2 Environment-Forward Test for the “Potential Term” in Strong-Gravitational-Lens Time Delays

Home ／ Appendix-Prediction and Falsification

This chapter follows the publication template for the falsification program. It uses plain language, avoids equations, and keeps the structure fixed. Definitions are given at first mention for general readers.

I. One-Sentence Goal

Use only environmental information—for example, the level of voids, filaments, and nodes along the line of sight; group or cluster membership near the lens; and proxies for external convergence and shear—to forecast in advance the direction and relative strength of the “potential term” in the time delays of strong gravitational lenses. Then compare those text-only forecasts with the measured time delays between multiple images. If the forecasts repeatedly land on the correct sign and ordering with a clear, monotonic dependence on environment, the result supports the claim associated with Energy Filament Theory (EFT). If the forecasts perform no better than chance or show no environmental link, the claim is disfavored.

Reader notes:

• The potential term means the part of the total delay—geometric plus gravitational (Shapiro-like)—that should be predictable from environmental mass structure outside the main lens model.
• Environment-forward means we make prior, blinded predictions from the environment template alone, without looking at any time-delay data.

II. What to Measure

• Environment-forward “potential-level” labels: For each lens system, assign a unified qualitative label—strong / medium / weak—and a sign expectation (for example, higher external convergence implies a net lengthening of delays).
• Within-system pairwise predictability: For common image pairs in a system (brightest pair, widest-separation pair, and any pair that includes a saddle-point image), state in words which pair should be more affected by environmental enhancement or suppression.
• Cross-system consistency and environmental gradient: Across corridors dominated by filaments or nodes versus voids, evaluate whether the hit rate, wrong-sign rate, and null rate of the forecast-versus-measurement comparison change monotonically with environment class.

III. How to Do It

1. Sample and environment templates:
   Build a set of strong-lens systems along several “sky corridors” that intentionally span field galaxies, group environments, and cluster cores, with at least dozens in each category. For every system construct a two-layer environment template: a lens-neighborhood layer (host morphology, companions, local mass proxies) and a line-of-sight layer (void level, filament strength, distance to nodes, and proxies for external convergence and shear). Select control systems in nearby sky areas but with contrasting environment grades, and assemble a lens-like non-lensed control sample (similar morphologies but without multiple imaging) to probe blind-judgment biases.
2. Forward prediction and blinding:
   An independent prediction team uses only the environment templates to produce a text forecast card per system. Each card states: the qualitative potential level (strong/medium/weak), which image-pair type should be most affected, and the expected increase/decrease and relative ordering of delays. A separate measurement team—without access to the cards—derives final time delays using its standard pipelines (optical, radio, or high-energy light curves). A third-party arbiter aligns forecast cards with measured delays under pre-registered rules and computes hit, wrong-sign, and null rates, split by target vs. control, environment grade, and image-pair category.
3. Positive/negative controls and removal of false positives:
   Treat as positive controls those systems in group/cluster or filament-dominated environments where the forecasted “potential enhancement” is confirmed, especially for pairs containing saddle-point images; field/void corridors should show weaker effects. As negative controls, randomly permute forecast cards across systems or swap environment templates; performance should then drop to near chance. If one instrument or band drives the signal alone, require cross-instrument replication before counting it as positive. To separate path terms, test frequency independence of delay differences across bands; any frequency-dependent (dispersive) behavior is attributed to path or pipeline effects and excluded from potential-term conclusions.
4. Systematics and safeguards (three items):
   • Mass–shape degeneracy: Different mass profiles can mimic similar image positions and delays. Safeguard: use model ensembles to report stable intervals rather than single values, and constrain free parameters with the environment layers.
   • Source variability and microlensing: Intrinsic variability and stellar microlensing can add time-dependent features. Safeguard: use multi-band, long-baseline monitoring and down-weight high microlensing-probability periods.
   • Incomplete environment templates: Line-of-sight mass may be undercounted. Safeguard: assign a template confidence grade per system and stratify statistics; treat low-confidence templates as qualitative only.
5. Execution and transparency:
   Pre-register the forecast rules, grading standards, metrics, and exclusion criteria. Hold out a confirmation subset within each environment class. Repeat across seasons or years to test stability. Publicly release the forecast cards, environment-template summaries, and comparison results (in plain language) to enable external replication. This chapter forms a closed loop with the chapters on flux-ratio and odd-image statistics, saddle-point image demographics, and environment-predictable residuals in time-delay cosmology; results should cross-reference across those chapters.

IV. Pass/Fail Criteria

1. Support (passes):
   • In at least two environment classes, the environment-forward forecast achieves a hit rate significantly above chance for the sign and magnitude ordering of time delays.
   • Forecasts about which image-pair category (especially those including saddle-point images) should be most affected are consistently validated.
   • The results replicate across instruments, bands, and processing pipelines, and remain independent of observing frequency.
2. Refutation (fails):
   • Hit rates remain near chance over time, or are driven mainly by a single institution or single pipeline.
   • Apparent “hits” flip with frequency or re-scale in a dispersive manner, pointing to non-potential origins.
   • Differences between target vs. control and across environment grades are not significant, preventing any attribution to environment.