47-PTN Template v1.0 | Chapter 4 — Testable Predictions & Falsification Plan

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Chapter 4 — Testable Predictions & Falsification Plan

I. Predictions

Unified format: PRED-ID | Scenario | Control Eq./Interface | Metric | Direction/Magnitude | Preconditions

• PRED-01 | Medium perturbation → arrival-time offset | see S20-1 | Metric: ΔT_arr (vs. baseline) | Direction: same sign as ∫ δ n_eff d ell; Magnitude: |ΔT_arr| ≥ τ_T | Preconditions: explicit gamma(ell), d ell, and recorded delta_form.
• PRED-02 | Phase-window consistency | see S21-2 | Metrics: ΔPhi, correlation r(Phi_ref, Phi_obs) | Direction: r ≥ r_phi; Magnitude: |ΔPhi| ≤ τ_phi | Preconditions: λ_ref matches observation band.
• PRED-03 | Paraxial propagation conservation | see S31-1 | Metric: cross-sectional flux error ε_flux | Direction: ε_flux → 0 at O(θ^2) | Preconditions: |θ| ≪ 1, slowly varying medium.
• PRED-04 | Continuity (mass conservation) | see S30-1 | Metric: ΔM = |∫ ρ dV|_t2 - |∫ ρ dV|_t1 | Direction: ΔM ≤ τ_M | Preconditions: source-free, lossless, stable boundaries.
• PRED-05 | Statistical robustness (observation noise) | see S40-1 | Metrics: residual normality/robust quantile gap Q_res, entries of Σ | Direction: Q_res within admissible band | Preconditions: H[·] linear or piecewise-linear.
• PRED-06 | Metrological closure | see S50-1 | Metric: dimension-check pass rate p_dim | Direction: p_dim = 1.0 | Preconditions: complete unit/dimension registry.
• PRED-07 | Detectable first-order background drift | n_eff = n_0 + ε n_1 | Metric: regression coefficient significance/effect β_ε | Direction: |β_ε| ≥ τ_ε and p ≤ α | Preconditions: ε within declared working range.

II. Criteria & Thresholds

1. Binary decision (positive/negative)

• Positive: at least m of k primary predictions pass their thresholds (pre-registered m/k), with all preconditions satisfied.
• Negative: any CORE prediction fails, or key dependencies (path/measure/version) are missing.

2. Statistical thresholds

• Significance: α = 0.01 (core metrics), α = 0.05 (secondary); multiple testing via Benjamini–Hochberg, FDR ≤ 0.1.
• Effect size: Cohen’s d ≥ d_min (default d_min = 0.35); correlation threshold r_phi ≥ 0.6.

3. Engineering thresholds (actionable)

• τ_T (arrival time): from metrology uncertainty u(T_arr), τ_T = k · u(T_arr), k = 3.
• τ_phi (phase): τ_phi = 3·σ(Phi_ref − Phi_obs) on calibration set.
• τ_M (mass conservation): relative error ≤ 0.5% on stable segments.

4. Pre-registration

• Pre-register the prediction list, thresholds, analysis formulas, exclusion rules, and stopping criteria; lock references[] and version.

5. Failure triggers

• Violation of metrology closure (p_dim < 1) or non-compliant citations (missing version/anchor) triggers package-level rejection.

III. Sample Size & Power

1. Continuous endpoints (difference in means)

• Assumptions: two-sided test, independent samples, approximately homogeneous variance.
• Target power: 1 − β = 0.9; α = 0.01 (core), α = 0.05 (secondary).
• Approximation:
  N_per_group ≈ 2 · ( z_{1−α/2} + z_{1−β} )^2 / d^2.
  With d ∈ {0.35, 0.5, 0.8} ⇒ N_per_group ≈ {≈190, 94, 38}.

2. Correlation tests (r)

• N ≈ ( z_{1−α/2} + z_{1−β} )^2 / ( atanh(r_target) )^2 + 3, for r_target = 0.6 ⇒ N ≈ 41.

3. Non-parametric/robust plans

• Use permutation/bootstrap for power; inflate sample size by ×(1.15–1.35) to cover skew/heavy-tails.

4. Stratification & pairing

• With batch/device/sky-region strata, use stratified randomization and paired designs; compute power with effective degrees of freedom df_eff.

5. Missingness & exclusions

• Predefine missing rate p_miss and exclusion rate p_excl; retain ≥ 85% effective samples; report a CONSORT-style flow diagram.

IV. Null & Permutation Tests

1. Null definitions

• H0-T: ΔT_arr = 0
• H0-Phi: r(Phi_ref, Phi_obs) = 0
• H0-flux: ε_flux distributed as zero-difference
• H0-M: ΔM distributed as zero-difference

2. Permutation schemes

• Free permutation: shuffle condition labels; compute permutation distributions for ΔT_arr, r, ε_flux; repetitions B ≥ 10^4.
• Stratified permutation: permute within strata (batch/device/sky-region) holding stratum sizes fixed.
• Phase-aligned permutation: align within the reference phase window before permutation to avoid band-mixing artifacts.

3. Robust intervals & reporting

• Report permutation p_perm and 95% CIs; if p_perm ≤ α and effect size meets threshold, mark as positive.

4. Multiplicity & composite predictions

• Control FDR for multi-endpoints; use sequential gatekeeping (core first, then secondary) to avoid power dilution.

5. Exclusion rules

• Only pre-registered rules allowed: missing path/measure, check_dim failure, sampling interruptions; freeze exclusion lists/thresholds before permutation.

V. Deliverables

• predictions.csv: fields id, metric, threshold, direction, alpha, beta, effect_size, core_flag, see[].
• falsify.md: includes null definitions, permutation schemes, exclusion rules, thresholds, and stopping criteria.
• power_plan.yaml: power/sample-size settings and stratified design.
• audit.jsonl: hash per analysis run with references[], version, and parameter snapshots.

Appendix: Machine-Readable Template (drop-in)

version: "1.0.0"

predictions:

- id: PRED-01

metric: "ΔT_arr"

threshold: "tau_T = 3*u(T_arr)"

direction: "sign(ΔT_arr) = sign(∫ δ n_eff d ell)"

alpha: 0.01

effect_size: "d>=0.35"

core: true

see:

- "EFT.WP.Core.Equations v1.1:S20-1"

- id: PRED-02

metric: "r(Phi_ref, Phi_obs)"

threshold: "r>=0.6; |ΔPhi|<=tau_phi"

alpha: 0.01

core: true

see:

- "EFT.WP.Core.Equations v1.1:S21-2"

- id: PRED-03

metric: "ε_flux"

threshold: "≈0 at O(θ^2)"

alpha: 0.05

core: false

see:

- "EFT.WP.Core.Equations v1.1:S31-1"

power:

alpha_core: 0.01

alpha_secondary: 0.05

target_power: 0.9

effect_sizes: [0.35, 0.5, 0.8]

null_tests:

permutations: 10000

scheme: ["free", "stratified", "phase-aligned"]

exports:

must_include: ["references", "version"]