4 Rotation Curves—Weak-Lensing Consistent Fit (No Object-Specific Halo Profiles)

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. Terms that may be unfamiliar to general readers are defined at first mention.

I. One-Sentence Goal

Test whether a single, environment-indexed, frequency-independent path term—combined with ordinary baryons—can fit both galaxy rotation curves and galaxy–galaxy weak-lensing shear without assigning a bespoke dark-halo profile to each object. If one shared term explains the inner kinematics and the outer shear with a clear, monotonic dependence on large-scale environment, it supports the claim made by Energy Filament Theory (EFT). If the fit only works by fine-tuning object-specific halos or shows no environmental trend, the claim is disfavored.

Reader notes:

• A rotation curve is the orbital speed as a function of radius inside a galaxy.
• Weak lensing means tiny, coherent distortions of background galaxy shapes by foreground mass, often summarized as tangential shear around the lens.
• Object-specific halo profiles are bespoke mass profiles per galaxy (for example, Navarro–Frenk–White–like shapes with per-object concentration and scale parameters).

II. What to Measure

• Joint-fit consistency across probes: Using the same galaxies, ask whether one shared additive term (indexed by environment class rather than by object) improves baryons-only predictions in a way that simultaneously reduces residuals for rotation curves and for stacked tangential shear. The key signal is that the same parameter values help both probes.
• Residual correlation across radii: After subtracting baryons-only predictions, check whether inner-curve residuals co-vary with outer-shear residuals in the same sense for the same galaxies. This cross-probe residual linkage indicates a common, non-bespoke contribution.
• Environmental monotonicity: Compare void-dominated, filament-dominated, and node/cluster environments. The amplitude of the shared term should increase from voids to filaments/nodes, and this trend should appear in both the kinematic and lensing residuals.

III. How to Do It

1. Sample and data channels:
   Build a sample with high-quality, two-dimensional kinematics (for example, Hα, CO, or H I velocity fields) and overlapping weak-lensing shape catalogs. Span void, filament, and node/cluster corridors with matched redshift and stellar-mass distributions. Exclude strong bars, major mergers, or severe warps that dominate non-circular motions. Keep a control subset in each environment for later hold-out tests.
2. Modeling design and blinding:
   • Baryons-first baseline: From stellar light and gas maps, derive a baryons-only prediction for circular speed and for the expected weak shear from luminous matter alone (using a registered range of mass-to-light ratios).
   • Add one shared term: Introduce a single environment-indexed path term that is common across galaxies within the same environment class (not per object). This term feeds both the kinematics and the lensing predictions with the same parameter(s).
   • Independent pipelines: A kinematics team and a lensing team work separately, without sharing residuals or parameter values. Each team fits its data under pre-registered rules that restrict parameters to be shared at the environment level.
   • Arbitration: An independent arbiter opens the envelopes, checks whether the same shared-term values reduce residuals in both probes, and quantifies gains relative to baryons-only and to object-specific halo fits.
3. Controls and falsification drills:
   • Positive controls: In filament/node environments, the shared term should measurably improve both rotation and shear fits; in voids, the improvement should be smaller. The improvement should persist when switching kinematic tracers (Hα vs CO vs H I).
   • Negative controls: Randomly permute environment labels or scramble the mapping between galaxies and the shared-term values; the joint-fit advantage should collapse to near the baryons-only baseline. Allowing bespoke halos per galaxy should raise apparent fit quality for a single probe but reduce cross-probe predictivity, revealing overfitting.
   • Separation from frequency-dependent effects: Rotation curves measured at different wavelengths (for example, Hα versus radio lines) should yield consistent shared-term inferences. Any band-dependent sign flips or rescalings point to medium or instrument effects, not to a genuine shared term.
4. Systematics and safeguards (three items):
   • Beam smearing and non-circular motions: Poor resolution or bars/outflows can bias inner speeds. Safeguard: use 2D velocity fields with beam-smearing corrections, mask obvious non-circular sectors, and verify stability under controlled PSF degradation.
   • Inclination, distance, and mass-to-light uncertainties: Misestimated geometry or stellar scaling can masquerade as extra mass. Safeguard: adopt multi-band photometric inclinations, cross-check distances, and stratify results over a registered mass-to-light range to demonstrate that the shared term is not a tuning artifact.
   • Weak-lensing calibration (shear and photo‑z): Shear biases and photometric-redshift errors can shift the shear profile. Safeguard: apply standard shear calibration, propagate redshift uncertainties, and down-weight low-confidence shape measurements; report environment-stratified confidence grades.
5. Execution and transparency:
   Pre-register the model families, parameter-sharing rules, environment grading, and metrics (for example, residual reduction and cross-probe concordance). Keep hold-out galaxies in every environment class for final confirmation. Replicate results across independent surveys and instruments. Publicly release environment grades, plain-language summaries of residuals, and a registry of which parameters were shared or locked. This chapter forms a closed loop with the chapters on flux-ratio/odd-image statistics and on environment-predictable residuals in time-delay cosmology; results should be cross-referenced.

IV. Pass/Fail Criteria

1. Support (passes):
   • A single environment-indexed shared term, combined with baryons, jointly fits rotation curves and weak-lensing shear in the same sense for most galaxies in an environment class.
   • The amplitude of the shared term increases from voids to filaments/nodes in both probes.
   • The joint success survives changes of tracer, instrument, resolution, and processing pipeline; allowing per-object halos does not materially improve cross-probe predictivity.
2. Refutation (fails):
   • Acceptable fits require per-object halos with bespoke parameters, and the shared term fails to produce cross-probe gains.
   • Rotation–shear residuals do not correlate within the same galaxies, or the improvement disappears when environment labels are scrambled.
   • Results are not reproducible across tracers or instruments, or show band-dependent sign flips that argue for medium/instrument origins rather than a shared path contribution.