27 Cosmic-Scale “Path Redshift” 4D Tomography (Sky × Redshift × Environment × Common Term)

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: we reconstruct a four-dimensional tomographic cube—sky pixels × redshift slices × environment strata × common-term indicator—under unified frequency/time standards and source-end calibration. After removing source-side velocities and gravitational redshift, instrument/calibration drift, Galactic/ionospheric/Faraday effects, and foreground/selection biases, we test whether a frequency-independent common term (the “path redshift” residual) is (i) non-dispersive across lines and bands, (ii) monotonic with environment (void → filament → node; external convergence/shear; weak-lensing κ), and (iii) reproducible across probes, teams, and facilities with environment-forward blind hits. Otherwise—if residuals trace instruments, line-ensemble biases, resolution mixing, or dispersive media—the claim is disfavored.

I. One-Sentence Goal

Build and test a 4D tomography of path-redshift residuals that is band-insensitive, environment-monotonic, and cross-probe replicable, using unified standards and blinded predictions based only on environment templates.

II. What to Measure

• Common-term indicator (text grades):
  In each sky-pixel × redshift-slice, assign the path-redshift residual a direction (uplift/depression) and strength (strong/medium/weak). Require same-sign behavior across adjacent sub-bands and multiple spectral lines.
• Non-dispersion consistency:
  Across same-source multi-lines (narrow forbidden lines, molecular/atomic transitions, 21 cm, radio pulses) and multiple bands (radio/millimeter/optical/NIR), compare direction and strength ranking. Any λ² / 1/ν flip or Faraday-law rescaling indicates propagation/chain dispersion.
• Environment monotonicity:
  Grade linkage to the void–filament–node skeleton, external convergence κ_ext / external shear γ_ext / weak-lensing κ_weak, galaxy density, and distance to nodes as enhanced / plateau / uncorrelated. Test for void → filament/node and increasing κ/γ/κ_weak monotonic strengthening.
• Angular scale and correlation length:
  After smoothing at 10–30–100 Mpc, record angular correlation length and anisotropy (e.g., large-scale sign flips or hemispheric contrasts) and compare to the morphology of environment templates.
• Redshift layering and tomographic continuity:
  Using thin and thick slices, evaluate same-sign continuity and stable strength ordering along the redshift axis. Signals confined to only thick slices flag radial mixing risk.
• Multi-probe closure:
  Check direction/strength consistency with strong-lens time-delay/flux-ratio statistics, Fast Radio Burst (FRB) post-de-dispersion arrival-time residuals, 21 cm intensity-mapping common term, same-source multi-line rigid-shift (ratio-invariant), and rotation-curve—weak-lensing joint fits.

III. How to Do It

1. Data domains and sample families:
   • Spectroscopic/imaging surveys: choose large-area spectroscopic redshifts overlapping deep imaging to ensure wide area × depth.
   • Radio/millimeter/21 cm/pulse samples: include cleaned FRB/pulsar sequences and 21 cm residual products.
   • Environment templates: reconstruct the void–filament–node network from 3D galaxy maps and weak-lensing κ, producing κ/γ/κ_weak projections.
2. Pixelization and tomography:
   • Sky pixels: build an equal-area pixel grid for pixel × slice indexing.
   • Redshift slicing: set thin/thick slices from spectroscopic z and high-quality photometric z.
   • Environment stratification: assign each pixel–slice an environment grade and distance-to-node/voidness.
3. Unified calibration and de-systematics:
   • Frequency/time standards: anchor optical/radio frequency/time to a single external standard across facilities; publish stability logs.
   • Source-side removal: use same-source multi-line ratio-invariance and narrow-line/stellar anchors to subtract intrinsic velocity/gravitational terms.
   • Medium/foreground: remove Galactic foregrounds, Faraday/ionosphere, and atmosphere/bandpass effects; apply de-RM and polarization-leakage corrections for radio.
   • Masks and coupling: adopt a unified sky mask and run mask rotation/substitution to bound template–mask coupling.
4. Common-term construction and normalization:
   • Aggregate multi-line/multi-band path-redshift residuals per pixel–slice and text-grade them.
   • Normalize by noise floors, window functions, and instrument stability to form a dimensionless common-term indicator, yielding the 4D tomographic cube.
5. Forward prediction, blinding, arbitration:
   • Environment team (forward): using only environment templates and masks, issue prediction cards per pixel–slice with expected direction/strength/scale and monotonicity.
   • Measurement teams (independent pipelines): using ≥2 cleaning paths and two pixel/slice schemes, produce the common-term cube and non-dispersion/monotonicity verdicts without access to predictions.
   • Arbitration: align prediction cards and results; compute hit / wrong / null rates across sky/redshift/environment/method strata.
6. Cross-consistency:
   • Cross-correlate with weak-lensing κ, galaxy density, and Cosmic Microwave Background (CMB) lensing κ; report any band-dependent sign flips.
   • Verify directional agreement against saddle-image ablation/flux-ratio smooth-field statistics, 21 cm common-term–environment correlations, and absolute cross-environment speed-of-light comparisons.

IV. Positive/Negative Controls and Removal of Artifacts

1. Positive controls (supporting a path-redshift common term):
   • Indicators align in sign across adjacent sub-bands / multiple lines and multiple facilities, and strengthen monotonically with void → filament/node and κ/γ/κ_weak.
   • Non-dispersion holds: no λ² / 1/ν / Faraday flips or rescalings.
   • 4D continuity: structures are morphologically continuous across sky and redshift and follow the environment skeleton’s shape and orientation.
   • Forward-prediction hit rates exceed chance and the direction/strength ordering matches strong-lens/FRB/21 cm probes.
2. Negative controls (against a path-redshift common term):
   • Residuals co-vary with instrument zeros/bandpass/thermal drift or Galactic/Faraday templates, or follow dispersive laws.
   • Significance is confined to one survey/pipeline/field, or is highly mask/slice-thickness sensitive.
   • Template rotation/label shuffles/redshift scrambling still give “detections,” indicating method/selection bias.

V. Systematics and Safeguards (Three Items)

• Frequency-zero and survey-stitch drifts: forge large-scale residuals. Safeguard: overlap-region closures and laboratory line anchors; publish zero-point stability/stitch curves; recompute with dual pipelines and held-out line sets.
• Environment-template × mask coupling: skeleton edges create spurious correlations. Safeguard: mask rotation/randomization and pixel re-sampling; report template–mask interaction bounds; down-weight edge pixels.
• Redshift uncertainty and radial mixing: photo-z tails and thin slices distort layering. Safeguard: build a redshift-error kernel, compare thin/thick slices with deconvolved/raw conventions, and separately report high-error subsets.

VI. Execution and Transparency

Pre-register pixel resolution and slice thickness, line/band lists, unified calibration/de-systematics workflows, text-grade rules for the common term, and criteria for non-dispersion / environment monotonicity / 4D continuity, plus all controls/exclusions and arbitration scoring. Reserve held-out pixels/line groups per redshift/environment/sky block for final confirmation. Enable cross-team/facility replication by exchanging raw photometry/spectra/visibilities and scripts, and run down-sampling/noise/mask-variant/template-rotation robustness tests. Publicly release prediction cards, the 4D common-term cube (text-graded), non-dispersion and environment-monotonicity summaries, and calibration/mask/redshift-error logs, with key intermediates. This chapter closes the loop with Chapters 1 (cross-probe non-dispersive common term), 9/21 (strong-lens statistics), 23 (21 cm common term—environment correlation), and 24 (absolute cross-environment light-speed comparison).

VII. Pass/Fail Criteria

1. Support (passes):
   • In two or more pipelines and two or more facility/survey classes, a cross-line/cross-band non-dispersive 4D common-term field strengthens monotonically with void → filament/node and κ/γ/κ_weak.
   • Directions agree with strong-lens/FRB/21 cm probes; forward-prediction hits exceed chance.
   • Results are robust to pixel/slice/mask/calibration choices and to template rotation/label shuffles.
2. Refutation (fails):
   • Residuals are dispersive or strongly tied to instrument/foreground templates, and fail cross-facility/team replication.
   • No environment monotonicity or poor 4D continuity; arbitration hit rates near chance.
   • High sensitivity to slice thickness/mask or disappearance in held-out samples.