 2.000793e+02 bps |
| Initialization max difference | 0.0 bps |
| pyrevm afterSwap peak | 215,414 (seed 70016, callback 911) |
| pyrevm afterSwap mean | 118244.01706 |

The raw maximum is one callback (seed 70026, callback 1702): the reference's unclamped bid fee was 0.1200 and the contract returned 0.1000, its MAX_FEE clip; replica2.py applies `clampf` (cap 10%) to the reference's return before use, so the fee in force is identical and the applied-fee row is the gate. Excluding that callback the raw and applied figures coincide. **Gate F: pass on applied fees (max 5.83e-6 <= 0.01 bps); the raw-return comparison shows the single clip case.** The 190,000 pyrevm peak target is not met (215,414).

## Gas of one ask+bid lookup

Measured on seed 70022's first 1,000 callbacks by compiling a variant whose `fees` skips both `interp` calls (filter, table fill and Bernstein weights kept): variant mean 45,987, peak 95,820 at callback 996; shipped contract mean 118,642, 191,638 at callback 996. So one ask+bid lookup costs **about 72,700 gas on average and 95,800 at that peak callback**. A second variant with the lane arithmetic stubbed showed the 8 basis evaluations plus loop skeleton at about 34,000 and the lane arithmetic at about 76,000 on the peak callback. The lookup is 25-30 lanes per side for dense knot regions; gas per lane is far above the opcode count, which points at the compiler's handling of the inner loop. Two alternative inner-loop shapes (per-lane accumulator writes; a separate row-sum function) measured 200,114 and 205,707 on the same callback and were not shipped.

## Engine

Each strategy alone vs the 30 bps VanillaStrategy via `amm_sim_rs.run_batch(X, VanillaStrategy, configs, 8)`, configs per MatchRunner._build_configs (seed-i default_rng; size, rate, sigma draws). Seeds **9534000..9534127 exactly once**, four chunks of 32 per strategy (12 engine processes, 11-19 s each), gas_stats_reset before and gas_stats after each chunk. The contract was frozen (sha256 below) before the first engine chunk. t03 = zlib(base64(code_zb64)) of the named artifact, sha256 cb6d9455... matched; IB sha256 30194182... matched.

| Pair | n | mean_diff | se | wins | worst5 (seed, diff) |
|---|---:|---:|---:|---:|---|
| port - t03 | 128 | +2.137129915 | 0.113017454 | 124 | (9534024, -0.526433), (9534018, -0.470212), (9534100, -0.129994), (9534015, -0.128967), (9534071, +0.067457) |
| port - IB | 128 | +2.911653450 | 0.123620669 | 126 | (9534096, -1.505829), (9534071, -0.681756), (9534100, +0.091754), (9534008, +0.123777), (9534020, +0.475773) |
| t03 - IB | 128 | +0.774523535 | 0.082867464 | 110 | (9534096, -3.272258), (9534051, -2.754364), (9534009, -1.322586), (9534020, -1.257404), (9534071, -0.749213) |

Mean submission edges: port 546.243653746, t03 544.106523831, IB 543.332000296. SE = sample SD of differences / sqrt(n).

Submission afterSwap gas (engine accounting, all chunks; revert, other_halt, tx_error, bad_return all zero):

| Strategy | Calls | OOG | Peak | Mean |
|---|---:|---:|---:|---:|
| port | 1,408,559 | 0 | 245,140 | 155,143.08 |
| t03 | 1,397,792 | 0 | 235,456 | 135,469.86 |
| IB | 1,386,857 | 0 | 234,853 | 135,139.06 |

Per chunk (first seed: calls, peak): port 9534000: 356,623, 245,140; 9534032: 350,091, 240,220; 9534064: 355,217, 239,631; 9534096: 346,628, 244,809. afterInitialize peak: port 140,543; t03 243,234; IB 243,227 (128 calls each, none failed).

**Caveat on gas.** The engine peak 245,140 exceeds the 245,000 figure while every call succeeded, so the engine's reported gas and its limit are not on the same scale as the nominal cap (trim-1 recorded the same accounting). The margin is nevertheless thin: on these 128 seeds the heaviest callback used 29,000 more than the heaviest t03 callback (its engine peak 235,456 is 9,700 under the port's), and a different seed set could produce an OutOfGas. To make this robust the lookup needs roughly 25,000-30,000 gas less: the measured costs say the 8 Cox-de Boor evaluations (about 34,000 with the loop skeleton) and the lane arithmetic (about 76,000 at the peak) are both far above their opcode counts, so the next step is restructuring those loops (fewer live variables, fewer memory-array bounds checks, or precomputed per-span basis weights) rather than changing the tables; the fee logic itself is verified.

### Per-seed engine edges (9 decimals)

```csv
seed,port,t03,IB
9534000,588.174360077,585.240384638,584.677130778
9534001,727.523189379,725.475190935,724.538143837
9534002,514.314199123,512.729266634,510.948757674
9534003,478.965486570,477.086903554,475.897964311
9534004,682.179507537,679.404565249,678.304913377
9534005,712.678712679,708.911646369,707.988160504
9534006,627.566448243,622.746063967,621.512912293
9534007,560.357631045,556.243101180,555.678514591
9534008,390.967959658,390.802615970,390.844182328
9534009,648.934002121,644.718757367,646.041343123
9534010,495.867098582,493.111955718,493.768286137
9534011,598.507748981,595.501467093,594.759796771
9534012,636.635264340,636.249394465,636.054672192
9534013,554.371153316,553.462425424,552.285969226
9534014,383.766416778,382.705763262,382.269302085
9534015,467.046493012,467.175460194,464.223315095
9534016,681.100278930,678.499239153,676.605480604
9534017,541.606681492,539.890756810,538.543246444
9534018,479.387696880,479.857909277,477.570912008
9534019,637.129734630,632.689705845,631.402005826
9534020,542.366355614,540.633178436,541.890582361
9534021,522.271669672,520.586226865,518.965989845
9534022,648.436592491,646.982935226,646.333087661
9534023,661.048649169,657.434525590,655.923797246
9534024,463.115412397,463.641845699,462.116584693
9534025,469.147471588,467.483356911,466.211753528
9534026,560.024942677,556.600363602,555.846990920
9534027,463.046640458,462.233791749,461.379561221
9534028,464.398665553,462.948502657,462.174811266
9534029,426.772532171,424.349425341,423.801790835
9534030,668.453627885,666.461693201,664.443527270
9534031,417.583899855,416.063207407,414.900982442
9534032,544.501589324,540.698506484,540.858559112
9534033,567.726369261,566.384756025,563.594793821
9534034,518.884274536,516.308336166,515.795348437
9534035,560.215563430,559.444299084,557.160298104
9534036,384.817887712,383.577669075,383.283997665
9534037,591.894807749,589.351476824,589.245444751
9534038,469.868300695,469.005861668,467.907341165
9534039,555.938573922,553.035899703,552.628602197
9534040,676.694528518,672.030279218,671.528980837
9534041,662.419655816,659.055869159,658.243295651
9534042,389.644387801,388.583765740,387.934173776
9534043,502.773979633,500.652230211,500.415183744
9534044,743.855031566,740.413510197,738.556422232
9534045,692.162369784,691.038804760,688.679926892
9534046,550.591869371,548.136760645,548.321595892
9534047,587.709168361,586.299929033,584.225799095
9534048,446.011788531,444.350991840,443.218637735
9534049,549.099103901,546.033272573,546.324247241
9534050,469.671373950,469.422925707,468.249117570
9534051,521.813629667,517.830298950,520.584662787
9534052,705.929291736,703.401425184,703.178277742
9534053,406.450114869,406.253069029,405.381326236
9534054,493.229326454,489.288201322,489.802039443
9534055,502.614992370,501.122411171,499.422664913
9534056,688.932708497,686.759113853,685.238385164
9534057,758.149320369,753.410870448,752.932790058
9534058,608.658525270,608.333531478,607.738597904
9534059,484.879927067,483.613222046,482.826733745
9534060,652.545432327,648.575774007,647.469476699
9534061,519.091498079,517.969550995,517.397959800
9534062,399.543212294,397.639402939,397.038471073
9534063,668.295026148,665.508894669,665.938261540
9534064,524.697506546,523.524533236,522.666522247
9534065,467.978550504,467.229594779,466.116631228
9534066,518.603061339,517.463871936,516.504636846
9534067,573.647578265,572.436544070,571.245742593
9534068,544.990053662,540.537128627,540.281918957
9534069,594.377522250,591.179534878,590.729921346
9534070,689.896713582,685.725025295,684.647283352
9534071,434.230328516,434.162871925,434.912084567
9534072,651.387647279,648.418462104,647.532272698
9534073,382.075458144,381.346560859,377.261910143
9534074,592.712909702,589.747707335,589.278496721
9534075,553.735800689,550.271339360,549.476360860
9534076,519.888717087,518.730435288,517.193797808
9534077,575.342497058,572.957722838,572.453992385
9534078,595.934631901,593.189677981,591.112165585
9534079,563.456203208,559.237813349,558.773236598
9534080,434.315125331,433.590092411,432.876739701
9534081,419.802057223,419.361420544,419.179624496
9534082,648.606898810,644.646019296,645.244958097
9534083,570.876743652,566.087792660,564.678838539
9534084,350.864463651,350.110240253,348.973640226
9534085,499.892306030,498.646749994,498.522902398
9534086,439.661729853,437.973802089,437.327093566
9534087,583.949087068,579.979029863,579.143672712
9534088,506.437501512,505.179071089,504.428010241
9534089,588.899880649,585.957488219,584.697651675
9534090,623.683187086,621.242760246,621.389642799
9534091,564.502443943,561.647394057,560.351973328
9534092,601.905522984,600.377975955,599.282967384
9534093,463.706804315,463.185613297,461.942818039
9534094,657.340409398,655.364870012,654.515929505
9534095,508.761967544,507.012264967,505.950691776
9534096,485.558781039,483.792351900,487.064609875
9534097,526.054463274,523.555207365,522.658840969
9534098,380.092263843,379.432031760,378.829713878
9534099,571.105286561,567.905627638,567.770163172
9534100,529.768224476,529.898218151,529.676470221
9534101,539.246739048,537.642534276,537.490300911
9534102,475.115277247,473.222667372,472.145414517
9534103,614.368049053,610.928268300,609.360685402
9534104,630.502048649,628.378016828,627.349050976
9534105,495.606116742,492.736706584,492.802369581
9534106,396.840852919,395.915850443,395.717810782
9534107,475.929789379,473.023325058,472.352071574
9534108,641.478730195,639.755214969,639.034324388
9534109,680.180547966,676.514335708,676.145382275
9534110,596.638545210,593.390442800,592.191567229
9534111,574.149266489,570.744857159,569.803936820
9534112,647.883302612,643.712394862,643.184185835
9534113,646.793614656,646.423127106,644.812145047
9534114,375.529179958,374.593792692,374.304112520
9534115,610.001580238,607.502688227,606.586966313
9534116,396.395651097,395.848246004,395.656073893
9534117,472.448461476,470.763796083,468.629383871
9534118,556.253488732,554.284210258,554.643013643
9534119,640.088476873,638.991857671,637.829315164
9534120,433.920548888,431.230836769,431.292487611
9534121,599.434272102,596.416298943,594.876592299
9534122,447.094458081,445.598873238,444.981603571
9534123,527.113243715,525.392845560,524.481428050
9534124,639.350684839,637.606257196,635.615019363
9534125,425.760310272,423.678207486,423.436045984
9534126,506.753514518,505.134806242,503.739700330
9534127,547.210448613,543.647210876,544.315241853
```

## Strategy.sol (complete, verbatim)

UTF-8 source **sha256 50a741b97a7ed79237ecbd34a4bc20b5c70b7872d8916f31dc3fe3f72b4fe859**, 40606 bytes including the final newline. Base contracts are the unchanged pinned challenge files. No external table or library is needed at runtime.

```solidity
// SPDX-License-Identifier: MIT
pragma solidity ^0.8.24;
import {AMMStrategyBase} from "./AMMStrategyBase.sol";
import {TradeInfo} from "./IAMMStrategy.sol";
// Policy: Taylor's (gtmcknight) solved game, amm-sim 4812a638, research/solved: Gaussian filter (real.py) + qm tables
// (qm_k1e-5.json, +3.77 vs HA on Taylor's replica). Filter, state and timing from trim-1 / port-3.
// Fees: real.py RealPolicy.lookup over the qm-spline decoded tables (QMS1 blob sha256 3ac7cb88..., 10,167 B):
// per table one 3-D cubic B-spline (row, column, sd level), clamped knots, coefficients int16 in 1/8 bps.
// The contract evaluates the spline at the 8 surrounding grid nodes (2 rows x 2 columns x 2 sd levels) with
// Cox-de Boor bases computed at runtime, clamps each node fee at 0, then interpolates bilinearly and linearly in sd
// exactly as real.py reads the decoded JSON. Bid side = point mirror of the ask surface (row 320-i, column last-j).
// Coefficient words are PUSH32 immediates written to memory: 4 lanes x 64 bits per word, lane = 4 int16 sd coefficients.
// Knot words: interior knots as uint16 in 1/64 grid units (W[0..1] rows, W[2] columns).
// Taylor source: https://github.com/gtmcknight/amm-sim/tree/4812a6382311624e1506aca131ca35c378f930d9/research/solved
contract Strategy is AMMStrategyBase {
    int256 constant U=1e18;
    int256 constant LN2=693147180559945309;
    struct Belief { int256 m; int256 v; int256 fb; int256 fa; int256 sp; int256 r; uint256 t; uint256 cnt; bool sy; }
    function getName() external pure override returns(string memory) { unchecked {return "TaylorGaussianQmSpline";} }
    function mul(int256 a,int256 b) internal pure returns(int256){ unchecked {return a*b/U;} }
    function divi(int256 a,int256 b) internal pure returns(int256){ unchecked {return a*U/b;} }
    function ab(int256 x) internal pure returns(int256){ unchecked {return x<0?-x:x;} }
    function clip(int256 x,int256 a,int256 b) internal pure returns(int256){ unchecked {return x<a?a:(x>b?b:x);} }
    function ln(int256 x) internal pure returns(int256 z){ unchecked {
        require(x>0); int256 k;
        while(x< U){x*=2;k--;}
        while(x>=2*U){x/=2;k++;}
        // Reduce to [1/sqrt(2),sqrt(2)]; |y| <= 0.171573.
        if(x>1414213562373095048){x/=2;k++;}
        int256 y=divi(x-U,x+U); int256 y2=mul(y,y); int256 q=y; z=y;
        q=mul(q,y2);z+=q/3;
        q=mul(q,y2);z+=q/5;
        q=mul(q,y2);z+=q/7;
        q=mul(q,y2);z+=q/9;
        q=mul(q,y2);z+=q/11;
        q=mul(q,y2);z+=q/13;
        q=mul(q,y2);z+=q/15;
        q=mul(q,y2);z+=q/17;
        q=mul(q,y2);z+=q/19;
        return 2*z+k*LN2;
    } }
    function exp(int256 x) internal pure returns(int256){ unchecked {
        if(x< -42*U)return 0; require(x<60*U);
        int256 k=(x>=0?x+LN2/2:x-LN2/2)/LN2; int256 r=x-k*LN2;
        int256 r2=mul(r,r);
        int256 even=665280*U+mul(r2,75600*U+mul(r2,840*U+r2));
        int256 odd=mul(r,332640*U+mul(r2,10080*U+mul(r2,42*U)));
        int256 s=divi(even+odd,even-odd);
        return k>=0?s*int256(uint256(1)<<uint256(k)):s/int256(uint256(1)<<uint256(-k));
    } }
    function root(int256 x) internal pure returns(int256){ unchecked {
        if(x<=0)return 0; uint256 n=uint256(x)*1e18;uint256 z=n;uint256 r=1;
        if(z>>128>0){z>>=128;r<<=64;} if(z>>64>0){z>>=64;r<<=32;}
        if(z>>32>0){z>>=32;r<<=16;} if(z>>16>0){z>>=16;r<<=8;}
        if(z>>8>0){z>>=8;r<<=4;} if(z>>4>0){z>>=4;r<<=2;} if(z>>2>0){r<<=1;}
        for(uint256 i=0;i<7;i++)r=(r+n/r)/2;
        return int256(r<n/r?r:n/r);
    } }
    function pdf(int256 x) internal pure returns(int256){ unchecked {if(ab(x)>10*U)return 0;return divi(exp(-mul(x,x)/2),2506628274631000200);} }
    // Hart rational normal tail approximation; erf(x)=2*Phi(x*sqrt(2))-1.
    function cdf(int256 x,int256 density) internal pure returns(int256){ unchecked {
        int256 y=ab(x);if(y>8*U)return x>0?U:int256(0);
        int256 n=35262496599891000;
        n=mul(n,y)+700383064443688000;n=mul(n,y)+6373962203531650000;
        n=mul(n,y)+33912866078383000000;n=mul(n,y)+112079291497871000000;
        n=mul(n,y)+221213596169931000000;n=mul(n,y)+220206867912376000000;
        int256 d=88388347648318400;
        d=mul(d,y)+1755667163182640000;d=mul(d,y)+16064177579207000000;
        d=mul(d,y)+86780732202946100000;d=mul(d,y)+296564248779674000000;
        d=mul(d,y)+637333633378831000000;d=mul(d,y)+793826512519948000000;
        d=mul(d,y)+440413735824752000000;
        int256 tail=(density*2506628274631000200/U)*n/d;return x>0?U-tail:tail;
    } }
    function trunc(int256 m,int256 v,int256 lo,int256 hi) internal pure returns(int256,int256,int256){
        return truncSD(m,v,lo,hi,root(v));
    }
    function truncSD(int256 m,int256 v,int256 lo,int256 hi,int256 sd) internal pure returns(int256 Z,int256 mt,int256 vt){ unchecked {
        if(v<=0)return (lo<=m&&m<=hi?U:int256(0),m,0);
        int256 a=divi(lo-m,sd);int256 b=divi(hi-m,sd);
        int256 pa=pdf(a);int256 pb=pdf(b);
        Z=cdf(b,pb)-cdf(a,pa);
        if(Z<1e6)return(Z,m<lo?lo:hi,v/4);
        int256 r=divi(pa-pb,Z);
        mt=m+mul(sd,r);vt=mul(v,U+divi(mul(a,pa)-mul(b,pb),Z)-mul(r,r));if(vt<0)vt=0;
    } }
    uint256 constant K64=15625000000000000; // U/64: knots are stored in 1/64 grid units
    uint256 constant SL=0x221a14100c080400;    // S levels 0,4,8,12,16,20,26,34 bps, byte k
    uint256 constant W2=0xa08264503c32281e1914110e0c0b0a090807060504030201;  // |W2L[i]| for i<24 (grid units), byte 23-i
    function S(uint256 k) internal pure returns(int256){ unchecked {return int256((SL>>(8*k))&0xff)*U;} }
    function w2l(uint256 idx) internal pure returns(int256){ unchecked {
        if(idx==24)return 0; bool neg=idx<24; if(!neg)idx=48-idx;
        int256 v=int256((W2>>(8*(23-idx)))&0xff)*U; return neg?-v:v;
    } }
    function pos2(int256 l2) internal pure returns(int256){ unchecked {
        if(l2<= -160*U)return 0; if(l2>=160*U)return 48*U;
        uint256 lo;uint256 hi=48;
        while(hi-lo>1){uint256 mid=(lo+hi)>>1;if(w2l(mid)<=l2)lo=mid;else hi=mid;}
        int256 wl=w2l(lo);return int256(lo)*U+(l2-wl)*U/(w2l(hi)-wl);
    } }
    function kn(uint256 w0,uint256 w1,uint256 ni,int256 lastU,uint256 idx) internal pure returns(int256){ unchecked {
        if(idx<4)return 0;if(idx>=4+ni)return lastU;idx-=4;
        return int256(((idx<16?w0:w1)>>((idx&15)<<4))&0xffff)*int256(K64);
    } }
    // Clamped cubic B-spline at integer x: span m (coefficients m-3..m) and the 4 nonzero bases N (WAD), Cox-de Boor.
    function basis(uint256[87] memory W,uint256 base,uint256 ni,uint256 last,uint256 x,int256[4] memory N) internal pure returns(uint256 m){ unchecked {
        int256 xu=int256(x)*U;int256 lastU=int256(last)*U;uint256 w0=W[base];uint256 w1=W[base+1];
        uint256 q;
        while(q<ni&&int256(((q<16?w0:w1)>>((q&15)<<4))&0xffff)*int256(K64)<=xu)q++;
        m=3+q;
        int256 l1=xu-kn(w0,w1,ni,lastU,m);int256 r1=kn(w0,w1,ni,lastU,m+1)-xu;
        int256 l2=xu-kn(w0,w1,ni,lastU,m-1);int256 r2=kn(w0,w1,ni,lastU,m+2)-xu;
        int256 l3=xu-kn(w0,w1,ni,lastU,m-2);int256 r3=kn(w0,w1,ni,lastU,m+3)-xu;
        int256 d=r1+l1;int256 n0=r1*U/d;int256 n1=l1*U/d;int256 n2;
        int256 tmp=n0*U/(r1+l2);n0=r1*tmp/U;int256 sv=l2*tmp/U;
        tmp=n1*U/(r2+l1);n1=sv+r2*tmp/U;n2=l1*tmp/U;
        tmp=n0*U/(r1+l3);n0=r1*tmp/U;sv=l3*tmp/U;
        tmp=n1*U/(r2+l2);n1=sv+r2*tmp/U;sv=l2*tmp/U;
        tmp=n2*U/(r3+l1);n2=sv+r3*tmp/U;N[3]=l1*tmp/U;
        N[0]=n0;N[1]=n1;N[2]=n2;
    } }
    // sd axis has no interior knots: Bernstein cubics in u=k/7.
    function bern(uint256 k,int256[4] memory B) internal pure { unchecked {
        int256 a=int256(k);int256 c=7-a;
        B[0]=c*c*c*U/343;B[1]=3*a*c*c*U/343;B[2]=3*a*a*c*U/343;B[3]=a*a*a*U/343;
    } }
    // Node sums carry U^3/8 scale: divide, clamp each grid node at 0 (as the decoder), then bilinear in (f,g).
    function bl(int256 n00,int256 n10,int256 n01,int256 n11,int256 f,int256 g) internal pure returns(int256){ unchecked {
        int256 D=8*U*U;n00/=D;n10/=D;n01/=D;n11/=D;
        if(n00<0)n00=0;if(n10<0)n10=0;if(n01<0)n01=0;if(n11<0)n11=0;
        return ((n00*(U-f)+n10*f)/U*(U-g)+(n01*(U-f)+n11*f)/U*g)/U;
    } }
    struct Ctx { uint256 nr; uint256 nc; uint256 last; int256[4] B0; int256[4] B1; int256 tt; }
    // Fee surface interpolated at rows i,i+1 (weight f), columns j,j+1 (weight g), sd levels k,k+1 (weight tt).
    function interp(uint256[87] memory W,Ctx memory X,uint256 i,uint256 j,int256 f,int256 g) internal pure returns(int256){ unchecked {
        int256[4] memory Nr0;int256[4] memory Nr1;int256[4] memory Nc0;int256[4] memory Nc1;int256[8] memory A;
        uint256 mr0=basis(W,0,X.nr-4,320,i,Nr0);uint256 dr=basis(W,0,X.nr-4,320,i+1,Nr1)-mr0;
        uint256 mc0=basis(W,2,X.nc-4,X.last,j,Nc0);uint256 dc=basis(W,2,X.nc-4,X.last,j+1,Nc1)-mc0;
        uint256 nc=X.nc;bool two=X.tt!=0;
        int256 b00=X.B0[0];int256 b01=X.B0[1];int256 b02=X.B0[2];int256 b03=X.B0[3];
        int256 b10=X.B1[0];int256 b11=X.B1[1];int256 b12=X.B1[2];int256 b13=X.B1[3];
        for(uint256 bb=0;bb<=dr+3;bb++){
            int256 s0;int256 s1;int256 s2;int256 s3;
            uint256 Li=(mr0-3+bb)*nc+mc0-3;
            for(uint256 cc=0;cc<=dc+3;cc++){
                uint256 lane=W[3+(Li>>2)]>>((Li&3)<<6);Li++;
                int256 c0=int256(int16(uint16(lane)));int256 c1=int256(int16(uint16(lane>>16)));int256 c2=int256(int16(uint16(lane>>32)));int256 c3=int256(int16(uint16(lane>>48)));
                int256 wc0=cc<4?Nc0[cc]:int256(0);int256 wc1=cc>=dc?Nc1[cc-dc]:int256(0);
                int256 d=b00*c0+b01*c1+b02*c2+b03*c3;s0+=wc0*d;s1+=wc1*d;
                if(two){d=b10*c0+b11*c1+b12*c2+b13*c3;s2+=wc0*d;s3+=wc1*d;}
            }
            int256 wr0=bb<4?Nr0[bb]:int256(0);int256 wr1=bb>=dr?Nr1[bb-dr]:int256(0);
            A[0]+=wr0*s0;A[1]+=wr1*s0;A[2]+=wr0*s1;A[3]+=wr1*s1;
            if(two){A[4]+=wr0*s2;A[5]+=wr1*s2;A[6]+=wr0*s3;A[7]+=wr1*s3;}
        }
        int256 v0=bl(A[0],A[1],A[2],A[3],f,g);
        if(!two)return v0;
        return (v0*(U-X.tt)+bl(A[4],A[5],A[6],A[7],f,g)*X.tt)/U;
    } }
    function table(uint256 ph,bool sy) internal pure returns(uint256[87] memory W,uint256 nr,uint256 nc){ unchecked {
        if(ph==0){T0c(W);return(W,24,10);}
        if(ph==1){if(sy){T1s(W);return(W,24,14);}T1c(W);return(W,18,12);}
        if(sy){T2s(W);return(W,24,14);}T2c(W);return(W,12,8);
    } }
    // real.py RealPolicy.lookup: w, r, sd in WAD bps; returns (bid fee, ask fee) in WAD.
    function fees(int256 w,int256 r,uint256 ph,bool sy,int256 sd) internal pure returns(int256 fb,int256 fa){ unchecked {
        uint256 k;while(k<6&&sd>=S(k+1))k++;
        int256 tt=(sd-S(k))*U/(S(k+1)-S(k));if(tt<0)tt=0;if(tt>U)tt=U;
        int256 pw=w/5+160*U;if(pw<0)pw=0;if(pw>320*U)pw=320*U;
        int256 p2;uint256 last;
        if(ph==0){int256 l2=r/5;if(l2< -6*U)l2=-6*U;if(l2>6*U)l2=6*U;p2=l2+6*U;last=12;}else{p2=pos2(r/5);last=48;}
        uint256 i=uint256(pw/U);uint256 j=uint256(p2/U);if(i>319)i=319;if(j>last-1)j=last-1;
        int256 f=pw-int256(i)*U;int256 g=p2-int256(j)*U;
        Ctx memory X;(uint256[87] memory W,uint256 nr,uint256 nc)=table(ph,sy);X.nr=nr;X.nc=nc;X.last=last;X.tt=tt;bern(k,X.B0);bern(k+1,X.B1);
        int256 Fa=interp(W,X,i,j,f,g);
        int256 Fb=interp(W,X,319-i,last-1-j,U-f,U-g);
        int256 base=(pw-160*U)*5;
        int256 aw=base-w+Fa;int256 bw=w-base+Fb;
        fa=aw>0?U-exp(-aw/10000):int256(0);fb=bw>0?U-exp(-bw/10000):int256(0);
        fb=clip(fb,0,U/10);fa=clip(fa,0,U/10);
    } }
    function T0c(uint256[87] memory W) internal pure { unchecked {
        W[0]=0x2b762af92aaf2a6729f0290226fe2610259925512507248a23b422211e0711d1;
        W[1]=0x0000000000000000000000000000000000000000000000003e2f31f92ddf2c4c;
        W[2]=0x000000000000000000000000000000000000000002f002e0024800b8001b000b;
        W[3]=0x1813181b1872184717ef18cf17c618390000000000000000180018521836183e;
        W[4]=0x183218f917db18521850184518ac1880185417c919441863180c191d17aa1846;
        W[5]=0x0000000000000000144c14ad148114891849187d185a18620000000000000000;
        W[6]=0x143e17811158146514c512ff164b14cb143a158113b0147814801389158914aa;
        W[7]=0x14721595144a14a4000000000000000014a214b6152614f314591622136e141c;
        W[8]=0x0e500d880f2d0e780dd60ff00cdf0e1f00000000000000000e1e0e600e450e50;
        W[9]=0x0e560ed60e090e0e0ebd0cf810210f570eb20be6120b0ec90dde10c20c330e15;
        W[10]=0x0000000000000000074f075e074407460e9d0dbc0f040e970000000000000000;
        W[11]=0x077d0928059a072f07d20630085c078f073b07fe06d407320783068907e3076a;
        W[12]=0x077c08ee06b4073d000000000000000007a40882070a07b10767092f064e069b;
        W[13]=0x042a03e803ec03d703fe044d037903a00000000000000000041203fa03ba03ba;
        W[14]=0x047604cc053a054604db03cb062d0606044d04d904ca040d042804d9036903bd;
        W[15]=0x000000000000000002c802ed028702810492047105ba05cd0000000000000000;
        W[16]=0x03af02c30425052d03200309030c026f02c00331027802a502f202a702a6026f;
        W[17]=0x038803ee03ae0339000000000000000003a5038e041103ee032d0488032002eb;
        W[18]=0x0266025d023801d50232029c01db02540000000000000000026a024b020f0201;
        W[19]=0x033d022d048e06c7031202e90390035802c5031e037c025802b2026f02500326;
        W[20]=0x0000000000000000022e021201da01cc0305031f033f059b0000000000000000;
        W[21]=0x028e02a002f403a5028401e8026c021401fe024801c801f3027201a201ff01b5;
        W[22]=0x02d20301033a01d9000000000000000002800392032300ac02e202f702b30355;
        W[23]=0x0238018801e80195018a02b7017001ea0000000000000000022d01cb01c301b2;
        W[24]=0x03070232039f03aa025702c602ab02f802b60225033d01cc01e8025901c60294;
        W[25]=0x000000000000000001ec01c801b6019702ab02860340035d0000000000000000;
        W[26]=0x0203028702690308024a014a020c01b5019e022e017101c402ac00a0021b0178;
        W[27]=0x027902ed020303130000000000000000026d02660255033102ac0276031f026f;
        W[28]=0x0233009801de013300ff034f00660176000000000000000001d5016801660162;
        W[29]=0x026d026703a5017301d9025e016b02a202a0016902a401c201140270012b0200;
        W[30]=0x00000000000000000142010200f600fa024601ec037301bc0000000000000000;
        W[31]=0x00e302de006901ed01aefef60225009400f50236ffad01250233fdca03690205;
        W[32]=0x01a101d601a801ed0000000000000000015a0255008d029701f100fa02aa0182;
        W[33]=0x0121fe76021c017eff9404e5fad7fe17000000000000000000b9ffc80022016b;
        W[34]=0x01400044023700c4008f019c003900e70134ff34022400e2fff001e6fe96ffab;
        W[35]=0x0000000000000000000c00b8ffdffd4100f500ff011400f80000000000000000;
        W[36]=0xff960187fea6ffcf00b2fc7e02c8001effd102a8fc0e002d00eaf6730df70125;
        W[37]=0x0063005c00740084000000000000000000120121ff56008a008effaf011a0073;
        W[38]=0x005df6c704e8ffd6fe5514b4eccbff0300000000000000000010fbd9023801ea;
        W[39]=0x0089fea501250020ffb40140ff66fff90055fc9b00c3ffdfff3305d9fe12ffaf;
        W[40]=0x0000000000000000ffb2017efecafdb2000d004a000a001f0000000000000000;
        W[41]=0xffe20183ff6a00140043fd5c010cfff4ff9b0354fd6b000a00a2f92d0672000e;
        W[42]=0xffe0ffcaffd000000000000000000000ffe900cbff0d000c000bff290071ffee;
        W[43]=0x002efe7b007c0008ff8a0351fdc5ffc300000000000000000008ff64005a004d;
        W[44]=0xffddffc6005cfffefffe0062ffce00030010ff230047fffdffd70130ffa6fffb;
        W[45]=0x0000000000000000fffa0016ffebffc20001001e0014fffd0000000000000000;
        W[46]=0x00000038ffe800020006ffb0001ffffefff4004fffda00010010ff5e005ffffb;
        W[47]=0x0005ffc5fff400010000000000000000001dffcaffcf0002fffbffe50012fffe;
        W[48]=0x0002ffe100090000fffd0037ffe900000000000000000000fffffffe00020001;
        W[49]=0xffe10028000f000000040006fff90000fffeffe6000affffffff001efff60000;
        W[50]=0x000000000000000000000001ffff0000fffb00160004ffff0000000000000000;
        W[51]=0x0004000efff90000fffcfff0000700000002000efffa0000fffcffe9000b0000;
        W[52]=0x0005ffd2fffe000000000000000000000018ffaafff20000fffafffd00050000;
        W[53]=0xffffffea0008000000030024fff2000000000000000000000000ffff00010000;
        W[54]=0xffda003d002900000004fffefff70000fffeffe5000c00000001001bfff50000;
        W[55]=0x000000000000000000000001ffff0000fff8002c000700000000000000000000;
        W[56]=0x00000025ffea00000001ffda00120000ffff001afff300000001ffd500160000;
        W[57]=0x0008ff96ffe60000000000000000000000170009ff4f0000fffdfff300160000;
        W[58]=0x0001ffee00090000ffff001dfff1000000000000000000000000ffff00010000;
        W[59]=0xfff4ffe7006e00000001000cffec00000001ffe500110000ffff0019fff30000;
        W[60]=0x00000000000000000000000000000000fff7003c0013ffff0000000000000000;
        W[61]=0x00000008fffa00000001fff800040000ffff0005fffd00000001fff800050000;
        W[62]=0x0001ffd3fffa0000000000000000000000010014ffc40000fffffffb00050000;
    } }
    function T1s(uint256[87] memory W) internal pure { unchecked {
        W[0]=0x2a892a072989291128a0283427cc276026ef267725f9257724df23f420d812c5;
        W[1]=0x0000000000000000000000000000000000000000000000003d3b2f282c0c2b21;
        W[2]=0x000000000000000000000000091007a906f6068d063005d00573050a045702f0;
        W[3]=0x17d8189617bc18221739180117d6180a16ed186b17d91806173b181917fb1806;
        W[4]=0x1841176019be186217c318c217e6180f17d018f91765182c17cd18371856182c;
        W[5]=0x181b18b917a517de181418c517b61893182c179a193f1875182f17171a31184e;
        W[6]=0x131a150f13a114351349140e142e141b185218bc17c1182b185d1875182d1809;
        W[7]=0x148d12be161214721404147513c8142c1405139314e7146113571439149d141b;
        W[8]=0x13f515fd12b2124313df173a10131345136517ce0f0c13b414a9139c148714c9;
        W[9]=0x1481145715ad14b0145a158713bb15bf145d140f155b1565144f13fc14ed1396;
        W[10]=0x0d3e0e610c400d3c0c620dfd0c5a0d6d0c710c910dcd0d320c810dcf0d030d5b;
        W[11]=0x0eae086515670e300c490f810c600cdd0cae0f8b0b3c0d440d000dcf0ddd0d91;
        W[12]=0x0d6e0e7f0be80c020d660ee40c430f150dec0ad6112f0fe70dc00a4313220f14;
        W[13]=0x05f4066105e805ff058f05e105c605f00db00ed20a900d020dfd0cba0df30abb;
        W[14]=0x06e3053807240636064405ff05a905ff0610059706f506240558059805e905df;
        W[15]=0x0646085c033e049e0646086402e6050f05a2094e019905b7074604d306ef0683;
        W[16]=0x06e8069b08f4079806ad081c06d2098a06d00601074e06b406a60633060304be;
        W[17]=0x028f0269022b0260034602de0284028102610252021802450290020f02930262;
        W[18]=0x03a5028a04a002da028004a601f8024902a603c4022c02870310034002a8029f;
        W[19]=0x038e0450047e04ae03750453046204ca0394035204d4043c0363035e044b0398;
        W[20]=0x014b00a5015401830169018effbc01350417049b04990477042d04e5049b042a;
        W[21]=0x0257023a025d01e601fc022d01d501c5021101d801b701bb0256022001d30193;
        W[22]=0x02c4031702e4031402b102bb02ca0331027802b6026e032a02c901c402f40319;
        W[23]=0x03bc03b50379039203c703c103bd03ec031e03a4032b033d031003140378030c;
        W[24]=0x01ee019b018b01a501b101ac01e2011d017d01ebffde01430165ffcf02e700f1;
        W[25]=0x024002220298022b022b01dd024b020e01f801fb01b1023801d201c101bf0198;
        W[26]=0x02c703490306032d02af02fd02f8032b027e028b02eb02c7025c024302ce0279;
        W[27]=0x00fa00a90052ffe000c50021ffff0194038a037b0387034203ae035b0390033d;
        W[28]=0x01d8019301cd01aa01b3017a017b018401b101810147010301d7016501280186;
        W[29]=0x0242025802730276022a0203024b025601fb01e10216024201e901bc01e50212;
        W[30]=0x0385034e0276031403a303460311033f02bd02ee02b602b20291029102bf0295;
        W[31]=0x019f010e012f00f90116012500ff00d80113011a000d002000b3ff38ff850112;
        W[32]=0x01c501c501e801d201c5016101cb0196017e0184013d01ae0176012e012d013d;
        W[33]=0x029c02b9027e0293024c02790267026e020d021f0237023201db01f5020601f4;
        W[34]=0x006a00caff3dfff60066fe3403be0044039302d103e502b4038c030e02e202b6;
        W[35]=0x015c01040144012f015700c7012d00f6013d012200af00e10150008600fd0087;
        W[36]=0x01d501dd01f901e101a101bd01b901c701a3015001b301910154016901320190;
        W[37]=0x03a2035c00df029403af033b02a903190282024c026b02020211027e01fd0230;
        W[38]=0x011e0084012d004800ba0114ff4b00d3002effdf0233feb200080479ee7800d2;
        W[39]=0x013f014301450158014300eb0138012700fc00ef00e3010a011800d1006e010e;
        W[40]=0x021502ef01ac024701d901f3024701bd017a01e3017501b80175015b01ae0164;
        W[41]=0x000702c8fb4cffd80020f7aa0ba9fe3903d4034d05c201b103fe025a02e20247;
        W[42]=0x00fc005e00de008300e7007a0079013a00e200e5ffb200a100cdff4501faffca;
        W[43]=0x0155014201a90161013c0131014e0141011800da012000fc00e200e600d20106;
        W[44]=0x03e5061105a80a39042504f701ad023501a3019a028d021e01bc0215017401a3;
        W[45]=0x0051005e00290124010f00c3ff58fdbbff40fcaf025e032aff58057af7c1fb4b;
        W[46]=0x00ce00c500d400c600cf006b00ba00e20083008b0054008500de0052002400b4;
        W[47]=0x01b80233018300ca0164015f01b901bd011b014f014600e400f600ee00fb0120;
        W[48]=0x0099fad90478fd14ff6400eeff4502650468072203a0073a0418019d04460686;
        W[49]=0x008b000ffff500c0008a0019005300b400d5ff57ffc900d8ffdc0198fe7300ac;
        W[50]=0x00e400ec012a009200ce0083010600e600900093006d0081007f002c00bb0015;
        W[51]=0x047e03330a8e044603f2007d056f068d01af01ee0128008001310147013601a9;
        W[52]=0x0024ffd9ffe3006300d8fbf1020d007dfde208bbfaebff95007cfcf2011afefd;
        W[53]=0x006b0021008f007d0046fffd002d0058004d0040ffe700310088fef200fe004f;
        W[54]=0x014300d801c0ffe500ed013200f9018e00ae00bb00c800700066008d0041006f;
        W[55]=0x015bf8220412000bff60025dff0afed7040e0775085e051d0311052a016306e9;
        W[56]=0x0091fd99010a004bffd1009cff90003e0043fdce0158003dff510426fd850042;
        W[57]=0x0062007200740068005afffa00b40052001f0002fff8fff0fff5ffd8ffe3ffc7;
        W[58]=0x03e5061e0831030702fb016702ff05f30152014d0125004b00aa00c600c700e3;
        W[59]=0xffc50114ff49002c00b4fd57017effd8fecd0471fdcbffdf0043fe6d0080002a;
        W[60]=0xff8afff60029ffee00110002ff78001fff580171ff85ffea002eff4a0076ffe6;
        W[61]=0x0032019affceff42009b003a00e600f70020006b00120010ffbeffccffa7ffc6;
        W[62]=0x006bfda50136fffdffd00091ffb0fff2030e089f093703be044401190457070e;
        W[63]=0x002dff3d0012fffcfff2005affc0000c0013ff6a0065ffd7ff990167ff2a0018;
        W[64]=0xffbcffa1ffbaffca0055ff82007a0015ffd900a9ff89fff7001dffb9006cfff3;
        W[65]=0x030102c90d0100d1027e014501b804ff00c2ffff0128fff0002d009d00200033;
        W[66]=0xfffa001bffe700050016ffae003cffffffde0089ff9dfff70007ffd800120003;
        W[67]=0x002eff21003bfff6ffe0004fffd70007ffe60022fffafff90006fff00012ffff;
        W[68]=0xff790098ffb3ff570000ffb0ffb60010ffd3ffdb0065fff6ffa600b1ffa90002;
        W[69]=0x000dffcbffe8fff9fffa000cfffc0002029e0375140e02f403c002f20279058c;
        W[70]=0x0010ffdcffb8fff6fff7000d000a00060006fff2ffecfff7ffee0027000d0008;
        W[71]=0xffebfe7afeceffc80006ffdb00ce0019000c0068fec9ffe5fffbffe30062000a;
        W[72]=0x01d5fd0304df0112018705cd031602dcfed3faf4fe77fecd00a8020a0146005e;
        W[73]=0x0003ffed000500090002fffa0009fff6fffc0014fffb00080002fff90004fffc;
        W[74]=0x00b7fd3b00780021ff7b0121ffc4fff0000dffae001a000cfff5001a000afff8;
        W[75]=0x022bfd8200bafd77fba7086cff94ff9a013ffb720107003efe9402bdff6fffd7;
        W[76]=0x0004ffe20024fff7fffe000afff20003fe3601df019cfc26fe2cfdfffaa30046;
        W[77]=0xfff1004c006dffee0008fffeff850009fffc000a0025fff2fffd0015ffbf000c;
        W[78]=0xfe5506300445ffa10130fdd0fb850032ff3202db023affcc0078fef4fe5f0013;
        W[79]=0x00fdfc2e00a5010e021004e8fe7201e1fceb01b317e503bb03fff3e1eddd0070;
        W[80]=0x0003fff9ffdc00080001fff50030fff9fffe0010ffe300050001fffb0009fffe;
        W[81]=0x008afe3afca50019ffa400aa0172fff6000cffccff6e0009fffa00030056fffb;
        W[82]=0x025c029fdd65fb5bfd2f05641178ffe70122fca6f7c30029ff22013904a9ffe7;
        W[83]=0x0001fff9000cfffeffff0002fffc0000ff860199ffce0008fe63fb0a01c8fcd8;
        W[84]=0xfffc000e0028fffd00010001ffd30002ffff0000000ffffcffff0006ffea0003;
        W[85]=0xffab01540169ffed0021ff9dfe2d0007ffd9008400c4fff6000fffb6ff600003;
        W[86]=0x000fffa20006ffe7002d009effe1009aff7d009f065e00660074fd27f94a0011;
    } }
    function T1c(uint256[87] memory W) internal pure { unchecked {
        W[0]=0x0000000037dd2d9b2af42a2d298e28f7285527ab2709267225d3250c22651823;
        W[2]=0x000000000000000000000000000000000781074006bc064505bb054404c0047f;
        W[3]=0x17da180c18131828179c1808183f181e172018b817cf183617631852180e1826;
        W[4]=0x18a614971b5b185d1855182718161843180717731927182917b0185e1807182c;
        W[5]=0x17e618e917c9185c185b168319b217e1174a1b82163618aa18a7163a18d3184e;
        W[6]=0x12f913f0131d1327128c135912d5131a1245131b133513281275137812f41328;
        W[7]=0x119d19ad0e6d129611ac16c10f5a12e0127a17640ec1134413a5128a14101339;
        W[8]=0x1336127e13bd12f21202186c0f1c13a514ce0c561768129c113e1887119112cb;
        W[9]=0x0b960b900be20bfc0b950af30c9b0bee0ae40d7b0b190c1b0b3c0bec0c010bf4;
        W[10]=0x0d8407240fad0cdc0dad07c410470c780c4209690f220be20aaf0c5f0b980bf2;
        W[11]=0x0bb60d7c0b1c0c930d0506ee11a60b7a09fc13c105f30d330dd108000e170c9c;
        W[12]=0x051004910484047a03f1059d03f7045b048b039c04c00481046504be04510473;
        W[13]=0x046206fa02f903f1041b07160269043b04b405c0037404a205b40481048c0487;
        W[14]=0x050d04d8048a041404d9071c015204f2055602e506f7039004b20641035b0418;
        W[15]=0x0210029801cf01e202170095021b01a902fa034a01450264025d01c7020701a8;
        W[16]=0x02ea027a041b0399030d01e8043f0282025602d102a501c201c602f101e801ee;
        W[17]=0x02ba034c03eb047c035b011806c50233024304cd021e05fe02e902b1045a0406;
        W[18]=0x01df017701a0019901f30216015c00d501d10098040b020401da01f8008b0148;
        W[19]=0x023d021502f2012b01e602c8005f04fa01c4025a00b80438020b017a01f4016c;
        W[20]=0x027f026d02bf01ed021d0353005903e802c801f6043b00c702480260029101f7;
        W[21]=0x01a9012a014900f001a7014000e601e6019dffbc0202005101ad01c000e0016c;
        W[22]=0x01e6020201f8029f01dc018202850127020700cc02efff8a016f0177011c01d5;
        W[23]=0x02160275026102bc0281012903a401a601d703700180037b020d0213025b027a;
        W[24]=0x015400ea00cf0168015101ca00f6fde101bcfebf01140388012e01be0107fffb;
        W[25]=0x01b8016e0265010401670190015501900156016200730350013a00e101500031;
        W[26]=0x021501f3020701d900d3032a00d701c602f20143031c01e80191020101c601c6;
        W[27]=0x011100c40060009c00d8ff9101e6024600f7042efd38ffdb0129ff7600ee015b;
        W[28]=0x011b01a0011a0196015000db013a01a700e200db0148ffc60141002d00e20135;
        W[29]=0x016b01e3020701870215012700aa0209011f027c020601730177013c01eb0139;
        W[30]=0x00a2007b002900af01b3ff7fff2f016e0006fee5038f000d00ed009aff7600e5;
        W[31]=0x011000a001b2006600be00d10085010900bd003a00b100d700c80030004e002d;
        W[32]=0x01800144013b01340047020a0160013a025300ad023b010b00e5016000f7016f;
        W[33]=0x008d0008ffde0125ff910142fdd2ff56015dff5e012f012d0051ffd0ffe4003d;
        W[34]=0x007b00e2002201060095001600d000560065fff80083ff930087ffefffbe011e;
        W[35]=0x00d600fe00eb012c016500c0014800020093020a001f01fa00c100b101090093;
        W[36]=0x0003ffcdfeee016e00e7fdeb04a7ffa0ff92ff1bfd850073004eff4101160040;
        W[37]=0x007fffd2011cffc4fff10042ffc4002f0039ff2e004effbc005aff61001000b7;
        W[38]=0x00d7004800e700aa002c0215ff8b00990130007c01ae00ca0062007c006300ba;
        W[39]=0x0013ffdc006aff90ff560154fe22ff040069ff5f012100ceffd20042ff98ffaf;
        W[40]=0xfff4ffccff3c0019fff6ffc40010ffee00020037ffb0002affcd003bffedffc8;
        W[41]=0x008d0038ff400079ffe500ce02c0ff0900f60066ff6e00deffcf0074001dffe3;
        W[42]=0xfff7fff80005ffff0047ffa600620006ffba0045ffadfff1000effed00150000;
        W[43]=0xff5000df0034fff10047fff5ffbd0008ffd100060035fff3002affd4ffec0006;
        W[44]=0xffa70067006bffc700a6fd2cfe2b004cfed00331ff9fffad0086fdaf0039000a;
        W[45]=0x000e0009fff60000ffa10067ff92ffff0048ffb600560004ffe9001affe50000;
        W[46]=0x010cfd8fffe10009ff66002d0050fffa0054ffe9ffc10006ffb7004a0008fffd;
        W[47]=0x0038fe8dfff50018fedc05fdffc5ffca0182f7b701360035ff1003c2ffa1fff9;
        W[48]=0xffddffd90022ffff0069ffcb00570001ffac0031ffacfffb0013ffef0017ffff;
        W[49]=0xfdb2044400c9ffef00fa001cfef9000aff4500330091fff5008bff17ffed0005;
        W[50]=0xff9c01a0fffeffdc01a1f64f0068005cfd830e01fe79ff97019bf5c6008a000d;
        W[51]=0x0010004cffba0001ffc1ffedffee00000031ffff00230002fff30005fff40000;
        W[52]=0x017dfcf7fe15000cff53fecd018cfff800710059ff090007ff8b0090003bfffc;
        W[53]=0x0039fedefffe0016fed50650ffbfffbc01d9f68800b20048fe76073d0013fff6;
        W[54]=0xfffaffd400170000000d001cfffdfffffff4fff3fffa00000001000000020000;
        W[55]=0xff5b01770059fffc002fffbcff9b0002ffd80036001efffd0017ff8500000001;
        W[56]=0xffe4004afff5fff80056fd3e0032000fff700411ffeaffed004efc34ff4c0002;
    } }
    function T2s(uint256[87] memory W) internal pure { unchecked {
        W[0]=0x2a992a092989291328a2283627ca275e26ed267725f7256724b42395204c12b7;
        W[1]=0x0000000000000000000000000000000000000000000000003d492fb42c6b2b4c;
        W[2]=0x000000000000000000000000094507e30720069d063305cd056304e0041d02bb;
        W[3]=0x17bb187d17b4180d172818051806180f1704187817c91803172f182817f51807;
        W[4]=0x17a4196a170a17f717b918be176f182b17dc1844181b181f17a5184618281821;
        W[5]=0x17cd196f1686196717f717f1193f183e17e0183e18f4183e17d7181818db1828;
        W[6]=0x133214c813e014361358141414201426180618a117ec1870182217ea19301795;
        W[7]=0x13f1148913f7143713f1146c13b4143013f2139315011457130f145e13f9141a;
        W[8]=0x14211496121f13a5141c151212231423149a11f8164f1515143b134e15cc1429;
        W[9]=0x1442144c144413ee140a1591122c151a14761245172311de13f0159c112213c7;
        W[10]=0x0d510e2b0c4b0d5f0c7a0e5d0d330d790cbd0cc80d9f0d660ca00e1a0d230d6e;
        W[11]=0x0c69118d09f80c8c0cdd0fa10b060d930d7c0d540dbf0d9d0ce10dfa0e020d8b;
        W[12]=0x0cf010d20a9a10000dae0c3912260ed20d420db011030e410d330d2d10760da2;
        W[13]=0x05e906da05ea062c05af05c5062b062a0d880e400ddc0e110dd80cf90f800dd4;
        W[14]=0x064b0694060e062f068d05f20601063606040629069e0644059c057d06360617;
        W[15]=0x06e105f5045a05cb06c7069604830658076203c0088206e706d00569079d0640;
        W[16]=0x06e1068105b605b50688081103cc0608073e03f7083804430665082701e40563;
        W[17]=0x02cb02b402780297032702f8029c02a602d702980275028a02c2023d02a1028b;
        W[18]=0x02ee05680136022f0306045501be02c2034b032a02c502cc0318035b02ba02b5;
        W[19]=0x02a206d303a50515040601c0059a0390030c050b03a70321036f03c90361028f;
        W[20]=0x01980134012b01820183006700fd017603a004d8054a0543046502140673051f;
        W[21]=0x021d02ee01a401b6020f021701d401d2022801e401b701bc0261020d01d901b8;
        W[22]=0x02fb02a3039f03d90261034102f503b3032b0140044f030a020e02d1029901b1;
        W[23]=0x035d037c03530302031e04dc02df03420353029e03b2035f027c0457032a03ce;
        W[24]=0x01f4019f01a1018501b001a101c0016f014201e2009100dc01ac01590192017e;
        W[25]=0x01f602d001d40276028e014e025e02e0020501c10206019e01d201c101a9019f;
        W[26]=0x026003be02c1031302b10289032d02d1026702d202db028a028701fc02fd0253;
        W[27]=0x0171017eff53004b00f5ff0301a4012602fa036f034503820365026204350381;
        W[28]=0x01bc01bf017001e901b301670179014601b701750139010d01bd014c01450142;
        W[29]=0x023f026b027602b50219021f025b027a020e019f026b022301ea019e021301ab;
        W[30]=0x030302f802b7029c03000364029f02d4029702b002bb02ad025e02d802ab02cd;
        W[31]=0x017d0123012b00e5015500c000eb00f3008d01d800be0007011c0074ff3800b3;
        W[32]=0x01b301da01d601df01c80160019f01d3017b015a0157016d019400fb013700ff;
        W[33]=0x0266029502a1026502510279024a0256020a0213026c022301e601e6020501f5;
        W[34]=0x0110012dff52ff570086feda01e200c002f7029802f40293030e0307026f02ff;
        W[35]=0x015300f901240142014500e700e500f8014800f700c2009b011c00af00cc00e7;
        W[36]=0x01de021401d30207019c01e201c901cf01aa013201c90191015c0153014c0159;
        W[37]=0x02ea02b70216025d02fa02d602b70211027f02610207023a020002770226022a;
        W[38]=0x00dc00dd009f009a0158000300590029ff8d015900fefff300f00056fb34ff66;
        W[39]=0x0146013901540157013700f0012c013d010200dd00be00f9013f007b00e0008c;
        W[40]=0x02340225022b01c901c4027201ea01d9017601c401d601a6016e0183018f017e;
        W[41]=0x0148ff420045fd81fff0fef100e000dc0334024a0292020e0303031e0253030f;
        W[42]=0x010e002800f900a900b00113ffae01200129ffef00ceffd40031012fff790125;
        W[43]=0x013b01c8015201860144011f0187013c011c00dc011e011200e300e000c700e7;
        W[44]=0x028902db01cd01be04260331018900e7011f02d701c1027c0203015c02320193;
        W[45]=0xffcd017dfeec008701f7fd9e02c0fe86fe1d03abfad302430112fdb00297fe0c;
        W[46]=0x00d000c400de00c700d0005200c800cf00810092001700740104ffda007f00d0;
        W[47]=0x023501b1013a0117013001db017501c6013401340180011800fb00ff00fb0111;
        W[48]=0x01a1f6720aaafdb1fe7204d0f9a10133033a048d0819092201c101d7053704b3;
        W[49]=0x0090fffd000900b800cfffdf000c00ac00cbff25003400dfff5f0383faff009b;
        W[50]=0x00ce0128010300f800d50093011200d400980081007a00a50087002100a30029;
        W[51]=0x020d051d079702190556fc760562066f011b02b8013b00bd015f01320157016e;
        W[52]=0x009cfed30002003c002efde302ec0073ff25069bf7d100970088fca50412fea3;
        W[53]=0x006c0029008000720044fff100350043007effd1ffec00340059fedd00ff0054;
        W[54]=0x01b800100276000500c7018a00b9017c00cb00ac010a00a200730093005f004e;
        W[55]=0x00b3f944042effeeff7c027bfd9a00a4033305d906c3037600a20759fdb206f5;
        W[56]=0x0074fe0e00d10066ffd000d0ff9d001e0053fe4d0104007effc40246fe69ffea;
        W[57]=0x0067008900830075004b0031008800710025ffd80026fffafff9ffeeffd1ffd5;
        W[58]=0x0293032c047a05ca03defdb004400604012201f200c2004600c000ca00d30108;
        W[59]=0xfff800c0ffc4ffb6ffcdfeb30061008dff9403d4fe4afef70027fe000129ffcf;
        W[60]=0xffc20019fff1ffebffcbffe1ff69001bffb30126ffcdffd00010ff3d00280018;
        W[61]=0x006601410037ff620091007e00a801200032005a00690030fff6ffd6ffa1ffbf;
        W[62]=0xfffffe7700de0047ffee006fffadfff10239091f0852022f029200e9042006f5;
        W[63]=0x0021ff570032000bfffb0053ffcdfff9fffcff9f003f0010001a009fff93ffd2;
        W[64]=0xffd5fff0ff9effc8ffb9ff620020001000370052003400030002ffdd001ffff7;
        W[65]=0x028fff2b04ce013702fbff8c020b057800bc00cd00a7fff50032004e004c0031;
        W[66]=0xfff70029ffcafff5fffdffc800520013fff3007fff61ffe00003ffdd00180003;
        W[67]=0xff9aff54ff84ffea001800250016000affd10046ffbbfff2000affdf00260006;
        W[68]=0xff8e0072ff19ff15ffddffd5fff9001cffd7ffcc002fffea005d00a00007000c;
        W[69]=0x0006ffc800190006fffd000dfff6ffff01dd08fd0ee501fe0348011f052a0632;
        W[70]=0x0024ffb6fff4fffcfff20022fff700010007ffd9000a0000fff9002bffeffffc;
        W[71]=0xffaafe73fee2ffc9ff81fff50084001800530031ff8effeaffd9fffd002c0005;
        W[72]=0x01a1ff0a02df0051005702f801eb0279ff19fd39ff0ffedd00cb011500b8005c;
        W[73]=0xfffa000dfff7ffff0004ffef000f0005fffc0019ffeffff90001fff700070001;
        W[74]=0xffb4fe81005a000a000300a7ffb9fffeffe3ffff000200010007fff9000c0000;
        W[75]=0x00ff03f90024fedafef0ff9bfe5dffeaff43ff1b01c8001c007301a8ff3fffef;
        W[76]=0x0002ffed0019000affff0007fff7fffdfe150460ffbcfed7ffbcfb37fdb3ff3f;
        W[77]=0x00080039003f0005fffefff2ffc3fffc0002fffe00180004ffff0004ffdffff7;
        W[78]=0xffd0047fff2e0020ffe8fc28fdcfffdfffec02ce00db000c0019fe91ff7fffee;
        W[79]=0x00d6faf0012d00030156056ffec90128fd7dfd4d0c1203d5017dfdebfafafee4;
        W[80]=0x0000fff9ffeafffc000000050017000800000004fff0fff80000fffd00050002;
        W[81]=0x0025fda3fe65ffdeffe30113009600140001ffb6ffaefff6ffff0013002c0005;
        W[82]=0x01eafff3ec85fadcfecd02a506b6012c005ffb78fdedffb9ffdb030c0265002b;
        W[83]=0x0000ffff0005000200000001fffeffffffb9019affc50025fea3fd6e0175fe7a;
        W[84]=0xfffd001b001100010001fff3ffeffffeffff0004000600010001fffcfff9fffd;
        W[85]=0xffbf0187001c000e001bfe71ff3cfff1ffe600ca00470006000aff71ffd2fffa;
        W[86]=0xfff5ffc20001fff3006a007fffe3004eff2d00ac031700ad0056feeefdb4ffa7;
    } }
    function T2c(uint256[87] memory W) internal pure { unchecked {
        W[0]=0x00000000000000000000000000000000314e2bc82a0228c4273c25fe24381eb2;
        W[2]=0x00000000000000000000000000000000000000000000000007710685057b048f;
        W[3]=0x180b17571da2181b18211669159b182016911a8818541808176717b8180b1828;
        W[4]=0x17e1185a15eb180b17b518161fe3183a17681b230e2a180a19d9120a16bc1842;
        W[5]=0x112b0fd90c0111cf0fea176c13f8119e12160c3012b711b210d813d810d911c9;
        W[6]=0x114513c814691238125c0d1c0722118f121a10f822cd121f0bcb23f50f0b1161;
        W[7]=0x0b6409120cca0a2109ba0b24093c0a3b08b80c3307990a60094409630b3e0a1d;
        W[8]=0x09e00a4d08b309bf0a9b0a730fb70a44086c0f9cfc1209f80e6601120d050aab;
        W[9]=0x029d03e901f502b202f6016b02a8026702a001e8046e02950314034401a40292;
        W[10]=0x03b5029302d102d602280547014e031f0474ff8208e502bd027e05d80138025d;
        W[11]=0x01d301a801ea00f501ae011e014d01f7039602b8ff93fc77012400d002480357;
        W[12]=0x016c051f039b058904a9fae8064cfc82010b06d3fde7092a02a100a804630289;
        W[13]=0x00c70196ff6904220210ff50019afba3febe05ff07b806a60212ff34fd4bfea0;
        W[14]=0x0261fe10033c0025ffec0b95fb03047a0275004204eeff440134021400cf0111;
        W[15]=0x00ceff230120fe7f002b018dfebd02b800c0f659fe67fe0700c102d200ea01b7;
        W[16]=0x00be04a4013e01f201e3f9c1028effcf010102a50073031900d9001f01a50078;
        W[17]=0xffd9ff8300110013ffd2fffbff9a0028004cff920340003bff7bff29fedcff89;
        W[18]=0x0068fe9e017fff9500040371fbfa00ee00af00fe01eeffb3ff9f0015fecbfffe;
        W[19]=0x004c0022ffbefffaffd4003c009bffea0036fde1fd87000fffe000470068fffa;
        W[20]=0xff99fdd4ff3f001b004d06060165ff6ffe92f7f0003f001affeb00ac007cfffc;
        W[21]=0xfdd7fed2012c0008015600c3fdcb0019ff0d00a80437fff60018ffe8fed0ffff;
        W[22]=0xff9e049500beffaa0192e88ffe550104fedb160a009fffc7005ffcf0fdf10006;
        W[23]=0x0194057afd1bfff6ff42fc2a039bffeb006101b0fb5f0009fff7ffa800f90002;
        W[24]=0x003dfb8fff24003bfee610d4046fff1600cfee1cfb4b002fff2901c70382fffb;
        W[25]=0xff24f9c902090002003a03e2fe230000fff2fde40180fff6fff10058ff6bffff;
        W[26]=0xffd3005e0074ffec006cfc70fc6c0020ff88017a0463fff6fffc00f8fc56ffff;
    } }
    function save(Belief memory b) internal { unchecked {
        slots[0]=uint256(b.m);slots[1]=uint256(b.v);slots[2]=uint256(b.fb);slots[3]=uint256(b.fa);
        slots[4]=uint256(b.sp);slots[5]=uint256(b.r);slots[6]=b.t;slots[7]=(b.cnt<<1)|(b.sy?1:0);
    } }
    function afterInitialize(uint256 x,uint256 y) external override returns(uint256,uint256){ unchecked {
        Belief memory b; b.sp=int256(y*1e18/x);b.r=b.sp;b.m=ln(b.sp);
        b.fb=2823606104386589;b.fa=2823606104386589;save(b);return(uint256(b.fb),uint256(b.fa));
    } }
    function update(Belief memory b,int256 s,bool buy,uint256 dt) internal pure returns(bool arb){ unchecked {
        int256 lo;int256 hi;int256 m;int256 v;int256 lp;int256 la;int256 Z;int256 mt;int256 vt;int256 d;int256 pi;
        lo=ln(mul(b.sp,U-b.fb));hi=ln(divi(b.sp,U-b.fa));m=b.m;v=b.v;
        uint256 n=dt-1;if(n>12)n=12;
        for(uint256 i=0;i<n;i++){v+=893025000000;(,m,v)=trunc(m,v,lo,hi);}
        v+=893025000000;
        lp=ln(buy?mul(s,U-b.fb):divi(s,U-b.fa));
        int256 sd=root(v);la=divi(pdf(divi(lp-m,sd)),sd);
        (Z,mt,vt)=truncSD(m,v,lo,hi,sd);
        d=ab(ln(divi(s,b.sp)));if(d<1e9)d=1e9;
        pi=pdf(divi(ln(d)+6801595083153746944,1176000000000000000));
        int256 lr=divi(mul(mul(Z,260000000000000000),pi),mul(d,1176000000000000000));
        pi=la+lr>0?divi(la,la+lr):int256(0);arb=pi>U/2;
        if(pi>5e15&&pi<995e15){
            b.m=mul(pi,lp)+mul(U-pi,mt);
            b.v=mul(pi,mul(lp-b.m,lp-b.m))+mul(U-pi,vt+mul(mt-b.m,mt-b.m));
        }else if(arb){b.m=lp;b.v=0;}else{b.m=mt;b.v=vt;}
        int256 pe=exp(b.m);b.r=clip(b.r,mul(pe,997e15),divi(pe,997e15));b.cnt=0;b.sy=false;
    } }
    function afterSwap(TradeInfo calldata t) external override returns(uint256,uint256){ unchecked {
        Belief memory b=Belief(int256(slots[0]),int256(slots[1]),int256(slots[2]),int256(slots[3]),int256(slots[4]),int256(slots[5]),slots[6],slots[7]>>1,(slots[7]&1)==1);
        int256 s=int256(t.reserveY*1e18/t.reserveX);bool arb;
        if(t.timestamp>b.t)arb=update(b,s,t.isBuy,t.timestamp-b.t);
        uint256 ph;
        if(!arb){
            int256 c=t.isBuy?divi(mul(s,U-b.fb),997e15):divi(mul(s,997e15),U-b.fa);
            if(t.isBuy?c<b.r:c>b.r){b.r=c;b.sy=true;}
            b.cnt++;ph=b.cnt==1?1:2;
        }
        int256 w=(ln(s)-b.m)*10000;int256 r=(ln(b.r)-b.m)*10000;
        (b.fb,b.fa)=fees(w,r,ph,b.sy,root(b.v)*10000);b.sp=s;b.t=t.timestamp;save(b);
        return(uint256(b.fb),uint256(b.fa));
    } }
}
```

## Harness notes

check.py and replay.py are trim-1's with the reference swapped to `real.RealPolicy(T, {'soft': 1.0, 'qm': <decoded json>})` (sys.path gets Taylor's solved directory; T = pol_mid.json); engine.py is trim-1's with the seed assertion moved to 9534000..9534128 and `port` pointing at this Strategy.sol. Commands: `. scratch/env.sh && timeout 470 python scratch/replay.py scratch/Strategy.sol 70000 70025` (and 70025 70050); `timeout 470 python scratch/engine.py KIND START END` for KIND in port, t03, IB and START in 9534000, 9534032, 9534064, 9534096; `amm-match validate scratch/Strategy.sol`. No command ran longer than eight minutes (longest: a Gate F chunk, about 2.5 minutes).