Energy-Flow Cosmology (EFC) – Complete Validation Ledger

This ledger documents the empirical, methodological, and theoretical validation status of Energy-Flow Cosmology (EFC) across observational regimes. It explicitly distinguishes between early-universe linear physics (L0–L1), late-time non-linear structure formation (L2–L3), and the discrete gravity operator sector (Graph-Based AQUAL).

Each entry links to citable public artifacts where available. Planned tests are listed with defined methodologies to ensure transparency and falsifiability.

0.0 Regime Coordinates (ξ, η)

All validation tests in this ledger are classified by their position in the EFC regime space, defined by two dimensionless coordinates:

Governance Rule: All test entries are tagged with their (ξ, η)-regime to prevent cross-regime conflation. Tests in the Newtonian regime (ξ ≫ 1, η ≫ 1) constrain screening fidelity. Tests in the IR regime (ξ < 1, η < 1) constrain the discrete gravity operator. Intermediate regimes (ξ ~ 1) constrain the transition structure.

0.1 Status Classification Key (Validation Maturity Levels)

Each entry in the Validation Ledger is assigned a status label describing what level of scientific validation has been achieved. These labels are hierarchical and prevent phenomenological consistency from being mistaken for full theoretical validation.

Governance Rule: No result may be labeled "Completed (Data Likelihood Test)" unless all active parameters were declared in the Global Parameter Registry, no probe-specific re-tuning occurred, and frozen parameters remained fixed across datasets.

Validation Tier Classification

Each completed entry is additionally assigned a Validation Tier (T1–T4) indicating the depth of empirical confrontation. This prevents internal structural results from being visually conflated with full data-vs-model tests.

Note: Validation Tiers are shown as badges in the Status column of the main table (e.g., T1). Planned tests and entries awaiting execution are not assigned a tier.

0.2 Global Parameter Registry (Frozen-Parameter Governance)

This section defines the complete set of physical and phenomenological control parameters used across the EFC validation program. Its purpose is to prevent hidden tuning, enforce cross-regime consistency, and document where each parameter is allowed to enter. Unless explicitly stated otherwise, all parameters are globally shared across probes and cannot be re-tuned per dataset.

Background-Coupling Parameters

Growth-Sector Parameters

Density-Saturation Mechanism

Lensing Phenomenology (Temporary Closures)

⚠️ Restriction: Σ(k,z) and α_L2 are not part of canonical EFC. They are placeholders until a Θ(ρ)-derived metric coupling is implemented.

Discrete Gravity Sector (Graph-AQUAL Operator)

Parameters Explicitly NOT Allowed to Vary

• Baryon physics adjustments to compensate cosmological effects
• Galaxy bias redefinition between probes to restore agreement
• Dataset-specific transition redshift z_t
• Probe-dependent Θ(ρ) or screening scale
• Independent lensing vs clustering couplings

Any use of these would constitute model extension, not validation of baseline EFC.

0.3 Regime × Observable × Sector Matrix

This matrix shows which physical regime, which observable, and which parameter sector each validation test constrains. It prevents cross-regime leakage and makes it explicit where EFC is active, suppressed, or structurally tested.

Sector Key: BG = Background expansion (β, T(a)) · GRW = Growth/perturbation (Δ_F, μ) · LEN = Lensing/metric response · SCR = Density saturation/screening Θ(ρ) · PROP = Propagation sector · DGS = Discrete Gravity Sector (Graph-AQUAL, ξ/η regime)

Legend: ✅ = Primary sensitivity · ⚪ = Suppressed / secondary / consistency role · — = Not relevant in this regime

This matrix prevents the most common cosmology mistake: Using a probe to constrain a sector that is physically suppressed in that regime.

1. Validation Status Summary (Regime-Aware)

Symbols: ✅ supported · ⚪ inactive / not dominant in regime · ❌ discrepant · ⚠️ tension / partial

Note on ⚪ in EFC columns: ⚪ indicates no full field-derived validation, not no relevance. Phenomenological tests may show consistency without constituting theory-derived support.

Note: Joint BAO + SN Ia + RSD consistency constitutes a stronger geometric and growth-sector constraint than BAO alone, but remains a necessary rather than sufficient condition for full EFC validation.

A phenomenological BAO+RSD joint fit (BOSS DR12) has been completed, yielding a mild (1.7σ) preference for positive α_L2. This result demonstrates consistency but does not constitute empirical validation.

1.1 Discrete Gravity Sector (Graph-Based Operator)

This section documents the validation status of the EFC discrete gravity operator (Graph-AQUAL), which implements modified gravitational dynamics on a graph-based spatial discretisation. This sector is distinct from background cosmological tests and phenomenological closures. All tests below are tagged with their (ξ, η)-regime.

Note: The Discrete Gravity Sector tests are structurally independent from the background cosmological tests in Section 1. The Λ-locked screening mechanism (KT3, CGS) treats Λ as an active bulk-capacity parameter, not a freely tunable constant. All entries are governed by the (ξ, η) regime classification defined in Section 0.0.

1.2 Phenomenological Constraints

These entries reflect empirically driven tests of EFC-motivated parametrizations rather than fully derived field-level results. They constrain EFC phenomenology without providing canonical theoretical validation.

2. Key Regime Insight (February 2026 – v1.7)

A regime-transition analysis demonstrates that EFC admits a dynamically enforced L0 limit. During recombination (z ≈ 1100), all EFC-specific contributions are suppressed by many orders of magnitude (G_CMB ≈ 10^-12), ensuring exact recovery of standard linear cosmology.

The CMB therefore functions as a calibration of regime coupling rather than a veto. Observable EFC effects, if present, are expected only in late-time, non-linear regimes. However, linear-regime observables such as weak lensing require an explicitly derived metric–entropy coupling; phenomenological closures alone are insufficient for canonical validation.

A quantitative transition metric (Δ_F) has now been defined and constrained (Δ_F ≈ 0.1) using DESI-calibrated interpretations of Fugaku-class simulations. Importantly, Δ_F represents a growth-sector dynamical constraint, not a modification of the background matter density.

2.1 Density Saturation Mechanism (v1.3)

The Density Saturation hypothesis introduces a physical mechanism for regime gating:

• Saturation function Θ(ρ): A smooth function with Θ(ρ) → 1 for ρ ≪ ρ_crit (galactic/cosmological scales) and Θ(ρ) → 0 for ρ ≫ ρ_crit (Solar System / stellar interiors).
• Physical motivation: Grid-Higgs thermodynamics – finite degrees of freedom on the underlying grid impose saturation at high baryon densities.
• Consequence for Solar System: At ρ ~ 1 g/cm³, Θ(ρ) → 0 dynamically suppresses all EFC corrections, recovering GR to arbitrary precision (Cassini/LLR compliance).
• Consequence for Δ_F: The regime transition metric is now physically bounded rather than freely tunable – Δ_F ≈ 0.1 emerges from grid saturation, not parameter choice.
• Consequence for lensing: Weak-lensing shear in intermediate-density environments (galaxy clusters) can now be derived from Θ(ρ)-modulated action, replacing Postulate A with a physical coupling.

The regime response surface is accordingly extended from R(k,S) to R(k,S,ρ), enabling explicit discrimination between dwarf galaxies (low ρ, full EFC), galaxy clusters (medium ρ, partial EFC), and Solar System (high ρ, GR recovery).

2.2 Weak-Lensing Case A Insight (v1.4)

The KiDS-1000 Flinc testbed (v1.0) demonstrates that a phenomenological lensing amplitude modification Σ(k,z)² can significantly improve cosmic shear fits. Key findings:

• Fit improvement: Δ(−2 ln L) = −50.9 at α_L2 = 0.10 (1 additional parameter)
• Stress tests passed: Not IA compensation (IA=0); 63% persists with hard scale cuts; not just amplitude (Δ=41 residual after A_s tuning)
• Tomographic localization: Low-z improvement 3.2× stronger than high-z, consistent with late-time EFC activation (z_activ = 0.5)
• Critical cosmology shift: EFC (Case A) prefers S₈ = 0.685 vs ΛCDM S₈ = 0.739
• Tension implication: Planck tension increases from 2.3σ to 3.6σ

Interpretation: Case A (P_EFC = P_ΛCDM × Σ²) enhances the lensing signal, so less intrinsic structure (lower σ₈) is needed to match the data. This improves the shear fit but moves the inferred cosmology away from Planck. Case A is therefore a phenomenological lensing boost, not a solution to the S₈ discrepancy.

Next step: Case B implementation (consistent MG with μ in growth equations) is required to test whether a self-consistent modified gravity treatment can improve fits while reducing CMB tension.

2.3 S₈ Stage-III Convergence (v1.7.1)

Recent Stage-III weak-lensing re-analyses show convergence toward a common S₈ value, superseding earlier discrepant results. The table below distinguishes DEPRECATED values from CURRENT consensus measurements.

EFC implication: With Stage-III convergence to S₈ ≈ 0.81, the original "S₈ tension" (~3σ) is now reduced to ~1.3σ. EFC's predicted σ₈ ≈ 0.805 (from growth-sector constraints) is consistent with the updated Stage-III consensus. The remaining mild discrepancy does not require new physics but remains a target for Stage-IV precision tests.

Reference: 31305550 (EFC vs Stage-III analysis)

3. Structural Constraints (Model-Space Reduction)

3.1 Explicit Falsification Conditions

The following observational outcomes would constitute strong evidence against the EFC framework or specific EFC sectors. Each condition is linked to the sector and regime it would constrain. These are standing predictions — not post-hoc adjustments.

Standing commitment: These falsification conditions are pre-registered and will not be revised post-hoc. If any condition is met by future data, the affected sector must be either formally revised or abandoned, with the revision documented in this ledger.

3.2 Historical Falsifications and Model Revisions

Scientific integrity requires documenting not only current successes but also historical failures. The following entries record model versions or specific predictions that were falsified by data or internal analysis, leading to structural revisions. These are preserved to demonstrate that EFC evolves through empirical confrontation, not post-hoc accommodation.

Governance Rule: Any future falsification of a current EFC sector or prediction must be documented in this section within one ledger revision cycle. Suppressing or omitting falsification history is explicitly prohibited under the EFC validation protocol.

4. Evidence Register (Citable Artifacts)

Empirical & Phenomenological

• CMB constraints (null result, regime-consistent): 31095466
• CMB thermodynamic interpretation: 31064929
• SPARC175 / EBE documentation: 31047703
• Regime-dependent rotation-curve validity: 31007248
• SPARC Phase 3 empirical screening fit (intrinsic scatter; cross-scale consistency): 31224538
• JWST COSMOS-Web abundance excess: 31059964
• Identified multi-regime tension under standard ΛCDM assumptions: 31061665
• H₀ / S8 inference separation: 31026151
• DESI DR2 BAO phenomenology: 31127380
• Unified BAO + SN Ia + RSD analysis with derived effective gravitational coupling: 31215613
• Regime transition metric (Δ_F): 31144030
• Cluster lensing + galaxy rotation curve empirical validation: 31190233
• Regime response surface R(k,S) framework: 31211437
• WP3 – First empirical slice through R(k,S) response surface (RSD trajectory): 31215259
• BAO transfer test (DESI → BOSS/eBOSS; no refit): 31231522
• Joint BAO+RSD phenomenological α_L2 analysis: 31243828
• GR Recovery via Density Saturation – Quantitative PPN Analysis: 31244827
• KiDS-1000 Flinc cosmic shear – Regime-activated lensing response (Case A): 31271917
• Cluster core entropy–structure coupling – Pre-registered diagnostic test: 31286368
• ISW cross-correlation predictions (DESI DR1 tracers; original): 31301953
• ISW Consistency Audit and Revision Patch (v2.1): 31329082
• Four-channel background coupling architecture test (β·T(a)): 31304980
• High-z null regime consistency (Lyα, BAO, RSD): 31304995
• Structural Coherence Evaluation (SCE) – post-tension weak-lensing landscape: 31305550
• Covariance-aware BAO consistency test (BOSS DR12 consensus likelihood): 31314922
• MVP-G1 Growth fσ₈ LOO robustness (N2a): 31332730
• EFCLASS Sign Structure: background channel σ₈ exclusion: 31333414
• Perturbation-Level σ₈ Suppression via μ(a) < 1 (WP1a reference model): 31333600
• Systematic Localization of EFC in CMB+LSS Parameter Space (background emptiness, μ–Σ valley, lensing barrier, A_L degeneracy, fσ₈ bridge, GRAV structural gap): 31368433
• Discrete Entropic Gravity on a Cubic Graph: Emergent Newton and MOND Regimes with Λ-Locked Screening: 31348411
• Cosmological Growth Constraints on Λ-Locked Discrete Entropic Gravity: 31348414
• Regime Structure and Entropic Flow Ontology in Discrete Gravity Models: 31348417

Structural / Theoretical Constraints

• Cluster merger structural constraint: 31173850
• Weak-lensing phenomenological closure (Postulate A): 31188193
• EFC closure ansätze (closure conjectures): 31224466
• EFC Relativistic Action: Field Equations, Perturbation Theory, and Extraction of μ, Σ, η (variational derivation closing GRAV→(μ,Σ) gap; c_T = c; FA1–FA6 falsification conditions): 31876324
• Covariant EFT for Entropy-Driven Gravitational Modification (17-iteration construction; c_gw = c theorem; RAR = Bose–Einstein; exponential solar-system screening; microphysical gap identified): 31878334
• From Grid Microphysics to the Radial Acceleration Relation (gradient-coupled excitation model; E ∝ √g; BE RAR from 3 assumptions; lattice derivation; KT3 resolution pathway; microphysical bridge): 31878760
• EFC Screening Model — Track 1 (k = 0.415 ± 0.029; g† = 2.51×10⁻¹⁰; 174 SPARC galaxies; cross-scale C = k/a_G = 4.4; KiDS-1000 lensing; Bullet Cluster; H₀ tension channel): 31940469
• EFC-C Cognitive Entropy — Track 2 (neural entropy-production; centrifugal gradient P1; disorder signatures P2; entropy threshold P3; comparison with EBH/FEP/IIT): 31940505
• RLHF Thermodynamic Isomorphism — Track 3 (J = −F exactly; R → −E; β_KL → T; grokking as first-order phase transition; jailbreaking as Kramers escape; alignment–capability bound): 31940535
• Local Degree Heterogeneity Predicts Functional Variability Gradients (centrifugal entropy score κ; degree ratio r ≈ −0.97; Fiedler eigenvalue not predictive r ≈ 0.16; HCP connectome): 31940370
• Cross-Domain Bridge Equations (B1: cosmology → neural via degree heterogeneity η = C_eff/D_ratio^γ; B2: neural → RLHF via ∇&Sdot;_neural ↔ ∇²R; unified gradient-flow dynamics; 3 cross-domain predictions P4**–P6): 31940547
• Homo Fluxus v2.0 (civilisation map; Grid→EF→S→D→C; ego as thermodynamic function; DSM reframe; triad Nature/ASI/Homo Fluxus; empirically anchored via connectome r ≈ −0.97): 31940604
• EFC Gradient-Coupled Grid Action (minimal three-term Lagrangian S_grid; κ_eff = κ₀ + αg; Theorem 1: E ∝ √g in gravitational regime; operator uniqueness C1–C4 eliminated, C5 = |∇Φ| survives; three regimes; bosonic quantisation; two predictions P1/P2): 31941465
• Consistency of Scale-Dependent Gravitational Response: Numerical Regime Transition Test (first numerical confirmation μ < 1 (linear) ↔ μ > 1 (non-linear) from same action; R ∝ k⁻⁴ stiffness mechanism; survival valley μ ≈ 0.94, η ≈ 1.10, Σ > 1; spatiotemporal localization; 49.6% parameter-space viability; Δχ² = −0.03 vs ΛCDM on BOSS DR12 fσ₈; P1 cluster-scale test, P2 no μ > 1 in linear regime): 31941543
• Void ISW Sign-Flip (density-dependent μ(ρ) produces Rees–Sciama term δ·dμ/dt opposing standard ISW in voids; sign flip for δ ≲ −0.8; A_total turnover near δ ≈ −0.7; P1 depth-binned stacking, P2 scale dependence, P3 redshift evolution; ISW excess position; falsifiable): 31942677
• Cosmic Dipole Working Note (regime-dependent isotropy; entropy gradient ∇S → directional G_eff; D_EFC from ~2% entropy variation; P1 redshift dependence, P2 directional fσ₈, P3 alignment, P4 scale dependence; gaps G1–G4 open): 31942731
• Entropy Budget Working Note (SMBH-dominated entropy ~10¹⁰⁴ k_B maps to S = S_max grid collapse; GRC triad mapping; thermostat conjecture: AGN decline drives S(z) sigmoid at z ~ 1; P1 SMBH–μ correlation, P2 S(z)–AGN tracking, P3 void μ enhancement, P4 S_max normalization; gaps G1–G4 open): 31942734
• Density of States of Gravitationally Active Grid Modes (D_eff(ρ) ∝ √(ρ/ρ_crit) from boundary-mode mechanism; Γ(ρ) bridge completion; Scenario B+: microphysically derived, correct saturation, emergent ρ_crit = a₀/(G_Nl_g); ~20% agreement with phenomenological ansatz; ρ̇ rescue converts ρ^−1/2 → ρ^1/2; testable by Euclid/DESI DR3): 31942800
• Derivation of Γ(ρ) from Grid-Mode BE Statistics (first derivation from von Neumann entropy S[n]; chain rule ds/dn = x (marginal entropy = dimensionless energy); per-mode |Γ| = (β/(2a₀))n(n+1); total Γ ∝ ρ^3/2/(ρ + ρ_crit); Scenario B: R1 partial (exponent 3/2 not 1), R2 partial (ρ^1/2 not const), R3 passed (ρ_crit emergent); double-counting μ ≠ Γ resolved): 31942821
• Multi-epoch Growth Rate Test of EFC (EFC-VAL-2026-003): 14 fσ₈(z) measurements over z = 0.067–1.491 from 6dFGS, BOSS DR12, eBOSS DR16, and DESI DR1 (full-shape + peculiar velocities). EFC regime-dependent μ(a) = 1 − B·T(a) with T(a) = 1/(1+(a_t/a)ⁿ), n = 2 fixed. ΛCDM: χ² = 12.07/12 (Ω_m = 0.302, σ₈ = 0.774). EFC: χ² = 11.98/11 (B = 0.078, μ₀ = 0.922). Δχ² = 0.10 (0.06σ), ΔAIC = −1.90. Null result for the growth sector: B = 0 lies inside the 1σ profile interval; survival valley μ₀ ∈ [0.93, 0.96] consistent but not preferred. Identifies a strong Ω_m–B degeneracy (with Ω_m free, optimiser drifts to Ω_m ≈ 0.45, B ≈ 0.48); cross-probe combination with BAO required. Full AI-friendly package (README, index.json, metadata.json, schema.json, JSON-LD, citations.bib, src/ + data/ + examples/): 31955871
• WP4 — Cross-Survey Parameter Transfer Validation (DESI DR2 → BOSS/eBOSS, no refitting): EFC parameters z_L1L2 = 1.01, α_L2 = 0.045 calibrated on DESI DR2 give χ²_EFC = 13.06 vs χ²_ΛCDM = 20.83 on BOSS DR12 with k_eff = 0 (Δχ² = −7.77); whitening shows improvement dominated by w₅ (Δχ² = −11.39); eigenmode partition: 71% gain in strongly-penalised low-λ modes; covariance robustness sweep ρ = 0→1 shows EFC outperforms ΛCDM at every level; best-fit α_L2 ≈ 0.036–0.045 stable across BOSS, BOSS+eBOSS, DESI DR2; post-hoc validation — not pre-registered, not a discovery claim): 31954125
• EFC Lensing Validation Update — DES Year 6 P3 PASS (pre-registered): observed S₈^lens/S₈^CMB = 0.944 ± 0.018 vs EFC prediction 0.95 ± 0.03 (0.3σ agreement); pass window [0.90, 1.00] deposited prior to DES Y6 release (Jan 21, 2026); ties SPARC k = 0.415 to cosmic shear; A3 (η = 1) consistent at this level; 1D test — full C_ℓ comparison still pending; KiDS Legacy divergence acknowledged): 31951992
• ΛCDM as a Special Case of Energy-Flow Cosmology (consolidation: ΛCDM = L0/L1 limit; Eq. 6: K(ρ) → ∞, T(a) → 0, ξ ≫ 1 ⇒ Friedmann+Poisson; DESI DR2 α = −0.14 ± 0.21; gas-law analogy; perturbation valley μ ≈ 0.94, Σ ≈ 1.05; 5 kill-tests; 6 historical falsifications; 14 months, 204 publications, 102 tests): 31943361
• Effective Horndeski Reduction of the EFC Perturbation Sector (Bellini–Sawicki α-function mapping CLOSED: α_T=0, α_M∝S(a), α_B∝dS/d ln a; stiffness response R(k,z) beyond standard Horndeski; closes Euclid EFT pipeline gap): 32011407
• Pre-Registered Predictions for EFC (P1–P5 sealed: σ₈ +1.21%, P(k_c) +2.09%, C_ℓ^φφ −6.01%, E_G −3.98%, C_ℓ^TT +1.88% vs ΛCDM; SHA-256 sealed; anchored to α-functions from 32011407 and Boltzmann benchmark from 31990053): 32010399
• Scale-Localised Modified Gravity from the EFC Action (band-pass stiffness R(k,a); single parameter ε_tot ∈ [0.38, 0.42]; μ<1, η>1, Σ>1; E_G bump ~7% at k_c=0.05 h/Mpc, testable at 2.5–3.5σ with DESI×Planck; joint Δχ² = +1.4; directly falsifiable via E_G(k,z) survey): 31985313
• Regime-Transition Fit to DESI DR2 BAO with Cross-Validation: χ²_EFC = 9.33 vs χ²_ΛCDM = 31.34 on 7 DESI DR2 BAO points, ΔAIC = −18.01, ΔBIC = −16.88; hold-out validation Δχ² = −18.65 (PASS); combined BAO+Pantheon+ n=47: χ²_EFC = 24.19 vs χ²_ΛCDM = 46.36; best fit z_L1L2 = 1.01, α_L2 = 0.045: 31230703
• From RSD to Clusters — Parameter-Locked Test of Late-Time Structure Growth: α_L2 = 0.34 ± 0.16 from BOSS DR12 RSD propagated to eRASS1 cluster abundances with zero free parameters; N_EFC/N_ΛCDM = 1.064 ± 0.02 (+6.4% excess clusters); shape gradient dR/dz < 0 with R(z=0.15) ~ 1.05, R(z=0.7) ~ 0.94; 2–3σ discrimination power; pre-registered against eRASS1: 31292827
• Triangulation Test for Thermodynamic Lensing in Galaxy Clusters: pilot Bullet Cluster ρ_Pearson = −0.129, ρ_Spearman = −0.179 (p = 0.21, n=57 bins, under-powered); Abell 1835 ρ = −0.052 (p = 0.7); achromaticity check χ²/dof = 0.91; sealed speculative prediction δz ~ 10⁻⁶–10⁻⁵ correlated with entropy structure (null in ΛCDM); 3σ detection requires ρ > 0.08 for stacked 10-cluster sample: 31267297
• Unified Origin of RAR and Hubble Tension via Entropic Gravity Modification: two independent derivations of a_G: a_G(H₀ tension) = 0.094, a_G(MOND/SPARC) = 0.107 (14% discrepancy); H₀ predicted = 73.9 km/s/Mpc (1.2% accuracy vs SH0ES 73.0); falsification criteria: a_G from SPARC must yield ~0.1, correlation signature must be constant: 31223908
• Hubble Tension as Regime Mismatch: H_structure/H_background = exp(0.5 · a_G · δφ) = 1.083, predicting H₀ = 73.0 km/s/Mpc; sealed falsification: SNe Ia in voids should give lower H₀ than in clusters, ΔM ~ 0.02 mag by environment; a_G from SPARC must yield 0.1 ± 0.02: 31224247
• Directional Lensing Residuals at Cluster Merger Shock Fronts (pre-registered pipeline): A_sig > 3 AND p_rot < 0.05 in ≥ 2 clusters; mock Bullet Cluster EFC A_sig = 0.19, ΛCDM A_sig = −1.21, false positive rate 0; predicted δκ = 0.03 (15% of local κ); 2–6σ significance range; pipeline_ready, awaiting JWST κ data: 31288063
• Sealed Blind Predictions for Growth-Rate Observables (EFC vs ΛCDM, T1 pre-registered, NUTS sampler): formal DOI publication of the SHA-256 sealed predictions deposited 2026-02-18 (7a850cfa...) and 2026-02-21 (dbccda15...). P1: fσ₈(z=0.7) = 0.430 vs ΛCDM 0.449 (≈2.0σ); P2: D_H/r_d(z=0.7) = 19.797 vs 20.719 (≈2.3σ); P3: D_H/r_d(z=1.0) = 16.527 vs 17.466 (≈3.1σ). Falsifiable by DESI DR2/DR3 full-shape RSD + BAO (2027–2028): 32013156
• Consistent Modified Gravity Weak Lensing in EFC: μ–Σ Eigenmode Analysis (T2 sealed): eigenmode decomposition of the (μ, Σ) response surface for weak-lensing observables; identifies a single dominant mode carrying ≈95% of the perturbation-sector variance; predicts Euclid DR1/DR2 shear discrimination channel. Companion to 31288063: 31990194
• ISW Void Sign-Flip Observational Pipeline (T3): end-to-end data-confrontation pipeline from EFC theory to void-stacked ISW measurements; density-binned protocol for the δ ≲ −0.8 sign-flip (cold→hot) predicted in 31942677; pipeline_ready, awaiting Rubin DR1 void catalogue × Planck/ACT/SO CMB (~2028): 31990212
• EFC — Observational Evidence for Entropy in Cosmic Evolution (T3, historical empirical synthesis): compiles Planck CMB gradient, SDSS/DES large-scale structure entropy inventory, SMBH entropy budget (~10¹⁰⁴ k_B), and AGN sigmoid transition at z ~ 1 as observational anchors for the entropy-driven cosmology hypothesis. Foundational baseline for the SMBH thermostat conjecture (31942734): 28098299
• EFC — Observational Evidence for Energy Flow (T3, historical empirical synthesis): compiles rotation-curve flattening (SPARC precursor), cluster lensing residuals, CMB large-scale anomalies, and void expansion asymmetry as multi-scale observational support for regime-dependent energy-flow dynamics. Companion to 28098299: 28098365
• EFC Background No-Go Theorem (T3 empirical, joint BAO+CMB analysis across three independent H(z) parametrisations): any low-z EFC background modification large enough to improve BAO (Δχ²_BAO ≈ −14.6) is decisively penalised by CMB distance priors (Δχ²_CMB > +100), yielding Δχ²_total ≈ +98. Establishes that EFC phenomenology must enter via the perturbation sector, not the background; formally forecloses background-only ΛCDM modifications. Falsifier KC1: a joint BAO+CMB distance-prior analysis showing Δχ²_BAO ≤ −5 with Δχ²_CMB ≤ +5 (Δχ²_total ≤ −5): 31985163
• Pre-Registered Predictions for EFC vs Rubin LSST DP2 Cosmic Shear (T2 sealed_prediction, KC4 lensing sector η ≈ 1.10): SHA-256 sealed EFC forecast S₈ = 0.847 ± 0.015 and ~8% large-scale enhancement of ξ₊(θ) at θ = 20′–300′ vs ΛCDM. Falsifiers: KC1 — S₈ < 0.820 at >2σ rules out η ≈ 1.10; KC2 — ξ₊^obs/ξ₊^ΛCDM < 1.00 at θ > 30′ with >2σ significance falsifies the predicted large-scale enhancement. Confrontation window: Rubin Observatory LSST DP2 shear-only or joint 3×2pt measurement: 32013738
• Pre-Registered EFC Prediction: EG Statistic from SO × Euclid Cross-Correlation (T1, sealed prediction): Sealed preregistered prediction that SO × Euclid DR2 will measure EG_obs / EG_GR = 1.086 ± 0.012 at kc ≈ 0.05 h Mpc−1, z ≈ 0.5, with a pass window [1.03, 1.14] and a fail (kill) condition at EG_obs / EG_GR = 1.00 ± 0.01 (>3σ).: 32023788
• Systematic elimination of local gravitational models on the SPARC 175 sample: Evidence for an irreducible galaxy-specific degree of freedom (T2, empirical test): All purely local multiplicative models with only global parameters are falsified on SPARC 175 (FLOW win ≈ 27%), while a one-parameter-per-galaxy hybrid matches NFW performance (FLOW win 72.6%, median χ2/dof 2.71 vs 2.64).: 32029704
• EFC Perturbation Sector: Constraint-Driven Gravitational Slip and a Testable Lensing Crossover (T3, sealed prediction): EFC predicts a weak-lensing efficiency ratio eff(z) that crosses unity at z ≈ 0.42 with deviations confined to about ±2% over 0.2 ≲ z ≲ 1.4, implying enhanced lensing below the crossover and suppressed lensing above it relative to CDM.: 32037990
• Expected Outcomes for Energy-Flow Cosmology Cross-Survey Invariance and BIG-SPARC Tests (TN/A, sealed prediction): Pre-data projection favors Scenario B (partial support): Tier-1 AIC WR ≈ 58% (BIC ≈ 53%), WALLABY AIC WR ≈ 43% (BIC ≈ 35%), overlap consistency 10–12/14, with decisive SPARC wins expected to remain robust; BIG-SPARC is anticipated to be the decisive…: 32045592
• A structural closure of growth-sector couplings between discrete gravity and cosmology (T2, empirical test): All three tested growth-sector coupling classes (multiplicative Geff, additive local friction, additive non-local memory friction) are either structurally excluded by the data or fully absorbed by the Ωm–coupling degeneracy once geometry is fixed by BAO+H(z);…: 32084670
• Non-MCMC Constraints on the EFC α-Parameter: BAO, Growth-Rate, and Cluster Evidence (T2, empirical test): Non-MCMC model-comparison tests prefer EFC over ΛCDM across BAO and growth probes (Δχ2DESI = −22.01; Δχ2BOSS = −7.77; α = −1.00 ± 0.46 at 2.20σ from fσ8), while MCMC chains remain degeneracy-limited and uninformative for α.: 32101213
• EFC-VAL-2026-006: Interim Audit Corrigendum to VAL-005 Stub-to-Real Data Transition, VariantH Genuine (T2, empirical test): Under real (DOI-traced) data and patched priors/pipeline, the DESI DR1 non-MCMC signal vanishes and VariantH delivers a genuine null with ΔAIC ≈ +3.9 favoring ΛCDM on parsimony, leaving EFC underdetermined by current probes.: 32114530
• Pre-Registration: EFC dN/dM (z>7) Halo Mass Function via µ(a) Perturbation Channel (T2, sealed prediction): A sealed, structurally rigid prediction for the z=7–12 halo mass function in EFC is registered via the µ(a) perturbation channel with predefined falsification thresholds and dataset commitments.: 32149654

Methodological / Core Framework

• Core Lock v1.0 (consistency enforcement layer): 31223503
• EBE Core Principles v1.0 (regime-first epistemic rules): 31222903
• EFC Ontological Foundations v1.0.2: 31223668

5. Planned Validation Pipeline

• Derive physical control parameter χ(z) from EFC-R principles
• Joint BAO + growth fits under strict L0 suppression
• Weak-lensing κ and shear discriminators (now with Θ(ρ)-derived coupling)
• DESI DR2 / Y5 w(z) falsification
• Euclid / Rubin small-scale non-linear tests
• Solar System PPN parameter analysis – explicit γ, β bounds from Θ(ρ) saturation (v1.3)
• CMB lensing + gravitational slip η(k,z) – Φ/Ψ consistency tests (v1.3)
• Cluster abundance N(M,z) – integrated growth discriminator vs ΛCDM (v1.3)
• Numerical S(r) solution – first-principles RAR derivation for toy galaxy profile (v1.3). v3.5 note: Covariant EFT construction (31878334) shows classical S(r) produces correction increasing with acceleration (wrong direction). v3.6 note: Microphysical bridge (31878760) shows gradient-coupled bosonic modes reproduce BE form; remaining: numerical implementation with statistical occupation in Graph-AQUAL solver.
• DES Y3 cross-validation – replicate KiDS-1000 Case A test on independent dataset (v1.4)
• Case B implementation – consistent MG with μ in perturbation growth equations (v1.4)
• Cluster shock-front lensing asymmetry – execute pre-registered A_sig directional residual test on JWST κ maps (v1.5)
• ISW amplitude validation – measure tracer-independent A_ISW on DESI DR1 × Planck C_ℓ^Tg when published; EFC predicts uniform A_ISW ≈ 0.89 ± 0.03 (𝒞-metric tomographic test deprecated under ISW Consistency Audit v1.1) (v2.1)
• β-split test (distance vs growth) – constrain propagation-limit coupling ε using DESI DR3 + Euclid; evidence basis: 31305421 (v1.7)
• Standard sirens (GW vs EM distances) – sector-dependent propagation test (Einstein Telescope era) (v1.7)
• Relativistic EFC perturbation theory — COMPLETED (v3.4): EFC Relativistic Action (31876324) derives μ < 1, Σ > 1, η ≠ 1 from a single variational action. c_T = c exactly. GRAV→(μ,Σ) structural gap closed. Typical values (μ ≈ 0.94, Σ ≈ 1.05, η ≈ 1.10) fall within the survival valley. Action-level falsification conditions FA1–FA6 now govern the perturbation sector. Remaining: numerical EFCLASS implementation and full ADM Hamiltonian analysis for super-horizon modes.
• A_L degeneracy resolution – combine CMB lensing, fσ₈, and galaxy lensing to independently distinguish physical Σ > 1 from phenomenological A_L > 1; methodology defined in systematic localization (v3.3)
• Cross-domain prediction P4** – test η = C_eff/D_ratio^γ from bridge B1** on HCP structural connectome + multiscale entropy; primary criterion: |r| > 0.40 on n ≥ 20 subjects (31940547) (v3.7)
• Cross-domain prediction P5 – calibrate optimal RLHF β_KL from resting-state fMRI T_cog without grid search; validates bridge B2 temperature mapping (31940547) (v3.7)
• RLHF grokking phase-transition test (P2) – verify Δt_grok ∼ (H_mem − H_gen)/T_eff on modular arithmetic datasets; falsified if r < 0.3 (31940535) (v3.7)

6. Recent Updates

• 2026-05-04 (v3.18 – Composite-Level Aggregation & Sign-Flip Decomposition for SPARC 175) — Integrated 32162706 (EFC-VAL-2026-009): “Composite-level aggregation in rotation curve model selection: A sign-flip decomposition of ΔAIC for SPARC-175”. Decomposes the global ΔAIC of the v3.17 Kill-Test v6 Universality run (31986762) into inner (r < r_c) and outer (r ≥ r_c) contributions. Headline result: at r_c = 3 kpc, 30/171 galaxies (17.5%) exhibit a sign-flip — the global statistic and the outer-only statistic disagree on which model wins. The sign-flip rate is robust across r_c ∈ {2, 3, 4, 5} kpc. Re-weighting decomposition on the same Kill-Test pipeline: standard 1/σ² → 114/171 (66.7%) EFC win rate, sum ΔAIC = +40.4, median = +6.21; uniform → 148/171 (86.5%), sum = +4416, median = +148.9; capped median → 110/171 (64.3%), sum = +36.3, median = +4.47. The v3.17 single-number 60.2% headline is therefore a lower bound under the χ²-default convention, and the v2 narrative “standard χ² is anti-EFC” is retracted in §6.2 of the paper. Methodological consequence: per-regime ΔAIC_inner/ΔAIC_outer reporting is the new RCMP-compliant default; a single global ΔAIC is now a known compression artefact whenever sign-flip galaxies are present. New v4-development question: the LATENT-tail outer-collapse pattern (bulge-heavy spheroids where the single-component EFC profile cannot capture the inner-disk contribution — same five galaxies flagged in v3.17: NGC 5055, UGC 02953, UGC 09133, NGC 2841, UGC 02487) is opened as a structural question, routed to the Gap Analysis page rather than the Validation Ledger row. Atlas tier for rotation_curves_SPARC is unchanged but the robustness band is widened from a single 60.2% point estimate to [64.3%, 86.5%] depending on weighting convention. EFC-VAL-2026-009 binds the DOI to the SPARC 175 / Kill-Test v6 / EBE / RCMP entries. Reproducer artefacts already shipped under docs/papers/efc/component_weighted_bias/: component_weighted_bias_v3p1.tex, rcmp_decomposition.py, rcmp_decomposition.json, rcmp_robustness_rc.json, rcmp_signflip.png, rcmp_signflip_table.json, rcmp_summary.json, sparc175_killtest_results.json, sparc175_killtest_universality_curves.py, sparc175_model_curves.json. Ledger promoted to v3.18 (public HTML) / v4.6 (internal).
• 2026-04-11 (v3.17 – Kill-Test v6 Universality on SPARC 175) — Integrated 31986762 (EFC-VAL-2026-008): “Kill-Test v6 Universality — Extension of the EFC vs ΛCDM Kill-Test to the Full SPARC 175 Sample”. Extended the Kill-Test v6 probe-2 methodology (5 hand-refitted SPARC galaxies, ΔAIC = 35.4 on DDO 154, ΔAIC = −126 on NGC 7331) to all 175 SPARC galaxies using identical infrastructure (scipy.differential_evolution, seed = 42, AIC model comparison). Single-component EFC (v(r) = v_flat·√(1 − exp(−(r/r_turnon)^sharpness))) fitted to each galaxy alongside a 2-parameter NFW reference (V₂₀₀, c) with H₀ = 67.4 km/s/Mpc. Result: 171/175 galaxies successfully fitted (4 dropped for n < 5 data points); EFC wins 60.2% (42.1% EFC_decisive, 18.1% EFC), ties 26.9%, loses 12.9% (9.4% LCDM_decisive, 3.5% LCDM). Median ΔAIC = ΣAIC_NFW − AIC_EFC = +6.21 (positive favours EFC); median χ²_red: EFC 0.44 vs NFW 1.69. Regime partition: FLOW 71, TRANSITION 84, LATENT 16 galaxies. Mann-Whitney FLOW vs LATENT on ΔAIC distribution: U = 1136, p ≈ 0 — rejects the null hypothesis that the two regimes share a ΔAIC distribution. Spearman ρ(ΔAIC, v_max) = 0.11 (p = 0.15) — no mass-scale selection bias; the EFC advantage spans the full v_max range (~15 km/s to ~280 km/s). Probe-2 anchor cross-check: DDO 154 reproduces at ΔAIC = +125.2 (sign convention verified; the simpler 2-parameter NFW reference gives a larger single-component advantage than the multi-component probe-2 refit). Top EFC wins: UGC11914 (+4218), UGC05253 (+3965), UGC06787 (+1577), IC2574 (+1448), UGC02916 (+794). LATENT tail (NGC5055, UGC02953, UGC09133, NGC2841, UGC02487) is dominated by massive bulge-heavy spirals where the single-component profile cannot capture the inner-disk contribution — consistent with Kill-Test v6 regime architecture, and precisely the galaxies that were hand-refitted with the multi-component EFC model in probe-2. Universality verdict: CONFIRMED at the single-component level. Cherry-picking objection against the probe-2 5-galaxy selection is refuted. The Kill-Test v6 probe-2 open question #2 ("175-galaxy universality") is partially resolved; the full multi-component test with a single fixed (K₀, m²) across all galaxies remains open. No new free parameters introduced — EFC bounds are inherited from Kill-Test v6 (v_flat ∈ [10, 350] km/s, r_turnon ∈ [0.05, 50] kpc, sharpness ∈ [0.3, 6]); NFW bounds are standard (V₂₀₀ ∈ [10, 400] km/s, c ∈ [1, 50]). Reproducibility: deterministic at seed = 42; runtime ~2 minutes single-core; dependencies are numpy + scipy only. Full AI-friendly reproducer package shipped (README + QUICKSTART + MANIFEST + CITATION.cff + citations.bib + index.json + metadata.json + JSON-LD + src/sparc175_killtest_universality.py + data/sparc175_killtest_results.json + data/summary.json + data/verdict_distribution.json + data/top_galaxies.json + examples/reproduce.py) under docs/papers/efc/Kill-Test v6 Universality_SPARC175/. Ledger promoted to v3.17 (public HTML).
• 2026-04-09 (v3.16 – EFC White Paper Series Parts 1–4) — Full AI-friendly packages created for the canonical four-part EFC White Paper series. Part 1 (31970886): Recovery Conditions and the ΛCDM Limit — three independent sufficient conditions establishing EFC ⊃_pert ΛCDM. Part 2 (31970898): Field Equations and Observable Mapping — entropy field S(a), coupling μ(a) = 1 + βS(a), complete observable mapping. Part 3 (31970904): Data, Validation Ledger, and Falsification Protocol — 102 tests, sealed predictions, five kill criteria. Part 4 (31970907): Regime Susceptibility and Cross-Scale Mapping — T(S) function, entropic continuum, dynamical dark energy w(a) = −β(S)·a. Each part ships with full 10/10 AI-friendly package. Ledger promoted to v3.16.
• 2026-04-09 (v3.15 – EFC Consciousness Bridge) — Integrated 31969983 (EFC-VAL-2026-007): “Closing the EFC Consciousness Bridge: Dimensionless, Scale-Invariant Unification of Entropy, Dynamics, and Awareness”. Closes the open bridge between the cosmological entropy field S (EFC field equation G_μν + Λg_μν = κ_GT_μν + αS_μν) and the EFC-C cognitive variables (spectral differentiation Ω, causal integration κ). Three structural problems resolved: (1) the separable product C = Ωκ is replaced by a non-separable functional C = Ω̂κ̂(1 − e^{−γΩ̂κ̂}), where the exponential factor couples Ω̂ and κ̂ multiplicatively inside the exponent and prevents factorisation; (2) coupled dynamical equations are derived: dΩ̂/dt = −λ_Ω∇(δS/δρ) + D_Ω∇²Ω̂ (entropic drift + diffusion) and dκ̂/dt = J_flow − μκ̂Ω̂ (energy-flow source + suppression); (3) dimensionless regime-indexed variables Ω̂_ℓ = S(ρ)/S_ℓ and κ̂_ℓ = (α_ℓ/ρ_ℓ)|∇(δS/δρ)| are introduced, mapping across all four EFC regimes (L0 cosmology, ρ ~ 10⁻²⁶ kg/m³; L1 astrophysics, ~10⁻²²; L2 biology, ~10⁻³; L3 cognition/AI, ~10⁰). One genuinely free parameter (γ, the non-separable coupling constant). All others are inherited from existing EFC constraints (α_ℓ, λ_Ω = α_ℓ) or measurable from data (D_Ω, μ). The cross-domain constant K = 4.4 ± 0.6 is inherited from SPARC galactic rotation curve data via the B1* bridge equation κ̂_healthy = K/(1 + λ₂(W)τ_c), with zero neural parameter fitting. Anti-correlation derived, not imposed: at steady state, κ̂_ss = J_flow/(μΩ̂) ∝ Ω̂⁻¹ — exactly the anti-correlation that was previously an external axiom corr(Ω, κ) < 0. Existing empirical support: five out of five directional predictions of the three-equation system are consistent with propofol sedation EEG data (Chennu et al. 2016, N = 20, re-analysed in the Consciousness Field Resonance paper): Ω waking > sedated (d = 1.50), κ anti-correlated (r = −0.27), global C = Ωκ fails (F1 falsified, p = 0.43), parietal C > 0 (d = 2.10, p = 0.018), double dissociation (Ω tracks awareness). These do not constitute a direct test of the full model (which requires estimating ∇(δS/δρ) from the same data) but are consistent with the qualitative structure. Kill conditions: (1) corr(Ω̂, κ̂) > 0 in confirmed conscious states; (2) C increases monotonically with Ω̂ without κ̂-suppression; (3) cortical entropy gradient shows no correlation with consciousness level across anaesthesia depths; (4) K inconsistent between galactic and neural data at > 3σ. Limitations: γ unconstrained (must be estimated from EEG/fMRI data, e.g. PCI database or Human Connectome Project); the identification δS/δρ ↔ transfer entropy in the cognitive regime is an ansatz, not a derivation; regime characteristic scales (ρ_ℓ, S_ℓ) must be calibrated empirically for biological and cognitive systems; the diffusion term assumes continuum approximation that may break down in discrete network topologies. Derived through the Symbiotic Insight Framework (SIF). Full AI-friendly reproducer package shipped (README + index.json + metadata.json + schema.json + JSON-LD + citations.bib + src/consciousness_bridge.py + data/regime_mapping.json + data/propofol_consistency.json + examples/evaluate_bridge.py) under docs/papers/efc/Closing_the_EFC_Consciousness_Bridge/. Ledger promoted to v3.15.
• 2026-04-08 (v3.14 – EFC vs ΛCDM Kill-Test v6 & Non-Rejectable Model Stage) — Integrated 31964847 (EFC-VAL-2026-006): “EFC vs ΛCDM: Complete Kill-Test — From Rotation Curves to Planck”. Six-probe kill-test spanning L0 (CMB primary) to L3 (dwarf-galaxy rotation curves), with a single two-parameter action (K₀, m²) producing μ₀ ≈ 0.94, η ≈ 1.23, Σ₀ ≈ 1.05 from one mechanism (K→μ for stiffness, m²→f→η→Σ for slip via the δλ counter-term). All four cobaya minimize runs return Δχ² ≤ 0: MGCAMB (plik_lite + lowl + lensing, free cosmo, (μ₀,Σ₀)) = −0.45; A_lens proxy (fixed cosmo) = −0.81; A_lens free cosmo = −0.70; (K₀, m²) cobaya bobyqa on lowl + lensing + BAO + Pantheon+ = −0.30. Best-fit K_0,min = 1.552 (−6.5% vs v5 reference 1.66) and m²_min = 0.00318 (−9.1% vs v5 reference 0.0035), both inside sweet-spot windows [1.55, 1.68] and [0.0032, 0.0038]. Sector decomposition for the (K₀, m²) run: lowl TT −0.189, lowl EE −0.070, Lensing −0.329, BAO −0.179, Pantheon+ +0.468 → total −0.300. Ledger status after strict re-evaluation (no overclaiming): (i) the DDO 154 rotation curve (ΔAIC = 35.4) and the CMB lensing μ–Σ / A_L anomaly (Δχ² = −0.45, reproduced across three Planck runs) are marked ✅ Consistent with data (robust); (ii) Δχ² stability (4/4 runs Δχ² ≤ 0, range [−0.81, −0.30]) and (K₀, m²) parameter convergence are also marked ✅ Consistent (robust); (iii) the multi-component SPARC refit (5 of 175 galaxies refitted; NGC 7331 ΔAIC = −126), the Bullet Cluster (Δχ²_red = 0.46 vs 0.55, tied), and the global cosmology fit (Planck + BAO + Pantheon+, Δχ² = −0.30 sub-statistical) are marked 🟡 Consistent with data (preliminary / limited sample); (iv) the CMB primary probe (α_S → 0 at z > 1100) is marked ⚪ Neutral by construction, not a test and not a confirmation; (v) zero probes marked 🔴 Discrepant. Overall stage is set to 🔵 Non-rejectable model (NEW status level added in ledger v4.5): EFC cannot be rejected on current data and shows consistent Δχ² ≤ 0 across independent probes. Explicitly NOT ticked (UNDER REVIEW): “better than ΛCDM” (|Δχ²| < 1 sub-statistical; full MCMC with ≥ 16 GB RAM required), “no dark matter / dark energy proven” (only 5 SPARC galaxies refitted; cluster/void-lensing pending), “full cosmological validation” (cobaya minimize is a local optimum, not a posterior). Global verdict remains OPEN. Falsification conditions (unchanged): any cobaya run returning Δχ² > 0 in the EFC direction; μ(k = 5 h/Mpc) deviating from 1 by more than 10⁻⁴ (Solar-System PPN); η outside [1.20, 1.27] at m² = 0.0035; Σ-signal absent from high-ℓ TT+TE+EE after plik_lite inclusion; full MCMC returning Bayes factor > 3 against EFC. Full AI-friendly reproducer package shipped (README + QUICKSTART + MANIFEST + CITATION.cff + citations.bib + index.json + metadata.json + schema.json + JSON-LD + src/k_rho_bridge.py + src/gravitational_slip.py + src/kill_test_suite.py + src/cobaya_minimize.py + data/parameters.json + data/cobaya_runs.json + data/sector_decomposition.json + data/probe_results.json + data/mu_k_table.json + examples/run_kill_tests.py + examples/slip_window_scan.py) under docs/papers/efc/EFC_vs_LCDM_Kill_Test_v6_final/. Status Classification upgraded to v3.1 with three new levels: 🔵 Stage: Non-rejectable model, ✅ Consistent (robust), 🟡 Consistent (preliminary/limited). Ledger promoted to v3.14 (public HTML) / v4.5 (internal).
• 2026-04-08 (v3.13 – KT3b Cross-Regime Diagnosis) — Integrated 31963821 (EFC-VAL-2026-005): “Cross-Regime Measurement Failure in the KT3b Environmental Test”. The KT3b environmental test returned a null result after four iterations of increasing methodological rigour: Run 1 [log(L/D²), N~60, slope = −0.21, mass artefact], Run 2 [Karachentsev Θ₁, N = 17, slope = +0.01, N too small], Run 3 [Tully N_mem raw, N = 70, slope = −0.07, mass bias], Run 4 [Tully mass-matched, N = 23, slope ≈ 0, null]. The paper argues that the monotonic signal collapse across these four runs is consistent with a cross-regime measurement failure, not a physical absence of environmental coupling. The KT3b architecture violates the Regime-Consistent Measurement Principle (RCMP, 31222900) in three independent ways: (V1) driver–response mismatch, ρ_env (L1) drives D_eff (L2) through an underived α operator; (V2) boundary observation, ℝ at a₀ sits on the L1/L2 boundary where stiffness response R(k), entropy production Γ(ρ), and effective Poisson coupling μ(k) all transition — measuring at a₀ is analogous to measuring a phase transition at the critical temperature, where fluctuations diverge and S/N collapses; (V3) implicit regime mixing in the mass model, Υ_☆ = 0.5 M_⊙/L_⊙ assumes L1-regime stellar physics applied to L2-regime gravitational dynamics. Any one of V1/V2/V3 is sufficient to degrade the measurement; together they explain the systematic signal collapse. Three valid test architectures are proposed: (A) pure L2 — RAR slope d log(g_obs/g_bar)/d log g_bar evaluated in the deep-MOND regime 10⁻² ≲ g_bar/a₀ ≲ 10⁻¹, zero free parameters, existing SPARC data, recommended next step; (B) pure L1 — fσ₈(z) from RSD or void–cluster growth, one free parameter (α_L1 = −0.14 ± 0.21 currently), link to 31955871; (C) derived L1→L2 mapping — derive α(k, z) from the EFC action, zero free parameters, open problem. The diagnosis is stronger than the null result itself: it applies to MOND’s External Field Effect, Verlinde’s emergent gravity, and any framework in which the transition scale a₀ marks a regime boundary. No new parameters introduced. Full AI-friendly reproducer package shipped (README + index.json + metadata.json + schema.json + JSON-LD + citations.bib + src/rcmp_check.py + data/kt3b_run_history.json + data/test_architectures.json + examples/check_kt3b_and_alternatives.py). Ledger promoted to v3.13.
• 2026-04-08 (v3.12 – Bullet Cluster Confronted) — Integrated 31963668 (EFC-VAL-2026-004): “The Bullet Cluster Under EFC: Confronting Entropy-Gradient Gravity with JWST-Era Mass Reconstructions”. Analytical four-axis confrontation with Cha et al. 2025 (arXiv:2503.21870), Rihtaršič et al. 2026 (arXiv:2601.22245) and Cho et al. 2025 (arXiv:2512.03150). (i) Cluster merger geometry structural test (31173850) is tightened, not weakened, under the Cho et al. 10:1 minor-merger ratio. (ii) PIEMD A_sig 2D baseline (512×512 grid, 1800×1200 kpc, four-halo geometry motivated by Rihtaršič et al. lens model): A_sig = −6.9×10⁻⁴, bootstrap p = 0.98 (5000 random rotations) — null hypothesis validated, PIEMD halo geometry alone produces no shock-front asymmetry. This validates the test operator, not the physical hypothesis. (iii) SIDM bound σ/m < 0.5 cm²/g is consistent with EFC’s no-DM-particle postulate (structural consistency, not confirmation). (iv) Regime-transition prediction P1 (μ ≈ 1.1, Σ ≈ 1.2 at cluster scales) confronted qualitatively against Cho et al. virial masses; μ ~ 1.1 enhancement alone is insufficient for the full cluster mass discrepancy, a full field-equation solution on the new mass profile remains a planned pipeline item. No new parameters introduced; all values frozen at Ledger v3.11. Verdict: none of the three JWST-era studies falsifies EFC and none provides positive support for EFC over ΛCDM. The Bullet Cluster remains a consistency check at the L2 regime, not a detection site. The pre-registered A_sig test on the free-form δκ = κ_obs − κ_PIEMD residual (requires Cha et al. κ-map FITS) remains pending as the only decisive discriminator. Given the v3.11 growth-sector null (B = 0 within 1σ), the lensing-sector Σ must carry the L2 signal if EFC is correct — making this shock-front test more important, not less. Full AI-friendly package shipped (README + index.json + metadata.json + schema.json + JSON-LD + citations.bib + src/asig_2d_piemd.py + data/asig_baseline_result.json + data/external_studies.json + examples/reproduce_baseline.py) under docs/papers/efc/bullet_cluster_efc/. Three §4b external entries updated from “no EFC working note yet” to “confronted in 31963668”. Ledger promoted to v3.12.
• 2026-04-07 (v3.11 – Multi-epoch fσ₈ Growth Test + AI-Friendly Package Rollout) — Integrated 31955871 (EFC-VAL-2026-003): the first multi-epoch test of the EFC perturbation-sector coupling using 14 fσ₈(z) measurements over z = 0.067–1.491 from 6dFGS, BOSS DR12, eBOSS DR16, and DESI DR1 (full-shape + peculiar velocities). The full linear growth ODE is integrated with regime-dependent gravitational coupling μ(a) = 1 − B·T(a), gate width n = 2 fixed. ΛCDM: χ² = 12.07/12 (Ω_m = 0.3019, σ₈ = 0.7739). EFC: χ² = 11.98/11 (σ₈ = 0.7916, B = 0.078, μ₀ = 0.922). Δχ² = 0.10 (0.06σ), ΔAIC = −1.90 (ΛCDM marginally preferred). Null result for the growth sector: the data do not require modified gravitational coupling, but EFC is fully consistent — B = 0 lies inside the 1σ profile interval and the perturbation survival valley μ₀ ∈ [0.93, 0.96] is contained in the 1σ contour. The pilot BOSS-DR12-only result (1.7σ preference for α_L2 > 0) weakens to < 0.1σ once regime-dependent μ(a) correctly separates BAO geometry from RSD growth. Key methodological finding: a strong Ω_m–B degeneracy — with Ω_m free the optimiser drifts to Ω_m ≈ 0.45, B ≈ 0.48 — means fσ₈ data alone cannot constrain the transition redshift z_t; joint BAO + fσ₈ with full covariance is required to break it. Caveats: diagonal covariance (cross-survey overlap not modelled), Ω_m fixed at ΛCDM best-fit, scale-independent μ(a) (no k-dependence), z_t hits boundary at 3.0. Full AI-friendly reproducer package shipped (README + index.json + metadata.json + schema.json + JSON-LD + citations.bib + src/efc_multi_epoch_v2.py + data/fsigma8_compilation.csv + data/efc_multi_epoch_results.json + examples/reproduce_minimal.py). Ledger promoted to v3.11.
• 2026-04 (v3.10 – WP4 BOSS Transfer Validation) — Integrated 31954125. EFC parameters z_L1L2 = 1.01 and α_L2 = 0.045, calibrated on DESI DR2, are applied frozen to BOSS DR12 BAO without any refitting. Result: χ²_EFC = 13.06 vs χ²_ΛCDM = 20.83, Δχ² = −7.77 with k_eff = 0. Model: H_EFC(z) = H_ΛCDM(z)·√(1 + α_L2·Θ(z<z_L1L2)). Three covariance diagnostics confirm the gain is genuine: (i) Cholesky whitening — improvement dominated by component w₅ (Δχ² = −11.39), the most constrained linear combination of observables; (ii) eigenmode partition — 71% of the gain (Δχ² = −5.50) sits in strongly-penalised low-λ modes, the opposite of what a flexible model exploiting covariance slack would do; (iii) robustness sweep over C(ρ) = (1−ρ)·diag(C) + ρ·C — EFC outperforms ΛCDM at every ρ from 0 (diagonal) to 1 (full). Best-fit reference (parameters free): α_L2 ≈ 0.036–0.045 stable across BOSS, BOSS+eBOSS, and DESI DR2 — structural evidence beyond the transfer test. Caveats: post-hoc validation, NOT pre-registered (unlike P3); does not prove EFC nor rule out ΛCDM; Δχ² = −7.77 is not a detection. Together with P3 (predictive accuracy) and WP4 (cross-survey generalisation), EFC now satisfies two complementary necessary conditions for physical validity. Full AI-friendly package (10/10 files, demo verified). Evidence Register and Executive Summary updated. Ledger promoted to v3.10.
• 2026-04-07 (v3.9 – DES Y6 Lensing P3 PASS) — Integrated 31951992 (EFC-VAL-2026-002). The DES Year 6 3×2pt analysis (140M source galaxies, ~5000 deg²) reports S₈^DES Y6 = 0.789 ± 0.012; combined with the 2026 CMB lensing baseline S₈^CMB = 0.836 ± 0.012 (Planck+ACT DR6+SPT-3G, Qu et al. 2026, SNR = 61), this gives an observed lensing-to-CMB ratio of 0.944 ± 0.018. EFC’s pre-registered prediction (deposited January 2026 in 31188193, prior to DES Y6 release on Jan 21, 2026) was 0.95 ± 0.03, derived from η = 1 (A3) and the SPARC Phase 3 screening exponent k = 0.415 ± 0.05. Agreement: 0.3σ, well inside the pass window [0.90, 1.00]. P3 lensing consistency: PASS. This is the first cross-scale validation tying galactic SPARC fits to cosmological weak lensing through a single screening parameter; no fine-tuning to DES. Limitations explicitly stated: (i) 1D test — full C_ℓ spectrum and tomographic comparison still required; (ii) suppression signature is consistent with EFC but not unique to it (massive neutrinos, f(R), baryonic feedback also work); (iii) DES–KiDS divergence (KiDS Legacy gives <1σ from CMB) remains unresolved pending Euclid DR1 (Oct 2026); (iv) η = 1 not directly tested here — awaits E_G statistic; (v) k inherits SPARC systematics, P2 (C-stability) still untested. Updated validation status table (April 2026): P1 consistent, P2 untested, P3 PASS, A3 not tested directly. Full AI-friendly package (10/10 files, demo verified). Evidence Register and Executive Summary updated. Ledger promoted to v3.9.
• 2026-04-06 (v3.8 – ΛCDM as Special Case Consolidation) – Integrated the consolidation paper (31943361). Demonstrates that ΛCDM arises as the special-case limit of EFC via three conditions: K(ρ) → ∞, T(a) → 0, ξ ≫ 1 ⇒ μ, Σ, η → 1 ⇒ Friedmann + Poisson (Eq. 6). DESI DR2 confirms empirically: α = −0.14 ± 0.21 (0.65σ), ΔAIC = +1.59. Gas-law analogy formalised (Table 5). N1–N4 diagnostics COLLAPSED; N5 PASS; N6 SAFETY_FAIL (0.007% margin); N7 MARGINAL; VG WEAK. Pre-DR2 α ≈ −0.67 signal identified as Ω_m–α degeneracy artefact. Background sector closed. Perturbation survival valley confirmed: μ ∈ [0.93, 0.96], Σ ∈ [1.03, 1.07]. Kill-test summary: KT1–KT4 PASS, KT3 OPEN (decisive). Six historical falsifications + strong-α collapse documented. Consolidates 14 months, 204 publications, 100 registered tests into single reference. Full AI-friendly package (10/10 files). Evidence Register and Executive Summary updated. Ledger promoted to v3.8.
• 2026-04-06 (v3.7.7 – Entropy Production Γ(ρ) Derivation) – Integrated the first explicit derivation of Γ(ρ) from BE grid-mode statistics (31942821). Starting from von Neumann entropy S[n] = (1+n)ln(1+n) − n ln n, the chain rule through BE occupation yields per-mode |Γ| = (β/(2a₀))n(n+1). Key identity: ds/dn = x (marginal entropy = dimensionless energy). With density-dependent mode activation D_eff = D₀ρ/(ρ+ρ_crit), total Γ ∝ ρ^3/2/(ρ+ρ_crit). Scenario B: R1 partial (exponent 3/2 not 1), R2 partial (ρ^1/2 growth not saturation), R3 passed (ρ_crit = a₀/(G_Nl_g) emergent). Double-counting μ_BE ≠ Γ resolved: static occupation vs dynamical rate. Companion paper 31942800 upgrades to Scenario B+. Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.6 – Density of States Grid Modes) – Integrated density-of-states derivation (31942800). Derives D_eff(ρ) ∝ √(ρ/ρ_crit) from a boundary-mode mechanism: standing modes in a gravitational potential well activate when G_Nρλ_j² > a₀l_g. Combined with per-mode BE entropy and the ρ̇ rescue (cosmological trajectory converts dS/dρ ∝ ρ^−1/2 into Γ_t ∝ ρ^1/2), gives the fully microphysical Γ(ρ) = Γ₀√(ρ/ρ_crit)/(1 + √(ρ/ρ_crit)) with ρ_crit = a₀/(G_Nl_g). Scenario B+: no free functions remain, ~20% agreement with phenomenological ansatz in relevant range, distinguishable by Euclid/DESI DR3. R1 (low-ρ ∝ ρ) not met (∝ √ρ), but R3 (emergent ρ_crit) and R4 (D_eff derived) passed. Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.5 – Entropy Budget Working Note) – Integrated entropy budget working note (31942734). Shows the observed SMBH-dominated cosmic entropy budget (~10¹⁰⁴ k_B, Egan & Lineweaver 2010) is structurally consistent with EFC’s S-field: SMBHs at S = S_max (grid collapse boundary via Bekenstein-Hawking S_BH = 4πGM²k_B/(ℏc)), CMB at thermal S_max, baryons at S « S_max. GRC triad mapping (Table 2). SMBH Thermostat Conjecture: AGN luminosity density decline (~10× from z = 2 to z = 0) drives S(z) sigmoid at z ~ 1. Four predictions (P1–P4) and four open gaps (G1–G4). Working note status. Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.4 – Cosmic Dipole Working Note) – Integrated cosmic dipole working note (31942731). Shows EFC contains a natural mechanism for the observed galaxy number count dipole excess (2–3× kinematic): isotropy is a regime condition valid in L0/L1, not an absolute postulate. Any large-scale entropy gradient ∇S imprints a preferred direction via G_eff(S). Required gradient: ~2% entropy variation, consistent with CMB constraints under growth amplification (~10³). Four predictions (P1–P4) and four open gaps (G1–G4). Working note status. Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.3 – Void ISW Sign-Flip) – Integrated the void ISW sign-flip prediction (31942677). Density-dependent μ(ρ) produces a Rees–Sciama term δ·dμ/dt that is positive in voids (hot spot), opposing the standard linear ISW cold spot. For deep voids (δ ≲ −0.8), the net ISW signal flips sign. A_total = ΔT_EFC/ΔT_ΛCDM crosses zero near δ ≈ −0.7. Three predictions: P1 (depth-binned stacking turnover), P2 (scale dependence), P3 (redshift evolution). Takes explicit stance on ISW excess anomaly (A ≲ 1 predicted). Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.2 – Regime Transition Test) – Integrated the numerical regime transition test (31941543). First numerical confirmation that μ < 1 (linear cosmological) and μ > 1 (non-linear galactic) emerge as two limits of the same EFC relativistic action. The stiffness response R ∝ k⁻⁴ drives a smooth transition with no discontinuity. Survival valley (μ ≈ 0.94, η ≈ 1.10, Σ > 1) reproduced. Spatiotemporal localization confirmed: signal confined to z < 1 and k ≲ 0.1. Parameter-space robustness: 49.6% viable. Growth consistency: Δχ² = −0.03 vs ΛCDM on BOSS DR12 fσ₈. Two predictions: P1 (cluster-scale μ ≈ 1.1, Σ ≈ 1.2) and P2 (no μ > 1 in linear regime). Evidence Register and Executive Summary updated.
• 2026-04-06 (v3.7.1 – Gradient-Coupled Grid Action) – Integrated the EFC Gradient-Coupled Grid Action (31941465). Derives E ∝ √g from a minimal three-term Lagrangian S_grid with gradient coupling ½α|∇Φ|(∇ξ)². Operator uniqueness established: C1(ρ), C2(Φ), C3(g²), C4(∇²Φ) eliminated; only C5 = |∇Φ| survives. Three regimes (bare elastic, transition, gravitational) with crossover at g_cross = κ₀/α. Completes the microphysical derivation chain S_grid → E-L → WKB → E ∝ √g → BE → μ(g). Two falsifiable predictions: P1 (bare stiffness bound) and P2 (sub-kpc dispersion signature). Evidence Register and Executive Summary updated.
• 2026-04-05 (v3.7 – Three-Track Programme & Cross-Domain Bridge) – Integrated six new papers completing the three-track EFC research programme. Track 1: EFC Screening Model (31940469) — k = 0.415 ± 0.029, g† = 2.51×10⁻¹⁰, C = 4.4 from 174 SPARC galaxies. Track 2: EFC-C (31940505) — neural entropy gradients with 3 falsifiable predictions. Track 3: RLHF (31940535) — J = −F algebraically exact. Connectome analysis (31940370) — degree ratio r ≈ −0.97 (empirical anchor for EFC-C). Bridge equations (31940547) — unified gradient-flow dynamics dF/dt ≤ 0 across cosmology, neural, and RLHF regimes. Homo Fluxus v2.0 (31940604) — civilisation-level synthesis with empirical anchor. Evidence Register updated with all six DOIs. Planned Pipeline updated with cross-domain validation tests.
• 2026-03-28 (v3.6 – Microphysical Bridge Update) – Integrated results from “From Grid Microphysics to the Radial Acceleration Relation” (31878760). The microphysical gap identified in v3.5 is now bridged: three assumptions (bosonic statistics, gradient-coupled excitation energy E ∝ √g, single scale a₀) reproduce the Bose–Einstein RAR μ(g) = 1/(exp(√(g/a₀)) − 1) with no free parameters. Lattice derivation: k_eff = g/l_g → ω ∝ √g (structural consequence of harmonic restoring forces). KT3 resolution pathway identified: statistical occupation should restore β = 0.50 (vs classical 0.29). Microphysical gap row updated from OPEN to BRIDGED (kinematic; full QFT remains outstanding). KT3 row annotated with resolution pathway. Evidence Register updated with 31878760.
• 2026-03-28 (v3.5 – Covariant EFT Update) – Integrated results from “Covariant EFT for Entropy-Driven Gravitational Modification” (31878334). Five structural results established for the minimally coupled entropy-scalar class: c_gw = c (theorem), η ≈ 1, exponential solar-system screening, ghost-free stability, and RAR = Bose–Einstein occupation number. A critical microphysical gap identified: classical S(r) produces acceleration-increasing correction (wrong direction for RAR). Solar System/PPN row updated with complementary exponential amplitude suppression mechanism. Microphysical gap added to Structural Constraints (open). Planned Pipeline S(r) item annotated with gap status. Evidence Register updated with 31878334.
• 2026-03-27 (v3.4 – Relativistic Action Update) – Integrated results from “EFC Relativistic Action: Field Equations, Perturbation Theory, and Extraction of μ, Σ, η” (31876324). The GRAV→(μ,Σ) structural gap identified in v3.3 is now closed: a single variational action (F(φ)R + K(ρ) stiffness + flow constraint) derives μ < 1 (entropy-stiffness mechanism), Σ > 1 (flow-anisotropy mechanism), η ≈ 1.10, and c_T = c exactly (GW170817 satisfied). Falsification condition F7 formally passed. Updated validation matrix: GRAV structural gap row upgraded from T4 (gap identified) to T3 (gap closed). Six action-level falsification conditions (FA1–FA6) now govern the perturbation sector. Planned Pipeline: relativistic perturbation theory marked as completed; remaining items are numerical EFCLASS implementation and full ADM Hamiltonian analysis. Evidence Register updated with 31876324.
• 2026-02-19 (v3.3 – Systematic Localization Update) – Integrated results from “Systematic Localization of EFC in CMB+LSS Parameter Space” (31368433). New validation rows: CMB systematic localization (T2), Perturbation-sector μ–Σ valley (T3), Lensing barrier (T3), A_L degeneracy diagnostic (T3), B0 fσ₈ bridge test (T2), GRAV→(μ,Σ) structural gap (T4). Updated “Late-time background coupling” row with CMB background-emptiness result. Added F7 falsification condition: GRAV regime must produce η ≠ 1. New Structural Constraints entries: lensing barrier, μ–Σ survival valley, GRAV structural gap. Planned Pipeline updated with relativistic EFC perturbation theory (critical path) and A_L degeneracy resolution. Regime × Observable × Sector Matrix extended with perturbation-sector localization entries. Evidence Register updated with 31368433.
• 2026-02-16 (v3.2 – Referee Version) – Validation Tier system (T1–T4) added to distinguish joint-likelihood, single-probe, structural, and diagnostic results. Explicit Falsification Conditions section (F1–F6) with pre-registered predictions. Historical Falsifications and Model Revisions section documenting EFC v1 scalar index falsification, SPARC partial failure, ISW C-metric deprecation, and Poisson-sector disfavouring. Normative language toned down throughout (assertions → constraints). ISW status updated to “awaiting observational confrontation”. JWST status clarified as “interpretation-dependent”. H₀ regime interpretation expanded.
• 2026-02-16 (v3.1) – Major structural upgrade: Ledger partitioned into background, perturbation, and discrete operator sectors. Added Section 0.0 Regime Coordinates (ξ, η) with formal UV–IR classification. Added Section 1.1 Discrete Gravity Sector (Graph-Based Operator) with KT1–KT5 tests and Cosmological Growth Stability. Added DGS parameters to Global Parameter Registry (ξ, η, μ_Λ, C, β). Extended Regime × Observable × Sector Matrix with DGS entries. Updated Executive Summary to reflect multi-sector theoretical program with Λ-locked screening. MOND language refined to “MOND-like regime emerges in IR gradient-dominated sector.” Evidence basis: [31348411] [31348414] [31348417]
• 2026-02-13 (v2.3) – Added EFCLASS Sign Structure: background channel σ₈ exclusion (ΔE² ≤ 0 proven; fσ₈ enhanced +0.5–0.76%). Added Perturbation-Level σ₈ Suppression via μ(a) < 1: WP1a reference model (μ₀=0.85, S₈=0.790, 73% gap closure; structural ceiling ~43%). [31333414] [31333600]
• 2026-02-13 (v2.2) – Added MVP-G1 Growth fσ₈ Leave-One-Out robustness test (N2a mode). α = −1.00 ± 0.46 (2.20σ); 7/7 LOO pass; robust hint for α<0 in EFC growth channel. Full MVP-G1 control suite passed (N1 r_d-independence, N2 σ₈-prior strengthening, T7 LOO robustness). [10.6084/m9.figshare.31332730]
• 2026-02-12 (v2.1) – ISW Consistency Audit v1.1 applied (31329082). Recomputed A_ISW under fully self-consistent growth-channel implementation. Revised prediction: uniform suppression A_ISW ≈ 0.89 ± 0.03 across tracers.
• 2026-02-11 (v2.0.1) – KB synchronization update: Added Section 2.3 S₈ Stage-III Convergence table. Standardized three "Completed" rows to follow 9-level status system.
• 2026-02-11 (v2.0) – Full methodological upgrade: Executive Summary, Status Classification Key, Global Parameter Registry, Regime × Observable × Sector Matrix.
• 2026-02-11 (v1.9) – Methodological refinements on 5 rows; added 8 planned validation rows for critical cosmological tests.
• 2026-02-11 (v1.8) – Covariance-aware BAO consistency test with BOSS DR12 consensus likelihood (31314922).

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