{
 "approach": "NP with fBase (BASE+SIGMA+LAMBDA+FLOW) replaced by a closed-form expected-edge maximiser over a fixed 12-fee grid (2,4,6,8,10,12,15,18,22,27,33,40 bps). Objective J(f)=flow*share(f)*f - 0.5*sigma^2*g(f/sigma); share = clamp(1/2+(30bps-f)/(KM*sizeRatio)); g(u)=1/(1+0.24u^2+5.36u^4) approximates E[(|Z|-u)+^2]; sizeRatio is a slow EMA of trade/reserveY kept in new slot 11 (the existing sizeHat decays 0.53/step so is unusable here). Chosen fee minus an empirical OFFS bps (floor 2bps). tox/stale/dir/tail layers unchanged. RESULT: NEGATIVE - frozen candidate is 3.39 below NP on fresh seeds; do not promote.",
 "tried": [
  {
   "variant": "c1 sizeHat, KM4, loss.5, off0",
   "seeds": "30000..30499",
   "n": 500,
   "mean_diff": -14.5351,
   "se": 0.1194
  },
  {
   "variant": "c3 sizeHat, KM1, off0",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -14.4858,
   "se": 0.0614
  },
  {
   "variant": "c4 slow size, KM4, off0",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -14.5062,
   "se": 0.0638
  },
  {
   "variant": "c5 slow size, KM1, off8",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -7.5123,
   "se": 0.0348
  },
  {
   "variant": "c6 slow size, KM1, off14",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -35.8952,
   "se": 0.0894
  },
  {
   "variant": "c7 slow size, KM1, off4 (frozen)",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -3.3866,
   "se": 0.0214
  },
  {
   "variant": "c8 slow size, KM1, off0",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -9.602,
   "se": 0.0423
  },
  {
   "variant": "c9 slow size, KM2, off13",
   "seeds": "30000..31999",
   "n": 2000,
   "mean_diff": -3.7136,
   "se": 0.0256
  }
 ],
 "next": "Closed form with my router-share/arb-loss model picks fees well above NP's fBase (loss term is negligible at f/sigma>2), so an empirical offset was needed and the optimum offset is sharp (4bps ok, 0 or 8 much worse). Better: measure the true share elasticity and arb loss in-sim (fit J on data) or use the maximiser only as a small additive correction to the NP fBase (blend w*fOpt+(1-w)*fBase, w~0.1-0.3), or add the loss term from the realized tox/stale layers.",
 "code": "// SPDX-License-Identifier: MIT\npragma solidity ^0.8.24;\n\nimport {AMMStrategyBase} from \"./AMMStrategyBase.sol\";\nimport {TradeInfo} from \"./IAMMStrategy.sol\";\n\ncontract Strategy is AMMStrategyBase {\n    uint256 constant ELAPSED_CAP = 8;\n    uint256 constant SIGNAL_THRESHOLD = WAD / 650;\n    uint256 constant DIR_DECAY = 750000000000000000;\n    uint256 constant ACT_DECAY = 530000000000000000;\n    uint256 constant SIZE_DECAY = 530000000000000000;\n    uint256 constant TOX_DECAY = 910000000000000000;\n    uint256 constant SIGMA_DECAY = 690000000000000000;\n    uint256 constant LAMBDA_DECAY = 990000000000000000;\n    uint256 constant SIZE_BLEND_DECAY = 828000000000000000;\n    uint256 constant TOX_BLEND_DECAY = 51000000000000000;\n    uint256 constant ACT_BLEND_DECAY = 985000000000000000;\n    uint256 constant PHAT_ALPHA = 270000000000000000;\n    uint256 constant PHAT_ALPHA_RETAIL = 60000000000000000;\n    uint256 constant DIR_IMPACT_MULT = 4;\n    uint256 constant GATE_SIGMA_MULT = 7200000000000000000;\n    uint256 constant MIN_GATE = 30000000000000000;\n    uint256 constant TRADE_TOX_BOOST = 3500 * BPS;\n    uint256 constant STALE_ATTRACT_FRAC = 1050000000000000000;\n    uint256 constant RET_CAP = WAD / 35;\n    uint256 constant TOX_CAP = WAD / 5;\n    uint256 constant TRADE_RATIO_CAP = WAD / 5;\n    uint256 constant LAMBDA_CAP = 1820000000000000000;\n    uint256 constant STEP_COUNT_CAP = 64;\n    uint256 constant BASE_FEE = 4 * BPS;\n    uint256 constant SIGMA_COEF = 200000000000000000;\n    uint256 constant LAMBDA_COEF = 12 * BPS;\n    uint256 constant FLOW_SIZE_COEF = 4842 * BPS;\n    uint256 constant TOX_COEF = 300 * BPS;\n    uint256 constant TOX_QUAD_COEF = 15000 * BPS;\n    uint256 constant ACT_COEF = 91843 * BPS;\n    uint256 constant DIR_COEF = 20 * BPS;\n    uint256 constant DIR_TOX_COEF = 100 * BPS;\n    uint256 constant STALE_DIR_COEF = 6850 * BPS;\n    uint256 constant SIGMA_TOX_COEF = 500 * BPS;\n    uint256 constant TAIL_KNEE = 600 * BPS;\n    uint256 constant TAIL_SLOPE_PROTECT = 930000000000000000;\n    uint256 constant TAIL_SLOPE_ATTRACT = 850000000000000000;\n\n    function afterInitialize(uint256 initialX, uint256 initialY) external override returns (uint256, uint256) {\n        slots[0] = BASE_FEE; slots[1] = BASE_FEE; slots[2] = 0; slots[3] = WAD; slots[4] = 0;\n        slots[5] = initialX > 0 ? wdiv(initialY, initialX) : 100 * WAD;\n        slots[6] = 950000000000000; slots[7] = 800000000000000000; slots[8] = 2000000000000000;\n        slots[9] = 0; slots[10] = 0;\n        return (BASE_FEE, BASE_FEE);\n    }\n\n    function afterSwap(TradeInfo calldata trade) external override returns (uint256, uint256) {\n        uint256 prevBidFee = slots[0]; uint256 prevAskFee = slots[1]; uint256 lastTs = slots[2];\n        uint256 dirState = slots[3]; uint256 actEma = slots[4]; uint256 pHat = slots[5];\n        uint256 sigmaHat = slots[6]; uint256 lambdaHat = slots[7]; uint256 sizeHat = slots[8];\n        uint256 toxEma = slots[9]; uint256 stepTradeCount = slots[10];\n\n        bool isNewStep = trade.timestamp > lastTs;\n        if (isNewStep) {\n            uint256 elapsedRaw = trade.timestamp - lastTs;\n            uint256 elapsed = elapsedRaw > ELAPSED_CAP ? ELAPSED_CAP : elapsedRaw;\n            dirState = _decayCentered(dirState, DIR_DECAY, elapsed);\n            actEma = wmul(actEma, _powWad(ACT_DECAY, elapsed));\n            sizeHat = wmul(sizeHat, _powWad(SIZE_DECAY, elapsed));\n            toxEma = wmul(toxEma, _powWad(TOX_DECAY, elapsed));\n            if (stepTradeCount > 0 && elapsedRaw > 0) {\n                uint256 lambdaInst = (stepTradeCount * WAD) / elapsedRaw;\n                if (lambdaInst > LAMBDA_CAP) lambdaInst = LAMBDA_CAP;\n                lambdaHat = wmul(lambdaHat, LAMBDA_DECAY) + wmul(lambdaInst, WAD - LAMBDA_DECAY);\n            }\n            stepTradeCount = 0;\n        }\n\n        bool firstInStep = stepTradeCount == 0;\n        uint256 spot = trade.reserveX > 0 ? wdiv(trade.reserveY, trade.reserveX) : pHat;\n        if (pHat == 0) pHat = spot;\n\n        uint256 feeUsed = trade.isBuy ? prevBidFee : prevAskFee;\n        uint256 gamma = feeUsed < WAD ? WAD - feeUsed : 0;\n        uint256 pImplied;\n        if (gamma == 0) { pImplied = spot; }\n        else { pImplied = trade.isBuy ? wmul(spot, gamma) : wdiv(spot, gamma); }\n\n        {\n            uint256 ret = pHat > 0 ? wdiv(absDiff(pImplied, pHat), pHat) : 0;\n            uint256 alpha = firstInStep ? PHAT_ALPHA : PHAT_ALPHA_RETAIL;\n            uint256 adaptiveGate = wmul(sigmaHat, GATE_SIGMA_MULT);\n            if (adaptiveGate < MIN_GATE) adaptiveGate = MIN_GATE;\n            if (ret <= adaptiveGate) { pHat = wmul(pHat, WAD - alpha) + wmul(pImplied, alpha); }\n            if (firstInStep) {\n                if (ret > RET_CAP) ret = RET_CAP;\n                sigmaHat = wmul(sigmaHat, SIGMA_DECAY) + wmul(ret, WAD - SIGMA_DECAY);\n            }\n        }\n\n        if(!firstInStep&&feeUsed>30*BPS&&gamma>0){uint256 nS=trade.isBuy?wdiv(wmul(spot,gamma),WAD-30*BPS):wdiv(wmul(spot,WAD-30*BPS),gamma);pHat=wmul(pHat,9*WAD/10)+wmul(nS,WAD/10);}\n\n        uint256 tradeRatio = trade.reserveY > 0 ? wdiv(trade.amountY, trade.reserveY) : 0;\n        if (tradeRatio > TRADE_RATIO_CAP) tradeRatio = TRADE_RATIO_CAP;\n        uint256 slowSz = slots[11];\n        if (slowSz == 0) slowSz = 2000000000000000;\n        if (tradeRatio > SIGNAL_THRESHOLD) slowSz = wmul(slowSz, 980000000000000000) + wmul(tradeRatio, 20000000000000000);\n        slots[11] = slowSz;\n\n        if (tradeRatio > SIGNAL_THRESHOLD) {\n            uint256 push = tradeRatio * DIR_IMPACT_MULT;\n            if (push > WAD / 4) push = WAD / 4;\n            if (trade.isBuy) {\n                dirState = dirState + push;\n                if (dirState > 2 * WAD) dirState = 2 * WAD;\n            } else {\n                dirState = dirState > push ? dirState - push : 0;\n            }\n            actEma = wmul(actEma, ACT_BLEND_DECAY) + wmul(tradeRatio, WAD - ACT_BLEND_DECAY);\n            sizeHat = wmul(sizeHat, SIZE_BLEND_DECAY) + wmul(tradeRatio, WAD - SIZE_BLEND_DECAY);\n            if (sizeHat > WAD) sizeHat = WAD;\n        }\n\n        uint256 tox = pHat > 0 ? wdiv(absDiff(spot, pHat), pHat) : 0;\n        if (tox > TOX_CAP) tox = TOX_CAP;\n        toxEma = wmul(toxEma, TOX_BLEND_DECAY) + wmul(tox, WAD - TOX_BLEND_DECAY);\n        uint256 toxSignal = toxEma;\n\n        stepTradeCount = stepTradeCount + 1;\n        if (stepTradeCount > STEP_COUNT_CAP) stepTradeCount = STEP_COUNT_CAP;\n\n        uint256 flowSize = wmul(lambdaHat, sizeHat);\n        uint256 fBase = _baseFee(sigmaHat, slowSz, wmul(lambdaHat, slowSz));\n        uint256 fMid = fBase + wmul(TOX_COEF, toxSignal) + wmul(TOX_QUAD_COEF, wmul(toxSignal, toxSignal)) + wmul(ACT_COEF, actEma);\n        fMid = fMid + wmul(SIGMA_TOX_COEF, wmul(sigmaHat, toxSignal));\n\n        uint256 dirDev; bool sellPressure;\n        if (dirState >= WAD) { dirDev = dirState - WAD; sellPressure = true; }\n        else { dirDev = WAD - dirState; sellPressure = false; }\n\n        uint256 skew = wmul(DIR_COEF, dirDev) + wmul(DIR_TOX_COEF, wmul(dirDev, toxSignal));\n        uint256 bidFee; uint256 askFee;\n        if (sellPressure) {\n            bidFee = fMid + skew;\n            askFee = fMid > skew ? fMid - skew : 0;\n        } else {\n            askFee = fMid + skew;\n            bidFee = fMid > skew ? fMid - skew : 0;\n        }\n\n        {\n            uint256 staleShift = wmul(STALE_DIR_COEF, toxSignal);\n            uint256 attractShift = wmul(staleShift, STALE_ATTRACT_FRAC);\n            if (spot >= pHat) {\n                bidFee = bidFee + staleShift;\n                askFee = askFee > attractShift ? askFee - attractShift : 0;\n            } else {\n                askFee = askFee + staleShift;\n                bidFee = bidFee > attractShift ? bidFee - attractShift : 0;\n            }\n        }\n\n        {\n            bool tradeAligned = (trade.isBuy && spot >= pHat) || (!trade.isBuy && spot < pHat);\n            if (tradeAligned) {\n                uint256 tradeBoost = wmul(TRADE_TOX_BOOST, tradeRatio);\n                if (trade.isBuy) { bidFee = bidFee + tradeBoost; }\n                else { askFee = askFee + tradeBoost; }\n            }\n        }\n\n        if (sellPressure) {\n            bidFee = clampFee(_compressTailWithSlope(bidFee, TAIL_SLOPE_PROTECT));\n            askFee = clampFee(_compressTailWithSlope(askFee, TAIL_SLOPE_ATTRACT));\n        } else {\n            askFee = clampFee(_compressTailWithSlope(askFee, TAIL_SLOPE_PROTECT));\n            bidFee = clampFee(_compressTailWithSlope(bidFee, TAIL_SLOPE_ATTRACT));\n        }\n\n        slots[0] = bidFee; slots[1] = askFee; slots[2] = trade.timestamp;\n        slots[3] = dirState; slots[4] = actEma; slots[5] = pHat;\n        slots[6] = sigmaHat; slots[7] = lambdaHat; slots[8] = sizeHat;\n        slots[9] = toxEma; slots[10] = stepTradeCount;\n\n        return (bidFee, askFee);\n    }\n\n\n    uint256 constant LOSS_C = 50 * WAD / 100;\n    uint256 constant KM = 1;\n    uint256 constant OFFS = 4;\n    bytes constant GRID = hex\"02040608\" hex\"0a0c0f12\" hex\"161b2128\";\n\n    function _baseFee(uint256 sigmaHat, uint256 sizeHat, uint256 flow) internal pure returns (uint256 best) {\n        uint256 kk = KM * sizeHat; if (kk < 2 * BPS) kk = 2 * BPS;\n        uint256 sg = sigmaHat < BPS ? BPS : sigmaHat;\n        uint256 s2 = wmul(wmul(sg, sg), LOSS_C);\n        int256 bestJ = type(int256).min;\n        uint256 fN = 30 * BPS;\n        for (uint256 i = 0; i < 12; i++) {\n            uint256 f = uint256(uint8(GRID[i])) * BPS;\n            uint256 sh;\n            if (f <= fN) { sh = WAD / 2 + wdiv(fN - f, kk); if (sh > WAD) sh = WAD; }\n            else { uint256 d = wdiv(f - fN, kk); sh = d >= WAD / 2 ? 0 : WAD / 2 - d; }\n            uint256 u2 = wmul(wdiv(f, sg), wdiv(f, sg));\n            uint256 den = WAD + wmul(240000000000000000, u2) + wmul(5360000000000000000, wmul(u2, u2));\n            int256 j = int256(wmul(flow, wmul(sh, f))) - int256(wmul(s2, wdiv(WAD, den)));\n            if (j > bestJ) { bestJ = j; best = f; }\n        }\n        best = best > OFFS * BPS + 2 * BPS ? best - OFFS * BPS : 2 * BPS;\n    }\n\n    function _compressTailWithSlope(uint256 fee, uint256 slope) internal pure returns (uint256) {\n        if (fee <= TAIL_KNEE) return fee;\n        return TAIL_KNEE + wmul(fee - TAIL_KNEE, slope);\n    }\n\n    function _powWad(uint256 factor, uint256 exp) internal pure returns (uint256 result) {\n        result = WAD;\n        while (exp > 0) {\n            if (exp & 1 == 1) result = wmul(result, factor);\n            factor = wmul(factor, factor);\n            exp >>= 1;\n        }\n    }\n\n    function _decayCentered(uint256 centered, uint256 decayFactor, uint256 elapsed) internal pure returns (uint256) {\n        uint256 mul = _powWad(decayFactor, elapsed);\n        if (centered >= WAD) { return WAD + wmul(centered - WAD, mul); }\n        uint256 below = wmul(WAD - centered, mul);\n        return below < WAD ? WAD - below : 0;\n    }\n\n    function getName() external pure override returns (string memory) { return \"no_cubic\"; }\n}\n",
 "code_sha256": "22a8614abe85a0aef4f042f8f1ad45ab65804228173de569046f189f9249642f",
 "delta": "vs NP: new slot 11 slowSz (EMA 0.98 of tradeRatio>SIGNAL_THRESHOLD, init 0.002); fBase=_baseFee(sigmaHat,slowSz,lambdaHat*slowSz); added constants LOSS_C=0.5, KM=1, OFFS=4, GRID and function _baseFee (12-iteration constant loop, no history loop). NP sha256 e93f6ea4b83679b7d7122532fde450fbf89a7f6e000fa79b91d0c194f88ae3f3.",
 "score": {
  "seed_start": 0,
  "n": 1000,
  "mean_edge": 522.6124281482237,
  "se": 2.7856040753597746,
  "note": "CH 525.5258821780834 and NP 526.0337302497852 reproduced exactly on seeds 0..999"
 },
 "paired": {
  "reference": "NP",
  "seed_start": 800000,
  "n": 2000,
  "n_covered": "800000..801999 in 4 chunks of 500",
  "mean_diff": -3.3923391342968547,
  "se": 0.023875275613050268,
  "worst": [
   {
    "seed": 800366,
    "diff": -17.146
   },
   {
    "seed": 800840,
    "diff": -8.253
   },
   {
    "seed": 800843,
    "diff": -7.088
   },
   {
    "seed": 801115,
    "diff": -7.021
   },
   {
    "seed": 801051,
    "diff": -6.87
   }
  ]
 },
 "seconds_per_simulation": 0.3,
 "validate_ok": true,
 "blocked": null,
 "citations": []
}