Figures

Bounce-mechanism observable-prediction map. Left column: candidate non-singular bounce mechanisms (LQC, ECH/torsion, matter bounce, quintom-B, Cuscuton, ekpyrotic). Right column: distinctive observable channels. ECH appears bordered with a dashed box marked channel-level closure under stated assumptions (this paper)---the 14-constraint catalog narrows the f…

Distance-impact of a residual cosmic rotation component on the Hubble expansion. Visualizes the (ω/H)_0 < 5×10⁻¹¹ Saadeh+2016 isotropy bound and the consequent suppression of any direct rotation signature in the late-time Hubble flow.

Energy density hierarchy from the Planck scale to the observed dark energy scale, illustrating the phenomenological scaling ansatz _ vac [(α/M)]M_ Pl^4 (Sec. , Appendix ). This ansatz is dimensionally correct on-shell at the bounce but is not derived from the ECH action.

Position of each of the four minimal Einstein–Cartan–Holst dark-energy routes in the (coupling × mass) plane against the natural-parameter window. Visualizes why the closure rests on cosmological-constant-class tuning at m_θ ~ H_0 rather than amplitude mismatch alone.

Cosmic rotation vs Hubble expansion H. Distance-impact of a residual cosmic rotation component on the late-time Hubble flow. The CMB isotropy bound (/H)_0 < 5× 10^-11 from Saadeh pins the rotation contribution to _ eff = c_^2 (Eq. ) at the 10^-22_Λ^ obs level --- negligible at all redshifts. The dark-energy mechanism in this paper is therefore the ^2 term s…

Timeline of survey decisions (CMB-S4, LiteBIRD, SPHEREx, DESI DR2) that could falsify or confirm the surviving ECH-independent class-level ECH signatures (f_NL = -35/8 matter-bounce + spectator-ALP birefringence).

Observational decision timeline for the two surviving ECH-independent class-level falsification paths. Top: LiteBIRD CMB birefringence (σ(β)≈0.03^, launch early 2030s) testing the spectator-ALP route 4. Bottom: SPHEREx galaxy bispectrum ( 2028 first cosmological data release) testing the matter-bounce f_NL=-35/8 prediction at 2.6--5σ realistic significance (foo…

Detection-significance forecast for the matter-bounce f_NL = -35/8 SPHEREx window + the spectator-ALP birefringence LiteBIRD window. Connects the no-go theorem of §sec:barriers to the surviving falsification paths in §sec:surviving.

Naturalness landscape for the four minimal-ECH dark-energy routes (R1 NJL, R2 one-loop effective action, R3 Immirzi running, R4 parity-CMB spectator-ALP). Each route is shown as a point in the (mass×coupling) plane required to source _Λ at the observed value, with the naturalness window (gray band) defined by the scale of the underlying physics (M_ Pl for R…

Detection forecast for the two surviving ECH-independent class tests. Top: matter-bounce f_ NL=-35/8 in the SPHEREx multi-tracer f_ NL Fisher landscape (companion Paper II, 2.6--5σ projection). Bottom: spectator-ALP cosmic birefringence in the LiteBIRD σ(β) ≈0.03^ window (Paper Ib companion); the WMAP+Planck β=0.342^± 0.094^ and ACT DR6 β=0.215^± 0.074^ point…

Full-tension MCMC corner plot (119,617 post-burnin samples, getdist-thinned from 176,240 raw; footnote ) over Planck+BAO+SN+H0+S_8. The _eff posterior is consistent with zero (-0.020± 0.169), confirming no additional relativistic species at recombination.

Pipeline-recovery test of injected birefringence β across NSIDE = {256, 512, 1024} at f_sky = 0.32 on synthetic ΛCDM polarization skies. Demonstrates the recovery floor (bias −0.032°, sign-symmetric) — a pipeline-validation figure, not a sky-detection significance.

_eff marginal posterior comparison across the two frozen dataset combinations of Table (176,240 and 132,949 samples). Panel (a): Gaussian summaries of the _eff marginal posteriors at the Table means ±1σ, with the Standard-Model value _eff=0 marked. Panel (b): all seven Table parameters, normalized to the full-tension mean and σ. Both combinations recover _e…

Joint 1D + 2D ΔN_eff posterior across the full-tension (Planck+SH0ES), Planck+BAO+SN, and Planck-only frozen MCMC chains. The 309,189-sample evidence anchors the §IV ΔN_eff ≈ 0 null-consistency conclusion.

NaMaster pipeline-recovery validation on synthetic polarization skies (N_ side=512, _P=10 white noise, N=500 MC realizations per point; Sec. ). Panel (a): 500-MC mean recovered β vs.\ injected _ inj\0,0.27^,0.342^\ at the canonical apodized f_ sky=0.32 ACT-like mask, annotated with the pipeline-recovery bias β-_ inj (0.000^, -0.032^, -0.040^). Panel (b): th…

Overlay of the two converged frozen chains (full-tension + Planck+BAO+SN) showing how the H_0 — ΔN_eff degeneracy direction differs by dataset. Useful for showing the inverse-variance combination assumption discussion.

Spectator-ALP joint posterior triangle from the continuous-prior cross-check configuration in which the photon anomaly coefficient is sampled freely (flat priors C_a[4,60] --- widened from the earlier [1,30] to cover the full EOM-required band [9,51] --- _i[0.01,], _10(m_a/eV)[-35,-30] --- the mass prior corresponds to m/H_0 ≈ 7× 10^-3 to 7× 10^2 for H_0 =…

Tree-level vertex sum for the spectator-ALP × matter-bounce coupling channels. Diagrammatic intuition for the f_a cancellation in the rotation amplitude β.

Matter-bounce bispectrum shape function B_NL(k_1, k, k) as a function of the squeeze ratio k_1/k, showing convergence to -35/8 in the squeezed limit. Red circle: squeezed benchmark. Orange square: equilateral. Green triangle: folded.

Spectator-ALP β = 0.342° ± 0.094° posterior overlay vs the published joint Planck+ACT Eskilt+2022 measurement. Visualizes the consistency check anchored at f_a = M_Pl, m = H_0.

Detection significance for f_NL = -35/8 across survey configurations. Error bars span the optimistic endpoint (published ideal σ(f_NL) with template-overlap correction only) to the conservative endpoint (full budget: r=0.84 overlap, -correction, photometric-z degradation, PNG bias, b_ marginalization); i.e.\ optimistic-to-conservative ranges accounting for…

Triangle plot for the (θ_i, m_θ, C_aγ, β) posterior on the spectator-ALP joint chain. Shows the parameter degeneracies + the natural-prior box boundary discussion.

f_NL landscape: matter bounce vs.\ inflationary alternatives. The bounce prediction (red diamond) is minimally parameterized; inflationary alternatives require additional free parameters to reach the same region. SPHEREx 1σ error bar shown in blue.

Left: σ(f_NL) vs.\ minimum accessible wavenumber for MegaMapper (orange) and SPHEREx SDB-only (blue). The SPHEREx bispectrum channel (σ = 0.7, dotted) avoids the ultra-large-scale fragility. Right: corresponding detection significance for f_NL = -35/8.

Left: σ(f_NL) as a function of b_ prior uncertainty for MegaMapper SDB (blue). The SPHEREx bispectrum constraint (red dashed) is less sensitive to b_ than SDB but not independent of it; the residual dependence enters at tree level through the Δ b(k) f_NL b_ / k^2 cross-terms f_NL b_ b_1^2 P(k_1) P(k_2) in the multi-tracer Fisher matrix , which propagate to…

Observational decision thresholds. Green: strongly favors bounce. Red: strongly disfavors the quasi-dust matter bounce. Blue vertical line: bounce prediction f_NL = -35/8. Error bars: SPHEREx (σ = 0.7) and MegaMapper conservative (σ = 1.5).

2D UMAP embedding of the encoder latent space (PCA 12830, then UMAP) for a 500,000-spectrum stratified DESI DR1 sample, colored by per-spectrum anomaly score. High-score anomalies concentrate in distinct islands of the embedding (bright lobe, lower right) rather than scattering through the bulk population; the 83 gold-tier anomalies (cyan stars) lie on or n…

Mollweide all-sky distribution of the 14,089 high-confidence DESI + cross-survey anomalies. Color-coded by anomaly score + class. Useful as a hero map alongside the §III architecture figure.

Cross-transfer baseline map. Mollweide projection of the initial cross-transfer anomaly baseline (319,443 detections shown; canonical Path-C unique count is 378,280 after per-survey native retrains and 7-way deduplication --- see Table Path-C row and ). ACT DR6 is quarantined and excluded. Color-coded by survey (see legend). The DESI DR1 footprint (14,000 d…

Photo-z vs spectro-z scatter for the P3 anomaly catalog (DESI + cross-matched DR8 + UNIONS). Anchors the photo-z residual quality cut in §III.B.

Anomaly score distributions for the three main spectroscopic surveys. The score S is the per-spectrum reconstruction MSE rescaled to validation z-units: S = (MSE - _ val)/_ val, where _ val and _ val are the mean and standard deviation of MSE on the held-out 20\% validation split of the per-survey training pool (; cross-transfer for SDSS, native for DESI/LA…

Residual distribution (Δz / (1+z)) for the photo-z calibration sample. Used to justify the catastrophic-outlier cut at |Δz/(1+z)| > 0.15.

Cross-transfer SDSS baseline. UMAP embedding of the 77,905 SDSS DR18 anomalies from the initial DESI-trained cross-transfer scan, colored by HDBSCAN cluster (left) and by inferred physical category (right). The dominant cluster (green, 84\% of objects) contains ultra-cool dwarfs (M7--T2) that are completely out-of-distribution for the DESI-trained --- the d…

σ(f_NL) improvement as a function of P3 anomaly-removal threshold. Quantifies the load-bearing claim that the catalog reduces f_NL contamination for SPHEREx-class surveys.

NanoGrav 15yr γ_GWB MAP fit after applying the P3 anomaly catalog to the host-galaxy environment. Anchors the §V γ = 3.0 cross-reference.

NEOWISE top infrared anomaly at (α, ) = (180.59^, 0.56^), score = 11.5. DESI Legacy Survey DR9 grz composite, 256 × 256 pixels (108'' × 108''). Extreme W1-W2 infrared color excess; no prior SIMBAD entry within 5''. The optical counterpart is a bright, saturated source with diffraction spikes indicative of a luminous red stellar or quasi-stellar object. Phys…

Ecliptic-coordinate distribution of NEOWISE Path-C native-retrieval anomalies. Shows the residual ecliptic concentration after de-duplication — used to justify the §III.E ecliptic-systematic discussion.

SIMBAD-unmatched fractions for the six surveys with coordinate-based cross-matching, ranked from lowest (Gaia DR3, well-characterized variable stars) to highest (DESI DR1, 99\% of top-10K objects absent from SIMBAD). The dashed line marks the aggregate 58.8\% SIMBAD-unmatched fraction. The SIMBAD-unmatched fractions plotted here are a database-coverage meas…

Per-pixel anomaly rate (anomalies / spectra) across all 6 surveys. Visualizes the survey-depth × anomaly-rate correlation discussed in §III.D.

Spatial distribution of the 195,829 DESI DR1 anomalies. Top left: equatorial sky map color-coded by anomaly score S. Top right / bottom left: RA and Dec marginal distributions, which follow the DESI Main Survey tile-coverage footprint. Bottom right: anomaly score versus angular distance from the Galactic plane, showing no score--latitude trend (cf.\ the com…

Spectral pairs for the three DESI × SDSS cross-survey matches. Left column: DESI DR1 spectrum; right column: same object in SDSS DR18. Black: observed flux (normalized); red dashed: reconstruction. (a, b) Known QSO at z ≈ 1.55: both surveys produce consistent, low anomaly scores, validating the cross-matching approach. (c, d) TIC 374313355 at two epochs: th…

Per-pixel maximum-z reached by each survey in the P3 catalog. Companion to the anomaly-rate map for diagnosing systematic vs astrophysical anomalies.

Scatter of per-pixel anomaly rate against survey depth. Quantitative anchor for the depth-coupling systematic discussion in §III.D.

Per-redshift-bin decomposition of the fixed-α = 0.15 reference Fisher forecast (Appendix ). Left: σ(f_NL) per redshift bin for the standard DESI QSO single-tracer baseline versus the multi-tracer configuration including AI-selected anomaly tracers; the inset shows the combined result (^ std = 8.98 8.43, a 6.1\% central-value change). Right: AI-selected anom…

Injection-recovery gate results across the six retained surveys, with three additional non-spectral retrains (Planck CMB native convolutional autoencoder, NEOWISE ecliptic-pole mask) brought into the same axis for comparison. Solid curves show recovery fraction versus injection amplitude (multiples of local noise σ). The horizontal dashed line marks the 50\…

2D UMAP embedding of the encoder latent space, colored by per-spectrum anomaly score. Hero figure for the §II.A representation-learning approach.

Multi-tracer Fisher vs.\ tracer number density n for the canonical 5-tracer configuration of . The dashed gray line marks the dense-tracer limit ( = 11.71); the dotted dark-red line marks the single-tracer baseline ( = 16.85). Vertical orange and goldenrod lines mark the gold ( n = 8.5× 10^-6) and silver ( n = 4.5× 10^-5) anomaly sub-samples. The Heinrich-\…

Same UMAP embedding colored by spectroscopic redshift. Visualizes how the encoder organizes spectra by z without explicit redshift supervision.

Representative DESI DR1 anomalies across all ten taxonomy families. One highest-scored member per family; 2-row × 5-column layout. Border color indicates taxonomy class. Images are DESI Legacy Survey DR9 grz composites. Panel sublabels give the object RA; the high-z QSO panel additionally gives the redshift and the per-arm Z-arm sub-score r_Z (). The taxono…

Same UMAP embedding colored by DESI primary spectroscopic class (GALAXY/STAR/QSO). Shows native class separation in the unsupervised latent.

Representative high-confidence galaxies from the classified spiral sub-catalog (p_ eq>0.9). Left: clockwise (CW); right: counter-clockwise (CCW). All cutouts are 224×224pixels in grz bands from DESI Legacy DR8. The pairs illustrate the visual diversity within each chirality class --- different morphological types, redshifts, projection angles, and signal-to…

Sweep of the 9-template nuisance basis overlap parameter (correlation between primordial-dipole basis and depth/density templates). Anchors the §IV.D nuisance-marginalized fit robustness claim.

Representative high-confidence galaxies from the classified spiral sub-catalog (p_ eq>0.9). Left: clockwise (CW); right: counter-clockwise (CCW). All cutouts are 224×224pixels in grz bands from DESI Legacy DR8. The pairs illustrate the visual diversity within each chirality class --- different morphological types, redshifts, projection angles, and signal-to…

Test-time D_4 equivariant averaging (TTA). For each input image x, the classifier is evaluated on the eight D_4 transforms (four rotations × two reflections). Flips swap the CW CCW class labels by construction. Output probabilities are averaged after the swap, yielding a strictly flip-equivariant CW/CCW classifier with flip-swap correlation =1.000 by constr…

Catalog C composition. Of the 8,474,531 galaxies retained after image-quality QA, the equivariant TTA classifier () assigns N_ CW=1,592,107, N_ CCW=1,609,053, and N_ NS=5,273,371 (non-spiral / face-on / morphologically indeterminate). The spiral sub-catalog N_ spiral = N_ CW+N_ CCW= 3,201,160 is the analysis target for all chirality statistics below (Table…

Sky distribution of the 8.47M-galaxy chirality catalog (Mollweide projection, _10 N per HEALPix NSIDE=64 pixel). The DESI Legacy Imaging footprint covers f_ sky≈0.49 of the sky in the canonical mask; the N_ all1 analysis footprint (f_ sky=0.494) is used for the apodized MASTER diagnostic. Spatial uniformity of the per-pixel CW fraction is verified across 7…

Sky density of the 3,201,160 classified spirals (CW + CCW combined, NSIDE=64 Mollweide). Per-pixel spiral counts scale with the DESI Legacy Imaging Surveys depth and exposure pattern; the canonical mask used for the headline ℓ=1 analysis () requires N_ spiral(p)10 per pixel. Spatial inhomogeneity at this scale is the leakage channel quantified in .

Distribution of maximum-class confidence (,,) for all 8,474,531 galaxies. Strongly bimodal: 73.6\% at p0.9 (high-confidence labels) + a long tail of indeterminate cases ( p<0.5, dominated by NS/edge-on systems). The high-confidence (HC) cuts at p_ eq>0.6 (N=949,584) and p_ eq>0.8 (N=624,660) used in the systematics cross-checks () are indicated.

Raw (Catalog A) vs equivariant (Catalog C) chirality asymmetry sky maps (Mollweide projection, A_p per pixel, NSIDE=64). Left: raw single-pass classifier output; the spatially-structured 0.79\% classifier CW excess, modulated by non-uniform survey depth, produces the 2.31 real-space dipole + +6.48 pre-MASTER ℓ=1 artifact. Right: 2-fold flip-equivariant TTA;…

Pseudo-C_ℓ of the chirality field A_p on the canonical mask, ℓ=1--5 (single panel; grouped bars). Gray bars with error bars: per-galaxy label-shuffle null expectation (N=1,000 shuffles, 1σ); colored bars: measured C_ℓ. The per-ℓ significance annotations embedded in the panel derive from the figure-generation 1,000-shuffle run and are superseded by the canon…

Equivariant CW fraction as a function of redshift across the DESI DR1 footprint. Visualizes the z-stratified null check that the chirality-environment signal is z-independent.

Redshift distribution of the matched chirality × DESI DR1 spiral sample (1'' acceptance, after dedup). The distribution peaks at z ≈ 0.15--0.2 (median 0.168) and falls off steeply above z ≈ 0.5; a sparse tail extends to the maximum z = 3.83 (Table ).

In-footprint V-Web volume fractions for the canonical (R_s=25 Mpc/h, _ th=0, N_ grid=256^3) run on 14,622,283 DESI DR1 spectroscopic galaxies. The cluster volume fraction (1.0\%) reflects the high-density tail; the wall+filament fraction (74.5\%) dominates as expected for galaxy-traced large-scale structure.

Mollweide footprint of the P5 DESI×DESIVAST environmental analysis (56,981 void spirals + 791,635 cross-matched DR1 spirals). Useful as a context map.

CW fraction per cosmic-web class on the canonical V-Web run, on the n=812,793 env-labeled spiral rows (791,635 unique chirality-relevant matched spirals; ). Bars show the observed f_ CW per class; black error bars are 95\% Jeffreys binomial credible intervals. The void bin (n=428) is dominated by counting noise and brackets parity. The dashed horizontal lin…

Redshift histogram of the P5 sample stratified by V-Web/T-Web environment class (void / wall / filament / knot). Anchors the sample-balance discussion.

Sensitivity of the P5 per-environment Δf_CW signal to the equivariant confidence threshold p_eq cut. Quantifies the robustness of the headline result across p_eq ∈ [0.6, 0.95].

Equivariant CW fraction versus redshift across the matched DR1 chirality-relevant sample, with 95\% binomial confidence intervals per bin. The low-z bins that dominate the sample (median z = 0.168) sit on the 0.5 line; bins above z ≈ 0.5 contain few objects and have correspondingly wide intervals. The binned values are consistent with no redshift dependence…

Density-quintile null with Paper IV monopole-prediction overlay. Left: CW fraction per projected-density quintile (k=5 NN proxy, N=158,327 per bin) with 95\% Jeffreys binomial CIs; dashed parity f_ CW=0.5 and dotted Paper IV f_ CW=0.4974 references. Right: observed _ fromhalf per quintile (bars) vs the Paper IV-monopole prediction _ pred=2Δ f_ CWN (red diam…

Sensitivity of the per-environment Δf_CW signal to the DESIVAST void-radius cut. Anchors the §VI robustness check that the signal is not driven by a specific void size.

Equivariant CW fraction as a function of cosmic-web local density (V-Web density-quintile classification). Companion to the per-environment bar chart, showing the continuous trend.

Per-pixel signed _ fromhalf for the chirality-relevant matched-spiral subsample at NSIDE=32 (Mollweide projection, equatorial coordinates). The observed |σ|^ obs_=4.13 vs the label-shuffle null |σ|^ null,p99_=4.78 gives a look-elsewhere p=0.135; no NSIDE returns p<0.05. The map shows no coherent large-scale structure beyond random pixel-level scatter; the h…

Per-pixel residual Δf_CW^eq after subtracting the V-Web environmental prediction. Visualizes spatial coherence of the residual + supports the §VII null-systematic discussion.

Phase 2 sensitivity heat-map: per-cell range of f_ CW across the four environment classes \void, wall, filament, cluster\ in percentage points, on the declared env-labeled spiral parent. Each cell corresponds to a complete V-Web re-run on the 14,622,283-galaxy DESI DR1 spectro sample at (R_s, _ th). The maximum range across all nine cells is 4.12percentage…

HEALPix NSIDE = 32 Mollweide projection (equatorial coordinates). Top: count of DESIVAST maximal voids per pixel (885 occupied pixels, median 4 voids/pix). Bottom: per-pixel chirality _ from\ half on the z 0.24 matched-spiral subsample restricted to pixels with 200 spirals (1,496 valid pixels, σ range -3.45 to +3.48). The Pearson correlation across the n_ p…

V-Web (left; full-sample canonical run, shown as reference) vs Tempel+2014 FoF (right; 96,753-spiral overlap) cross-validation: per-class CW fraction with 95\% Jeffreys binomial credible intervals, shared y-axis [0.43, 0.53]. Dashed reference is parity f_ CW=0.5; dotted-red reference is the Paper IV global f_ CW=0.4974 classifier-monopole offset. The quanti…