# Root merge 4: gap-oracle, rival-cut, and kappa-tell results ## Scope, provenance, and common checks This merges the six supplied artifacts: `b34-go-2`, `b35-rc-band`, `b35-rc-scale`, `b35-rc-joint`, `b36-kt-low`, and `b36-kt-high`. Their supplied SHA-256 values all matched when this merge was made. Numeric results below are local paired Taylor-replica measurements, not leaderboard scores, unless explicitly called a board figure. Source attribution after a claim names the input artifact; the underlying common source is Taylor (gtmcknight), `amm-sim` commit `4812a6382311624e1506aca131ca35c378f930d9`, `research/solved` (README read first), plus the recorded TCA and LB4 artifacts where applicable. All six inputs credit Taylor for the replica, Gaussian filter, solved policy, and qm tables. Each completed experiment reports the unmodified `python3 -I gates.py` P gate matching the README to six decimals: | seed | perfect-info | gaussian+plain | gaussian+qm | lattice+qm | | --- | ---: | ---: | ---: | ---: | | 10000 | 448.261189 | 443.455598 | 443.522608 | 443.663520 | | 10001 | 576.779199 | 571.463409 | 571.623551 | 571.977293 | | 10002 | 531.886497 | 525.168028 | 525.865859 | 526.411631 | Sources: all six inputs (P sections). `b34-go-2`, `b35-rc-band`, `b35-rc-joint`, and `b36-kt-high` also record matching TCA/LB4 downloads and materialized blob `3ac7cb8811ecfb6a17f499aa4e9b70dd55f7fb6432b3aec3b904e11a3dece38e`. Unless a row says otherwise, choose seeds are 64000--64255 (256 seeds), selection happens before the stated report seeds, each run has 10,000 steps, and paired SE is sample SD of per-seed differences divided by sqrt(n). ## Results at a glance | Work item | Frozen/report result | Decision and deployability | | --- | --- | --- | | Gap oracles | True own-path price (A4) is +1.193111 +/- 0.030145 over unchanged LEAN; A5 is +1.531476 +/- 0.031897. | Diagnostics using hidden truth, not feasible policies. | | Rival band | 40 bps/step: -177.527080 +/- 2.169463 vs LEAN. | Reject; active settings all failed. | | Run scale | disabled control: exactly 0 vs LEAN. | Keep LEAN; every live setting lost on choose. | | Joint filter | `rival64`: +0.145192 +/- 0.014521 vs LEAN; +3.277454 +/- 0.044521 vs qt1. | Positive replica result, but as built cannot meet contract gas/size limits. | | Kappa low | Undetermined; required DP solve timed out at 475 s. | No selection, report, or contract claim. | | Kappa high | FULL(1e-5): +0.111298 +/- 0.011764 vs compressed LEAN; +3.269334 +/- 0.044027 vs qt1. | Python full-table control only; no Solidity candidate. | Sources: `b34-go-2`, `b35-rc-band`, `b35-rc-scale`, `b35-rc-joint`, `b36-kt-low`, `b36-kt-high`. ## Gap-oracle diagnostic (`b34-go-2`) The 512-seed corrected block is 64512--65023, scored serially as A0,A1,A2,A3,A4,A5,qt1 per seed. Edge is total own-pool 10,000-step edge in Y, not bps or a board score. A0 is unchanged LEAN diagnostic bookkeeping; A1 replaces the first-own-callback mixture probability with the true own path; A2 supplies the true post-event rival spot; A3 adds a once-per-step hidden rival-price band; A4 supplies true own price; A5 combines A4 and A2. | arm | mean edge | paired vs A0: mean +/- SE | paired vs qt1: mean +/- SE | | --- | ---: | ---: | ---: | | A0 / unchanged LEAN | 542.734797 | 0 | +3.124007 +/- 0.042284 | | A1 / true first-own path | 542.740603 | +0.005805 +/- 0.006060 | +3.129812 +/- 0.042791 | | A2 / exact rival tracking | 542.947473 | +0.212675 +/- 0.018003 | +3.336682 +/- 0.045693 | | A3 / A2 plus rival band | 542.107603 | -0.627194 +/- 0.027765 | +2.496813 +/- 0.046163 | | A4 / true own price | 543.927909 | +1.193111 +/- 0.030145 | +4.317118 +/- 0.050101 | | A5 / true own price + rival spot | 544.266274 | +1.531476 +/- 0.031897 | +4.655483 +/- 0.051154 | The true first-own-trade flag alone is essentially neutral in this exact intervention (+0.005805), whereas true price is the large remaining diagnostic gap. Exact rival tracking is useful (+0.212675); the specified Gaussian projection band is harmful. Relative to the A4--A0 true-price gap, A3's signed effect is -52.568%, and its incremental effect beyond A2 is -0.839870. Sources: `b34-go-2` interpretation, arms, and additional paired results. Important limitations are retained: A3 processes hidden truth and is a specified projection diagnostic, not an optimized information policy; no contract, F gate, engine score, feasible policy, or board result is claimed. The original partial run of seeds 64512--64581 was invalidated after the band was found to be applied per callback rather than once per step; it was repaired and the full corrected block was then measured once. Thus the artifact explicitly says its strict global once-only condition is not met across the invalidated and corrected runs, although invalidated values were excluded and no choice was made from them. Source: `b34-go-2` execution and limitations. ## Rival-cut experiments ### RIVAL BAND (`b35-rc-band`) The policy cut LEAN's posterior around a last-sync rival spot after retail-classified callbacks; it kept the tell classifier, qm tables, fee rule, and LEAN rival tracking. The disabled-band exact control reproduced LEAN exactly over 256 choose seeds (2,772,898 callbacks). All active widening settings lost to LEAN: 0, 2, 5, 10, 20, and 40 bps/step were respectively -171.810, -171.668, -171.541, -171.457, -171.353, and -171.192 mean edge (SE about 3.01 each). The rule selected the least-bad active 40 bps setting; the artifact correctly recommends retaining LEAN instead. On report seeds 10000--10511, 40 bps/step had mean edge 362.590327 versus LEAN 540.117407 and qt1 536.990076; paired results were -177.527080 +/- 2.169463 vs LEAN and -174.399749 +/- 2.168858 vs qt1 (n=512). It made 4,553,525 cuts and encountered 295,438 numerical empty intersections. The four-seed explanatory audit found true price outside the band in 3,039--3,583 of 7,349--8,725 cuts per seed, mostly on sync steps. Belief MAE was 22.1863 bps for the pick versus 3.6526 bps for LEAN, descriptively, across their different callback populations. It is replica-only; no contract was built. Sources: `b35-rc-band` variant, choose, report, explanatory audit, belief error, recommendation, and limitations. ### RUN SCALE (`b35-rc-scale`) RUN SCALE estimated diffusion variance from tell-positive pin movements and an effective retail-first rate from `(1-pi)`, applying estimates only after a warmup. Its six live `(weight,warmup)` settings all lost on choose: (256,128) -0.023084 +/- 0.010814; (256,512) -0.036754 +/- 0.013453; (1024,128) -0.028882 +/- 0.012745; (1024,512) -0.024692 +/- 0.011407; (4096,128) -0.036591 +/- 0.015021; and (4096,512) -0.004138 +/- 0.009559. The zero-change control was therefore selected and exactly reproduced LEAN. On report seeds 10512--11023 it remained exactly zero vs LEAN and was +3.158036 +/- 0.043515 vs qt1 (n=512), the same as LEAN: levels were 531.808758 (control/LEAN) and 528.650723 (qt1). The observables check used raising garbage for hidden path, flags, prices, rival state, phase, previous reserves, and init values; all six settings had exact Python-float fee pairs on 247,985 callbacks over four seeds. This is a null result, not evidence that a live estimator has zero effect, and includes no contract/engine/board claim. Sources: `b35-rc-scale` variant, choose, report, observables check, limitations, and conclusion. ### JOINT FILTER (`b35-rc-joint`) The particle filter tracked log true-price and rival-spot hypotheses beside LEAN. It used public-game/Taylor constants, deterministic normal draws, own no-arbitrage constraints, rival clipping, a silent-order likelihood, and retail-fill likelihood. The selected `rival64` mode kept LEAN's analytic price mean and SD, taking only rival estimate and sync flag from 64 particles. Garbling all hidden callback and initialization data produced identical fees and edge for the control and every candidate in four proof seeds; this supports that the implementation's fee decisions use only permitted observables. Sources: `b35-rc-joint` variant-as-built and observables proof. Choose results vs LEAN were: full32 +0.064685 +/- 0.024375, full64 +0.103345 /- 0.024346, rival64 +0.160519 +/- 0.021828, and price32 -0.082718 +/- 0.019819. `rival64` was frozen by the rule. On report seeds 11024--11535, levels were 537.149580 (pick), 537.004387 (LEAN), and 533.872126 (qt1); paired results were +0.145192 +/- 0.014521 vs LEAN and +3.277454 +/- 0.044521 vs qt1 (n=512). LEAN was +3.132262 +/- 0.042881 vs qt1 on these same seeds. Worst paired values were -1.674416 vs LEAN (seed 11303) and -1.218613 vs qt1 (seed 11062). Source: `b35-rc-joint` choose, freeze, report, and headline. The result is small and is attributed to rival-state estimation, not a better price belief: the particle price mean did not improve true-price error versus LEAN in the supplied eight-debug-seed diagnostic. The mode is not deployable as built: 64 particles require roughly 64 packed slots plus substantial per-particle arithmetic/tables; the artifact says this exceeds the 245,000 gas limit and cannot fit the 24,576-byte target. No contract, F gate, engine run, or rival-estimate-error measurement was performed. Sources: `b35-rc-joint` belief error, contract needs, and notes. ## Kappa/tell table work ### Low weights: 0 and 5e-6 (`b36-kt-low`) This result is **blocked/undetermined**. The required unmodified DP solve (`DP_SAVEVN=1 DP_AMINL=-12 DP_NA=43 ...`) timed out after 475 seconds (exit 124), before a QMDP control reproduction or either low-weight table could be built. No substitute table was used. Consequently no choose selection, report score, policy improvement, contract, engine run, or board claim exists. Source: `b36-kt-low` solver, blocked, choose, report, contracts, conclusion. The supplied original 1e-5 table and TCA/LB4 bundles passed their recorded hash checks, and P passed. However, the task specifically flags one `e09-kappa-tell/kt-low` source as not holding on reread. Accordingly, this merge relies on the artifact's directly recorded timeout and local hash/P status, but gives no additional evidentiary weight to the unverified external source citation; there is no low-kappa performance claim to depend on it. ### High weights and full-table control (`b36-kt-high`) The DP control completed in 237.4 s and regenerated the original 1e-5 table byte-for-byte (`5ff53e42...ba1afd`). It also built 2e-5 (`2f7cb6b0...c01c5b9`) and 4e-5 (`4ca53aa...af90b16`) tables. Injecting the full tables into the LEAN tell policy matched Taylor interpolation exactly on 200 sampled states (maximum fee difference zero). Source: `b36-kt-high` solver and lookup fidelity. On choose seeds, FULL(1e-5) was selected at 529.992535 mean edge, +0.108388 /- 0.015628 vs compressed LEAN. FULL(2e-5) was -0.218768 +/- 0.023635 versus FULL(1e-5) and -0.110380 +/- 0.025148 versus LEAN; FULL(4e-5) was -1.454390 /- 0.044799 versus FULL(1e-5) and -1.346003 +/- 0.044187 versus LEAN. On report seeds 10512--11023, FULL(1e-5) scored 531.920056, LEAN 531.808758, and qt1 528.650723. Paired FULL(1e-5) results were +0.111298 +/- 0.011764 vs LEAN (worst seed 10602, -0.965501) and +3.269334 +/- 0.044027 vs qt1 (worst seed 10642, +0.335544), n=512. Its first-own arb-call share was 0.540308 (1,932,938 / 3,577,473); LEAN's was 0.541356. These are replica levels, not board scores. Sources: `b36-kt-high` choose, freeze, report, and scope note. This is not a new Solidity candidate: it is Taylor's existing 1e-5 control with the full Python JSON lookup rather than the compressed LEAN lookup. No contract gates were run. Source: `b36-kt-high` contract gates. ## Reconciliation and next implication There is no substantive numerical conflict between inputs. Different report blocks yield different absolute levels and qt1 deltas, as expected from their distinct seed ranges; they must not be compared as if they were one shared test. The apparent distinction between the joint filter (+0.145 vs LEAN) and the full 1e-5 table (+0.111 vs LEAN) is likewise not a head-to-head result: both use 10512--11023, but no combined policy or paired joint-vs-full run was reported. The oracle's much larger true-price gains are intentionally hidden- truth diagnostics and do not contradict the small observable-policy gains. The retained actionable evidence is therefore narrow: preserve LEAN as the baseline; reject rival-band and adaptive run-scale variants; treat full 1e-5 lookup as a modest Python-replica fidelity improvement; and, if pursuing a deployable improvement, reduce the joint filter to a closed-form or few-point rival posterior before measuring it again. That last reduction is an unmeasured proposal from `b35-rc-joint`, not an established result. None of these six inputs supplies a policy meeting the stated contract prerequisite of at least +0.5 over qt1 on its report seeds, so no contract can be advanced from this merge.