{ "blocked": null, "tail": [ { "seed": 941495, "avg_bid": 0.0042825, "avg_ask": 0.003025, "lean_bps": 12.58, "edge": 500.852, "config": { "retail_mean_size": 19.592, "retail_arrival_rate": 0.7789, "gbm_sigma": 0.0009075 }, "mean_spot_minus_fair_bps": 12.35, "fair_end": 89.63, "fair_min": 86.85, "fair_max": 100.01, "lean_by_quarter_bps": [ 10.2, 16.8, 4.4, 18.9 ], "frac_steps_lean_gt_10bps": 0.533, "frac_steps_lean_lt_minus10bps": 0.388 }, { "seed": 940487, "avg_bid": 0.0039077, "avg_ask": 0.0027642, "lean_bps": 11.44, "edge": 455.403, "config": { "retail_mean_size": 19.228, "retail_arrival_rate": 0.6076, "gbm_sigma": 0.000911 }, "mean_spot_minus_fair_bps": 11.48, "fair_end": 100.47, "fair_min": 98.1, "fair_max": 112.08, "lean_by_quarter_bps": [ 11.7, 10.8, 15.7, 7.6 ], "frac_steps_lean_gt_10bps": 0.516, "frac_steps_lean_lt_minus10bps": 0.396 }, { "seed": 941985, "avg_bid": 0.0044354, "avg_ask": 0.0033471, "lean_bps": 10.88, "edge": 694.225, "config": { "retail_mean_size": 19.492, "retail_arrival_rate": 0.9824, "gbm_sigma": 0.0009717 }, "mean_spot_minus_fair_bps": 9.0, "fair_end": 113.68, "fair_min": 98.06, "fair_max": 119.87, "lean_by_quarter_bps": [ 16.3, 6.7, -0.7, 21.2 ], "frac_steps_lean_gt_10bps": 0.506, "frac_steps_lean_lt_minus10bps": 0.41 }, { "seed": 940596, "avg_bid": 0.0043458, "avg_ask": 0.0033105, "lean_bps": 10.35, "edge": 568.696, "config": { "retail_mean_size": 19.737, "retail_arrival_rate": 0.8482, "gbm_sigma": 0.0009668 }, "mean_spot_minus_fair_bps": 10.4, "fair_end": 93.95, "fair_min": 88.68, "fair_max": 103.49, "lean_by_quarter_bps": [ 8.9, 4.4, 16.9, 11.2 ], "frac_steps_lean_gt_10bps": 0.523, "frac_steps_lean_lt_minus10bps": 0.412 }, { "seed": 941306, "avg_bid": 0.0039572, "avg_ask": 0.0029233, "lean_bps": 10.34, "edge": 519.468, "config": { "retail_mean_size": 19.93, "retail_arrival_rate": 0.6399, "gbm_sigma": 0.0009225 }, "mean_spot_minus_fair_bps": 9.79, "fair_end": 104.69, "fair_min": 97.51, "fair_max": 118.51, "lean_by_quarter_bps": [ 5.1, 11.1, 6.7, 18.5 ], "frac_steps_lean_gt_10bps": 0.518, "frac_steps_lean_lt_minus10bps": 0.398 }, { "seed": 940818, "avg_bid": 0.0046381, "avg_ask": 0.0036075, "lean_bps": 10.31, "edge": 770.896, "config": { "retail_mean_size": 20.849, "retail_arrival_rate": 0.9941, "gbm_sigma": 0.0009639 }, "mean_spot_minus_fair_bps": 10.71, "fair_end": 93.76, "fair_min": 90.28, "fair_max": 104.13, "lean_by_quarter_bps": [ -9.8, 10.2, 16.3, 24.6 ], "frac_steps_lean_gt_10bps": 0.531, "frac_steps_lean_lt_minus10bps": 0.405 }, { "seed": 941455, "avg_bid": 0.0029572, "avg_ask": 0.0039757, "lean_bps": -10.19, "edge": 485.107, "config": { "retail_mean_size": 19.703, "retail_arrival_rate": 0.6935, "gbm_sigma": 0.0009933 }, "mean_spot_minus_fair_bps": -9.64, "fair_end": 111.34, "fair_min": 97.29, "fair_max": 111.4, "lean_by_quarter_bps": [ -8.2, -14.2, -7.3, -11.0 ], "frac_steps_lean_gt_10bps": 0.389, "frac_steps_lean_lt_minus10bps": 0.524 }, { "seed": 940305, "avg_bid": 0.0038791, "avg_ask": 0.0028671, "lean_bps": 10.12, "edge": 397.454, "config": { "retail_mean_size": 19.42, "retail_arrival_rate": 0.6806, "gbm_sigma": 0.0009952 }, "mean_spot_minus_fair_bps": 9.22, "fair_end": 113.09, "fair_min": 96.07, "fair_max": 114.69, "lean_by_quarter_bps": [ 14.2, 5.2, 12.6, 8.4 ], "frac_steps_lean_gt_10bps": 0.524, "frac_steps_lean_lt_minus10bps": 0.404 }, { "seed": 941516, "avg_bid": 0.0044755, "avg_ask": 0.003464, "lean_bps": 10.12, "edge": 688.843, "config": { "retail_mean_size": 20.928, "retail_arrival_rate": 0.9017, "gbm_sigma": 0.0009689 }, "mean_spot_minus_fair_bps": 10.55, "fair_end": 95.19, "fair_min": 92.6, "fair_max": 106.4, "lean_by_quarter_bps": [ 7.7, 14.8, 5.2, 12.8 ], "frac_steps_lean_gt_10bps": 0.519, "frac_steps_lean_lt_minus10bps": 0.412 }, { "seed": 941244, "avg_bid": 0.0043587, "avg_ask": 0.0033507, "lean_bps": 10.08, "edge": 642.658, "config": { "retail_mean_size": 19.669, "retail_arrival_rate": 0.8913, "gbm_sigma": 0.0009079 }, "mean_spot_minus_fair_bps": 9.29, "fair_end": 116.38, "fair_min": 99.28, "fair_max": 116.53, "lean_by_quarter_bps": [ 10.0, 2.1, 21.1, 7.2 ], "frac_steps_lean_gt_10bps": 0.501, "frac_steps_lean_lt_minus10bps": 0.427 }, { "seed": 940712, "avg_bid": 0.0042275, "avg_ask": 0.0032515, "lean_bps": 9.76, "edge": 561.116, "config": { "retail_mean_size": 20.57, "retail_arrival_rate": 0.8369, "gbm_sigma": 0.0009539 }, "mean_spot_minus_fair_bps": 10.31, "fair_end": 88.61, "fair_min": 85.55, "fair_max": 106.14, "lean_by_quarter_bps": [ 15.6, 3.5, 11.9, 7.9 ], "frac_steps_lean_gt_10bps": 0.505, "frac_steps_lean_lt_minus10bps": 0.415 }, { "seed": 940327, "avg_bid": 0.004058, "avg_ask": 0.0030836, "lean_bps": 9.74, "edge": 457.169, "config": { "retail_mean_size": 20.903, "retail_arrival_rate": 0.639, "gbm_sigma": 0.0009668 }, "mean_spot_minus_fair_bps": 9.78, "fair_end": 90.63, "fair_min": 81.42, "fair_max": 99.95, "lean_by_quarter_bps": [ 7.1, 23.8, -7.6, 15.7 ], "frac_steps_lean_gt_10bps": 0.526, "frac_steps_lean_lt_minus10bps": 0.394 }, { "seed": 941176, "avg_bid": 0.0041236, "avg_ask": 0.0031515, "lean_bps": 9.72, "edge": 543.768, "config": { "retail_mean_size": 20.355, "retail_arrival_rate": 0.7363, "gbm_sigma": 0.0009222 }, "mean_spot_minus_fair_bps": 10.64, "fair_end": 103.46, "fair_min": 94.43, "fair_max": 104.51, "lean_by_quarter_bps": [ 9.7, 5.7, 14.8, 8.7 ], "frac_steps_lean_gt_10bps": 0.504, "frac_steps_lean_lt_minus10bps": 0.417 }, { "seed": 940667, "avg_bid": 0.003131, "avg_ask": 0.0040975, "lean_bps": -9.66, "edge": 547.4, "config": { "retail_mean_size": 19.36, "retail_arrival_rate": 0.9475, "gbm_sigma": 0.0009379 }, "mean_spot_minus_fair_bps": -9.65, "fair_end": 104.51, "fair_min": 99.81, "fair_max": 112.7, "lean_by_quarter_bps": [ -7.2, -6.8, -18.3, -6.3 ], "frac_steps_lean_gt_10bps": 0.403, "frac_steps_lean_lt_minus10bps": 0.521 }, { "seed": 941330, "avg_bid": 0.0042119, "avg_ask": 0.0032665, "lean_bps": 9.45, "edge": 547.791, "config": { "retail_mean_size": 20.664, "retail_arrival_rate": 0.7681, "gbm_sigma": 0.0009539 }, "mean_spot_minus_fair_bps": 8.87, "fair_end": 107.93, "fair_min": 93.46, "fair_max": 109.12, "lean_by_quarter_bps": [ 5.3, 8.7, 13.6, 10.1 ], "frac_steps_lean_gt_10bps": 0.515, "frac_steps_lean_lt_minus10bps": 0.412 }, { "seed": 940694, "avg_bid": 0.0039888, "avg_ask": 0.0030516, "lean_bps": 9.37, "edge": 461.577, "config": { "retail_mean_size": 20.454, "retail_arrival_rate": 0.6999, "gbm_sigma": 0.0008877 }, "mean_spot_minus_fair_bps": 9.18, "fair_end": 111.21, "fair_min": 99.89, "fair_max": 118.1, "lean_by_quarter_bps": [ 0.6, 17.3, 18.9, 0.7 ], "frac_steps_lean_gt_10bps": 0.505, "frac_steps_lean_lt_minus10bps": 0.408 }, { "seed": 940377, "avg_bid": 0.0040031, "avg_ask": 0.0030702, "lean_bps": 9.33, "edge": 456.637, "config": { "retail_mean_size": 19.914, "retail_arrival_rate": 0.6331, "gbm_sigma": 0.0008882 }, "mean_spot_minus_fair_bps": 10.38, "fair_end": 87.91, "fair_min": 85.41, "fair_max": 102.7, "lean_by_quarter_bps": [ 11.2, 10.8, -0.5, 15.9 ], "frac_steps_lean_gt_10bps": 0.501, "frac_steps_lean_lt_minus10bps": 0.419 }, { "seed": 940006, "avg_bid": 0.004355, "avg_ask": 0.0034418, "lean_bps": 9.13, "edge": 704.298, "config": { "retail_mean_size": 20.161, "retail_arrival_rate": 0.9471, "gbm_sigma": 0.0008987 }, "mean_spot_minus_fair_bps": 8.92, "fair_end": 99.1, "fair_min": 96.83, "fair_max": 104.08, "lean_by_quarter_bps": [ 6.0, 10.0, 14.1, 6.4 ], "frac_steps_lean_gt_10bps": 0.518, "frac_steps_lean_lt_minus10bps": 0.408 }, { "seed": 941343, "avg_bid": 0.0044364, "avg_ask": 0.0035254, "lean_bps": 9.11, "edge": 713.535, "config": { "retail_mean_size": 20.19, "retail_arrival_rate": 0.8724, "gbm_sigma": 0.001004 }, "mean_spot_minus_fair_bps": 8.96, "fair_end": 102.19, "fair_min": 94.48, "fair_max": 102.7, "lean_by_quarter_bps": [ 7.1, 2.6, 4.9, 21.9 ], "frac_steps_lean_gt_10bps": 0.494, "frac_steps_lean_lt_minus10bps": 0.436 }, { "seed": 940087, "avg_bid": 0.0042014, "avg_ask": 0.0032923, "lean_bps": 9.09, "edge": 564.593, "config": { "retail_mean_size": 20.425, "retail_arrival_rate": 0.7682, "gbm_sigma": 0.0009297 }, "mean_spot_minus_fair_bps": 8.08, "fair_end": 112.28, "fair_min": 98.13, "fair_max": 112.59, "lean_by_quarter_bps": [ 20.4, 10.2, 4.2, 1.5 ], "frac_steps_lean_gt_10bps": 0.514, "frac_steps_lean_lt_minus10bps": 0.415 } ], "finding": "On scan seeds 940000..941999 (n=2000) RS2 mean edge 526.21; run-average bid-ask lean mean +1.18 bps, sd 3.34, 1st/99th pct -6.68/+8.68 bps. 18 of the 20 largest-|lean| runs lean positive (+9.1..+12.6 bps), 2 negative (-9.7, -10.2). What they share: not config (corr of lean with size/rate/sigma -0.004/+0.026/+0.038) and not the price path's net move (corr with log fair_end/100 +0.022; lean is flat across return deciles, 0.95..1.57 bps). Lean is almost exactly the run-average pool mispricing: corr(lean, mean (spot-fair)/fair) = 0.98, and tail runs sit about 8-12 bps above (or below) fair on average. It is not a single stuck estimate: the sign holds in most quarters of each tail run, and per step the fees flip, with bid>ask+10bps on ~52% of steps vs ask>bid+10bps on ~40%. So it is a persistent tilt in how often the stale-estimate skew points each way, not one stuck episode. The tail runs are not losers for RS2: their mean edge is 564.1 vs 526.2 for all runs, and corr(lean, edge) = +0.065. Every governor that cut the lean lost money. Shrinking the staleShift lean term when the slow lean EMA passes L lost -11.8 to -18.6 (more arb loss). Amplifying it lost -6.9. Pulling pHat toward fastHat lost -0.03 to -1.08, scaling with r, and did not lower the lean: run-average lean rose from RS2's 1.00 to 1.09-1.53 bps on the screen block and from 8.05 to 8.4-9.5 bps on tail20. Only the weakest setting (K: decay 0.9995, L 1 bp, r 0.02) was positive on screen and re-check, and on held-out it is +0.010 +/- 0.011, which is zero. Conclusion: in RS2 the lean is arb protection that earns money, not a stuck-estimate fault. This governor family is not a path to the +5 the goal needs.", "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\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[15] = 400000000000000000 | (uint256(2000000000000000) << 160); 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 uint256 arbProbability;\n (fastHat, uncertainty, arbProbability) = _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 bool govOn = false;\n {\n uint256 le = slots[16]; if (le == 0) le = WAD;\n le = wmul(le, 999500000000000000) + wmul(WAD + prevBidFee - prevAskFee, 500000000000000);\n slots[16] = le;\n uint256 ex = le > WAD ? le - WAD : WAD - le;\n if (ex > 100000000000000) { govOn = true; pHat = _blend(pHat, fastHat, 20000000000000000); }\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 originalLambda = lambdaHat; uint256 originalSize = sizeHat;\n\n if (342406799193418304 > 0) {\n uint256 packedFlow = slots[15];\n uint256 rate = uint80(packedFlow);\n uint256 count = uint80(packedFlow >> 80);\n uint256 meanSize = uint80(packedFlow >> 160);\n if (rate == 0) rate = 400000000000000000;\n if (meanSize == 0) meanSize = 2000000000000000;\n if (isNewStep) {\n uint256 inst = count / silent;\n if (inst > 3 * WAD) inst = 3 * WAD;\n uint256 retention = _powWad(999000000000000000, silent > 64 ? 64 : silent);\n rate = wmul(rate, retention) + wmul(inst, WAD-retention);\n count = 0;\n }\n uint256 weight = firstInStep ? WAD-arbProbability : WAD;\n uint256 correctedWeight = weight;\n if (2 == 2) {\n uint256 missing = wmul(feeUsed, 196030880396139429888);\n if (missing > 333333333333333376) missing = 333333333333333376;\n correctedWeight = wdiv(weight, WAD-missing);\n }\n count += correctedWeight;\n uint256 learning = wmul(WAD-999000000000000000, weight);\n meanSize = _blend(meanSize, tradeRatio, learning);\n slots[15]=rate | (count << 80) | (meanSize << 160);\n uint256 retailRate = wmul(rate, 366383424913407168);\n uint256 retailSize = 2 == 0 ? sizeHat : wmul(meanSize,1883296158119298304);\n lambdaHat = _blend(lambdaHat,retailRate,342406799193418304);\n sizeHat = _blend(sizeHat,retailSize,342406799193418304);\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 lambdaHat = originalLambda; sizeHat = originalSize;\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 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, 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),0);\n }\n return (x.mu,x.v,0);\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,prob);\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 function getName() external pure override returns (string memory) { return \"censored_10\"; }\n}\n", "sha256": "9887279d59be0dcd00fa33029940ef287959b98143f662743ffd345b5cfab06e", "params": { "mode": "pull pHat toward fastHat", "lean_ema_decay_per_print": 0.9995, "L_bps": 1, "r": 0.02, "lean_ema_slot": 16, "lean_ema_storage": "WAD + EMA(prevBidFee - prevAskFee), updated every afterSwap before the pHat pull; L very large reproduces RS2 bit-for-bit (checked on 40 seeds, max |edge diff| 0.0)", "base": "RS2 sha256 f24b6ca198b744c380842c3a17964a6dc21f311bb04984b26733afa3fee8eaf2" }, "screen": [ { "name": "A", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.999, "L_bps": 3, "r": 0.1, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.167, "se": 0.0323, "worst": -1.414, "arb": 1.5873, "retail": -1.7544, "mean_lean_bps": 1.201 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.2104, "se": 0.1256, "worst": -1.368, "arb": 1.9764, "retail": -2.1868, "mean_lean_bps": 8.748 }, "afterSwap_max_ok": 224932, "oog": 0 }, { "name": "B", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.999, "L_bps": 3, "r": 0.3, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -1.0837, "se": 0.0572, "worst": -3.262, "arb": 3.3938, "retail": -4.4775, "mean_lean_bps": 1.529 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.9596, "se": 0.2453, "worst": -3.242, "arb": 4.135, "retail": -5.0946, "mean_lean_bps": 9.521 }, "afterSwap_max_ok": 224932, "oog": 0 }, { "name": "C", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.999, "L_bps": 6, "r": 0.1, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.0912, "se": 0.0266, "worst": -1.456, "arb": 1.0175, "retail": -1.1086, "mean_lean_bps": 1.126 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.1298, "se": 0.1077, "worst": -1.364, "arb": 1.5969, "retail": -1.7266, "mean_lean_bps": 8.676 }, "afterSwap_max_ok": 224932, "oog": 0 }, { "name": "D", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.999, "L_bps": 6, "r": 0.3, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.6742, "se": 0.048, "worst": -3.539, "arb": 2.192, "retail": -2.8662, "mean_lean_bps": 1.335 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.8286, "se": 0.193, "worst": -2.969, "arb": 3.2758, "retail": -4.1044, "mean_lean_bps": 9.261 }, "afterSwap_max_ok": 224932, "oog": 0 }, { "name": "E", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.995, "L_bps": 5, "r": 0.2, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.6611, "se": 0.0483, "worst": -2.657, "arb": 2.9409, "retail": -3.602, "mean_lean_bps": 1.398 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.6438, "se": 0.1897, "worst": -2.831, "arb": 3.2454, "retail": -3.8892, "mean_lean_bps": 9.184 }, "afterSwap_max_ok": 224964, "oog": 0 }, { "name": "F", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.9995, "L_bps": 2, "r": 0.05, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.0345, "se": 0.0238, "worst": -1.173, "arb": 0.8958, "retail": -0.9303, "mean_lean_bps": 1.091 }, "tail20_vs_rs2": { "n": 20, "mean_diff": 0.0256, "se": 0.096, "worst": -0.895, "arb": 1.2113, "retail": -1.1857, "mean_lean_bps": 8.412 }, "afterSwap_max_ok": 224964, "oog": 0 }, { "name": "G", "mode": 2, "action": "scale staleShift (stale-estimate lean term) by (1-r)", "ema_decay_per_print": 0.999, "L_bps": 4, "r": 0.5, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -18.5768, "se": 0.3862, "worst": -43.929, "arb": -25.1138, "retail": 6.5371, "mean_lean_bps": 0.4 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -24.8794, "se": 1.1431, "worst": -35.28, "arb": -33.8495, "retail": 8.9702, "mean_lean_bps": 2.362 }, "afterSwap_max_ok": 225034, "oog": 0 }, { "name": "H", "mode": 2, "action": "scale staleShift (stale-estimate lean term) by (1-r)", "ema_decay_per_print": 0.999, "L_bps": 6, "r": 0.5, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -11.8155, "se": 0.3202, "worst": -38.179, "arb": -15.7508, "retail": 3.9353, "mean_lean_bps": 0.475 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -17.9158, "se": 0.9254, "worst": -26.863, "arb": -24.2484, "retail": 6.3326, "mean_lean_bps": 3.054 }, "afterSwap_max_ok": 225041, "oog": 0 }, { "name": "I", "mode": 3, "action": "pull fastHat toward pHat at rate r per print", "ema_decay_per_print": 0.999, "L_bps": 3, "r": 0.1, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.1451, "se": 0.0204, "worst": -1.349, "arb": -0.6872, "retail": 0.5421, "mean_lean_bps": 0.921 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -0.1114, "se": 0.0715, "worst": -0.871, "arb": -0.8042, "retail": 0.6927, "mean_lean_bps": 7.705 }, "afterSwap_max_ok": 224946, "oog": 0 }, { "name": "J", "mode": 4, "action": "scale staleShift by (1+r)", "ema_decay_per_print": 0.999, "L_bps": 4, "r": 0.2, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": -6.9012, "se": 0.2935, "worst": -39.603, "arb": 8.6056, "retail": -15.5069, "mean_lean_bps": 8.03 }, "tail20_vs_rs2": { "n": 20, "mean_diff": -10.8693, "se": 1.113, "worst": -25.014, "arb": 11.2413, "retail": -22.1106, "mean_lean_bps": 29.443 }, "afterSwap_max_ok": 225034, "oog": 0 }, { "name": "K", "mode": 1, "action": "pull pHat toward fastHat at rate r per print", "ema_decay_per_print": 0.9995, "L_bps": 1, "r": 0.02, "seeds": "942000..942255", "paired_vs_rs2": { "n": 256, "mean_diff": 0.0177, "se": 0.018, "worst": -0.831, "arb": 0.4646, "retail": -0.4468, "mean_lean_bps": 1.052 }, "tail20_vs_rs2": { "n": 20, "mean_diff": 0.0635, "se": 0.0568, "worst": -0.276, "arb": 0.5355, "retail": -0.472, "mean_lean_bps": 8.268 }, "afterSwap_max_ok": 224964, "oog": 0 } ], "recheck": [ { "name": "K", "seeds": "943000..943255", "paired_vs_rs2": { "n": 256, "mean_diff": 0.0487, "se": 0.0178, "worst": -0.741, "arb": 0.4553, "retail": -0.4066, "mean_lean_bps": 1.191 } }, { "name": "F", "seeds": "943000..943255", "paired_vs_rs2": { "n": 256, "mean_diff": -0.0048, "se": 0.0229, "worst": -1.531, "arb": 0.8739, "retail": -0.8787, "mean_lean_bps": 1.23 } } ], "frozen": "K (best on re-check: +0.0487 se 0.0178; F -0.0048 se 0.0229), frozen before held-out", "held_out": { "seeds": "944000..944511", "vs_rs2": { "n": 512, "mean_diff": 0.009675396187513896, "se": 0.011377800721184739, "worst": [ { "seed": 944174, "d": -0.8883 }, { "seed": 944382, "d": -0.779 }, { "seed": 944207, "d": -0.7624 }, { "seed": 944089, "d": -0.7254 }, { "seed": 944317, "d": -0.7208 } ] }, "retail_edge_delta": { "mean": -0.4482334507402661, "se": 0.012365632677903863 }, "arb_edge_delta": { "mean": 0.4579088469277811, "se": 0.0057048934485681 }, "arm_mean_edge": { "K": 520.1498363814017, "RS2": 520.1401609852142 } }, "tail20_mean_diff": { "frozen_K": 0.0635, "se": 0.0568, "n": 20, "note": "the 20 tail seeds are from the scan block, run alongside the screen" }, "gas": { "max_ok": 224964, "init_max_ok": 243090, "oog": 0, "scope": "frozen K over screen+tail20, re-check and held-out runs; RS2 afterSwap peak 202491 on the same runs; afterInitialize 243090 is RS2s own init, unchanged and < 245k" }, "notes": [ "Method: each arm alone vs VanillaStrategy via amm_sim_rs.run_batch(bytecode, Vanilla, configs, 8) in 40-seed chunks with gas_stats_reset/gas_stats per chunk. Configs and the arb/retail split are taken from the harness_source of artifact 33f54e81 (default_rng(seed): size U(19,21), rate U(0.6,1), sigma U(0.000882,0.001008); seed=seed). se = sd(ddof=1)/sqrt(n).", "Avg bid/ask = mean over the 10000 steps of the submission fees recorded per step. Mean mispricing = mean over steps of (spot-fair)/fair.", "Setup ran without error at amm-challenge 6b6921f8; StarterStrategy and the frozen K validated with amm-match validate. The CH calibration (seeds 0..999) was not rerun in this item. No seed >= 5000000 was used, and no STACK comparison was made: everything here is relative to RS2.", "The arb/retail split reconciles to the edge diff by construction (retail = edge - arb). Held-out shows K trades a small amount of retail edge for slightly less arb loss, netting zero.", "Suggestion for the goal: the lean carries signal, and removing it costs arb protection. Future items should not spend more on lean governors." ], "screen_baseline_rs2_mean_lean_bps": { "screen_942000_942255": 1.0, "tail20": 8.05 } }