{
 "mechanism": "Expected-value fee: per side choose the fee maximizing E[retail edge]-E[arb loss] using the simulator's exact two-AMM retail split against the 30bps normalizer and the arbitrage rule (arb if fair beyond spot/gamma), with fair ~ N(pHat, sigma^2(1+1/lambda)) offset by spot-pHat.",
 "approach": "Derived from amm_sim_rs router.rs/arbitrageur.rs: linearizing the split, our retail share moves reserveY/4 Y per unit of (fee gap to 30bps + log mispricing); arb loss ~ (reserveY/4)E[(u-f)+^2]. Numerically (lognormal size mean 20, sigma 1.2, normalizer spot uniform within +-30bps) the per-side optimum is f* ~ 50.5bps + m (m = expected mispricing of that side, slope 1) + ~430*var; the arb term is negligible at sigma~0.1%/step because retail share sensitivity dominates. Closed form tried: center = fNorm/2 + EV_Q*E[size]/reserveY (+ variance term) per side shifted by spot-pHat. Replacing fMid/skew outright loses heavily: the variance term and raw gap shift feed back through sigmaHat/pHat gate (fees lock at 0/1000bps); without them -14.8 (30bps center). Partial blends toward the EV center lose monotonically in weight; best tested is a 5% blend (frozen) which is still negative. Conclusion: the static-routing EV model does not capture what NP2's adaptive fMid captures; mechanism does not help.",
 "delta": "Vs NP2 (51f1c1a9) only: add constants EV_HALF_NORM=15*BPS, EV_Q=3*WAD/8, EV_MEAN_SIZE=20*WAD, EV_W=WAD/20 (also EV_GAP_COEF=0, EV_GAP_CAP=20*BPS, unused). Directly before `uint256 dirDev; bool sellPressure;` insert: `fMid = wmul(fMid, WAD - EV_W) + wmul(EV_HALF_NORM + wmul(EV_Q, wdiv(EV_MEAN_SIZE, trade.reserveY)), EV_W);` i.e. fMid is blended 5% toward the EV-model center fNorm/2 + 0.375*E[size]/reserveY (~22.5bps at reserveY=1e4). dirState skew, stale shift, trade boost, tail compression and all other code are unchanged.",
 "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 PHAT_ALPHA_SAME = 120000000000000000;\n    uint256 constant REVERT_U = 0;\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    uint256 constant EV_HALF_NORM = 15 * BPS;\n    uint256 constant EV_Q = 3 * WAD / 8;\n    uint256 constant EV_MEAN_SIZE = 20 * WAD;\n    uint256 constant EV_GAP_COEF = 0;\n    uint256 constant EV_GAP_CAP = 20 * BPS;\n    uint256 constant EV_W = WAD / 20;\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; slots[11] = 0; slots[12] = 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 prePhat = pHat;\n            bool secondPrint = !firstInStep && stepTradeCount == 1;\n            bool sameDir = false; bool oppDir = false;\n            if (secondPrint) {\n                bool up2 = pImplied >= pHat;\n                uint256 d1 = slots[12];\n                if (d1 != 0) { sameDir = (d1 == 1) == up2; oppDir = !sameDir; }\n                if (sameDir) alpha = PHAT_ALPHA_SAME;\n            }\n            if (firstInStep) { slots[11] = pHat; slots[12] = pImplied >= pHat ? 1 : 2; }\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 (oppDir && REVERT_U > 0) {\n                uint256 pre = slots[11];\n                if (pre > pHat) pHat = pHat + wmul(pre - pHat, REVERT_U);\n                else pHat = pHat - wmul(pHat - pre, REVERT_U);\n            }\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        // Interior retail routing equalizes fee-adjusted marginal prices.\n        // Subsequent trades in a step cannot be the initial arbitrage trade.\n        if (!firstInStep && feeUsed > 30 * BPS && gamma > 0) {\n            uint256 normalSpot = trade.isBuy\n                ? wdiv(wmul(spot, gamma), WAD - 30 * BPS)\n                : wdiv(wmul(spot, WAD - 30 * BPS), gamma);\n            pHat = wmul(pHat, 90 * WAD / 100) + wmul(normalSpot, 10 * WAD / 100);\n        }\n\n        uint256 tradeRatio = trade.reserveY > 0 ? wdiv(trade.amountY, trade.reserveY) : 0;\n        if (tradeRatio > TRADE_RATIO_CAP) tradeRatio = TRADE_RATIO_CAP;\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 = BASE_FEE + wmul(SIGMA_COEF, sigmaHat) + wmul(LAMBDA_COEF, lambdaHat) + wmul(FLOW_SIZE_COEF, flowSize);\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        // Expected-value fee center: fNorm/2 + E[size]-term/reserveY (E[retail edge]-E[arb loss] optimum).\n        fMid = wmul(fMid, WAD - EV_W) + wmul(EV_HALF_NORM + wmul(EV_Q, wdiv(EV_MEAN_SIZE, trade.reserveY)), EV_W);\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    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": "e1745bc56dee6c4fde2b3f368a7b3a51479807c4496b49b707be0d4a05c3de7e",
 "validate_ok": true,
 "max_gas": 132763,
 "max_gas_note": "smallest GAS_LIMIT_TRADE at which afterSwap succeeds (pyrevm message_call, includes intrinsic), max over calls 1,2,3,6,15,40 of a synthetic trade sequence; NP2 same method 132444",
 "np2_check": {
  "sha256": "51f1c1a9d4999ee460eb8bbc66880da8074b173399a4f5d71f96671482e18ac0",
  "seeds": "0..999",
  "n": 1000,
  "mean": 526.115572,
  "ok": true
 },
 "tune": [
  {
   "variant": "A full replace fMid/skew: center=15bps+1.775*20/reserveY+430*sig^2(1+1/lam), side shift = |spot-pHat|/pHat (coef 1), stale kept",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -454.5014,
   "se": 3.3291
  },
  {
   "variant": "B as A, no side shift",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -454.7931,
   "se": 3.3295
  },
  {
   "variant": "C as A, NP2 stale shift removed",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -454.7562,
   "se": 3.3334
  },
  {
   "variant": "D 50/50 blend of NP2 fMid/skew with A",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -453.7465,
   "se": 3.3396
  },
  {
   "variant": "E as A without variance term (center ~50.5bps)",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -314.5781,
   "se": 2.427
  },
  {
   "variant": "I30 full replace, center 30bps (EV_Q=0.75), no variance term, no side shift",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -14.7804,
   "se": 0.1842
  },
  {
   "variant": "J30 as I30 + side shift 0.1*|spot-pHat|/pHat",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -8.1045,
   "se": 0.091
  },
  {
   "variant": "K30 50/50 blend NP2 fMid/skew with I30",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -3.9902,
   "se": 0.0713
  },
  {
   "variant": "S0 fMid 25% toward 30bps EV center, dirState skew dropped",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -1.0357,
   "se": 0.0384
  },
  {
   "variant": "S6 fMid 25% toward 30bps EV center, skew kept",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -1.1969,
   "se": 0.0399
  },
  {
   "variant": "S7 fMid 10% toward 30bps EV center",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -0.258,
   "se": 0.0218
  },
  {
   "variant": "S8 fMid 10% toward 45bps EV center",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -0.5817,
   "se": 0.0282
  },
  {
   "variant": "S9 fMid 10% toward 22.5bps EV center",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -0.3716,
   "se": 0.0251
  },
  {
   "variant": "S10 fMid 10% toward 15bps EV center",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -0.6391,
   "se": 0.0302
  },
  {
   "variant": "S11 fMid 5% toward 22.5bps EV center (frozen)",
   "seeds": "10000..10499",
   "n": 500,
   "mean_diff": -0.1468,
   "se": 0.0168
  }
 ],
 "paired": {
  "seeds": "2030000..2031999",
  "n": 2000,
  "mean_diff": -0.1166,
  "se": 0.0095,
  "worst": -1.9167,
  "np2_mean": 522.698707,
  "variant_mean": 522.5821
 },
 "next": "Do not pursue the static EV center. If revisited, the EV model must be dynamic: include the post-trade price displacement that arbs later reclaim and the normalizer's band state; the raw spot-pHat shift needs a bias correction (single-seed screen, seed 10000, with a 0.25 shift: median bid 42 vs ask 16bps, i.e. spot stays above pHat persistently). Other single-seed screens (n=1, not listed in tune) were also below NP2..",
 "blocked": null
}