{"item":"tail-b","blocked":null,"design_url":"https://api.imdnotdead.fun/artifacts/f0c84ebe66608025bb038a242030d6639f81ec87bf35146ee4495f572c7b74e7","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 // Size-tail residual: screen-fitted tables, signed tenths of a basis point.\n // Columns: spot/pEst -150..150 bps at 5 bps; rows: fair Y depth 7000..13000.\n uint256 constant TAIL_KAPPA = 0;\n uint256 constant TAIL_DMAX = 500000000000000;\n bytes constant TAIL_BID = hex\"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\";\n bytes constant TAIL_ASK = hex\"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\";\n\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\n function afterInitialize(uint256 initialX, uint256 initialY) external override returns (uint256, uint256) {\n slots[0] = BASE_FEE; slots[1] = BASE_FEE; slots[3] = WAD; \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]; \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 = uint64(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 // Process variance is propagated inside the censored update.\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 // S: infer a no-trade interval center from the last post-trade spot.\n uint256 silenceScale = WAD;\n if (firstInStep && silent > 1) {\n uint256 g = silent - 1;\n if (g > 1) g = 1;\n uint256 w = WAD - _powWad(800000000000000000, g);\n uint256 s = slots[2] >> 64;\n if (s == 0) s = pHat;\n uint256 c = (wmul(s, WAD - prevBidFee) + wdiv(s, WAD - prevAskFee)) / 2;\n if (c >= pHat) pHat += wmul(w, c - pHat);\n else pHat -= wmul(w, pHat - c);\n silenceScale = WAD - wmul(300000000000000000, 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 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[12] = pImplied >= pHat ? 1 : 2; }\n uint256 adaptiveGate = wmul(sigmaHat, GATE_SIGMA_MULT);\n if (adaptiveGate < MIN_GATE) adaptiveGate = MIN_GATE;\n alpha = wmul(alpha, silenceScale);\n adaptiveGate = wmul(adaptiveGate, silenceScale);\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 // Censored assumed-density posterior. No external or future inputs.\n (fastHat, uncertainty) = _posterior(PosteriorInput(\n fastHat, uncertainty, pImplied, spot, slots[2] >> 64,\n prevBidFee, prevAskFee, firstInStep, silent, trade.isBuy,\n trade.reserveY > 0 ? wdiv(trade.amountY, trade.reserveY) : 0\n ));\n slots[13] = uncertainty;\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 (!firstInStep && tradeBoost > 16 * BPS) tradeBoost = 16 * BPS;\n if (trade.isBuy) { bidFee = bidFee + tradeBoost; }\n else { askFee = askFee + tradeBoost; }\n }\n }\n\n {\n uint256 tailP = wmul(pHat,700000000000000000)+wmul(fastHat,300000000000000000);\n uint256 tailDepth = _isqrt(wmul(wmul(trade.reserveX,trade.reserveY),tailP)*WAD);\n bidFee = _tailAdd(bidFee,_tail(spot,tailP,tailDepth,bidFee,true));\n askFee = _tailAdd(askFee,_tail(spot,tailP,tailDepth,askFee,false));\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 if (firstInStep) {\n if (askFee > bidFee && bidFee > 30 * BPS) bidFee = 30 * BPS;\n else if (bidFee > askFee && askFee > 30 * BPS) askFee = 30 * BPS;\n }\n\n if (bidFee > askFee && askFee > 36 * BPS) askFee = 36 * BPS;\n else if (askFee > bidFee && bidFee > 36 * BPS) bidFee = 36 * BPS;\n\n slots[0] = bidFee; slots[1] = askFee; slots[2] = (spot << 64) | 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 struct PosteriorInput {\n uint256 mu; uint256 v; uint256 obs; uint256 spot; uint256 preSpot;\n uint256 bid; uint256 ask; bool first; uint256 silent; bool buy; uint256 size;\n }\n function _posterior(PosteriorInput memory x) internal pure returns (uint256, uint256) {\n if (x.mu == 0) x.mu = x.spot;\n if (x.v < 10000000000) x.v = 10000000000;\n uint256 lo = wmul(x.preSpot, WAD-x.bid);\n uint256 hi = wdiv(x.preSpot, WAD-x.ask);\n if (x.first) {\n if (x.silent>1 && x.preSpot>0) {\n uint256 elapsed=x.silent>8?8:x.silent;\n x.v += 902500000000*(elapsed-1);\n (x.mu,x.v)=_band(x.mu,x.v,lo,hi);\n }\n x.v+=902500000000;\n }\n uint256 mRetail = x.mu;\n uint256 vRetail = x.v;\n // A missing own-pool trade is a censored price observation.\n if (x.first && x.preSpot > 0) {\n (mRetail, vRetail) = _band(x.mu, x.v, lo, hi);\n }\n if (!x.first) {\n // A later print is retail. Only an interior route is an observation\n // of the other pool, with explicit observation uncertainty.\n uint256 fee = x.buy ? x.bid : x.ask;\n if (fee > 30 * BPS) {\n uint256 normal = x.buy ? wdiv(wmul(x.spot, WAD-fee), WAD-30*BPS)\n : wdiv(wmul(x.spot, WAD-30*BPS), WAD-fee);\n uint256 residual = wdiv(absDiff(normal, x.mu),x.mu);\n uint256 noise = 9000000000000 + wmul(residual,residual);\n uint256 k = wdiv(x.v,x.v+noise);\n return (_blend(x.mu,normal,k),wmul(x.v,WAD-k));\n }\n return (x.mu,x.v);\n }\n uint256 e = wdiv(absDiff(x.obs,x.mu),x.mu);\n uint256 totalV = x.v + 22500000000;\n uint256 z = wdiv(wmul(e,e),totalV);\n uint256 density = _powWad(wdiv(WAD,WAD+z/8),4);\n uint256 odds = wmul(density,wdiv(4000000000000000,_isqrt(totalV*WAD)));\n if (x.size > 3000000000000000) odds = wmul(odds,4*WAD/10);\n if (x.silent > 1) odds = wmul(odds,8*WAD/10);\n uint256 sd = _isqrt(x.v*WAD);\n uint256 width = wdiv(hi-lo,x.mu);\n uint256 distance = x.muhi?x.mu-hi:0);\n uint256 zBand = wdiv(wdiv(distance,x.mu),sd);\n uint256 mass = wmul(wdiv(width,width+2*sd),wdiv(WAD,WAD+wmul(zBand,zBand)));\n if (mass < WAD/100) mass = WAD/100;\n odds = wdiv(odds,mass);\n uint256 prob = wdiv(odds,WAD+odds);\n uint256 kArb = wdiv(x.v,totalV);\n uint256 mArb = _blend(x.mu,x.obs,kArb);\n uint256 mean = _blend(mRetail,mArb,prob);\n uint256 between = wdiv(absDiff(mArb,mRetail),x.mu);\n uint256 variance = wmul(prob,wmul(x.v,WAD-kArb)) + wmul(WAD-prob,vRetail)\n + wmul(wmul(prob,WAD-prob),wmul(between,between));\n if (variance > 100000000000000) variance = 100000000000000;\n return (mean,variance);\n }\n function _blend(uint256 a,uint256 b,uint256 w) internal pure returns(uint256) {\n return b>=a ? a+wmul(b-a,w) : a-wmul(a-b,w);\n }\n function _band(uint256 mu,uint256 v,uint256 lo,uint256 hi) internal pure returns(uint256,uint256) {\n // Soft interval likelihood avoids discontinuities at quadrature nodes.\n uint256 sd=wmul(mu,_isqrt(v*WAD));\n uint256[3] memory acc;\n for(uint256 j=0;j<7;j++) {\n uint256 p=j<3 ? (mu>(3-j)*sd?mu-(3-j)*sd:1) : mu+(j-3)*sd;\n uint256 w=j==3?72:(j==2||j==4?45:(j==1||j==5?10:1));\n uint256 distance=phi?p-hi:0);\n uint256 z=sd>0?wdiv(distance,sd):0;\n uint256 likelihood=wdiv(WAD,WAD+4*wmul(z,z));\n w=wmul(w*WAD,likelihood);\n acc[0]+=w;acc[1]+=wmul(w,p);\n uint256 d=wdiv(absDiff(p,mu),mu);\n acc[2]+=wmul(w,wmul(d,d));\n }\n uint256 m=wdiv(acc[1],acc[0]);\n uint256 d=wdiv(absDiff(m,mu),mu);\n uint256 second=wdiv(acc[2],acc[0]);\n return(m,second>wmul(d,d)?second-wmul(d,d):10000000000);\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 // Floor square root: bit-length seed and seven Newton steps (all uint256).\n function _isqrt(uint256 x) internal pure returns (uint256 r) {\n if (x == 0) return 0;\n unchecked {\n uint256 y = x;\n r = 1;\n if (y >> 128 != 0) { y >>= 128; r <<= 64; }\n if (y >> 64 != 0) { y >>= 64; r <<= 32; }\n if (y >> 32 != 0) { y >>= 32; r <<= 16; }\n if (y >> 16 != 0) { y >>= 16; r <<= 8; }\n if (y >> 8 != 0) { y >>= 8; r <<= 4; }\n if (y >> 4 != 0) { y >>= 4; r <<= 2; }\n if (y >> 2 != 0) { r <<= 1; }\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n r = (r + x / r) >> 1;\n uint256 t = x / r;\n return r < t ? r : t;\n }\n }\n\n\n function _tailAt(bytes memory t, uint256 row, uint256 col, uint256 feeCol) internal pure returns(int256) {\n uint256 i = (row * 61 + col) * 1 + feeCol;\n uint256 v = uint256(uint8(t[2*i])) * 256 + uint256(uint8(t[2*i+1]));\n return v >= 32768 ? int256(v) - 65536 : int256(v);\n }\n function _tail(uint256 spot, uint256 p, uint256 depth, uint256 fee, bool bid) internal pure returns(int256) {\n if (TAIL_KAPPA == 0 || p == 0) return 0;\n int256 coord = (int256(spot) - int256(p)) * 10000 / int256(p) + 150;\n if (coord < 0) coord = 0;\n if (coord > 300) coord = 300;\n uint256 c = uint256(coord) / 5;\n uint256 cw = uint256(coord) % 5;\n if (c == 60) { c = 59; cw = 5; }\n uint256 dv = depth / WAD;\n if (dv < 7000) dv = 7000;\n if (dv > 13000) dv = 13000;\n uint256 r = (dv - 7000) / 1000;\n uint256 rw = (dv - 7000) % 1000;\n if (r == 6) { r = 5; rw = 1000; }\n bytes memory t = bid ? TAIL_BID : TAIL_ASK;\n uint256 fc = 0;\n int256 a = (_tailAt(t,r,c,fc)*int256(5-cw)+_tailAt(t,r,c+1,fc)*int256(cw))/5;\n int256 b = (_tailAt(t,r+1,c,fc)*int256(5-cw)+_tailAt(t,r+1,c+1,fc)*int256(cw))/5;\n int256 d = (a*int256(1000-rw)+b*int256(rw))/1000;\n d = d * int256(BPS/10) * int256(TAIL_KAPPA) / int256(WAD);\n if (d > int256(TAIL_DMAX)) d = int256(TAIL_DMAX);\n if (d < -int256(TAIL_DMAX)) d = -int256(TAIL_DMAX);\n return d;\n }\n function _tailAdd(uint256 f, int256 d) internal pure returns(uint256) {\n if (d < 0) return f > uint256(-d) ? f - uint256(-d) : 0;\n return f + uint256(d);\n }\n\n function getName() external pure override returns (string memory) { return \"tail_b_identity\"; }\n}\n","sha256":"c4a9943e43670402fb004e49a5f12c491f55980e99b0ce0b0295ced7329a68c3","family":"S0 identity","params":{"kappa":0,"DMAX_bps":5,"caps":"V10 30/36 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unchanged V10 on screen seeds 790000..790255 with the pinned harness; retain every 25th post-step [fair_price, own spot, normalizer spot, bid, ask]. Quantiles of log(normalizer spot/fair) define u nodes. Median own bid/ask defines the opposite fee. M and sigma use config draws from 790000..793999 only. No re-check or held-out data fit any table.","onchain":"Causal TradeInfo and unchanged V10 state only. pEst=0.7*pHat+0.3*fastHat; fair Y-depth=sqrt(reserveX*reserveY*pEst). Bilinear lookup in depth and signed spot/pEst offset. S1 marginalizes three u nodes offline; S1-lite uses u=0. S3 interpolates in actual fee, with nonnegative whole-bps table entries. Delta is bounded before the original tail compression and caps; no additional storage.","harness_python":"\"\"\"Run from the pinned amm-challenge checkout, with the brief's environment active.\npython harness.py --start 5000000 --n 2000 --out measured.json candidate.sol STACK.sol\nDo not use confirmation seeds to choose or change a strategy.\n\"\"\"\nimport argparse,hashlib,json,math\nfrom pathlib import Path\nimport numpy as np\nimport amm_sim_rs as sim\nfrom amm_competition.evm.compiler import SolidityCompiler\n\n\ndef configs(start,n):\n configs=[]\n for seed in range(start,start+n):\n rng=np.random.default_rng(seed)\n size=rng.uniform(19.,21.)\n rate=rng.uniform(.6,1.)\n sigma=rng.uniform(.000882,.001008)\n configs.append(sim.SimulationConfig(n_steps=10000,initial_price=100.,\n initial_x=100.,initial_y=10000.,gbm_mu=0.,gbm_sigma=sigma,gbm_dt=1.,\n retail_arrival_rate=rate,retail_mean_size=size,retail_size_sigma=1.2,\n retail_buy_prob=.5,seed=seed))\n return configs\n\n\ndef decomposition(r,initial_fee=.0004):\n # Fees are outside reserves, so k stays fixed up to floating-point roundoff.\n # The arb acts before retail, using the preceding step's final spot and fees.\n steps=r.steps\n fair=np.array([s.fair_price for s in steps])\n spots=np.array([r.initial_fair_price]+[s.spot_prices['submission'] for s in steps[:-1]])\n bids=np.array([initial_fee]+[s.fees['submission'][0] for s in steps[:-1]])\n asks=np.array([initial_fee]+[s.fees['submission'][1] for s in steps[:-1]])\n x0,y0=r.initial_reserves['submission'];k=x0*y0\n x=np.sqrt(k/spots);y=np.sqrt(k*spots)\n ga=1-asks;gb=1-bids\n out=np.minimum(np.maximum(x-np.sqrt(k/(ga*fair)),0),.99*x)\n paid=(k/(x-out)-y)/ga\n buyprofit=np.where(out>0,np.maximum(out*fair-paid,0),0)\n gross=np.maximum((np.sqrt(k*gb/fair)-x)/gb,0)\n received=y-k/(x+gb*gross)\n sellprofit=np.where(gross>0,np.maximum(received-gross*fair,0),0)\n arb=-float(np.sum(np.where(spots=5000000 used, per this item assignment. No board claim.","Normalizer end-step quantiles are an approximate offline prior. Fair value, the opposite fee and next-arb charge are model approximations; future retail reversals are omitted. Tables clip the state/depth domain.","141-point lognormal quadrature, 1-bps argmax grid; representative 561-point checks differed by at most 1 bp in the residual. S3 uses whole-bps stored premiums and fee interpolation.","Identity/cap-only selections have trivial zero-strength matched ablations; any separate active diagnostic is clearly labeled."],"identity_won_nonzero_probe":{"freeze":{"arm":{"name":"lite_l0","family":"S1-lite","lam":0,"kappa":0.5,"dmax":5,"cap":0,"shuffle":false,"sha256":"984bc7c6ac52467cfed30fed865e047ece5aa5d283fbd767b1f9dbceb6f8cf16"},"ablation":{"name":"probe_F2","family":"F2-lite","lam":0,"kappa":0.5,"dmax":5,"cap":0,"shuffle":false,"sha256":"cbff99a0c0133fa09b8ad6c82ad60b957c7b1436148636cfac8f83d19b1a1df9"},"shuffled":{"name":"probe_F5","family":"S1-lite","lam":0,"kappa":0.5,"dmax":5,"cap":0,"shuffle":true,"sha256":"892ae279d90fc6075693019ae8ebb65a89e86cba87f28139e35eb015b30d6a18"}},"held_out":{"lite_l0":{"name":"lite_l0","sha256":"984bc7c6ac52467cfed30fed865e047ece5aa5d283fbd767b1f9dbceb6f8cf16","n":512,"mean_diff":-0.16149062615803778,"se":0.020287586181242993,"worst":[{"seed":795145,"d":-1.593401472387768},{"seed":795278,"d":-1.4805785608723454},{"seed":795052,"d":-1.3423491582939278},{"seed":795461,"d":-1.300612502112017},{"seed":795048,"d":-1.2371578787062845}],"retail_edge_delta":-0.8467929469069203,"arb_edge_delta":0.6853023207488843,"gas":{"max_ok":206258,"init_max_ok":220978,"oog":0}},"probe_F2":{"name":"probe_F2","sha256":"cbff99a0c0133fa09b8ad6c82ad60b957c7b1436148636cfac8f83d19b1a1df9","n":512,"mean_diff":-3.219862213129404,"se":0.04928605761089935,"worst":[{"seed":795496,"d":-8.509582558233149},{"seed":795098,"d":-6.476392953165487},{"seed":795321,"d":-5.861274573283367},{"seed":795459,"d":-5.7765283145214426},{"seed":795202,"d":-5.73697766920111}],"retail_edge_delta":2.0452373821523455,"arb_edge_delta":-5.26509959528175,"gas":{"max_ok":205534,"init_max_ok":220978,"oog":0}},"probe_F5":{"name":"probe_F5","sha256":"892ae279d90fc6075693019ae8ebb65a89e86cba87f28139e35eb015b30d6a18","n":512,"mean_diff":-0.205552129460561,"se":0.016303774211991234,"worst":[{"seed":795278,"d":-1.3897957860313},{"seed":795464,"d":-1.3429895347486536},{"seed":795145,"d":-1.1435788785113346},{"seed":795066,"d":-1.1096346987792458},{"seed":795280,"d":-1.1000980606494295}],"retail_edge_delta":-0.7815656303847561,"arb_edge_delta":0.5760135009241962,"gas":{"max_ok":206216,"init_max_ok":220978,"oog":0}}},"frozen_vs_ablations":{"probe_F2":{"paired":{"n":512,"mean_diff":3.058371586971366,"se":0.057327892442833456,"worst":[{"seed":795008,"d":-0.5746288108235831},{"seed":795500,"d":-0.5520816984837893},{"seed":795048,"d":-0.3969100830934167},{"seed":795145,"d":-0.3718235076756855},{"seed":795216,"d":-0.2367893829205059}]},"retail_edge_delta":{"mean":-2.892030329059266,"se":0.06253253965787223},"arb_edge_delta":{"mean":5.950401916030634,"se":0.021117160433582624},"reconciliation_max_abs":1.0658141036401503e-13},"probe_F5":{"paired":{"n":512,"mean_diff":0.04406150330252323,"se":0.017641526569757274,"worst":[{"seed":795448,"d":-1.6669701035758635},{"seed":795013,"d":-0.9573321016798673},{"seed":795456,"d":-0.9028236645641528},{"seed":795192,"d":-0.8995416016888385},{"seed":795337,"d":-0.8625626429389968}]},"retail_edge_delta":{"mean":-0.06522731652216418,"se":0.016435137526569988},"arb_edge_delta":{"mean":0.10928881982468801,"se":0.007154823287346858},"reconciliation_max_abs":1.1368683772161603e-13}}},"gas_scope":"Submission afterSwap/afterInitialize maxima over the frozen hash on scored screen, re-check and held-out blocks; OOG counts summed.","conclusion":"Identity retained; no active size-tail policy won the re-check. Zero-strength held-out comparisons are structural identities; the nonzero diagnostic is reported separately."}}