{
 "config": {
  "item": "nol-b",
  "batch": "ep9-loo",
  "candidate": "B + S + N (no L)",
  "seeds": "2980000..2981999",
  "n": 2000,
  "why_2000": "candidate probe 0.389 s/sim on first 20 seeds (>0.2 s/sim threshold), so the +1999 block was used",
  "pairing": "run_batch(X, VanillaStrategy, cfgs, 8) separately for X in {cand, NP2, B}, 20 seeds per call; default_rng(seed) draws size, rate, sigma; SimulationConfig seed=seed",
  "sim": "benedictbrady/amm-challenge @ 6b6921f8d463314aa7c96234e1be25a25245f704, Linux x86_64, setup script from brief completed; amm-match validate passed for cand and B"
 },
 "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 MIX_M = 625000000000000000;\n    uint256 constant MIX_E_SIXTEENTHS = 2;\n    uint256 constant ELAPSED_CAP = 8;\n    uint256 constant SILENCE_KEEP = 925000000000000000;\n    uint256 constant SILENCE_G_CAP = 4;\n    uint256 constant SILENCE_DAMP = 300000000000000000;\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\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[13] = 902500000000; slots[14] = slots[5]; slots[15] = 0; 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        uint256 sigmaForFast = sigmaHat;\n        uint256 fastHat = slots[14];\n        uint256 uncertainty = slots[13];\n        uint256 silent = trade.timestamp > lastTs ? trade.timestamp - lastTs : 0;\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            uncertainty += wmul(sigmaForFast, sigmaForFast) * elapsedRaw;\n            if (uncertainty > 100000000000000) uncertainty = 100000000000000;\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        // Silence: after g>=1 silent steps with no arb, fair price stayed inside\n        // our last no-arb band [s(1-bid), s/(1-ask)]; pull pHat to its centre.\n        uint256 silenceDamp = WAD;\n        if (firstInStep && silent >= 2 && slots[16] > 0) {\n            uint256 g = silent - 1;\n            if (g > SILENCE_G_CAP) g = SILENCE_G_CAP;\n            uint256 w = WAD - _powWad(SILENCE_KEEP, g);\n            uint256 sPrev = slots[16];\n            uint256 c = (wmul(sPrev, WAD - prevBidFee) + wdiv(sPrev, WAD - prevAskFee)) / 2;\n            if (c >= pHat) pHat = pHat + wmul(w, c - pHat);\n            else pHat = pHat - wmul(w, pHat - c);\n            silenceDamp = WAD - wmul(SILENCE_DAMP, w);\n        }\n\n        {\n            uint256 ret = pHat > 0 ? wdiv(absDiff(pImplied, pHat), pHat) : 0;\n            uint256 alpha = firstInStep ? PHAT_ALPHA : PHAT_ALPHA_RETAIL;\n            alpha = wmul(alpha, silenceDamp);\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            adaptiveGate = wmul(adaptiveGate, silenceDamp);\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        {\n            uint256 ret = fastHat > 0 ? wdiv(absDiff(pImplied, fastHat), fastHat) : 0;\n            uint256 alpha;\n            if (firstInStep) {\n                uint256 processSigma = sigmaForFast;\n                uint256 obsVar = uncertainty + wmul(processSigma, processSigma);\n                uint256 probArb;\n                uint256 z2 = wdiv(wmul(ret, ret), obsVar);\n                uint256 density = _powWad(wdiv(WAD, WAD + z2 / 16), 8);\n                uint256 sigmaObs = _isqrt(obsVar * WAD);\n                uint256 odds = wmul(density, wdiv(6700000000000000, sigmaObs));\n                bool arbFlag = ret <= wmul(sigmaForFast, 1220000000000000000);\n                uint256 prior = arbFlag ? 650000000000000000 : 400000000000000000;\n                uint256 tr = trade.reserveY > 0 ? wdiv(trade.amountY, trade.reserveY) : 0;\n                if (tr > 3000000000000000) prior = wmul(prior, 6 * WAD / 10);\n                if (silent > 1) prior = wmul(prior, 8 * WAD / 10);\n                odds = wmul(odds, wdiv(prior, WAD - prior));\n                probArb = wdiv(odds, WAD + odds);\n                alpha = wmul(probArb, 600000000000000000) + wmul(WAD - probArb, 80000000000000000);\n                slots[15] = pImplied >= fastHat ? 1 : 2;\n                uncertainty = wmul(uncertainty, WAD - alpha);\n            } else {\n                if ((slots[15] == 1) == (pImplied >= fastHat)) alpha = PHAT_ALPHA_RETAIL;\n            }\n            uint256 gate = wmul(sigmaHat, GATE_SIGMA_MULT);\n            if (gate < MIN_GATE) gate = MIN_GATE;\n            if (ret <= gate) fastHat = wmul(fastHat, WAD - alpha) + wmul(pImplied, alpha);\n            slots[13] = uncertainty;\n        }\n\n        if (!firstInStep && feeUsed > 30 * BPS && gamma > 0) {\n            uint256 normalObs = trade.isBuy\n                ? wdiv(wmul(spot, gamma), WAD - 30 * BPS)\n                : wdiv(wmul(spot, WAD - 30 * BPS), gamma);\n            if ((slots[15] == 1) == (normalObs >= fastHat)) {\n                fastHat = wmul(fastHat, 9 * WAD / 10) + wmul(normalObs, WAD / 10);\n            }\n        }\n        slots[14] = fastHat;\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        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 staleHat = wmul(pHat, 700000000000000000) + wmul(fastHat, 300000000000000000);\n            uint256 staleTox = wdiv(absDiff(spot, staleHat), staleHat);\n            if (staleTox > TOX_CAP) staleTox = TOX_CAP;\n            uint256 staleShift = wmul(STALE_DIR_COEF, staleTox);\n            uint256 attractShift = wmul(staleShift, STALE_ATTRACT_FRAC);\n            if (spot >= staleHat) {\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        // Mix the next retail quote with a defence against next-step arbitrage.\n        // The first observed print is our existing proxy for the step's arb;\n        // after it, k counts the subsequent observed retail prints.\n        {\n            uint256 q = _nextRetailProbability(lambdaHat, stepTradeCount - 1);\n            uint256 weight = _mixWeight(q);\n            uint256 defence = clampFee(_absLogRatio(spot, pHat) + wmul(MIX_M, sigmaHat));\n            if (spot >= pHat) {\n                bidFee = wmul(weight, bidFee) + wmul(WAD - weight, defence);\n            } else {\n                askFee = wmul(weight, askFee) + wmul(WAD - weight, defence);\n            }\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; slots[16] = spot;\n\n        return (bidFee, askFee);\n    }\n\n    // Poisson survival ratio P(N > k) / P(N >= k). Dividing both\n    // tails by P(N=k) avoids cancellation for large k. For k=0 this\n    // is exactly 1-exp(-lambda), up to fixed-point/series rounding.\n    function _nextRetailProbability(uint256 lambda, uint256 k) internal pure returns (uint256) {\n        uint256 term = WAD;\n        uint256 tail;\n        for (uint256 j = 1; j <= 24; ++j) {\n            term = wmul(term, lambda) / (k + j);\n            tail += term;\n            if (term == 0) break;\n        }\n        return wdiv(tail, WAD + tail);\n    }\n\n    // e is an exact multiple of 1/16. Successive square roots implement\n    // the fractional bits without an external fixed-point math library.\n    function _mixWeight(uint256 q) internal pure returns (uint256 weight) {\n        weight = _powWad(q, MIX_E_SIXTEENTHS / 16);\n        uint256 fraction = MIX_E_SIXTEENTHS % 16;\n        uint256 root = q;\n        for (uint256 bit = 8; bit > 0 && fraction > 0; bit >>= 1) {\n            root = _isqrt(root * WAD);\n            if ((fraction & bit) != 0) {\n                weight = wmul(weight, root);\n                fraction -= bit;\n            }\n        }\n    }\n\n    // 2*atanh((a-b)/(a+b)) = abs(ln(a/b)). Once z >= .05,\n    // the logarithm exceeds the 10% fee cap. In the remaining domain\n    // five positive terms give <1e-15 absolute truncation error.\n    function _absLogRatio(uint256 a, uint256 b) internal pure returns (uint256) {\n        if (a == b) return 0;\n        if (a == 0 || b == 0) return MAX_FEE;\n        uint256 z = wdiv(absDiff(a, b), a + b);\n        if (z >= WAD / 20) return MAX_FEE;\n        uint256 z2 = wmul(z, z);\n        uint256 power = z;\n        uint256 series = z;\n        for (uint256 j = 3; j <= 9; j += 2) {\n            power = wmul(power, z2);\n            series += power / j;\n        }\n        return 2 * series;\n    }\n\n    // Floor square root, value-identical to AMMStrategyBase.sqrt (Newton from\n    // any start >= isqrt(x) descends to isqrt(x)); a power-of-two start only\n    // saves gas, which the 250k afterSwap limit needs once N is stacked on B.\n    function _isqrt(uint256 x) internal pure returns (uint256 y) {\n        if (x == 0) return 0;\n        uint256 b = 1;\n        uint256 t = x;\n        while (t > 3) { t >>= 2; b <<= 1; }\n        y = b << 1;\n        if (y > x) y = x;\n        uint256 z = (x / y + y) / 2;\n        while (z < y) {\n            y = z;\n            z = (x / z + z) / 2;\n        }\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 \"nol_b\"; }\n}\n",
 "sha256": "930203f66137ac39fde8fc3545f6822c8cf74b2c935680dcc17851c92079ed8e",
 "b_sha256": "218426004abed3b6cb69f4ef133fea8e4789ddbeae37bd11e94e5ffc5f2a77cb",
 "np2_sha256": "51f1c1a9d4999ee460eb8bbc66880da8074b173399a4f5d71f96671482e18ac0",
 "paired_np2": {
  "n": 2000,
  "mean_diff": -92.35123790052272,
  "se": 4.2504250176890155,
  "worst": [
   {
    "seed": 2981839,
    "cand": 61.843078499278114,
    "np2": 779.9215735356358,
    "diff": -718.0784950363577
   },
   {
    "seed": 2981789,
    "cand": 64.78198974441229,
    "np2": 745.9738391258028,
    "diff": -681.1918493813905
   },
   {
    "seed": 2981807,
    "cand": 65.12488775330073,
    "np2": 737.1366892100934,
    "diff": -672.0118014567927
   },
   {
    "seed": 2981128,
    "cand": 77.59068294482755,
    "np2": 745.2034247080169,
    "diff": -667.6127417631893
   },
   {
    "seed": 2980459,
    "cand": 72.79874971913077,
    "np2": 729.4640906715439,
    "diff": -656.6653409524131
   }
  ]
 },
 "paired_b": {
  "n": 2000,
  "mean_diff": -92.84951066448808,
  "se": 4.2495846098953365
 },
 "paired_b_vs_np2": {
  "n": 2000,
  "mean_diff": 0.4982727639653548,
  "se": 0.01873129590323599
 },
 "means": {
  "cand": 431.5380495892046,
  "np2": 523.8892874897273,
  "b": 524.3875602536926
 },
 "seconds_per_sim": {
  "cand": 0.39098547184467314,
  "np2": 0.17013487613201142,
  "b": 0.23978533959388734
 },
 "gas": {
  "afterSwap_limit": 250000,
  "cand": {
   "max_successful_gas": 236395,
   "out_of_gas_calls": 6377412
  },
  "b": {
   "max_successful_gas": 242638,
   "out_of_gas_calls": 0
  },
  "np2": {
   "max_successful_gas": 196097,
   "out_of_gas_calls": 0
  },
  "note": "gas counted with a scratch-only counter added to the simulator's Rust EVM wrapper (call() Success gas_used max for 250k-limit calls, OutOfGas halts counted); simulation behavior unchanged. tx gas includes 21000 intrinsic + calldata."
 },
 "blocked": null,
 "notes": [
  "RESULT: the B+S+N stack as built is NOT a usable strategy. It exceeds the simulator's 250,000 afterSwap gas limit on a large share of calls (6377412 out-of-gas calls over 2000 sims). On OOG the engine keeps the previous fees and the call's storage writes are reverted, so the strategy's state freezes; that is why cand scores ~431.5 vs NP2 ~523.9. The paired numbers above measure that failure, not the mechanisms.",
  "Root cause: B alone already peaks at 242638 gas of 250,000 on this block (0 OOG); N's Poisson 24-term series, 3 square roots in _mixWeight and the atanh series add more gas than the ~7k headroom left. NP2 peaks at 196097.",
  "First build (literal B+S+N) failed validate: afterInitialize ran out of gas (250k) when S initialised slots[16]; removed that init, S is skipped until slots[16] holds a stored spot (first trade stores it).",
  "Gas mitigation tried (no constants changed): replaced AMMStrategyBase.sqrt with _isqrt, a floor Newton sqrt started from a power-of-two upper bound (same floor result; B with _isqrt gave bit-identical mean edge on seeds 10000..10007). That removed OOG on 4 tuning seeds 10000..10003 but not on the confirmation block. A further gas-equivalent rewrite (e.g. closed-form q for N, cheaper storage packing) was not attempted in the remaining time.",
  "S as implemented: slots[16]=post-trade spot s stored every afterSwap; on a step's first print with elapsed>=2, before the slow update: g=min(elapsed-1,4), w=1-0.925^g, c=(s*(1-prevBid)+s/(1-prevAsk))/2 with prevBid/prevAsk = slots[0]/slots[1] (the fees quoted after the previous trade), pHat+=w(c-pHat); that print's slow alpha (PHAT_ALPHA) and adaptiveGate multiplied by 1-0.30w. The fast estimator (B) is untouched by S.",
  "N copied verbatim from next-mix-a (MIX_M=0.625, MIX_E_SIXTEENTHS=2, block placed after the tail compression and before the slot writes, using the slow pHat; helpers _nextRetailProbability, _mixWeight, _absLogRatio).",
  "B+S without N (scratch BS.sol) did not OOG on seeds 10000..10003 (max gas 236,386) and scored 502.62 vs B 507.50 / NP2 507.25 on seeds 10000..10019 (20-seed check, not a confirmation). B+N without S (BN.sol) OOGs like cand. So the gas failure comes from N on top of B.",
  "The 20-seed probe on 2980000..2980019 ran inside a first attempt that timed out (10-min command limit) before writing results; the block was then run once in chunks (cand 700/700/600, NP2 2000, B 2000). No tuning was done on any seed of this block.",
  "Calibration: NP2 sha256 51f1c1a9... and B sha256 21842600... rebuilt and matched exactly. CH 0..999 / NP2 0..999 calibration was not re-run for lack of time; B's report gives NP2 526.1155722459862 on 0..999 with the same simulator commit."
 ]
}