Adaptive fusion — a derivation¶

Status: derivation under stated assumptions, not behaviour. Nothing here is implemented.

The equations were produced for the adaptive fusion section of the 2.6 design page, whose success condition they do not change. The transcribed facts about the code and the benchmark files were spot-checked against the tree, and the equations have not yet been re-derived independently. Every assumption in section 1 names the observation that would refute it, and section 9 lists the measurements that must come before an implementation.

The result in one paragraph. Fusing arms by their exceedance probability against each arm's own null is defensible, and a mixture likelihood ratio over those probabilities gives, without any fitted weight, a closed-form per-row influence for each arm and an identifiable limit that reproduces today's reciprocal rank fusion exactly.

What the mathematics does not give is the claim that a small null p-value means the arm is reliable, or that a confident dense arm suppresses lexical errors across a whole query. The first is an assumption about the relevant class; the second holds for the mixture rule and fails for Fisher's rule.

1. Assumptions, each with its refutation¶

The measured pipeline must be identified before a loss is attributed to fusion. Two premises in the record needed correction.

First, a row does not receive three votes. _recall_rrf fuses the vector list, the episode FTS list and the memory keyword list, but a memory row can only appear in the vector and memory-keyword lists, and an episode row only in the vector and episode-FTS lists, so each row has two applicable arms.

Second, CPERSONA_FUSED_GATE_ENABLED=false disables the calibrated gate only. do_recall still runs _apply_quality_gate with its heuristic threshold, so the benchmark regime is not a gate-free ranking experiment.

Refuted if a frozen-candidate replay shows that post-fusion filtering, duplicate handling or candidate coverage explains a material part of the Track B − Track A difference. 2. Null samples represent irrelevant query–document scores in the deployment scope: same model, same encoder roles, same corpus snapshot, same row type, same isolation scope, and a declared query population. Random documents approximate irrelevant ones, and random query–document pairs can contain positives.

Refuted if held-out irrelevant pairs show excess small-p rates, or calibration moves materially across query language, length, corpus group or retrieval eligibility. 3. Relevance changes the score distribution measurably. A valid null identifies \(f_0\), not \(f_1\), and any Bayesian reliability statement needs an alternative distribution, or a structural stand-in for one.

Refuted if gold and non-gold likelihood ratios do not increase with the fused statistic, or extremely small lexical p-values mostly identify distractors. 4. Bayes ranking is evaluated on a fixed eligible candidate universe. For expected DCG, rank by expected gain; for binary relevance the posterior probability suffices. Expected normalised DCG is a different quantity. The Track B metric uses binary membership in the relevant set. 5. Candidate-set independence is required; query conditioning is permitted. \(S(q,d;\mathcal C_1)=S(q,d;\mathcal C_2)\) whenever query, row, snapshot and scope are identical. A reference distribution may depend on \(q\), provided it is not estimated from the returned candidates.

Refuted if changing retrieval depth, adding an unrelated row, or activating another list changes an existing row's score. 6. Finite-sample guarantees use independent sampling units. All-pairs products over a document sample reuse vectors, so they are not independent observations. Refuted if an uncertainty calculation uses pair counts as sample counts. 7. Historical cross-model comparisons are hypotheses, not controlled interventions. The MiniLM per-task deltas were measured under an older pipeline than the jina and bge-m3 ones. Refuted if the QASPER sign flip disappears when both models run identical candidates, flags, code and labels.

First, a row does not receive three votes. _recall_rrf fuses the vector list, the episode FTS list and the memory keyword list, but a memory row can only appear in the vector and memory-keyword lists, and an episode row only in the vector and episode-FTS lists, so each row has two applicable arms.

Second, CPERSONA_FUSED_GATE_ENABLED=false disables the calibrated gate only. do_recall still runs _apply_quality_gate with its heuristic threshold, so the benchmark regime is not a gate-free ranking experiment.

Refuted if a frozen-candidate replay shows that post-fusion filtering, duplicate handling or candidate coverage explains a material part of the Track B − Track A difference.

2. Scores, nulls and the Bayes-optimal combination¶

For each applicable arm \(a\), define a higher-is-better score: the cosine for the dense arm, and the negated FTS5 bm25 for the lexical arms, since SQLite returns lower values for better matches. A short-query LIKE fallback is a separate discrete measurement, not a missing bm25 value.

Let \(F^0_a(t)\) be the null law of the arm's score and define the inclusive exceedance probability

\[ p_a(q,d)=\Pr_0\{S_a\ge s_a(q,d)\}=1-F^0_a\bigl(s_a(q,d)^-\bigr). \]

With a finite reference sample of size \(m_a\),

\[ \widehat p_a(s)=\frac{1+\sum_{i=1}^{m_a}\mathbf 1\{S^0_{ai}\ge s\}}{m_a+1}. \tag{1} \]

The added one prevents an unsupported zero. It does not repair domain shift or selection bias, and finite-sample validity needs exchangeability of the test score with the reference scores.

With class prior \(\pi\) and joint class densities \(f_1,f_0\),

\[ \Lambda(\mathbf s)=\operatorname{logit}\pi+\log\frac{f_1(\mathbf s)}{f_0(\mathbf s)}, \tag{2} \]

and sorting by expected gain is optimal for expected DCG. If the arms are conditionally independent in both classes,

\[ \Lambda=\operatorname{logit}\pi+\sum_a\log\frac{f^1_a(s_a)}{f^0_a(s_a)}. \tag{3} \]

This is a sum of likelihood ratios, not of raw scores and not of p-values. Two special cases name the classical rules. If \(p_a\mid R=1\sim \operatorname{Beta}(\alpha_a,1)\),

\[ \Lambda=C+\sum_a(1-\alpha_a)(-\log p_a), \tag{4} \]

which is Fisher's statistic when the \(\alpha_a\) are equal. If \(z_a=\Phi^{-1}(1-p_a)\) is Gaussian with a common covariance \(\Sigma\) and a shift \(\boldsymbol\delta\) under relevance,

\[ \Lambda=C+\boldsymbol\delta^\top\Sigma^{-1}\mathbf z, \tag{5} \]

which is weighted Stouffer. Writing the joint densities as \(f_y(\mathbf s)=D_y(\mathbf s)\prod_a f^y_a(s_a)\),

\[ \Lambda=C+\sum_a\log\frac{f^1_a}{f^0_a}+\log\frac{D_1(\mathbf s)}{D_0(\mathbf s)}. \tag{6} \]

Marginal null calibration says nothing about the last term. Equation (5) also shows a contract conflict: \(\Sigma^{-1}\boldsymbol\delta\) can have negative coordinates, so the unconstrained optimum can violate the requirement that raising an arm's score never lowers the fused score.

3. The candidate rule: a mixture likelihood ratio¶

Define the decreasing, normalised function

\[ B_{\kappa,\beta}(p)=\frac{(\kappa+p)^{-\beta}}{Z_{\kappa,\beta}},\qquad Z_{\kappa,\beta}=\int_0^1(\kappa+u)^{-\beta}\,du, \tag{7} \]

so that \(\mathbb E_0\,B(p_a)=1\) for a uniform null p-value. Let the relevant class be explained by one latent arm \(A\) with prior \(\pi_a\):

\[ f_1(\mathbf p\mid A=a)=B_{\kappa,\beta}(p_a)\,f_0(\mathbf p). \tag{8} \]

Marginalising \(A\),

\[ \frac{f_1(\mathbf p)}{f_0(\mathbf p)}=E(\mathbf p)=\sum_a\pi_a B_{\kappa,\beta}(p_a), \qquad \boxed{S(q,d)=\log E(\mathbf p(q,d)).} \tag{9,10} \]

Equal \(\pi_a\) over the arms applicable to the row type is an equal prior, not a learned weight. Equation (8) permits correlated null arms: under the alternative, the other coordinates keep their null conditional law given the explanatory one.

The unverified step is the single-coordinate tilt itself. Multi-arm agreement may carry extra evidence, and lexical distractors may not follow the tilt at all. Gold-conditioned joint distributions decide.

A first experimental choice is \(\kappa=0,\beta=\tfrac12\):

\[ E=\frac1{A}\sum_a\frac1{2\sqrt{p_a}}. \tag{11} \]

Validity survives dependence. Each \(B(p_a)\) is an e-value (decreasing, integrates to one), an average of e-values is an e-value, so \(\mathbb E_0 E\le1\) whatever the dependence between arms, and by Markov

\[ p_{\mathrm{comb}}=\min(1,1/E),\qquad\Pr_0(p_{\mathrm{comb}}\le\alpha)\le\alpha. \tag{12} \]

This is a conservative null statement, not a relevance posterior. Rank by \(S\); truncating \(p_{\mathrm{comb}}\) at one creates ties.

Emergent influence. With \(u_a=-\log(\kappa+p_a)\),

\[ \frac{\partial S}{\partial u_a}=\beta\,\omega_a,\qquad \boxed{\omega_a=\frac{\pi_a(\kappa+p_a)^{-\beta}}{\sum_b\pi_b(\kappa+p_b)^{-\beta}}} \tag{13} \]

and under the generative model \(\omega_a=\Pr(A=a\mid\mathbf p,R=1)\). For two equally weighted arms

\[ \omega_l=\frac1{1+\bigl((\kappa+p_l)/(\kappa+p_v)\bigr)^\beta}, \tag{14} \]

so with \(\kappa=0\) the lexical influence vanishes as \(p_v/p_l\to0\). Three limits: the influence is per row, not a weight shared across a query; a lexical distractor with its own extreme p-value still outranks a dense hit; and the finite reference caps suppression — with \(p_v\ge1/(m+1)\) and \(\beta=\tfrac12\), \(\omega_v\le\sqrt{m+1}/(\sqrt{m+1}+1)\).

For Fisher's rule, \(\partial S_F/\partial(-\log p_a)=1\). The dense arm can contribute more, but the marginal lexical influence never shrinks. "A confident dense arm silences lexical votes" is a property of the mixture rule, not of p-value combination in general.

Correction (September 2026). "Silences" describes the influence on one row's score, not the fused order. The rule is symmetric in the arms: a row that is extreme on the lexical arm alone outranks a row that is extreme on the dense arm alone, as the worked pair in the identifiability note shows. Read \(\omega_a\) as a per-row derivative, and nothing more.

4. When equal-weight RRF harms¶

Let \(X_a(d)=1/(K+r_a(d))\) if \(d\) occurs in arm \(a\) and \(0\) otherwise (the implemented vote). For a relevant \(g\) and an irrelevant \(b\) with \(M_a=X_a(g)-X_a(b)\), if the dense arm orders the pair correctly (\(M_v>0\)), equal-weight RRF reverses it if and only if

\[ \boxed{\sum_{a\ne v}M_a<-M_v,} \tag{15} \]

with equality decided by the tie order. For the mixture rule the exact reversal condition is

\[ \sum_{a\ne v}\pi_a\bigl[B(p_a(b))-B(p_a(g))\bigr]>\pi_v\bigl[B(p_v(g))-B(p_v(b))\bigr]. \tag{16} \]

No unconditional theorem follows from per-arm separations alone. Means, AUCs or marginal separations do not fix the alignment of the arms' errors, the near-top margins, candidate censoring or the position relative to the cutoff. Two systems with identical separations can fuse differently. The exact pairwise statement is \(\Pr(\sum_a M_a>0)<\Pr(M_v>0)\) (17).

Under a Gaussian approximation of the vote margins, \(\mathbf M\approx N(\boldsymbol\mu,\Sigma_M)\), with equal margin variances, correlation \(\rho\) and \(\delta_a=\mu_a/\sigma_a\), two-arm RRF harms pairwise accuracy exactly when

\[ \boxed{\delta_l<\bigl(\sqrt{2+2\rho}-1\bigr)\delta_v} \tag{18} \]

(For three equal arms: \(\delta_l+\delta_k<(\sqrt{3+6\rho}-1)\delta_v\), (19).)

Equation (18) matches the two-arms-per-row structure of the implementation. It explains how QASPER's sign can flip — lexical separation above the threshold for a weak dense arm, below it for a strong one — and how Gorilla can lose on every model. It does not establish that those separations occurred, which only a frozen replay can.

Pairwise loss is not NDCG@10 loss. For a judged query compute exactly

\[ \Delta_q=\frac{\sum_d G_d\bigl[w(r_{\mathrm{fusion}}(d))-w(r_v(d))\bigr]}{\operatorname{IDCG}_q}, \qquad w(r)=\frac{\mathbf 1\{r\le10\}}{\log_2(r+1)}. \tag{20} \]

5. RRF as a limit of the mixture rule¶

For complete rankings of the same \(N\) documents, identify the reference tail with the empirical rank tail \(p_a=r_a/N\), and set \(\beta=1\), \(\kappa=K/N\) in (7). Then

\[ B_{\kappa,1}(p_a)=\frac{N}{\log((K+N)/K)}\cdot\frac1{K+r_a}, \qquad E(\mathbf p)\propto\sum_a\frac1{K+r_a}, \tag{21} \]

and since the logarithm is increasing, (10) produces exactly the RRF order in this limit.

The path \(\beta(t)=1-t/2\), \(\kappa(t)=(1-t)K/N\) for \(t\in[0,1]\) connects RRF (\(t=0\)) to the square-root mixture on genuine null p-values (\(t=1\)). It is an analytic bridge, not a knob to tune on benchmark outcomes.

The identification does not cover truncated lists or unequal eligible universes. Today's RRF gives an absent list entry a zero vote, and a genuine non-match, an inapplicable arm and a censored score are three different observations.

Flat distributions. Identical raw scores get identical p-values, so the rule creates no artificial within-arm separation. Scores inside the null bulk get no exceptional evidence, and a degenerate finite reference gives \(p=1\) rather than a full-strength vote.

This removes the artificial promotion of flat or singleton lists that per-query min-max produces (_minmax_norm maps an all-equal list to one). It cannot fix contamination caused by a misspecified null, or by an alternative that mistakes topical unusualness for relevance.

6. Query–document versus document–document nulls¶

Let normalised query and document vectors have means \(\mu_Q,\mu_D\) and covariances \(\Sigma_Q,\Sigma_D\). For irrelevant, independent pairs,

\[ \mathbb E(Q^\top D)=\mu_Q^\top\mu_D,\qquad \mathbb E(D_1^\top D_2)=\|\mu_D\|^2, \tag{22} \]

\[ \operatorname{Var}(Q^\top D)=\operatorname{tr}(\Sigma_Q\Sigma_D)+\mu_Q^\top\Sigma_D\mu_Q+\mu_D^\top\Sigma_Q\mu_D, \qquad \operatorname{Var}(D_1^\top D_2)=\operatorname{tr}(\Sigma_D^2)+2\mu_D^\top\Sigma_D\mu_D. \tag{23} \]

No Gaussian assumption is needed. Under a location–scale approximation a doc–doc quantile transports to a query–doc one as

\[ t_{QD,\alpha}\approx\mu_{QD}+\frac{\sigma_{QD}}{\sigma_{DD}}\,(t_{DD,\alpha}-\mu_{DD}), \tag{24} \]

so a purely multiplicative factor additionally assumes the intercept is negligible. Today's RRF_THRESHOLD_FACTOR of 0.5 is a crude operating-point transport of this kind. The measured p95 ratios in the calibration note already vary from 0.64 to 0.86 across tasks, so no constant can be right for all of them.

Cold start. For \(m\) independent reference scores drawn from \(G\), with probability at least \(1-\delta\),

\[ \sup_s|\widehat G_m(s)-F^0(s)|\le \sqrt{\frac{\log(2/\delta)}{2m}}+\sup_s|G(s)-F^0(s)|, \tag{25} \]

the first term being the DKW–Massart bound (\(m=738\) gives \(0.05\) at \(\delta=0.05\)) and the second the unavoidable shift between the reference and the true null.

Reference	Recommendation	What invalidates it
Retained real query embeddings	Preferred steady state; freeze snapshots, stratify only on predeclared features	Query drift, scope mismatch, repeated sessions counted as independent
Document fragments encoded in the query role	Provisional only if those embeddings already exist outside the no-call path	Unknown fragment-to-query shift, near-duplicate leakage, wrong prompt
Doc–doc null with learned transport	Only after held-out quantile-transport validation	Non-affine shift, changed model/prompt/corpus
Current query against a fixed document panel	Cold-start estimator that needs no new model call: a query-conditional random-document null	Contamination \(\eta_q\) of the panel by relevant documents enters (25) as an extra term

The recommendation is retained real queries in the steady state, and the query-against-panel estimator at cold start. Using the whole small corpus as the panel reduces the score to a rank and erases the cross-query absolute separation that motivated adaptivity; a finite panel also saturates its tail.

7. Small corpora, proxy degeneration and starvation¶

A null-only fusion does not need a temporal positive proxy, so keep the proxy out of the p-value transform.

Where a separation threshold is still used, with \(J(t)=F_0(t)-F_1(t)\) estimated from independent samples of sizes \(m_0,m_1\), two DKW bounds and a union bound give \(|\widehat J(t)-J(t)|\le\epsilon_0+\epsilon_1\) uniformly, with \(\epsilon_y=\sqrt{\log(4/\delta)/(2m_y)}\). A defensible acceptance rule is

\[ \boxed{\widehat J(\widehat t)-\epsilon_0-\epsilon_1\ge J_{\min},} \tag{26} \]

with \(J_{\min}=0.2\) as declared policy, not a mathematical boundary. At \(m_0=m_1=220\) and \(\delta=0.05\) the uncertainty is about \(0.2\), so an observed \(\widehat J\ge0.4\) is needed to certify \(J\ge0.2\). Thousands of overlapping pairs are not thousands of observations.

Starvation. A fixed-p test cannot guarantee both precision and a non-empty result. Under an all-null query with \(n\) independent uniform p-values, \(\Pr(\text{no admission})=(1-\alpha)^n\). The policy \(\alpha(n)=\max\{\alpha_0,1-\eta^{1/n}\}\) (27) bounds that by \(\eta\) under those assumptions only.

The recommended fallback is deterministic: retain the top \(L_{\mathrm{reserve}}\) dense rows inside the declared candidate budget, mark rows failing the p criterion as fallback candidates, keep their calibrated scores, and separate candidate retention from any claim of relevance. Choose \(L_{\mathrm{reserve}}\) before evaluating; retaining at least the evaluation depth stops an admission rule from removing the raw dense top ten with nothing to show for it.

8. Invariants an implementation must keep¶

Monotone. (1) is non-increasing in the raw score, (7) and (10) are non-increasing in \(p\); the composition is non-decreasing in every higher-is-better raw score.
No zero tails. Floor at \(1/(m+1)\); no extrapolation past the observed tail without a validated tail model.
Stable arithmetic. Evaluate (10) as \(\operatorname{LSE}_a[\log\pi_a-\beta\log(\kappa+p_a)-\log Z]\); reject non-finite raw scores; validate dimensions and norms.
Ties. Total order \((-S,\,-s_v,\ \text{row-type code},\ \text{row id})\) with a fixed sentinel for a missing dense score. Equal raw scores get equal p-values; ids break ranking ties, never probabilities.
Fixed arm applicability from row type and configuration; never divide by the number of lists that happened to return rows.
Missing is not non-matching. Score every applicable arm on each candidate in the bounded union, or declare the candidate's score incomplete. A row outside a top-\(L\) list has a censored score, not \(p=1\); treating it as \(p=1\) and later learning its score would break candidate-set independence.
Frozen reference. Neither the null sample nor its stratum may depend on the returned candidates.
Lexical atoms. Include non-matches in the lexical reference; calibrating only returned FTS matches estimates a selected conditional law. Short-query LIKE fallbacks need discrete calibration; the FTS5 trigram tokenizer does not match strings shorter than three characters.
Bound the work. Bound candidate union, reference panel, cross-scoring and calibration storage separately. A SQL LIMIT bounds returned rows, not index traversal; the FTS and LIKE paths need an explicit work-budget audit before "no scan past declared limits" can be claimed.
No extra model calls. Fusion consumes stored embeddings and lexical scores; pseudo-query embeddings must be supplied beforehand.
Determinism scope. Freeze seed, sampled ids, calibration version, dtype, reduction order and tie policy. Cross-platform bitwise equality is a separate requirement.
Multiple comparisons. A per-row \(\alpha\) bounds expected false admissions by \(N_0\alpha\), not by \(\alpha\). Ranking quality, admission and contamination control are three measurements.

9. What to measure before implementing, in order¶

Each step gates the next.

Identify the measured pipeline and the candidate population. Per query: eligible corpus ids, stored row ids, vector candidates before and after admission, fused order, gate survivors, final order, effective configuration. The calibration record for QASPER lists 65,300 corpus rows while the null probe saw 20,508 unique ids; until that is reconciled the two instruments may not be scoring the same population. Gate: no loss is attributed to fusion alone until stages and populations agree.
Verify cache identity, normalisation and pair alignment through read-only connections. _null_distribution computes dot products without normalising, which is a cosine only while every stored vector is unit-norm. Gate: no cosine-distribution inference until norms and pairing are verified.
Separate admission, fusion and downstream stages on frozen embeddings and lexical scores: dense-only, dense with admission, RRF before post-processing, RRF after each stage; compute (20) at every transition. Gate: if the heuristic gate or the corpus population explains the DOWN family, revise the causal question before choosing fusion mathematics.
Test query–doc null calibration on held-out blocks at predeclared levels (0.01, 0.05, 0.1), stratified by model, language, query length, corpus group and row type, including lexical non-matches; record gold rejection and tail saturation. Gate: reject fixed-p admission wherever the null is anti-conservative or gold rejection is unacceptable.
Test the relevance model. Labelled joint p-value distributions, per-arm conditional error rates, empirical relevance as a function of \(S\); compare equal-\(\beta\) Fisher, the square-root mixture, dense-only and current RRF as one frozen, pre-registered experiment. Gate: if rare lexical distractors dominate the mixture or joint moderate evidence supplies most corrections, reject the single-tilt alternative.
Test emergent suppression directly with (13) and (16): count dense-correct reversals prevented and new reversals introduced. Gate: suppression counts only if harmful cross-document comparisons fall without destroying useful lexical corrections.
Check the Gaussian approximation (18) against empirical reversal probabilities; keep the exact replay equations if it mispredicts near-top reversals.
Evaluate cold-start references against held-out real query–doc scores, split by query group rather than by pair.
Apply the pre-fixed success condition: for both endpoint models, \(\overline\Delta_m=\frac1{22}\sum_t(\mathrm{NDCG}^{\mathrm{new}}_{m,t}-\mathrm{NDCG}^{A}_{m,t})>0\) on the same 22 tasks. Gate: an explicit weights layer is considered only after the declared p-only experiment fails this.
Verify the operational invariants of section 8 separately; ranking success does not authorise deployment if they fail.

10. Where it would slot in¶

All identifiers below are proposals, not settings the code reads today.

Location	Integration
`_recall_rrf`	Keep the mode. Add a sibling path that gathers a bounded union, cross-scores the applicable arms, applies frozen null transforms and computes (10).
`_minmax_norm`, `_recall_rsf`	Do not reuse min-max. Keep `rsf` unchanged as a comparison mode.
`_autocut`	Recognise the new score explicitly; its gaps are not cosine gaps.
`_apply_quality_gate`	Add a distinct signal type. Never compare log-evidence or a combined p-value with a cosine threshold. Make the gate's bypass observable in experiments.
`do_recall`	Explicit mode dispatch and trace fields. Resolve the final confidence re-sort before claiming the fusion reaches the output.
`_null_distribution`, `do_calibrate_threshold`	A bounded query–doc and lexical calibration artefact: store the empirical reference distributions with provenance, not one threshold. Keep the existing methods.
`_get_vector_threshold`	Keep the cosine getter; add an independent null-artefact lookup. A p threshold cannot be substituted into a cosine-valued accessor.
`config.py`	Reserve a value (for example `null_p`) for both the calibration method and the recall mode, default off, with explicit validation of compatible combinations.
benchmark runner	Add the mode only once implemented; record every effective gate, arm coverage, reference fingerprint and per-query stage outcome.

Stored artefacts need a fingerprint over: model identity and revision, dimension, and query/document prompts; normalisation, chunking, dtype, and scoring version; corpus and index generation, and isolation scope; row type and near/far eligibility; FTS tokenizer, query builder, bm25 configuration, and fallback policy; null definition, reference population, strata, sampled ids, seed, and effective sampling units; tail convention, clipping, combiner parameters, arm applicability, and candidate-depth contract; timestamp and schema version.

The existing sidecar checks dimension and scoring version. Dimension alone cannot detect a same-width model or a prompt replacement.

11. What this note would not do¶

Claim that null calibration estimates relevance probability.
Promise that dense confidence alone recovers Gorilla: cross-document lexical extremes defeat row-specific dominance.
Treat Fisher's rule as implicit arm selection.
Apply independent chi-square or normal references to correlated, discrete arms.
Declare admission solved by \(p=0.05\): it controls a null-tail event, not gold retention.
Use pair counts in i.i.d. error bounds, or an NN-maximum as an ordinary positive sample.
Treat \(\widehat J>0.2\) as sufficient.
Generate fragment embeddings inside the no-call path.
Use per-query candidate min-max, returned-list normalisation or top-result-dependent weights: each violates the score contract.
Collapse absent, inapplicable, failed and censored arms into one value.
Infer scan bounds from SQL output limits.
Read the historical sign table as proof of structural causation.
Build explicit weights before the declared p-only experiment has failed.