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Rank9 has a 50.35 bps run-average max; persistent shock state is a testable explanation, not an observation. Near-zero average lean favors symmetric fees.","falsifier":"Shock gain zero matches or exceeds positive gains on fresh paired seeds and removes no residual structure after conditioning on market volatility. Then aggregate nonlinearity rather than short path memory is sufficient."},{"entries":[20],"family":"mixed flow and volatility, positive lean","formula":"Using common_state: m=32+0.08*sqrtV+0.05*Q; bid=m+0.285; ask=m-0.285. Q is Y per timestamp unit; sweep Q coefficient through 0,0.025,0.05,0.10 and b around 32.","why":"Own-flow retail marginal 0.824 bps/10k Y exceeds other sampled competitors, but proxy still improves R2 from 0.437 to 0.532. Observed mean lean +0.571 bps and lower mid 33.994 motivate this distinct candidate family.","falsifier":"Direct Q coefficient zero is as good or better on fresh paired seeds across market-flow bands; or symmetric lean outperforms +0.57 bps. This rejects the flow/lean components."}],"residual_sd_rankings":{"own_flow":{"bid":[{"rank":10,"residual_sd":1.8183251701},{"rank":8,"residual_sd":1.8998354671},{"rank":20,"residual_sd":1.9401525792},{"rank":6,"residual_sd":1.9800488538},{"rank":9,"residual_sd":2.044192297},{"rank":29,"residual_sd":2.0645703004},{"rank":1,"residual_sd":2.1338343981},{"rank":7,"residual_sd":2.1967263707}],"ask":[{"rank":29,"residual_sd":1.7702159219},{"rank":10,"residual_sd":1.8120681911},{"rank":8,"residual_sd":1.8792528803},{"rank":6,"residual_sd":1.8985514861},{"rank":1,"residual_sd":2.0377974184},{"rank":20,"residual_sd":2.0505252025},{"rank":7,"residual_sd":2.0888306406},{"rank":9,"residual_sd":2.3791003609}],"mid":[{"rank":10,"residual_sd":0.7251474028},{"rank":29,"residual_sd":0.786955434},{"rank":8,"residual_sd":0.8494516634},{"rank":6,"residual_sd":1.0543432908},{"rank":7,"residual_sd":1.063336206},{"rank":20,"residual_sd":1.1152371197},{"rank":9,"residual_sd":1.1719559844},{"rank":1,"residual_sd":1.2201058385}],"lean":[{"rank":6,"residual_sd":3.2563205183},{"rank":20,"residual_sd":3.3109931955},{"rank":10,"residual_sd":3.3281286158},{"rank":8,"residual_sd":3.3757458237},{"rank":1,"residual_sd":3.3848341297},{"rank":29,"residual_sd":3.5092783375},{"rank":7,"residual_sd":3.7222177312},{"rank":9,"residual_sd":3.7661330343}]},"stack_arb_proxy":{"bid":[{"rank":10,"residual_sd":1.8098357056},{"rank":8,"residual_sd":1.8870007955},{"rank":20,"residual_sd":1.8959934757},{"rank":6,"residual_sd":1.9347190558},{"rank":9,"residual_sd":2.0200542674},{"rank":1,"residual_sd":2.0538313791},{"rank":29,"residual_sd":2.0645703004},{"rank":7,"residual_sd":2.1621335806}],"ask":[{"rank":29,"residual_sd":1.7702159219},{"rank":10,"residual_sd":1.7879773125},{"rank":6,"residual_sd":1.8334710256},{"rank":8,"residual_sd":1.8464159637},{"rank":1,"residual_sd":1.9280407061},{"rank":20,"residual_sd":1.9892036303},{"rank":7,"residual_sd":2.0280436505},{"rank":9,"residual_sd":2.3227462809}],"mid":[{"rank":10,"residual_sd":0.6796441163},{"rank":29,"residual_sd":0.786955434},{"rank":8,"residual_sd":0.7978945193},{"rank":6,"residual_sd":0.9474484253},{"rank":7,"residual_sd":0.9646102506},{"rank":20,"residual_sd":1.0164203662},{"rank":1,"residual_sd":1.0485817427},{"rank":9,"residual_sd":1.0922971497}],"lean":[{"rank":6,"residual_sd":3.258661317},{"rank":20,"residual_sd":3.3122504799},{"rank":10,"residual_sd":3.3312267716},{"rank":8,"residual_sd":3.375426665},{"rank":1,"residual_sd":3.3871867253},{"rank":29,"residual_sd":3.5092783375},{"rank":7,"residual_sd":3.7220523364},{"rank":9,"residual_sd":3.7655128209}]}},"notes":{"method":{"n":1000,"pairing":"Array index, as requester specifies; no common seed or market identifier exposed. Match id and submissionId are entry-specific and are not join keys.","fee_units":"Reported fee fractions multiplied by 10000 to bps. mid=(bid+ask)/2; lean=bid-ask.","fee_fit_design":["1","entry retailVolumeY/10000","arbVolumeY/10000 (own or STACK)","product of preceding two volumes"],"coefficient_units":["bps","bps per 10000 Y retail at arb=0","bps per 10000 Y arb at retail=0","bps per (10000 Y)^2 interaction"],"gap_fit_design":["1","(entry retail-STACK retail)/10000","(entry arb-STACK arb)/10000","entry mid fee-STACK mid fee in bps"],"gap_coefficient_units":["edge Y","edge Y per 10000 Y retail diff","edge Y per 10000 Y arb diff","edge Y per bps mid fee diff"],"solver":"numpy.linalg.lstsq, rcond=None; intercept included; no centering; finite values required","sd":"Fee sd uses ddof=1; residual sd=sqrt(SSE/(n-design_rank)); descriptive tercile sd uses ddof=1.","terciles":"STACK retail quantile boundaries at 1/3 and 2/3; equality assigned upward.","R2":"1-SSE/SST, in sample. With an intercept the fitted mean equals observed mean, so that mean share is trivially 1. R2 measures explained gap variance; mean slope share reported separately, can exceed 1 and is not causal.","interpretation":"Aggregated run averages cannot identify the underlying swap rule. Large residuals also admit omitted nonlinear aggregate effects, not only path dependence. Own flow is endogenous; STACK arb is a common-market proxy, not observed variance.","no_simulation":"No local simulation requested or performed. No source code, gas results, or holdout performance inferred.","missing":"No seeds, timestamps, reserves, swap paths, fee-weighting definition, or separate retail/arb edge fields exposed; cannot report separate retail and arb edge deltas. Volume deltas are not edge deltas.","tercile_sd":"Descriptive sd of per-run entry-minus-STACK fee differences, ddof=1."},"response_sha256":{"https://www.paradigm.xyz/puzzles/api/v1/submissions/52df1e42-4219-4931-b014-586d91720298":"831b7c7ed6da9a69b561618b0705be0a8528bb0077bfc42125dc09af3e2905b1","https://www.paradigm.xyz/puzzles/api/v1/submissions/8cdc1896-049f-4ae0-8e54-022d4866e4a1":"d2aa449815ff48d1cad10298fc1594559d941e45fc110840d104fb131f967511","https://www.paradigm.xyz/puzzles/api/v1/submissions/4f22492b-9643-47d2-82a2-bddb09edaccc":"399d46c62f24de4b892c5e6f5c0815ea59072716798621696e3b2b20aad86481","https://www.paradigm.xyz/puzzles/api/v1/submissions/ff0c41ee-376c-4869-90e4-52b937a2fb61":"f7564bf854d29f2b375e913497c2f1d600e48fa3257127f01864a4869afb8003","https://www.paradigm.xyz/puzzles/api/v1/submissions/2c1b6306-9e54-4f26-9c03-8c2f009ee6db":"3672fcbe0c74469f0434fb74a32c8a010d985cec708b97b71564e44248cd5321","https://www.paradigm.xyz/puzzles/api/v1/submissions/65573db6-d287-47b1-928a-bd65eaa4b7a9":"9209d4913024ed408047de8cbfe0cbdf53fae58daca8391459eb3aa5df2f4a2a","https://www.paradigm.xyz/puzzles/api/v1/submissions/a1731836-9dbc-4c10-8290-6631285030c6":"e9b5787fdd89ebbbc750a3df75bb018f49c01eea3043b8f0b75fd7e30e8669ac","https://www.paradigm.xyz/puzzles/api/v1/submissions/6438e80d-34a7-42f5-890f-b7d21f0ade41":"aa7c59bd050f9f644d4ce737452f2c32de500d3001c464d28d7c16e7e6e4642e"},"summary_lines":["1. These are aggregate statistical signatures and proposed tests, not recovered fee code; none of the sampled fees is constant.","2. Rank10 is the most nearly explained competitor: mid residual sd 0.725 bps own-flow, 0.680 with STACK-arb proxy; rank8 follows at 0.849/0.798.","3. Rank1/6/7/9 have volatility-related fees but retain roughly 0.95-1.09 bps proxy residuals, compatible with path dependence or omitted nonlinear state.","4. Rank20 mixes flow and volatility with +0.571 bps mean bid-over-ask lean; its proxy residual is 1.016 bps.","5. STACK mid fee is 36.716 bps versus competitors 33.994-35.570; STACK lean +1.010 bps versus -0.571 to +0.571 here.","6. Simple retail-only slopes are reported per entry; compare these with requester prior ~1.76 STACK/~1.1 leaders, not with interaction intercept slopes.","7. At mean STACK arb, partial retail slopes are 0.198-0.436 for competitors versus STACK 1.415 bps/10k Y; collinearity prevents causal attribution.","8. Competitors become cheaper relative to STACK in higher STACK-retail terciles; this supports reducing direct flow sensitivity.","9. Edge-gap models explain 13.6-42.5% of cross-run gap variance; mean decomposition is associative and intercept-dependent.","10. Test symmetric variance tracking for ranks8/10, variance plus shock memory for ranks1/6/7/9, and mixed flow/variance with small positive lean for rank20.","11. Every formula uses only afterSwap amounts, post-trade reserves, timestamp, and own state; neither hidden arb labels nor STACK volumes are runtime inputs."],"hypothesis_state_definition":"Maintain timestamp buckets: p=reserveY/reserveX (fixed-point); at a timestamp change finalize prior bucket close P, not individual within-timestamp swaps. d=10000*(P/Pprev-1); V=V+(d*d-V)/64; Q=Q+(bucket_abs_amountY/max(1,bucket_elapsed)-Q)/64. Accumulate bucket_abs_amountY=sum(abs(amountY)) using afterSwap amounts; timestamp/reserves and own stored state only. sqrtV=isqrt(V) in bps. Clamp output to [1,100] bps and convert to contract fee units. Initialize P at first observed price, V=0,Q=0; initialize output at b. These are candidate rules, not recovered code; Q mixes arb and retail. Coefficients below are proposed starting values, not fitted swap parameters.","source_attribution":"board_score/rank/id from urls_fetched[0]; fee_stats, fits, flow_stats, rankings and simple slopes are NumPy calculations from the entry source_url matches. Every gap/tercile calculation also uses STACK submission URL. Hypotheses and interpretation are explicitly inferences, not fetched values.","retrieval":"Fetched live during this task; source files contain exact response-prefix quotes; response hashes identify the analyzed submission payloads. Rankings reflect the fetched leaderboard.","validation":"Both fee designs recomputed for all 8 entries and all 4 fee outcomes; 9 source quotes re-fetched and matched exactly; all 1000-run counts, board mean edge identities, gap mean decompositions, finite numbers, JSON byte limit and 11-line summary checked."}}