34-EFT.WP.Astro.Acceleration v1.0 | Chapter 13 Inference, Criteria & Falsification

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Chapter 13 Inference, Criteria & Falsification

I. Abstract & Scope
This chapter defines cross-channel, cross-class workflows M72-* for unified inference, model comparison, and falsification: likelihood families and prior structures, evidence and evidence ratios, information criteria and cross-validation, dominance factors and energy/time masks, power analysis and sample sizing, comparator ablations and falsification-line recording, and uncertainty propagation for derived quantities such as alpha_loc(E), E_br, E_max, Pi, DM, RM. All symbols use English notation in backticks; SI units. Any time-of-arrival (ToA) quantity is handled in two parallel forms with explicit path gamma(ell) and measure d ell.

II. Dependencies & References

1. Unified symbols & units: Chapter 2 Tab. 2-1 and P12-*.
2. Kinematics & channels: Chapter 3 S20-; reconnection & shear: Chapter 4 S30-, Chapter 5 S40-; comparators & boundaries: Chapter 6 S45-.
3. Spectrum formation & transport: Chapter 7 S50-, Chapter 8 S52-.
4. GRB/FRB specifics: Chapter 10 M62-, Chapter 11 M64-.
5. Simulation & benchmarks: Chapter 12 M70-*.

III. Normative Anchors (added in this chapter, M72-*)

6. M72-0 (Model Families & Parameter Cards): define model family 𝓜 = {M_a} with parameter space Θ_a; parameter cards register {θ, bounds, transforms, priors, see}.
7. M72-1 (Likelihood Families):
   • Gaussian/continuous: L_G(D|θ) = N(y | μ(θ), Σ).
   • Poisson/counting: L_P(D|θ) = ∏_i Poisson(k_i | λ_i(θ)).
   • Poisson–Gaussian mixed: L_{PG} for counts with systematics.
   • Stokes/polarimetry: L_S over {Q,U,V} with covariance.
   • ToA residuals: L_{ToA} applied jointly to T_arr^A/T_arr^B, recording delta_form.
8. M72-2 (Prior Structures): non-informative, physically bounded, and hierarchical priors; hyperparameters φ shared across classes induce π(θ_c|φ) for class-level parameters θ_c.
9. M72-3 (Posterior & Evidence): p(θ|D,M) ∝ L(D|θ,M) * π(θ|M); evidence Z_M = ∫_{Θ} L(D|θ,M) π(θ|M) dθ; evidence ratio K_{ab} = Z_{M_a} / Z_{M_b}.
10. M72-4 (Information Criteria & CV): report {Z, logZ, K, WAIC, LOO-PSIS} and optionally ΔAIC/ΔBIC as approximations/ corroboration.
11. M72-5 (Dominance Factor & Masks): eta_dom(E[,t]) = A_channel(E[,t]) / max{A_other} (cf. Chapter 6); emit energy/time masks for spectrum/transport solvers.
12. M72-6 (Falsification Line): if k_STG → 0, beta_TPR → 0, gamma_Path → 0, sigma_shear → 0, xi_rate → 0, or chi_aniso → 0 and L/Z does not worsen (or K ≤ 1), the mechanism is falsified or nonessential; record as falsification_line.
13. M72-7 (Power Analysis & Sample Size): given effect size δ and thresholds K*/ΔWAIC*, find minimal N* achieving target power 1−β; output {N*, SNR*, band*}.
14. M72-8 (Uncertainty Propagation): sample {A_rec, A_shear, tau_esc, A_loss} and ToA delta_form, propagate to {alpha_loc, E_br, E_max, Pi, DM, RM}; report 68%/95% intervals and correlation matrices.
15. M72-9 (Comparator Ablations): with other conditions fixed, zero {A_rec, A_shear, A_dsa, A_turb} or a loss term; report {ΔlogZ, ΔWAIC, K} and curate a counterexample library.
16. M72-10 (Reproducibility & Logging): archive {code_hash, data_hash, rng_state, SimCfg, priors, delta_form, masks, environment}; all tables carry Unit/Dim and see: anchors.

IV. Body Structure

I. Inference Framework & Data-Item Composition

1. Observables packaged as O = { Φ(E), dN/dE, alpha_loc(E), Pi(E[,t]), PA(λ), tau_lag, T_arr, I(ν,t), DM, RM }, combined in block-diagonal or coupled likelihoods per dataset cards.
2. ToA handled in two forms:
   • T_arr^A = ( 1 / c_ref ) * ( ∫_{gamma(ell)} n_eff d ell ),
   • T_arr^B = ( ∫_{gamma(ell)} ( n_eff / c_ref ) d ell );
     residuals {r_A, r_B} are included jointly and delta_form recorded.

II. Implementation Notes for Likelihood–Prior–Evidence

• Use L_P/L_{PG} in count-dominated bands; L_G for continuous spectra; L_S for {Q,U,V}/PA; always include L_{ToA} for timing residuals.
• Hierarchical priors share hyperparameters across classes/regions for {E_cut, p, R_br, D(E)}; apply truncation or log transforms to physically bounded parameters.
• Prefer nested sampling or SMC; export evidence and posterior uncertainty.

III. Model Comparison, Criteria & Falsification

• Primary criteria: K_{ab} > K* (strong support for M_a) or ΔWAIC < −Δ* (M_a preferred over M_b).
• Execute falsification by driving control weights/parameters → 0; if logZ does not drop or WAIC does not rise, mark falsified/nonessential.
• Build dominance masks from eta_dom(E[,t]) to constrain Chapters 7–8 solvers and prevent cross-channel leakage fits.

IV. Comparator Design & Ablations

• Mechanism ablations: {only reconnection, only shear, reconnection+shear, +DSA, +turb}.
• Transport ablations: {diffusion only, +advection, +adiabatic, +collisional}.
• ToA ablations: {Form A only, Form B only, A+B with delta_form}.
• Metrics: ΔlogZ, ΔWAIC, K, SpecMAE, LagRMS, PA_RMS, ToAΔ.

V. Workflows & Deliverables (M-series)

• M72-1 (Dataset Aggregation): standardize dataset cards and covariances; unify units and conventions.
• M72-2 (Prior Setup): construct hierarchical priors under class physics and bounds; persist to parameter cards.
• M72-3 (Sampling & Evidence): run nested/SMC; output {posterior, Z, logZ} and posterior samples.
• M72-4 (Dominance Masks): compute eta_dom(E[,t]) from Chapter 6; feed back into Chapters 7–8.
• M72-5 (Ablations & Falsification): execute mechanism/transport ablations; emit falsification_line and counterexamples.
• M72-6 (Power & Sizing): estimate {N*, SNR*, band*} for targets K*/ΔWAIC*; provide sampling guidance.
• M72-7 (Uncertainty Propagation): push posteriors to {alpha_loc, E_br, E_max, Pi, DM, RM} and ToA residuals; report 68%/95% CIs.
• M72-8 (Reporting & Archival): deliver {posterior, evidence, masks, falsification_line, delta_form, repro_bundle}.

VI. Cross-References within/beyond this Volume

• Mechanisms & channels: Chapters 3–5; comparators & dominance: Chapter 6.
• Spectrum & transport: Chapters 7–8.
• GRB/FRB: Chapters 10–11.
• Simulation & benchmarks: Chapter 12.

VII. Validation, Criteria & Counterexamples

1. Positive criteria:
   • K_{ab} ≥ K* or ΔWAIC ≤ −Δ* supporting models with the target mechanism.
   • Dominance masks align with diagnostics (e.g., alpha_loc breaks, Pi evolution, tau_lag sign).
   • Dual-form ToA yields interpretable evidence differences or converges to a single preferred form given delta_form.
2. Negative criteria:
   • Driving a mechanism weight → 0 does not reduce logZ (or raise WAIC).
   • Dimensional/unit audits fail.
   • LOO/held-out CV shows degraded generalization.
3. Counterexample library: retain datasets/benchmarks where specific mechanisms are indistinguishable or falsified for regression and method improvement.

VIII. Summary & Handoff
M72-* closes the unified loop for inference, model comparison, and falsification, standardizing dominance masks, power analysis, and uncertainty propagation. Chapter 14 integrates data, pipelines, and benchmarks for release and audit trails.

V. Figures & Tables (this chapter)

• Tab. 13-1 Likelihood families vs data items (Φ/α_loc/Π/PA/tau_lag/T_arr/I(ν,t)/DM/RM → L_G/L_P/L_{PG}/L_S/L_{ToA}).
• Tab. 13-2 Priors & hierarchical structure (Param / Transform / Prior / Hyper / Bounds / See).
• Tab. 13-3 Evidence & decision criteria (Z, logZ, K, WAIC, LOO with threshold suggestions).
• Tab. 13-4 Falsification lines & ablations (k_STG, beta_TPR, gamma_Path, sigma_shear, xi_rate, chi_aniso → 0; outcomes).
• Tab. 13-5 Power & sample-planning (δ, K*, ΔWAIC*, 1−β → {N*, SNR*, band*}).
• Tab. 13-6 Deliverable fields (posterior, evidence, masks, falsification_line, delta_form, repro_bundle, code_hash, data_hash).