# Jump-interaction layer: the interaction hyper-mean theta_int (3360)

Script `07_analysis_code/3360_interaction_layer.py`; test `07_analysis_code/test_3360_interaction_layer.py`. Documented rule `qa/layers_20260929/3360_rule.json`, sha256 `194c1fe9785a5496ead529a01ffcedf39d85b5af62e1aac6e35c9678d52c629e` (written 2026-09-29 15:50:20 Central Daylight Time (2026-09-29T20:50:20Z), before any combined posterior was computed; verified at run time). Deterministic: no Monte Carlo in the layer.

theta is the autism-rate multiplier for a child who received both vaccines of a pair, beyond the product of the two single-vaccine rate ratios. theta = 1 means log-additive (no stacking); theta > 1 is synergy in the harmful direction. theta_int is the mean of log theta over exchangeable pairs. It is reported beside the main shared mean and is never multiplied into it.

## Headline

- **theta_int = 0.967 (0.881-1.063)** (posterior median, 95% CrI; 27 pairs from 3 sources). P(theta_int > 1) = 0.239; P(> 1.10) = 0.0046; P(> 1.25) = 2.69e-06; P(< 0.80) = 1.54e-04.
- **Synergy signal (rule: 95% CrI excludes 1 in the harmful direction): No.** The lower limit is 0.881; the upper limit, 1.063, is the largest mean stacking multiplier inside the 95% CrI.
- Between-pair heterogeneity: tau (log scale) 0.094 (95% CrI 0.025-0.192). A new exchangeable pair is predicted at 0.967 (0.762-1.229); P(new pair > 1.25) = 0.020.
- Information on the mean (share at the tau median): U.S. state pairs 97.5%, England 2.4%, U.S. core eta3 0.08%. Negative GLS weights at the tau median: us_state_pairs=4;england_pairs=0;us_core_eta3=0.
- Without the H1N1-season pair (influenza x Hib deferral): 0.952 (0.868-1.042), P(> 1.25) = 8.62e-07.
- Leave one source out: Without U.S. State Pairs 0.831 (0.429-1.602) (k = 3); Without England Pairs 0.971 (0.883-1.069) (k = 25); Without PCV × RotaShield 0.969 (0.882-1.064) (k = 26).
- Among the sensitivities that keep every source, the largest move comes from the U.S. dependence treatment, not from any single pair: treating the 24 U.S. pairs as independent instead of using 3305's estimated cross-pair correlation gives 1.019 (0.906-1.138) (P(> 1.25) = 2.4e-04). Under the estimated correlation (off-diagonal range -0.62 to 0.80, 51 state clusters), 4 rows carry small negative GLS weights (PCV7 × Hib Deferral -0.043, Varicella × Hepatitis A -0.021, Rotavirus × Hib Deferral -0.010, MMR Dose 2 × Hepatitis A -0.009; together -0.083) and positively correlated rows share weight. No synergy signal under either treatment.
- **Beside, not multiplied:** the 3320 main shared mean is 0.957 (0.599-1.517) (P(> 1.10) 0.275, P(> 1.25) 0.126). theta_int is a different estimand (departure from log-additivity of two effects, not a main effect) and does not enter it.
- **Contrast, older ecological hierarchy (2115/2116):** 1.009 (0.953-1.074) (ADDM-corrected 2026-09-25 value). Different estimand; see below.

## Inputs (27 pairs)

Published value = the source file's estimate and interval (U.S.: EIV-corrected, checked SE, t(50); England: calibrated, t(133); core eta3: posterior with its N(0, 0.709^2) prior). The pool uses log theta and SE as listed. Pair posterior = the pair's shrunken posterior under the primary layer (descriptive; no pair-level decision).

| Pair | Source | Published theta (95%) | log theta in pool | SE in pool | lambda | BH q | GLS weight | Pair posterior (95% CrI) |
|---|---|---|---|---|---|---|---|---|
| Varicella × PCV7 | U.S. states (3305) | 0.976 (0.325-2.924) | -0.0248 | 0.5466 | 0.710 | 0.965 | +0.0453 | 0.951 (0.757-1.163) |
| Varicella × Flu | U.S. states (3305) | 1.788 (0.632-5.055) | +0.5809 | 0.5175 | 0.710 | 0.965 | +0.0318 | 1.007 (0.837-1.299) |
| Varicella × Hepatitis A | U.S. states (3305) | 1.535 (0.500-4.706) | +0.4283 | 0.5579 | 0.710 | 0.965 | -0.0208 | 0.982 (0.790-1.256) |
| Varicella × Rotavirus | U.S. states (3305) | 0.759 (0.362-1.592) | -0.2755 | 0.3686 | 0.624 | 0.965 | +0.0324 | 0.924 (0.734-1.100) |
| Varicella × Hib Deferral | U.S. states (3305) | 1.181 (0.594-2.350) | +0.1664 | 0.3425 | 0.710 | 0.965 | +0.0116 | 0.992 (0.823-1.246) |
| Varicella × PCV13 | U.S. states (3305) | 0.931 (0.529-1.639) | -0.0714 | 0.2816 | 0.710 | 0.965 | +0.0136 | 0.967 (0.795-1.177) |
| PCV7 × Flu | U.S. states (3305) | 0.454 (0.150-1.377) | -0.7892 | 0.5522 | 0.710 | 0.965 | +0.0594 | 0.933 (0.731-1.128) |
| PCV7 × Hepatitis A | U.S. states (3305) | 0.389 (0.101-1.497) | -0.9454 | 0.6717 | 0.710 | 0.965 | +0.0199 | 0.968 (0.765-1.207) |
| PCV7 × Hib Deferral | U.S. states (3305) | 0.975 (0.456-2.086) | -0.0249 | 0.3785 | 0.710 | 0.965 | -0.0431 | 0.963 (0.766-1.198) |
| PCV7 × PCV13 | U.S. states (3305) | 0.608 (0.260-1.421) | -0.4977 | 0.4226 | 0.710 | 0.965 | +0.0826 | 0.904 (0.711-1.072) |
| Flu × Hepatitis A | U.S. states (3305) | 0.921 (0.642-1.320) | -0.0826 | 0.1794 | 0.710 | 0.965 | +0.0897 | 0.880 (0.741-1.012) |
| Flu × Rotavirus | U.S. states (3305) | 1.008 (0.696-1.460) | +0.0082 | 0.1843 | 0.710 | 0.965 | +0.0358 | 1.002 (0.865-1.167) |
| Flu × Hib Deferral | U.S. states (3305) | 1.501 (1.110-2.030) | +0.4063 | 0.1502 | 0.780 | 0.354 | +0.0728 | 1.078 (0.926-1.307) |
| Flu × PCV13 | U.S. states (3305) | 1.028 (0.782-1.352) | +0.0276 | 0.1362 | 0.710 | 0.965 | +0.0654 | 0.983 (0.854-1.144) |
| Hepatitis A × Rotavirus | U.S. states (3305) | 0.972 (0.687-1.376) | -0.0279 | 0.1728 | 0.673 | 0.965 | +0.0773 | 1.007 (0.885-1.160) |
| Hepatitis A × Hib Deferral | U.S. states (3305) | 1.248 (0.822-1.894) | +0.2217 | 0.2077 | 0.710 | 0.965 | +0.0928 | 0.973 (0.836-1.146) |
| Hepatitis A × PCV13 | U.S. states (3305) | 1.115 (0.594-2.091) | +0.1088 | 0.3131 | 0.710 | 0.965 | +0.0882 | 0.951 (0.808-1.131) |
| Rotavirus × Hib Deferral | U.S. states (3305) | 0.844 (0.558-1.277) | -0.1697 | 0.2062 | 0.710 | 0.965 | -0.0100 | 0.899 (0.724-1.063) |
| Rotavirus × PCV13 | U.S. states (3305) | 1.101 (0.650-1.864) | +0.0961 | 0.2623 | 0.710 | 0.965 | +0.0300 | 0.991 (0.843-1.201) |
| Hib Deferral × PCV13 | U.S. states (3305) | 1.115 (0.508-2.446) | +0.1086 | 0.3912 | 0.710 | 0.965 | +0.0810 | 0.982 (0.818-1.209) |
| MMR Dose 2 × Hepatitis B | U.S. states (3305) | 0.967 (0.688-1.360) | -0.0331 | 0.1695 | 0.811 | 0.965 | +0.0772 | 0.995 (0.858-1.173) |
| MMR Dose 2 × Varicella | U.S. states (3305) | 0.627 (0.254-1.549) | -0.4662 | 0.4497 | 0.710 | 0.965 | +0.0116 | 0.982 (0.791-1.226) |
| MMR Dose 2 × Hepatitis A | U.S. states (3305) | 1.059 (0.489-2.293) | +0.0572 | 0.3846 | 0.710 | 0.965 | -0.0093 | 0.975 (0.789-1.214) |
| Hepatitis B × Varicella | U.S. states (3305) | 1.687 (0.434-6.567) | +0.5231 | 0.6766 | 0.710 | 0.965 | +0.0403 | 0.968 (0.777-1.217) |
| Rotavirus → MenB (England) | England (3306) | 0.706 (0.238-2.101) | -0.3475 | 0.5511 | - | - | +0.0097 | 0.958 (0.754-1.195) |
| MMR → Rotavirus (England) | England (3306) | 0.947 (0.402-2.230) | -0.0549 | 0.4333 | - | - | +0.0141 | 0.964 (0.767-1.203) |
| PCV × RotaShield (U.S. Core) | U.S. core eta3 (p18) | 0.792 (0.224-2.848) | -1.4522 | 1.6476 | - | - | +0.0008 | 0.963 (0.755-1.216) |

GLS weight = the input's weight in the data part of E[mu | tau at its posterior median, y] (weights sum to 1; negative weights arise from the estimated cross-pair correlation). Pair posteriors borrow from correlated rows as well as from the mean, so under the estimated correlation a pair can move away from its own estimate (for example Flu x Hepatitis A).

- U.S. state pairs: the 24 `core` rows of `3305_interactions_us.csv`, log theta / lambda and teen-kappa SE / lambda (lambda pair-specific where 3305 simulated it, else 0.71), with 3305's own cross-pair correlation from state influence functions (24 x 24 sub-matrix, PD-projected as in 3305). These are exactly the inputs of 3305's derived pool (reproduced to machine precision, below; 3305 labels that combination of checker ingredients 'derived, not separately checked'). The influenza x Hib deferral pair is included; it is the only pair whose interval excludes 1 (p 0.009, BH q 0.354) and it coincides with the 2009 H1N1 season and the 2007-09 Hib shortage.
- England pairs: 3306 `england.pair_table` at the checker's reference NIS anchor, calibrated SEs, checker jackknife correlation -0.2606. The brief's 0.850 (0.498-1.457) is 3306's pool at the independence bound; it is reproduced exactly and run as the sensitivity `england_independence`.
- U.S. core eta3: the reported posterior 0.792 (0.224-2.848) (log mean -0.2270, log SD 0.6514) includes the core prior N(0, 0.7092^2). Dividing it out gives the likelihood N(-1.452, 1.648^2): the joint U.S. likelihood barely informs this interaction. Check against the draws: the U.S.-alone posterior reproduces exactly from both p18 core files (0.779 (0.218-2.820)); reweighted to the layer prior N(0, (log 2)^2) the exact draw posterior is 0.784 (0.230-2.717) (ESS 1305) against the Gaussian approximation 0.792 (0.229-2.747). Without its prior the draw weights have ESS 29, so a non-parametric likelihood from the draws is not usable. Independence from the state pairs is assumed (shared IDEA series, different contrast); `loso_us_core_eta3` shows its influence.

## Pooled theta_int: primary and every documented sensitivity

| Scenario | Category | k | theta_int (95% CrI) | P>1.10 | P>1.25 | tau (95% CrI) | New pair (95%) | Synergy |
|---|---|---|---|---|---|---|---|---|
| primary | primary | 27 | 0.967 (0.881-1.063) | 0.0046 | 2.7e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |
| only_us_state_pairs | per-source | 24 | 0.972 (0.884-1.071) | 0.0070 | 7.1e-06 | 0.095 (0.026-0.196) | 0.97 (0.76-1.24) | no |
| only_england_pairs | per-source | 2 | 0.871 (0.446-1.733) | 0.2381 | 1.4e-01 | 0.264 (0.012-0.943) | 0.87 (0.30-2.66) | no |
| only_us_core_eta3 | per-source | 1 | 0.815 (0.231-2.873) | 0.3203 | 2.5e-01 | 0.332 (0.015-1.105) | 0.81 (0.17-4.08) | no |
| no_h1n1_pair | H1N1 pair | 26 | 0.952 (0.868-1.042) | 0.0014 | 8.6e-07 | 0.077 (0.013-0.176) | 0.95 (0.77-1.17) | no |
| loso_us_state_pairs | leave-one-source-out | 3 | 0.831 (0.429-1.602) | 0.1899 | 1.0e-01 | 0.259 (0.012-0.924) | 0.83 (0.28-2.42) | no |
| loso_england_pairs | leave-one-source-out | 25 | 0.971 (0.883-1.069) | 0.0065 | 6.1e-06 | 0.095 (0.025-0.196) | 0.97 (0.76-1.24) | no |
| loso_us_core_eta3 | leave-one-source-out | 26 | 0.969 (0.882-1.064) | 0.0049 | 3.1e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |
| england_independence | England dependence | 27 | 0.966 (0.880-1.061) | 0.0041 | 2.3e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |
| england_max_dependence | England dependence | 27 | 0.968 (0.881-1.065) | 0.0050 | 3.3e-06 | 0.094 (0.025-0.193) | 0.97 (0.76-1.23) | no |
| england_plus_mmr5y_pcv | England added pair | 28 | 0.968 (0.882-1.064) | 0.0048 | 2.8e-06 | 0.094 (0.025-0.191) | 0.97 (0.76-1.23) | no |
| us_lambda_1 | U.S. attenuation | 27 | 0.975 (0.912-1.044) | 0.0006 | 3.4e-08 | 0.067 (0.017-0.139) | 0.97 (0.82-1.16) | no |
| us_lambda_0_62 | U.S. attenuation | 27 | 0.965 (0.870-1.069) | 0.0070 | 7.9e-06 | 0.105 (0.030-0.212) | 0.97 (0.74-1.26) | no |
| us_pairs_independent | U.S. dependence | 27 | 1.019 (0.906-1.138) | 0.0876 | 2.4e-04 | 0.078 (0.004-0.230) | 1.02 (0.78-1.30) | no |
| us_core_plus_hybrid | U.S. tier | 32 | 0.974 (0.886-1.070) | 0.0064 | 3.3e-06 | 0.129 (0.073-0.218) | 0.97 (0.73-1.31) | no |
| tau_prior_hn_log1_25 | prior | 27 | 0.967 (0.882-1.062) | 0.0042 | 1.8e-06 | 0.092 (0.023-0.184) | 0.97 (0.77-1.22) | no |
| tau_prior_hn_1 | prior | 27 | 0.967 (0.881-1.063) | 0.0046 | 2.9e-06 | 0.095 (0.025-0.194) | 0.97 (0.76-1.23) | no |
| mu_prior_x0_5 | prior | 27 | 0.968 (0.882-1.063) | 0.0044 | 2.3e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |
| mu_prior_x2 | prior | 27 | 0.967 (0.880-1.063) | 0.0046 | 2.8e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |
| eta3_us_alone | eta3 input | 27 | 0.967 (0.881-1.063) | 0.0045 | 2.7e-06 | 0.094 (0.025-0.192) | 0.97 (0.76-1.23) | no |

Priors: mu ~ N(0, (log 2)^2), tau ~ half-normal(0.5) unless the row says otherwise. Every row is exact on a 6,001-point tau grid (0-3); the tau mass within 0.25 of the grid edge is below 2.5e-08 in every row.

## Listed, not pooled

| Group | Item | Value (95%) | Reason |
|---|---|---|---|
| Japan sequence (3306) | Japan HIBPCV × VZV | - | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Japan sequence (3306) | Japan HIBPCV × HBV | - | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Japan sequence (3306) | Japan VZV × HBV | - | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Japan sequence (3306) | Japan JE_AGE3 × HIBPCV | 1.037 (0.786-1.368) | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Japan sequence (3306) | Japan JE_AGE3 × VZV | 0.726 (0.444-1.187) | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Japan sequence (3306) | Japan JE_AGE3 × HBV | 0.941 (0.634-1.398) | theta not identified: routine coverage saturates in every prefecture, joint receipt unobserved, NS fits run to the boundary (3306 checker); listed, not pooled |
| Sweden (3306) | Sweden PCV × HEPB_anchor (upper) | 0.446 (0.048-4.134) | theta not identified (permutation SD >= 1.07; imputation-dependent HepB); not eligible |
| Sweden (3306) | Sweden HEPB_anchor × ROTA_u85 (lower) | 0.301 (0.018-4.958) | theta not identified (permutation SD >= 1.07; imputation-dependent HepB); not eligible |
| Sweden (3306) | Sweden HEPB_anchor × rotavirus (R_min) (rotavirus recoded) | 0.276 (0.047-1.624) | theta not identified (permutation SD >= 1.07; imputation-dependent HepB); not eligible |
| Sweden (3306) | Sweden HEPB_anchor × rotavirus (R_mid) (rotavirus recoded) | 0.268 (0.046-1.553) | theta not identified (permutation SD >= 1.07; imputation-dependent HepB); not eligible |
| Germany (3306) | Germany VAR1 × PCV4 (lower) | 0.214 (0.029-1.605) | theta not identified: estimable only under the Frechet-lower rule, placebos equal the estimate; not eligible |
| Germany (3306) | Germany VAR1 × MENC (indep) | 0.291 (0.008-10.739) | theta not identified: estimable only under the Frechet-lower rule, placebos equal the estimate; not eligible |
| Germany (3306) | Germany VAR1 × MENC (lower) | 0.416 (0.070-2.484) | theta not identified: estimable only under the Frechet-lower rule, placebos equal the estimate; not eligible |
| Schedule load (3307) | Schedule load, 17 jurisdictions | - | exposure dataset (583 schedule-change events; 0-24 month load by birth cohort): no likelihood, theta, individual-scale r, placebo or schedule-load dependence test; the checker marked the theta = 1 test 'not applicable'. No identified interaction estimand |
| Hib booster deferral (3304): timing estimand | Hib booster on schedule vs late/never: corrected national primary | 1.055 (0.717-1.553) | a timing estimand (booster by 19 months vs late or never; primary series unaffected), not an interaction: reported separately, never pooled into theta_int |
| Hib booster deferral (3304): timing estimand | Hib booster deferral: cross-state dip-intensity pulse (implied r) | 0.992 (0.812-1.235) | a timing estimand (booster by 19 months vs late or never; primary series unaffected), not an interaction: reported separately, never pooled into theta_int |
| U.S. ADDM site panel (3305) | ADDM VAR × PCV7 | 0.654 (0.223-1.822) | separate panel, never pooled with the state panel (population overlaps it); corroborating context only (3305) |
| U.S. state rows excluded (3305) | 55 excluded state rows (not identified / not separable / superseded) | - | pool_tier 'excluded' in 3305; includes ROTASHIELD x PCV7 2.35 (0.26-21.08), not identified, whose contrast is the core eta3's |
| England pairs not eligible (3306) | PCV→HepB, Rotavirus→HepB, MMR→HepB, MMR→MenB | - | identified in V1 but ruled not eligible by the checker; fitted in 3306 only for its audit pools |

Japan values are the NS upper-bound runs (the run 3309 reads); none is identified. Sweden and Germany values are in-model estimates that the checker ruled not identified. The Hib booster deferral is a timing contrast (on schedule by 19 months vs late or never) and is reported only here, beside the layer.

## Contrast: the older ecological schedule-interaction hierarchy (2115/2116)

2116 `interaction_core_reference` = 1.009 (0.953-1.074) (P(> 1.10) 0.0088), pooling 5 area-level product terms (U.S. PCV x baseline schedule, ADDM PCV x prior schedule, Wales rotavirus x PCV, the U.S. IDEA pair panel, the England UTLA pair panel). Its unit is the departure from log-additivity per simultaneous +10pp coverage shift in both vaccines: a population-scale product term in area coverage. theta_int's unit is the per-child multiplier for joint receipt. A joint +10pp coverage shift moves only a fraction of children into joint receipt, so a population-scale product term near 1 with a narrow interval can coexist with a much wider range of per-child theta; the product term also absorbs any curvature of the coverage-rate relation that the mixture equation models explicitly. The two reuse the same U.S. IDEA state and England upper-tier-area outcome series. They answer different questions and are never combined.

## What this establishes and what it does not

- **Establishes:** across the 27 identified pairs (24 U.S. vaccine pairs, 2 England sequences and the U.S. PCV x RotaShield core term), the mean stacking multiplier is 0.97 (0.88-1.06); a mean supra-multiplicative penalty of 25% or more has posterior probability <0.1%, and of 10% or more 0.5%. Without the H1N1-season pair it is 0.952 (0.868-1.042); without England or without the core eta3 the medians are 0.969-0.971 with upper limits 1.064-1.069. The U.S. state pairs carry the result: without them only 3 weak inputs remain (0.831 (0.429-1.602), P(> 1.25) 0.10), consistent with the primary but unable to rule out moderate synergy on their own.
- **Pair level:** individual pairs are much less certain. The predictive interval for a new pair, 0.97 (0.76-1.23), is the layer's statement about an unstudied pair, and it holds only if that pair is exchangeable with the studied ones. It does not exclude a specific pair with theta near its upper end.
- **Does not establish same-child biology.** theta is identified from area-level (state, upper-tier area) variation in joint coverage, converted to the individual scale through the mixture equation. Joint receipt is measured in survey samples (U.S. NIS) or anchored/bounded (England, U.S. core) and is never linked to the same child's outcome. Pairs never studied here (for example DTaP or IPV with anything, or the Japanese, Swedish and German pairs whose theta is not identified) are covered only by the exchangeability assumption.
- **Does not test total cumulative load.** The layer is pairwise. It says nothing about three-way or higher-order stacking, dose spacing, or the total 0-24 month load. 3307 supplies load as an exposure dataset without any identified estimand, and the Hib deferral (3304) is a timing contrast. A load or timing claim needs its own estimand.
- The H1N1-coincident influenza x Hib deferral pair (1.50 (1.11-2.03), BH q 0.354) is a pair-level anomaly with a known calendar confound: its excess sits in the Hib-booster margin among children without influenza vaccine (3305). The layer neither hides it nor lets it decide the result.

## Reproduction and diagnostics

- 3305 core_empirical_R_teen_kappa_lambda0.71 mu_med: published 0.9719629868, reproduced 0.9719629868 (|diff| 1.1e-16; PASS).
- 3305 core_empirical_R_teen_kappa_lambda0.71 mu_lo: published 0.8836258707, reproduced 0.8836258707 (|diff| 1.1e-16; PASS).
- 3305 core_empirical_R_teen_kappa_lambda0.71 mu_hi: published 1.070598487, reproduced 1.070598487 (|diff| 4.4e-16; PASS).
- 3305 core_empirical_R_teen_kappa_lambda1.0 mu_med: published 0.9779214342, reproduced 0.9779214342 (|diff| 2.2e-16; PASS).
- 3305 core_empirical_R_teen_kappa_lambda1.0 mu_lo: published 0.9140203288, reproduced 0.9140203288 (|diff| 2.2e-16; PASS).
- 3305 core_empirical_R_teen_kappa_lambda1.0 mu_hi: published 1.048324441, reproduced 1.048324441 (|diff| 4.4e-16; PASS).
- 3306 England core pool independence (checker jackknife) theta_med: published 0.8499441531, reproduced 0.8499441531 (|diff| 0.0e+00; PASS).
- 3306 England core pool independence (checker jackknife) lo: published 0.4983499798, reproduced 0.4983499798 (|diff| 0.0e+00; PASS).
- 3306 England core pool independence (checker jackknife) hi: published 1.457308183, reproduced 1.457308183 (|diff| 0.0e+00; PASS).
- 3306 England core pool nis_anchor (checker jackknife) theta_med: published 0.8632011067, reproduced 0.8632011067 (|diff| 0.0e+00; PASS).
- 3306 England core pool nis_anchor (checker jackknife) lo: published 0.4834947788, reproduced 0.4834947788 (|diff| 0.0e+00; PASS).
- 3306 England core pool nis_anchor (checker jackknife) hi: published 1.545616137, reproduced 1.545616137 (|diff| 0.0e+00; PASS).
- U.S.-alone eta3 rr_median from the p18 core draws: published 0.7785505216, reproduced 0.7785505216 (|diff| 0.0e+00; PASS).
- U.S.-alone eta3 rr_low from the p18 core draws: published 0.2175937086, reproduced 0.2175937086 (|diff| 0.0e+00; PASS).
- U.S.-alone eta3 rr_high from the p18 core draws: published 2.820107958, reproduced 2.820107958 (|diff| 0.0e+00; PASS).
- U.S.-alone eta3 log_mean from the p18 core draws: published -0.2416211994, reproduced -0.2416211994 (|diff| 5.3e-16; PASS).
- U.S.-alone eta3 log_sd from the p18 core draws: published 0.646491521, reproduced 0.646491521 (|diff| 0.0e+00; PASS).
- Input files unchanged since the rule was written: yes.
- Rule sha256 194c1fe9785a5496ead529a01ffcedf39d85b5af62e1aac6e35c9678d52c629e matches `3360_rule.sha256`.

## Files

- `3360_interaction_layer.csv`: one table, `row_type` input / pooled / not_pooled / contrast / beside.
- `3360_interaction_layer.json`: rule hash, input hashes, settings, every scenario (with source information shares and negative GLS weights), pair posteriors, the eta3 draw check, reproduction.
- `08_figures/3360_interaction_layer.png`.
- `qa/layers_20260929/3360_rule.json` (+ `.sha256`, writer `write_3360_rule.py`).