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weak conditional flow, roughly symmetric mean","formula":"Using notes.hypothesis_state_definitions: m=clip(35.4+6*v+1*q,5,100); nextBid=nextAsk=m.","why":"Fit 2 mid-fee R2=0.6692/0.6746/0.6787, residual sd=0.8241/0.7383/0.8118 bps for ranks 3/4/5. Marginal retail slopes are only 0.255/0.209/0.302 versus STACK 1.415 bps per 10k Y. Test volatility-led symmetric quotes with a weak flow term; this is not a claim that actual leaders quote symmetrically per swap.","falsifier":"On disjoint paired simulations against STACK, require positive mean edge gain with 95% paired interval excluding zero and zero OOG calls. Reject the volatility mechanism if setting the coefficient of v to zero matches or improves edge; reject its symmetric implementation if learned state-dependent lean materially improves it.","sources":["https://www.paradigm.xyz/puzzles/api/v1/submissions/e9362d90-f6b7-4497-835c-dd538d030160","https://www.paradigm.xyz/puzzles/api/v1/submissions/4aad6b12-aa93-4593-98cc-642e0b9fd006","https://www.paradigm.xyz/puzzles/api/v1/submissions/88bbb4b5-6f95-45a4-8fd0-d1d8ba43ca52","https://www.paradigm.xyz/puzzles/api/v1/submissions/6438e80d-34a7-42f5-890f-b7d21f0ade41"]},{"entries":["52df1e42-4219-4931-b014-586d91720298","c38c9e1f-f458-4d13-963c-9294c3b91c62"],"family":"volatility-associated mid with greater unexplained/path-state variation","formula":"Using the same state: m=clip(35.4+6*v+1*q,5,100); ell=1.5*h; nextBid=clip(m+ell/2,5,100); nextAsk=clip(m-ell/2,5,100). Also test ell=-1.5*h as a prespecified sign ablation because API does not reveal causal trade direction.","why":"Fit 2 mid residuals 1.0486/0.9613 bps exceed ranks 3–5; all leaders have near-zero average lean but roughly 3.3–3.5 bps unexplained run-to-run lean. A signed transient lean can average near zero and exploit path information missing from volume aggregates; residuals alone do not prove such a rule exists.","falsifier":"Reject the added directional state if neither sign beats ell=0 on fresh paired seeds, or if gain comes with any OOG call; require a positive paired mean edge gain versus STACK with 95% interval excluding zero. Compare reproduced fee residuals as a secondary diagnostic, never as a substitute for edge.","sources":["https://www.paradigm.xyz/puzzles/api/v1/submissions/52df1e42-4219-4931-b014-586d91720298","https://www.paradigm.xyz/puzzles/api/v1/submissions/c38c9e1f-f458-4d13-963c-9294c3b91c62","https://www.paradigm.xyz/puzzles/api/v1/submissions/6438e80d-34a7-42f5-890f-b7d21f0ade41"]}],"notes":{"method":{"fit_targets":{"bid":"10000*averageBidFee","ask":"10000*averageAskFee","mid":"(bid+ask)/2","lean":"bid-ask"},"fee_design":["1","entry retailVolumeY/10000","arbVolumeY/10000 (entry in fit 1; STACK in fit 2)","product of previous two columns"],"coefficient_units":["bps","bps per 10000 Y retail at arb=0","bps per 10000 Y arb at retail=0","bps per (10000 Y retail)*(10000 Y arb)"],"gap_design":["1","(entry retailVolumeY - STACK retailVolumeY)/10000","(entry arbVolumeY - STACK arbVolumeY)/10000","entry mid fee - STACK mid fee (bps)"],"gap_coefficient_units":["edge Y","edge Y per 10000 Y retail difference","edge Y per 10000 Y arb difference","edge Y per bps mid-fee difference"],"ols":"numpy.linalg.lstsq; intercept included; unweighted; all 1000 runs; no missing or nonfinite fitted values; residual sd=sqrt(SSE/(n-rank(X))); fee sd and tercile sd use ddof=1; R2=1-SSE/SST.","terciles":"numpy.quantile STACK retail at 1/3 and 2/3; <= first cut, > first and <= second, > second.","tercile_cutpoints_Y":[69854.14367876071,81018.66749099916],"gap_share":"R2 is explained variation about the mean; mean-gap covariate share=sum(beta_j*mean(delta_j))/mean(edge gap), with intercept reported separately. With an intercept fitted mean equals observed mean by construction; neither measure is a causal attribution.","pairing":"Array position 0..999, as requester specifies; API exposes no seed or shared market id. Match UUIDs identify submission-specific matches, not common markets. Pairing cannot independently be certified from these responses.","dependencies":"Analysis uses numpy only for statistics; output is static JSON and requires no installed dependency. No simulation or gas measurement performed.","design_checks":"All fee and gap designs have rank 4. Uncentered fee-design condition numbers range 1515.59 to 2276.15; use marginal slopes at sample means, not isolated coefficients at zero flow, for interpretation.","tercile_sd":"Tercile table reports mean per-run entry-minus-STACK fee and SE=sample sd/sqrt(n), in bps."},"limitations":["Run-average fees and endogenous volumes cannot uniquely identify an afterSwap rule; low residuals describe these aggregates, not necessarily deterministic per-trade decisions. High residuals can also reflect omitted nonlinear terms or missing state, not just path dependence.","STACK arb volume is the requester-specified volatility proxy, not a measured variance; its own policy affects that proxy.","Retail and arb edge components, per-swap fees, amounts, reserves, timestamps, seeds and strategy code are absent from submission responses; retail_edge_delta and arb_edge_delta are unavailable, not zero.","Top-1 explicit id equals fetched rank 1; fetched once, no duplicate entry.","All numeric fits, rankings and tercile summaries derive from the cited submission URLs; board ranks/scores derive from urls_fetched[0]."],"summary_lines":["All five leaders have variable run-average mid fees (sd 1.29–1.61 bps), so a globally constant fee is inconsistent with these aggregates.","Ranks 3–5 form a relatively volatility-associated family: fit 2 explains 66.9–67.9% of mid-fee variation; propose m=clip(35.4+6*v+q,5,100), bid=ask=m.","Ranks 1–2 leave more unexplained mid variation (fit 2 R2 57.7%/55.2%); path state is a hypothesis, not identified code: add transient lean ell=1.5*h to the same m.","Mid residual order, low to high, fit 1: rank 4, STACK, rank 5, rank 3, rank 2, rank 1; fit 2: rank 4, STACK, rank 5, rank 3, rank 2, rank 1.","STACK is more flow-associated: fit 2 marginal retail slope 1.415 versus leaders 0.189–0.302 bps per 10k Y; proxy-arb slopes 3.214 versus 5.579–6.759.","The brief quotes ~1.76 versus ~1.1 retail-only slopes; these fetched runs instead give 1.817 versus 0.800–0.936, so the quoted approximations are not reused as measured values.","STACK mean bid-minus-ask lean is +1.010 bps; leaders range -0.131 to -0.022, but per-run lean remains variable and poorly explained by these flow fits.","Leaders gain 8.389–8.690 edge Y per paired run; gap-model R2 is 41.7–45.5%, while covariate contributions account algebraically for 20.1–30.4% of mean gap.","Online v,q,h are explicitly defined own-state proxies; hypotheses require fresh paired tests and OOG checks, which this data-fit item does not perform.","Low residuals support aggregate predictability only; endogenous flow, nonlinear effects and missing market/path variables prevent unique rule recovery."],"hypothesis_state_definitions":"Proposals only, not recovered code or fitted parameter estimates. All fees below are in bps; convert to protocol fee units in implementation. clip(x,l,h)=min(h,max(l,x)). Maintain last post-trade price p=reserveY/reserveX, current timestamp bucket volume Q=sum(abs(amountY)), and last completed bucket price p0. On the first callback with a newer timestamp, finalize the PREVIOUS bucket using dt=max(1,previous_bucket_timestamp-last_finalized_timestamp), x=abs(p_previous/p0-1), u=x*x/dt, z=Q/dt; do not use current trade in that bucket. Update fast and slow states F(s,a)=s+alpha*(a-s), alpha_f=1-(63/64)^dt and alpha_s=1-(1023/1024)^dt, capped dt at 4096 for arithmetic; V_f,V_s track u; Q_f,Q_s track z. Initialize states from first nonzero finalized observation, quote 35.4 bps both sides during warmup; floors 1e-18 for V and 1e-12 for Q prevent zero division. v=clip(sqrt(V_f/max(V_s,1e-18))-1,-1,1); q=clip(Q_f/max(Q_s,1e-12)-1,-1,1). These are realizable own-pool volatility and TOTAL-flow proxies, not observable retail/arb labels. Solidity can use fixed-point ratios, integer square root and exponentiation by squaring. Keep signed direction h=clip((p_previous/p0-1)/max(sqrt(V_s*dt),1e-9),-2,2) at bucket finalization. Hold resulting fees through the bucket; all quantities use own history only. Initialization, constants, caps, horizons and sign conventions are explicit experimental choices; fitted per-run volume coefficients cannot be plugged into these differently scaled online proxies."}}