9 Smooth-Statistical-Field Account of Flux-Ratio Anomalies and Odd-Image Fraction in Strong Gravitational Lensing

Home ／ Appendix-Prediction and Falsification

This chapter follows the publication template for the falsification program. It uses plain language, avoids equations, and preserves the fixed structure. For general readers: a “smooth statistical field” means a slowly varying, grain-free contribution along the line of sight that nudges the magnification of all images in a correlated way rather than adding spiky, image-by-image noise.

I. One-Sentence Goal

Explain two recurring facts in strong gravitational lenses—flux-ratio anomalies and the odd-image fraction (the detection rate of the central “odd” image)—with a smooth statistical field acting along the path. If one gently varying, grain-free contribution produces same-sign, correlated magnification residuals among images within the same system, grows monotonically in corridors dominated by filaments or nodes, and predictably modulates the visibility of central images, it supports the claim in Energy Filament Theory (EFT). If instead the anomalies are driven mainly by small-scale clumps, strong color/time dependence, or no link to environment, the claim is disfavored.

II. What to Measure

• Shape and correlation of flux-ratio residuals: Under a unified macro model, compare each image’s flux with model expectations. Look for same-direction offsets, image-type dependence (minima vs saddle images), and violations of fold/cusp relations that appear as a narrow distribution with correlated uplift, not as high-frequency scatter.
• Cross-band and multi-epoch stability: Between radio/mid-infrared (less affected by stellar microlensing and dust) and optical, test whether the sign and rank order of residuals remain stable. After aligning time delays, check that residuals do not vary excessively across epochs.
• Odd-image statistics and morphology: For the central image, record detection rate, angular size, and color behavior, and summarize any near-center brightening or suppression. Test for a monotonic link to environment class (void → filament/node).
• Environmental gradient: In filament/node corridors, expect larger amplitudes but stronger inter-image correlation and smaller scatter; the odd-image fraction should shift predictably in step with these trends.

III. How to Do It

1. Samples and environment templates:
   Assemble hundreds of quad/double quasar lenses and galaxy–galaxy arcs, including radio and mid-infrared subsets. Provide a shared macro-model family and image-level fluxes for each system. For every target, build a two-layer environment template: a line-of-sight layer (void fraction, filament strength, distance to nodes, proxies for external convergence and shear) and a neighborhood layer (group/cluster context of the lens host). Match each target to a nearby control with similar redshift and brightness but a different environment grade, and include “pseudo-lens” morphologies to check blind-judgment bias.
2. Forward prediction and blinding:
   • Environment-only forecast (prediction team): Without access to fluxes or image labels, issue a text prediction card per system that states three items:
     1. Residual pattern: “same-sign correlated uplift/suppression” or “saddle-dominated deviation”;
     2. Residual scatter: narrow / medium / broad;
     3. Odd-image tendency: easier / harder to detect the central image in that environment.
   • Measurement (independent pipelines): Under the shared macro model and with time delays aligned, reconstruct image-level fluxes and uncertainties in radio/mid-infrared/optical; consolidate across epochs to form a residual vector and inter-image correlation matrix. Report odd-image decisions (detected/upper limit), angular size, and color notes.
   • Arbitration: A third party, following pre-registered rules, aligns prediction cards with measured residuals/odd-image outcomes, computing hit, wrong-sign, and null rates and stratifying by target vs control and environment grade.

IV. Positive/Negative Controls and Removal of False Positives

1. Positive controls:
   • In filament/node corridors, flux-ratio residuals show same-sign correlation (higher inter-image correlation coefficients) and a narrower scatter, not point-like outliers.
   • The odd-image fraction shifts monotonically with environment (for example, higher external convergence makes the central image harder to detect or smaller in angle).
   • The direction and rank order of residuals match between radio/mid-infrared and optical.
2. Negative controls:
   • Shuffle environment labels or swap templates: hit rates should drop toward chance.
   • Inject small-scale substructure into simulations: this should create larger high-frequency scatter and weaker inter-image correlation, separating it from the smooth-field signature.
   • If residuals are strong in optical but absent in radio/mid-infrared and vary strongly with epoch, attribute them to stellar microlensing/dust, not to a smooth field.
3. Separating macro-model degeneracies: Use model-family ensembles and report stable intervals rather than single values; any apparent “hit” that occurs only under one macro model does not count as positive.

V. Systematics and Safeguards (Three Items)

• Intrinsic source variability and delay mis-alignment: Time variability can masquerade as anomalies. Safeguard: enforce strict delay alignment, use narrow time windows or simultaneous photometry, and validate with epoch differencing.
• Stellar microlensing and dust extinction: These introduce color dependence and strong time variability. Safeguard: anchor conclusions on radio/mid-infrared and narrow-line region ratios; treat optical as a comparison channel; down-weight or hold out targets with strong color/time swings.
• Macro-model degeneracy (mass sheets/anisotropy): Different macros can fit image positions yet predict different fluxes. Safeguard: use ensemble modeling and node-based reporting (strong/medium/weak hits), and stratify statistics to suppress model-selection bias.

VI. Execution and Transparency

Pre-register the macro-model families, environment grading, prediction-card format, metrics, and exclusion rules. Keep hold-out subsets at each environment grade for final confirmation. Replicate across independent surveys and teams. Publicly release environment labels, prediction cards, plain-language summaries of image-level fluxes and residuals, odd-image criteria, and key intermediate artifacts for outside review. This chapter forms a closed loop with the chapters on environment-forward time-delay potential, saddle-image “over-suppression” statistics, and environment-predictable residuals in time-delay cosmology; cross-referencing is required.

VII. Pass/Fail Criteria

1. Support (passes):
   • In two or more environment classes, environment-forward predictions of residual pattern (correlated/which image type) and scatter level achieve significantly above-chance hit rates.
   • In filament/node corridors, flux-ratio residuals show stronger inter-image correlation and less high-frequency scatter—not scattered point-like outliers—and the odd-image fraction changes monotonically with environment in the same direction as the flux-residual gradient.
   • Results are consistent across radio/mid-infrared and optical, and remain robust when macro-model families and processing pipelines are swapped.
2. Refutation (fails):
   • Hit rates hover near chance or are driven by a single pipeline/macro model.
   • Anomalies show strong color/time dependence or high-frequency scatter consistent with small-scale substructure or microlensing, not with a smooth field.
   • Differences between targets and controls and across environment grades are insignificant, offering no support for an environment-linked smooth statistical field.