Non-Additive Context Feature Registry

This page is the working backlog for proposed NAIL lineup-context features. It exists to preserve the basketball hypothesis, the exact mathematical contract, and the result of every test, including negative results.

The promoted NAIL-RAPM v1.2.1.3 currently retains two player-composition terms: usage_concentration and top_two_assists. Home-court advantage and back-to-back status are separate schedule controls, not player-composition features.

What Counts As Non-Additive

Let \(U\) be a five-player unit, \(O\) its opponent, and \(p_i\) the frozen, shrunken profile for player \(i\). A proposed side feature is \(\phi_k(U)\), and its matchup coordinate is

\[ x_k(U,O)=\phi_k(U)-\phi_k(O). \]

A feature belongs in this registry only when it cannot be written as

\[ \phi_k(U)=\sum_{i\in U}g_k(p_i). \]

Pure sums belong in the additive player-profile layer. Normalized shares, order statistics, thresholds, pairwise terms, cross-player products, and prior teammate relationships can be genuinely non-additive. A nonlinear formula is not sufficient by itself: a transformed statistic computed independently for each player and then summed remains additive.

All player inputs below must be frozen before the target season and use the same leakage-safe shrinkage contract as the incumbent model unless a candidate explicitly pre-registers a different source.

Status Key

[x] The proposed test is resolved. The accompanying decision says whether it was retained, rejected, or completed without promotion.
[ ] The candidate remains pending. Checking the box requires a persisted screen or model artifact and a documented decision.

Resolved Tests

[x] Usage concentration: retained. Share of all five usage events supplied by the two highest-usage players. This is the strongest retained non-additive term and a positive control for the screening harness.
[x] Top-two assists: retained. Sum of the two highest shrunken assist profiles. It is less influential than usage concentration but has stable incremental signal. See NAIL-RAPM v1.2.1.
[x] Low usage concentration: rejected at frozen screen. The share of usage supplied by the two lowest-usage players is almost a mirror of the retained high-usage share (weighted edge correlation -0.898 pooled). Its conditional residual correlations were -0.0017, -0.0004, and -0.0017 across 2023-24 through 2025-26 (-0.0013 pooled), so it does not justify a recursive fit.
[x] Primary-usage shooting alignment: rejected at frozen screen. This multiplies the top-two usage players' share of unit USG% by their mean shrunken 3PM per 100. Conditional residual correlations were -0.0049, -0.0011, and +0.0040 across the three frozen seasons (-0.0005 pooled), while the low-to-high decile residual direction reversed in 2025-26. It does not justify a recursive fit. Artifact: artifacts/models/analysis/frozen_feature_screen/primary_usage_shooting_alignment/frozen-feature-screen-primary_usage_shooting_alignment-20260908T164805Z-505d405e.
[x] Median three-point makes: rejected at frozen screen. The residual deciles were flat and the pooled weighted correlation was +0.0003. See Screen a Frozen Feature.
[x] Rim-protection ceiling: rejected at frozen screen. Maximum shrunken blocks per 100 did not produce a stable residual relationship. See Screen a Frozen Feature.
[x] Lower-tercile Critical Spacing: tested, not promoted. Its coefficient was directionally credible, but it did not improve pooled full-game RMSE. See NAIL Critical Spacing.
[x] Lower-quintile Critical Spacing: rejected. The stricter threshold was less stable and predictively worse. See Lower-Quintile Critical Spacing.
[x] Lower-quartile Critical Spacing with standard USG%: not promoted. The joint spacing-and-usage variant did not add sufficient predictive value. See Quartile Critical Spacing.

Priority Candidates

The definitions below are initial contracts. Any change to a formula after looking at outcomes creates a new candidate and receives a separate registry entry.

Creation-Spacing Alignment

[x] Frozen residual screen: rejected
[x] Recursive candidate fit: did not advance
[x] Coefficient and bootstrap decision: not applicable

Let \(u_i\) be usage events per 100 and \(s_i\) be shrunken three-point makes per 100. Define

\[ \phi_{\text{creation-spacing}}(U) =\sum_{i\in U}\frac{u_i}{\sum_{j\in U}u_j}s_i. \]

This asks whether a unit's shooting is concentrated in the players expected to use possessions. The normalization makes it a lineup allocation feature, not a sum of five independent shooting values.

The three-season frozen screen found no stable conditional residual signal: the weighted correlations were -0.0028, -0.0000, and +0.0070 for 2023-24 through 2025-26, respectively (+0.0017 pooled). The lowest-to-highest decile residual spread also reversed direction, from -2.07 in 2023-24 to +3.43 and +2.26 in the next two seasons. It therefore does not advance to a recursive fit. See Screen a Frozen Feature.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/creation_spacing_alignment/frozen-feature-screen-creation_spacing_alignment-20260830T195938Z-24613d9e.

Secondary-Creator Floor

[x] Frozen residual screen: rejected
[x] Recursive candidate fit: did not advance
[x] Coefficient and bootstrap decision: not applicable

Let \(a_{(2)}(U)\) be the second-highest shrunken assists-per-100 profile among the five players:

\[ \phi_{\text{secondary creator}}(U)=a_{(2)}(U). \]

This tests whether the current top_two_assists term is identifying a more specific mechanism: the presence of a credible second creator. The first test uses assists alone; adding a turnover conversion constant would be a separate candidate.

The three-season frozen screen did not support this narrower interpretation of the retained top_two_assists term. Weighted residual correlations were +0.0132, +0.0005, and -0.0037 from 2023-24 through 2025-26 (+0.0032 pooled); the endpoint decile spread similarly reversed from +4.45 to -1.58. The feature therefore does not advance to a recursive fit. See Screen a Frozen Feature.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/secondary_creator_floor/frozen-feature-screen-secondary_creator_floor-20260830T202300Z-2d67c5e4.

Rim Pressure By Spacing Floor

[x] Frozen residual screen: rejected
[x] Recursive candidate fit: did not advance
[x] Coefficient and bootstrap decision: not applicable

Let \(r_i\) be shrunken unassisted rim makes per 100 and let \(s_{(1)}(U)\) and \(s_{(2)}(U)\) be the two lowest shrunken three-point-make profiles. Define

\[ \phi_{\text{rim-spacing}}(U) =\left(\sum_{i\in U}r_i\right) \left(\frac{s_{(1)}(U)+s_{(2)}(U)}{2}\right). \]

The hypothesis is that rim pressure is more productive when the lineup's two weakest spacers still provide credible gravity. This is a continuous successor to the rejected hard Critical Spacing thresholds.

The three-season frozen screen did not support the interaction. Weighted residual correlations were +0.0038, -0.0065, and +0.0016 from 2023-24 through 2025-26 (-0.0001 pooled), while the lowest-to-highest decile spread changed direction in each season. It therefore does not advance to a recursive fit. See Screen a Frozen Feature.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/rim_pressure_by_spacing_floor/frozen-feature-screen-rim_pressure_by_spacing_floor-20260830T202922Z-0678e0aa.

Defensive Anchor By Perimeter Pressure

[x] Frozen residual screen: rejected
[x] Recursive candidate fit: did not advance
[x] Coefficient and bootstrap decision: not applicable

Let \(m\) be the player with the highest shrunken blocks per 100, \(b_m\), and let \(d_j\) be steals per 100 for the other four players. Define

\[ \phi_{\text{anchor-pressure}}(U) =b_m\sum_{j\in U,\,j\ne m}d_j. \]

Maximum blocks alone failed its frozen screen. This different hypothesis asks whether a rim protector becomes more useful when the other players generate perimeter disruption.

The three-season frozen screen did not support the interaction. Weighted residual correlations were -0.0067, +0.0004, and +0.0099 from 2023-24 through 2025-26 (+0.0021 pooled), and the endpoint decile direction reversed from -3.26 in 2023-24 to +4.92 in 2025-26. It therefore does not advance to a recursive fit. See Screen a Frozen Feature.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/defensive_anchor_by_perimeter_pressure/frozen-feature-screen-defensive_anchor_by_perimeter_pressure-20260830T203515Z-3b3a6ee0.

Offensive Role Redundancy

[x] Profile coordinates and scaling locked before outcomes are inspected
[x] Frozen residual screen: rejected
[x] Recursive candidate fit: did not advance
[x] Coefficient and bootstrap decision: not applicable

For target season (t), use each player's frozen, shrunken prior-season rates for usage events, assists, three-point attempts, unassisted rim makes, offensive rebounds, and free-throw attempts per 100 possessions. For each coordinate (k), let (c_{k,t-1}) be its possession-weighted 90th percentile across all source-season players, weighted by source RAPM possessions. Define

\[ z_{i,k,t}=p_{i,k,t-1}/c_{k,t-1}. \]

After player-level unit-length normalization, define average pairwise cosine similarity:

\[ \phi_{\text{redundancy}}(U) =\frac{1}{10}\sum_{i<j}\frac{z_i^\mathsf{T}z_j} {\lVert z_i\rVert_2\lVert z_j\rVert_2}. \]

High values represent five players with unusually similar offensive roles. The home-minus-away feature edge enters the screen. The pre-registered basketball hypothesis is negative: high role similarity may leave a unit with less complementary offensive coverage after player ratings and retained non-additive terms have been accounted for.

The frozen screen was directionally negative in 2023-24 and 2024-25, but reversed in 2025-26. Weighted residual correlations were -0.0096, -0.0144, and +0.0027 (-0.0069 pooled), so the candidate fails the pre-registered three-season stability gate and does not advance to a recursive fit. See Screen a Frozen Feature.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/offensive_role_redundancy/frozen-feature-screen-offensive_role_redundancy-20260830T204638Z-222eb12b.

Lead-Secondary Usage Gap

[x] Profile coordinate and direction locked before outcomes are inspected
[x] Frozen residual screen: advances with opposite-than-expected sign
[x] Superstar-proxy conditioning audit: survives
[x] Recursive candidate fit
[x] Coefficient and bootstrap decision: eligible, unpromoted

For a unit (U), sort its frozen, shrinkage-adjusted conventional usage profiles so that (u_{(1)}(U)\geq u_{(2)}(U)\geq\cdots\geq u_{(5)}(U)). Define

\[ \phi_{\text{handler gap}}(U)=u_{(1)}(U)-u_{(2)}(U). \]

This is a direct secondary-ball-handler hypothesis: a large gap means one player carries substantially more usage than the unit's next option. The home-minus-away feature edge enters the screen. The pre-registered expected direction was negative after player ratings and retained non-additive terms are accounted for. It is related to, but not algebraically equivalent to, the retained usage-concentration term: 30% and 28% top-two usage is concentrated but balanced, while 35% and 20% has a much larger handler gap.

The frozen screen produced a stable positive relationship instead: correlations were +0.0130, +0.0084, and +0.0085 from 2023-24 through 2025-26 (+0.0099 pooled), with positive decile slopes in all three seasons. The coordinate advances to a recursive fit unchanged, but not as evidence for the original "lack of a secondary ball handler" story. Conditional on the incumbent player and context model, it instead appears to identify a residual advantage from a clearer lead-handler allocation. See Screen a Frozen Feature.

A possession-weighted conditional audit also controls for the home-minus-away maximum frozen player prior. The standardized usage-gap coefficient remains positive in every target season: +1.27, +0.80, and +0.70 net-rating points per standard deviation (+0.94 pooled), versus unadjusted values of +1.33, +0.87, and +0.91. Its correlation with the superstar-proxy edge is only +0.12, +0.16, and +0.31. The feature is related to elite talent but is not reducible to it.

Screen artifact: artifacts/models/analysis/frozen_feature_screen/lead_secondary_usage_gap/frozen-feature-screen-lead_secondary_usage_gap-20260830T210743Z-2be1ccce.

Conditional audit artifact: artifacts/models/analysis/lead_secondary_usage_gap_conditioning/lead-secondary-usage-gap-conditioning-20260830T211716Z-6ba81f6c.

The direct three-season replay is marginally better on pooled possession, game margin, team net-rating, and Pythagorean-win RMSE, but worse on winner accuracy and not directionally consistent season by season. Its paired-bootstrap no-material-harm gate passes. See the complete lead-secondary usage-gap candidate record.

Secondary Candidates

Foul-Pressure Diversity

[ ] Frozen residual screen
[ ] Recursive candidate fit if supported

For \(f_i=\mathrm{FTA100}_i\), let \(q_i=f_i/\sum_j f_j\). Define

\[ \phi_{\text{foul diversity}}(U) =\left(\sum_i f_i\right)\left(1-\sum_i q_i^2\right), \]

with zero assigned when total FTA is zero. It distinguishes several credible foul-pressure sources from the same total generated by one player.

Size-Skill Coverage

[ ] Lock a single basketball-motivated contract
[ ] Frozen residual screen
[ ] Recursive candidate fit if supported

The candidate should combine a lower-tail size statistic with a middle or lower-tail perimeter-skill statistic. No formula is registered yet because an arbitrary product of height and shooting would be difficult to defend. The contract must specify the exact order statistics and season scaling before any outcome is inspected.

Defensive Weak Link

[ ] Identify a forward-safe defensive profile that is not derived from the target being predicted
[ ] Lock the order statistic
[ ] Frozen residual screen

The intended hypothesis is that a defense can be attacked through its weakest member. Blocks plus steals is not accepted as a general defensive-quality score, and using a fitted NAIL defensive split could double-count the rating state. This candidate remains a design problem rather than a registered formula.

Prediction-Only Relationship Features

These candidates may help real-team forecasting but are not intrinsic, portable descriptions of five-player composition. They should not silently penalize hypothetical or mixed-era units for never having played together.

Prior Teammate Continuity

[x] Build prior shared-possession pair table
[x] Frozen residual screen: advances
[x] Recursive candidate fit: tested, not promoted
[x] Coefficient and bootstrap decision: stable but no incremental lift

For the ten player pairs and strictly prior shared possessions \(c_{ij,t-1}\), define

\[ \phi_{\text{continuity}}(U,t) =\frac{1}{10}\sum_{i<j}\log\left(1+c_{ij,t-1}\right). \]

This tests repeatable coordination and familiarity. It belongs in a prediction layer or optional Lab control, not the portable lineup-composition score.

The v1.2.1.3 frozen screen found a positive residual relationship in all three target seasons. Weighted correlations were +0.0093, +0.0081, and +0.0106 for 2023-24 through 2025-26, with a pooled value of +0.0093. The residual spread from the lowest to highest continuity decile was positive in every season.

The full 30-season recursive fit retained a positive standardized coefficient in 27 of 29 states and a 99.75% positive one-sided mass share. The three-season frozen replay passed the paired non-inferiority gate but did not materially improve full-game RMSE; possession MAE and team net-rating RMSE worsened. The feature is therefore tested, not promoted: stable and probably real, but largely redundant with the incumbent forecast state. See the complete model result and the original screening result.

Prior Teammate Continuity Replacing Top-Two Assists

[x] Controlled recursive replacement fit
[x] Frozen three-season and playoff replay
[x] Coefficient and paired-bootstrap decision: not promoted

The follow-up candidate retains usage_concentration but replaces top_two_assists with prior teammate continuity. Continuity remains positive in 28 of 29 fitted states with a 99.999% positive directional-mass share and a median standardized weight of +1.12. Pooled full-game RMSE improves by only 0.0047, with a paired 95% interval of [-0.0364, +0.0272]. Possession MAE worsens by 0.000082, team NetRtg RMSE worsens from 3.2351 to 3.2853, and the feature remains unavailable for genuinely hypothetical teammate groups. The replacement is therefore stable and non-inferior, but not promoted. See the complete replacement result.

Shrunken Prior Pair Residual

[ ] Specify pair-effect shrinkage and minimum support
[ ] Frozen residual screen
[ ] Recursive candidate fit if supported

This is the direct chemistry candidate: estimate prior player-pair residuals, shrink them strongly toward zero as a function of pair exposure, and sum the ten frozen pair effects for a unit. It is more expressive than continuity but has the highest sparsity and multiple-testing risk in the registry.

Testing Contract

Every candidate follows the same sequence:

Pre-register the formula. Record the player inputs, shrinkage source, missing-value behavior, expected direction, portability, and support.
Check structural additivity. Reject or move the feature to the additive profile layer if it can be exactly decomposed into five player terms.
Run the frozen residual screen. Use the promoted model and 2023-24, 2024-25, and 2025-26 without refitting the candidate. Inspect weighted residual deciles and each season separately.
Run one controlled recursive fit. Add only the screened feature to the incumbent contract. The context correction must roll forward through every source season because it affects subsequent player states.
Audit coefficient history. Publish the full non-additive coefficient panel, directional one-sided mass, support, and displacement of the two incumbent terms.
Evaluate the frozen holdouts. Report the standard three-season regular and pooled playoff metrics and update the model tree for every completed full model, promoted or not.
Apply the paired game-block bootstrap gate. Report pooled and season-specific full-game RMSE differences. Passing the non-inferiority gate is necessary but does not establish improvement.
Record the decision here. Link the immutable screen, recursive fit, coefficient audit, bootstrap artifact, and model page before checking the candidate as resolved.

The residual screen is a compute-saving diagnostic, not a substitute for the recursive frozen replay. A feature can appear useful in a fixed residual and lose that signal after all player and context coefficients are refit jointly.

Result Template

Copy this block when a pending candidate is tested:

### Candidate Name

- [x] Frozen residual screen: `artifact path`
- [x] Recursive fit: `artifact path` or `did not advance`
- [x] Coefficient audit: `artifact path` or `not applicable`
- [x] Paired bootstrap: `artifact path` or `not applicable`
- **Decision:** retained / rejected / tested, not promoted
- **Reason:** one-sentence predictive and stability conclusion