33 Image-to-Image Non‑Dispersive Common‑Term Sequence in Lensed Fast Radio Bursts

Home ／ Appendix-Prediction and Falsification

This chapter follows the publication template for the falsification program. It uses plain language, no equations, and fixed structure. Under unified time–frequency standards and source‑end calibration, we analyze Fast Radio Bursts (FRBs) that are gravitationally lensed into multiple images. After per‑image de‑dispersion and macro‑lens time‑delay alignment, we test whether pairs of images share a frequency‑independent common term—in arrival time, phase, polarization angle, or envelope location—that forms a stable, reproducible sequence across bands, epochs, and polarization. If the sequence follows λ² / 1/ν dispersion, or tracks plasma‑lens/multi‑path/clock/model artifacts, or fails cross‑team/platform replication, the claim is disfavored.

I. One‑Sentence Goal

Establish, for each image pair of a lensed FRB, a non‑dispersive, polarization‑agnostic, time‑window‑stable sequence of common offsets that reproduces across frequency, epoch, and pipeline once images are individually de‑dispersed and aligned by the macro‑lens delay.

II. What to Measure

• Image‑pair “constant‑offset” sequence (text grades):
  For each image pair (A–B, A–C, …), after per‑image de‑dispersion and macro‑lens geometric + potential delay alignment, extract the common offset—arrival‑time shift, phase zero, envelope peak, or sub‑structure onset. Report strong / medium / weak; uplift / depression and index the sequence by epoch / sub‑band / polarization.
• Non‑dispersion consistency:
  Across low/mid/high sub‑bands and all Stokes channels, compare common offsets, image‑pair flux ratios, and intra‑burst shape similarity for direction and strength ordering. Any systematic λ² / 1/ν flip/scale or band‑edge sensitivity indicates a dispersive/chain origin.
• Zero‑lag co‑occurrence (after alignment):
  Within windows aligned to the macro delay, grade zero‑lag peaks and side‑lobe contrast (strong/medium/weak) for sub‑pulses, micro‑structure, or narrow‑band striae, and check whether these trends co‑evolve with the common offsets.
• Image‑pair polarization common terms:
  Compare polarization‑angle constants, post–Rotation Measure residuals, and circular‑polarization signs across images; accept only stable, multi‑epoch, multi‑band patterns.
• Orthogonality to dispersion and scattering differences:
  Record inter‑image differences in dispersion measure (DM) and pulse broadening/trailing index. Test their correlation/non‑correlation with the common offsets; strong correlation implies a medium artifact.
• Cross‑epoch reproducibility for repeaters:
  For repeated bursts from the same lensed source, test ordering and sign stability of the common‑term sequence; across different lensed sources, assess statistical commonality (same‑direction / plateau / uncorrelated).

III. How to Do It

1. Targets and facilities:
   • Candidate selection: prioritize repeating FRBs and events with suspected multi‑image delays from seconds to days; include millisecond‑scale micro/milli‑lensing candidates.
   • Observing setup: baseband capture (microsecond–nanosecond), parallel sub‑bands (e.g., 0.4–8 GHz continuous coverage), full polarization, multi‑station coordination including Very Long Baseline Interferometry (VLBI) or long‑baseline electric‑field correlation.
   • Time/frequency standards: a single external reference (atomic clock) with one‑pulse‑per‑second (1PPS) and white‑noise injection, plus station‑to‑station clock‑offset closure and delay calibration.
2. Calibration and de‑systematics:
   • Per‑image de‑dispersion: fit DM and intrinsic drifts for each image separately; publish residual upper bounds.
   • Macro‑model alignment: align by macro‑lens model (e.g., mass ellipsoid + shear) or model‑free image‑position/time alignment; propagate model degeneracies in posteriors.
   • Bandpass/sidelobes/dynamic range: publish the bandpass kernel, band‑edge hold‑outs, and receiver nonlinearity logs; unify to a common point spread function (PSF) and common bandpass kernel.
3. Image‑to‑image sequence construction:
   • Co‑located, co‑window alignment: for image pair × epoch × sub‑band × polarization, align to the macro delay, compute common offsets and shape‑similarity / cross‑correlation peaks.
   • Text‑graded curves: generate strong/medium/weak; uplift/depression text curves and sequence indices for offsets and polarization constants.
   • Differencing and orthogonality tests: analyze DM/scattering differences against common offsets to enforce orthogonality and avoid confusion.
4. Forward prediction, blinding, arbitration:
   • Environment forward team: using only lens environment priors—convergence/shear (κ/γ), nearby galaxies/clusters, and line‑of‑sight structure—and masks, issue prediction cards (direction/strength of common term, non‑dispersion expected, zero‑lag expected or not).
   • Measurement teams (independent pipelines): run ≥ 2 cleaning paths (time‑first / frequency‑first), two pixel/mask apertures, and two alignment strategies in parallel.
   • Arbitration: match predictions to measurements and report hit / wrong / null rates by pair / sub‑band / epoch / method family.
5. Cross‑consistency:
   • Check image positions/parities/saddle‑point sensitivities against Chapters 9 and 21 (strong‑lens flux‑ratio and saddle‑image statistics).
   • Align non‑dispersion/zero‑lag definitions with Chapter 1 (cross‑probe common terms).
   • Share environment slices and common‑term directions with Chapter 27 (four‑dimensional path‑redshift tomography).

IV. Positive/Negative Controls and Artifact Removal

1. Positive controls (support the image‑to‑image common‑term sequence):
   • After independent de‑dispersion and macro alignment, multiple images show a same‑direction, stable sequence of common offsets across sub‑bands and polarizations, with significant zero‑lag co‑occurrence.
   • Common terms are nearly orthogonal to DM/scattering differences and do not flip/scale with λ² / 1/ν.
   • The sequence repeats across epochs and exhibits statistical commonality in another lensed FRB.
   • Forward‑prediction direction/strength hit rates exceed chance, robust across stations/teams/pipelines.
2. Negative controls (argue against a common‑term sequence):
   • The sequence flips/scales with λ² / 1/ν / band edges, or co‑varies with plasma lensing / multi‑path scattering.
   • Significance occurs only in one epoch / one sub‑band / one path, or is highly sensitive to bandpass kernels / alignment strategy / windowing.
   • Label swaps, image‑order shuffles, mask rotations, alignment‑parameter randomization still “detect” signals—evidence of selection/method bias.
   • Clock/delay calibration anomalies reproduce the “common term,” or macro‑model degeneracy fabricates a pseudo‑sequence.

V. Systematics and Safeguards (Three Items)

• Clock and delay‑calibration drift: can mimic constant offsets. Safeguard: 1PPS + white‑noise injection, bi‑directional links or inter‑satellite timing, and pulsar/synthetic‑pulse live monitors; hold out anomalous epochs.
• Plasma lensing and multi‑path scattering: introduce frequency‑dependent shifts and distortions. Safeguard: sliding‑window de‑dispersion within sub‑bands, broadening–tail joint fits, inject–recover tests, band‑edge hold‑outs, and down‑weight high‑scattering fields.
• Macro‑model degeneracy and micro/milli‑lensing perturbations: can be misread as common terms. Safeguard: cross‑check with model‑free image‑position/time alignment, include micro‑lensing priors, and use VLBI to resolve image‑level structure.

VI. Execution and Transparency

Pre‑register candidate lists and image pairs, baseband/sub‑band/polarization settings, unified calibration/de‑systematics workflows, criteria for common‑offset / zero‑lag / non‑dispersion, all controls/exclusions, and arbitration scoring. Define held‑out sub‑bands/epochs/high‑scattering regions for final confirmation. Enable cross‑facility/team replication via raw baseband electric fields, delay solutions, and alignment scripts; run down‑sampling/noise/kernel‑variant/alignment‑perturbation robustness tests. Publicly release prediction cards, image‑pair common‑term tables, zero‑lag and similarity summaries, and de‑dispersion/bandpass/clock logs, with key intermediates. This chapter closes a lensing–path–environment loop with Chapters 9/21/1/27.

VII. Pass/Fail Criteria

1. Support (passes):
   • In two or more pipelines, two or more facilities, and two or more lensed FRBs, recover a non‑dispersive image‑to‑image common‑term sequence that shows zero‑lag co‑occurrence under aligned windows.
   • The sequence is robust to bandpass kernels / alignment strategies / windows and across epochs/polarizations, and remains nearly orthogonal to DM/scattering differences.
   • Prediction‑card hit rates exceed chance; signals replicate in held‑out units.
2. Refutation (fails):
   • Results follow dispersion laws or track medium/chain systematics, and do not replicate across teams/facilities.
   • The sequence is parameter‑fragile or vanishes in held‑out units.
   • Arbitration hits are near chance, or signals are indistinguishable from clock/calibration anomalies.