42 Consistency of the Common Term in Gravitational Microlensing “Image-Sequence Jigsaw”

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. With unified time/frequency standards and source-end calibration, we decompose unresolved microlensing light curves into a jigsaw of micro-image episodes. After removing dispersion/instrument/sampling effects, we test for a cross-band, cross-method, cross-station consistent non-dispersive common term—a smooth baseline in arrival time/phase. If the term scales with frequency dispersion, depends strongly on sampling/method choices, or fails to replicate across events and reobservations, the claim is disfavored.

I. One-Sentence Goal

Show that a non-dispersive common baseline exists across micro-image episodes and remains consistent across frequency, station, and pipeline—otherwise reject the hypothesis.

II. What to Measure

• Common-term consistency index (text grades):
  After common bandpass/timescale/alignment, extract constant and slow-slope terms from arrival-time/group-delay/phase residuals. Grade strong / medium / weak; uplift / depression; stable / unstable on a grid of event (source × epoch) × sub-segment (micro-image episode) × band × station/pipeline × window.
• Cross-band non-dispersion:
  Across multiple bands/edges, require insensitivity to carrier and band edge. 1/f² scalings or edge flips indicate ionospheric/plasma dispersion or color-calibration residue.
• Zero-lag co-occurrence (cross-band/station/method):
  Using a single external timescale, align residual series from bands/stations/pipelines and grade zero-lag peak and side-lobe contrast. Same-direction common terms within the same micro-episode count as supportive evidence.
• Order stability of the jigsaw:
  Segment light curves by derivative peaks/convexity changes/structured pulsations and verify that the sign and relative magnitude of the common term are stable across the sub-segment order, independent of window length/smoothing kernel.
• Ratio robustness and shape profiles:
  Validate “rigid shift, ratio unchanged”: amplitude/polarization ratios stay stable within a sub-segment while absolute arrival/phase exhibits a smooth constant. Profile the term as plateau / monotonic / threshold versus sub-segment time and event strength.
• Environment–structure forward predictability:
  Using only lens-plane environment (stellar surface mass density, shear direction, relative transverse speed) and source structure (size–wavelength scaling; disc/jet orientation), issue prediction cards (expected sign/magnitude/threshold), and score hit / wrong / null against measurements.
• Cross-epoch and cross-source transferability:
  For the same source across epochs and different sources in similar environments, check sign/order reproducibility; in reference windows away from micro-image activity the term should weaken/disappear.

III. How to Do It

• Observing and facilities:
  Global relay cadence to achieve minutes–hours sampling; parallel channels for multi-band photometry/narrow-line and continuum/polarization time series; ≥ 2 stations and ≥ 2 independent pipelines (difference imaging/curve fitting/frequency-domain) in parallel.
• Unified calibration and de-systematics:
  Publish bandpass transfer kernels, color terms, timestamp-queue logs; map to a common kernel with band-edge hold-outs. Build dispersion templates via dual/tri-band differencing with inject–recover tests; set upper bounds using ionosphere/troposphere tomography. Address source size–wavelength scaling and neighbor/host blending with leave-out–refill and inject–recover checks.
• Jigsaw construction:
  Tag micro-episodes via derivative/curvature/pulsation dictionaries. For each sub-segment, build multi-band–multi-station–multi-pipeline residuals and publish consistency and zero-lag index tables (text grades only).
• Stratification and stacking:
  Stack by stellar surface density/shear strength/transverse speed/source angular size, and identify plateau/monotonic/threshold patterns. Use method hold-outs (smoothing kernels/windows/detrending) to avoid overfitting.
• Forward prediction, blinding, arbitration:
  The forward team uses only environment/structure layers to issue prediction cards; the measurement team independently reports common-term and zero-lag summaries; the arbitration team reconciles them and scores hit / wrong / null by event/sub-segment/band/station/pipeline.

IV. Positive/Negative Controls and Artifact Removal

1. Positive controls (support a common term):
   • Within the same sub-segment, multi-band–multi-station–multi-pipeline data show same-direction, similar-amplitude smooth constants, robust to edge/kernel/window choices.
   • Non-dispersion: the term does not scale with frequency.
   • The term varies monotonically/plateaus with environment/structure and shows threshold boosts in high-shear/high optical depth or specific source orientations.
   • Prediction-card hit rates exceed chance and held-out sources/epochs replicate.
2. Negative controls (argue against a common term):
   • Residuals follow dispersion laws or correlate with color/ionosphere/troposphere.
   • Significance restricted to one band/pipeline/station/sub-segment, or fragile to bandpass/alignment/window/kernel choices.
   • Label swaps/time reversal/method swaps/parameter shuffles still “detect” signals—selection/method bias.
   • Signals vanish after stricter dispersion removal, band-edge hold-outs, blending controls or are reproduced by sampling aliasing/color calibration/background drift.

V. Systematics and Safeguards (Three Items)

• Sampling/window aliasing: uneven cadence and window choices create pseudo-constants. Safeguard: global relays, multi-window parallels, and inject–recover audits.
• Color calibration and neighbor blending: color drift or nearby variables project as a common term. Safeguard: common bandpass kernel, difference imaging, neighbor template regression, and field hold-outs.
• Chromatic source structure: size–wavelength changes yield chromatic smooth lags. Safeguard: explicit modeling in stratification and orthogonality tests to keep the target term distinct.

VI. Execution and Transparency

Pre-register sources/epochs, sampling and bands, common bandpass/timescale settings, criteria for non-dispersion/zero-lag/profile shapes, environment/structure variables and bins, positive/negative controls, exclusions, and arbitration scoring. Define held-out windows (quiet/active/reference). Enable cross-team replication by exchanging raw photometry/basebands, timestamp logs, calibration/color files, and scripts; run down-sampling/noise/kernel-variant/alignment-perturbation tests. Publicly release prediction cards, common-term consistency tables, zero-lag/non-dispersion summaries, bandpass/color/atmosphere/sampling logs, and key intermediates.

VII. Pass/Fail Criteria

1. Support (passes):
   • In ≥ 2 pipelines, ≥ 2 stations, ≥ 2 bands and across multiple epochs/sub-segments, recover a non-dispersive, zero-lag common term.
   • The term shows monotonic/plateau trends with environment/structure and is robust to bandpass/alignment/window/kernel/blending choices.
   • Arbitration significantly exceeds chance, and held-out units replicate.
2. Refutation (fails):
   • Results are dominated by dispersion/color/sampling/blending, or do not replicate across band/station/pipeline/epoch/sub-segment.
   • High parameter fragility or disappearance/reversal in held-out units.
   • Arbitration near chance and indistinguishable from system/method artifacts.