{ "credit": "Taylor (gtmcknight), github.com/gtmcknight/amm-sim@4812a6382311624e1506aca131ca35c378f930d9 research/solved: the replica, Gaussian filter, fee rule, qm tables and exact-arithmetic arb/retail tell are his. fastlean port from artifact a88ff982eba7e078bc6ccb1555b90a211c00c9fd76ec90a62ab13a4074f2c3d0; LEANBASE reference (leanbase.py, stalerival.py) from artifact efb8e653f8fd061cecf3e92b186470c9c96c5188523364b60ed101e9672869aa.", "mech": "mixture-branch", "already_present": false, "already_present_where": "Not for fees. real.py (RealPolicy.on_trade, soft=1) already mixes the two hypotheses in the price BELIEF: for 0.005 < pi < 0.995 it keeps the moment-matched mixture m2 = pi*lp_arb + (1-pi)*mt. The FEE is not mixed: is_arb = pi > 0.5 picks one branch (arb: table 0c, rival estimate only band-clipped; retail: tables 1s/1c, router bind and, in stalerival.py, the staleness pull) and one lookup is returned. tellpol.py / leanbase.py (variant A) set pi = 0 on a tell miss and treat over/unsupported as a miss; variant B keeps the ordinary posterior there but still takes one branch. So the mechanism was tested as stated, not a neighbour.", "definition": { "formula": "On a step's first own trade (t > t_last), after LEANBASE has updated its state and looked up its fees (fb, fa): let p = pi (the tell-cut arb posterior) when the lean tell is positive, or p = the ordinary posterior la/(la+lr) with chance = 1 when the tell returns over/unsupported (computed for the fee only; the belief stays variant A's). If eps < p < 1-eps: arb-branch fees (fb_A, fa_A) = lookup(table 0c, w = ln s - m_A, w2 = ln clip(old_s2h, e^m_A*G2, e^m_A/G2) - m_A, sd_A, unsynced); retail-branch fees (fb_R, fa_R) = lookup(table 1s/1c, w = ln s - m_R, w2 = ln r_R - m_R, sd_R), r_R = old_s2h band-clipped at m_R, then the router bind (cand = s(1-fb0)/G2 on a buy, s*G2/(1-fa0) on a sell) and the staleness pull a = a0 + (ainf-a0)*age/(age+tau) exactly as stalerival.py does them. q = p^gamma / (p^gamma + (1-p)^gamma). Returned fee, each side: f + w*(q*f_A + (1-q)*f_R - f), then LEANBASE's own wad rounding. own=0: m_A = m_R = the belief LEANBASE keeps (so one of the two branch fees is exactly LEANBASE's; checked, replication error 0). own=1: (m_A, v_A) = (lp_arb, 0), (m_R, v_R) = (mt, vt), each branch at its own conditional belief. A tell miss (pi = 0, certain retail) and later trades of a step are left alone.", "parameters": { "wU": "mixing weight w when the tell is positive", "wN": "mixing weight w when the tell cannot run (over/unsupported)", "eps": "undecided band: eps < p < 1-eps", "own": "0 shared belief, 1 per-branch belief", "gamma": "posterior exponent (1 = plain posterior, >1 toward the hard rule, <1 hedged)" }, "state_kept": "none beyond LEANBASE: belief, rival estimate, phase, router time and counters follow the branch LEANBASE takes (pi > 0.5); only the two returned fees change (and so the fee wads LEANBASE already stores).", "inputs": "only afterSwap inputs (side, amounts, timestamp, own reserves) and LEANBASE state; no price, flag or rival data." }, "footprint": "The tell-cannot-run case is empty on the replica: 0 'over' and 1 'unsupported' label in 14399902 first trades (2,048 report seeds, LEAN), and that one was not in the band, so wN never acted in any run. Undecided positives (0.005 < pi < 0.995) are 53912 of 7806753 tell-positive first trades in the LEAN runs; with the chosen setting on, 26.3 callbacks per 10,000-step run are mixed, mean own-side fee change 0.141 bps on those.", "selection": { "seeds": "64000..65023", "rounds": "8 rounds of 128, every setting paired with LEAN (a0=0.35 ainf=0.35 tau=2) on the same seeds", "drop_rule": "mean_diff + 3*se < 0 after any round", "dropped": 0, "lean_mean_edge_1024": 539.018328, "note": "Before the formal rounds a 128-seed probe on 64000..64127 ran settings s01 (twice, once with wN=0: identical), s02, s03, s04, s05; those seeds were then rerun as round 1. No setting outside the 12 was run." }, "settings_tried": [ { "id": "s01", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.005, "own": 0.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.001397, "se": 0.000945, "dropped_early": false, "seeds_positive": 533, "mixed_callbacks": 26637, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.1419 }, { "id": "s02", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.005, "own": 1.0, "gamma": 1.0 }, "n": 1024, "mean_diff": -0.000296, "se": 0.001877, "dropped_early": false, "seeds_positive": 510, "mixed_callbacks": 26604, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 2.2511 }, { "id": "s03", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 1e-06, "own": 0.0, "gamma": 1.0 }, "n": 1024, "mean_diff": -0.003522, "se": 0.002609, "dropped_early": false, "seeds_positive": 475, "mixed_callbacks": 3733780, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.0017 }, { "id": "s04", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 1e-06, "own": 1.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.00014, "se": 0.002783, "dropped_early": false, "seeds_positive": 517, "mixed_callbacks": 3733479, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.0176 }, { "id": "s05", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 1e-06, "own": 1.0, "gamma": 0.5 }, "n": 1024, "mean_diff": -0.002281, "se": 0.003993, "dropped_early": false, "seeds_positive": 506, "mixed_callbacks": 3736108, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.0743 }, { "id": "s06", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.005, "own": 0.0, "gamma": 0.5 }, "n": 1024, "mean_diff": 0.000458, "se": 0.001184, "dropped_early": false, "seeds_positive": 506, "mixed_callbacks": 26603, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.6535 }, { "id": "s07", "parameters": { "wU": 0.5, "wN": 0.5, "eps": 0.005, "own": 0.0, "gamma": 1.0 }, "n": 1024, "mean_diff": -0.000462, "se": 0.001072, "dropped_early": false, "seeds_positive": 541, "mixed_callbacks": 26613, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.0713 }, { "id": "s08", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.05, "own": 0.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.000443, "se": 0.000407, "dropped_early": false, "seeds_positive": 484, "mixed_callbacks": 3163, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.6393 }, { "id": "s09", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.05, "own": 1.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.000694, "se": 0.000763, "dropped_early": false, "seeds_positive": 479, "mixed_callbacks": 3139, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 5.0862 }, { "id": "s10", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.005, "own": 0.0, "gamma": 2.0 }, "n": 1024, "mean_diff": -0.000397, "se": 0.00094, "dropped_early": false, "seeds_positive": 547, "mixed_callbacks": 26613, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.0292 }, { "id": "s11", "parameters": { "wU": 0.5, "wN": 0.5, "eps": 0.005, "own": 1.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.000646, "se": 0.001344, "dropped_early": false, "seeds_positive": 497, "mixed_callbacks": 26630, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 1.1262 }, { "id": "s12", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.2, "own": 0.0, "gamma": 1.0 }, "n": 1024, "mean_diff": 0.000484, "se": 0.000248, "dropped_early": false, "seeds_positive": 318, "mixed_callbacks": 921, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 1.0549 } ], "chosen": { "id": "s01", "parameters": { "wU": 1.0, "wN": 1.0, "eps": 0.005, "own": 0.0, "gamma": 1.0 }, "why": "highest mean_diff of the 12 after 1,024 selection seeds (+0.0014, se 0.0009: 1.5 se, not distinguishable from 0; none of the 12 exceeds 0.0015 in absolute mean except s03 and s05, both negative)" }, "report": { "vs_lean": { "n": 2048, "mean_diff": 0.000526, "se": 0.000736, "seeds_positive": 1052, "seeds_negative": 996, "seeds": "10000..12047", "base": { "a0": 0.35, "ainf": 0.35, "tau": 2.0 }, "base_mean_edge": 536.603391, "mech_mean_edge": 536.603917, "min_diff": -0.3359, "max_diff": 0.2674, "mixed_callbacks": 53795, "mixed_per_run": 26.27, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.1408, "base_run_first_trades": 14399902, "base_run_tell_pos": 7806753, "base_run_tell_over": 0, "base_run_tell_unsupported": 1, "base_run_tell_pos_by_pi": { "pi<=0.005": 5016, "0.005=0.995": 7747825 } }, "on_age": { "n": 2048, "mean_diff": -0.000455, "se": 0.000733, "seeds_positive": 978, "seeds_negative": 1070, "seeds": "10000..12047", "base": { "a0": 0.0, "ainf": 0.7, "tau": 1.0 }, "base_mean_edge": 536.6231, "mech_mean_edge": 536.622646, "min_diff": -0.3203, "max_diff": 0.2865, "mixed_callbacks": 53764, "mixed_per_run": 26.25, "mean_abs_own_side_fee_change_bps_per_mixed_callback": 0.1422, "base_run_first_trades": 14407398, "base_run_tell_pos": 7814201, "base_run_tell_over": 0, "base_run_tell_unsupported": 1, "base_run_tell_pos_by_pi": { "pi<=0.005": 4801, "0.005=0.995": 7755598 } }, "se_definition": "sd of the paired differences (n-1) over sqrt(n)", "note": "vs_lean: chosen setting on LEAN (a0=0.35 ainf=0.35 tau=2) minus LEAN; on_age: chosen setting stacked on AGE (a0=0 ainf=0.7 tau=1) minus AGE. Each run once." }, "promote": false, "conclusion": "No effect: +0.0005 (se 0.0007) on LEAN and -0.0005 (se 0.0007) on AGE over 2,048 replica seeds each, against a +0.03 bar. The policy already mixes the belief, and the fee branch is undecided on about 0.37% of first trades, so there is nothing for a fee mixture to recover. Do not promote.", "contract_cost": { "measured": false, "storage_words": 0, "gas": "Not measured: no Solidity was written and this item has no engine profile. Work added to a mixed callback (a tell-positive first trade with eps < pi < 1-eps, about 26 per 10,000-step run), setting s01: one more qm table lookup (16 table entries: 2 tables x 2 sigma levels x 4 corners; a binary search over the 49-point W2L grid when the added branch is the retail one; 14 linear interpolations; 2 exp for the fee conversion), about 2 ln for the other branch's rival estimate, and 4 multiplications for the mix. Whether that fits the 37,000 gas of headroom (208,000 of 245,000), and whether a tell-positive first trade is the contract's peak path at all, is not known here.", "characters": "Estimate, not measured: the mechanism lines of the Python patch (without counters and comments) are 2548 characters; a Solidity port reusing the contract's lookup, ln and exp would be of that order, against 2,770 characters of headroom (47,230 of 50,000).", "runtime_bytes": "Not measured; headroom is 2,136 bytes (22,440 of 24,576)." }, "checks": { "mechanism_off_is_fastlean_bit_for_bit": "128 of 128 seeds (64000..64063, 10000..10063) bit-identical to the unmodified fastlean.py for LEAN and for AGE; LEAN mean 527.8724874783702 equals the artifact's parity.mean_edge_fast", "shared_mode_branch_replication_error": 0.0, "patch_applies": true, "fastlean_source_sha256": "d9c758bb280fe24e09c039496f84ab0804173ff70cec24c332f8683f28efa582", "patched_sha256": "37d794853e4a78ba7cdd1899e61d77317bdd183eae999bcc5c68876ef98746e3", "environment": "CPython 3.12.3, numpy 2.2.6, llvmlite 0.44.0, numba 0.61.2 (PyPI wheels, sha256 checked), Linux x86_64, one CPU" }, "run": [ "unpack code_zb64 of artifact a88ff982eba7e078bc6ccb1555b90a211c00c9fd76ec90a62ab13a4074f2c3d0, then: python3 -c \"import json,base64,zlib;open('mix.diff','wb').write(zlib.decompress(base64.b64decode(json.load(open('result.json'))['patch_zb64'])))\" && git apply mix.diff # or: patch -p1 < mix.diff", "import fastlean; tabs = fastlean.load(amm_dir, a0=0.35, ainf=0.35, tau=2.0, mech=(1.0, 1.0, 0.005, 0.0, 1.0)); edges = fastlean.run_batch(tabs, range(10000, 12048)) # mech=None is LEANBASE; AGE: a0=0, ainf=0.7, tau=1" ], "patch_zb64": "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", "not_done": [ "Replica only: no Solidity port, no gas, size or peak measurement, no engine run (gate E), no submission.", "The tell-cannot-run half of the mechanism could not be exercised: over/unsupported labels do not occur on the replica (1 in 14.4 million first trades), so every number here is for the undecided-positive half. On the engine the label mix may differ; not tested.", "Only the returned fees are mixed; the state still follows the hard branch. Mixing the state (rival estimate, phase) is a price-belief variant, which the brief lists as spent, and was not tried.", "The drop rule never fired, so all 12 settings ran the full 1,024 selection seeds; the chosen setting leads by less than 1 se over several others, so the choice among them is noise.", "The kit's setup_fast.sh was not used; the replica artifact's own run lines were followed (Taylor's research files via tell/fetch.sh at 66604b3426d8e97a73e2c39940b9b4dece51cc73, 34 files, 0 hash mismatches; LEANBASE modules from artifact efb8e653). pip is absent on this machine, so the three pinned wheels were downloaded from PyPI, sha256-checked and unpacked.", "Contract cost is an operation count and an estimate, not a measurement." ] }