Energy-Flow Cosmology (EFC) – Complete Validation Ledger

Executive Summary: EFC now contains a discrete gravity sector with explicit UV–IR regime structure and Λ-locked screening. All validation entries are partitioned into background, perturbation, and discrete operator sectors. Each observable is mapped to the specific regime (L0–L3) and physical sector it can constrain, ensuring that early-universe, late-time, and strong-field tests are not conflated. All results are governed by a global parameter registry and status hierarchy, so that empirical likelihood tests, consistency checks, phenomenological probes, and structural constraints are clearly distinguished. v3.4 update: The GRAV→(μ,Σ) structural gap identified in v3.3 is now closed by the EFC Relativistic Action derivation (31876324). A single variational action — coupling entropy-flow scalar field φ to gravity via F(φ)R with density-dependent kinetic stiffness K(ρ) and a Lagrange-multiplier flow constraint — produces μ < 1 (entropy-stiffness mechanism), Σ > 1 (flow-anisotropy mechanism), and η ≠ 1, consistent with the survival valley (μ ≈ 0.94, Σ ≈ 1.05, η ≈ 1.10). c_T = c exactly (GW170817 satisfied). Falsification condition F7 (η = 1) is formally passed. Six action-level falsification conditions (FA1–FA6) supersede the phenomenological F7 for the perturbation sector. v3.5 update: A complementary covariant EFT construction (31878334) establishes five structural results for the minimally coupled entropy-scalar class: (i) c_gw = c exactly (theorem); (ii) η ≈ 1 with O(Φ·μ) suppression; (iii) solar-system constraints satisfied by exponential amplitude suppression; (iv) ghost-free, tachyon-free, hyperbolic; (v) the RAR is formally identical to a Bose–Einstein occupation number. A critical negative result identifies a microphysical gap: the classical field equation produces a correction increasing with acceleration, whereas the observed RAR requires a decreasing correction. v3.6 update: The microphysical gap is now bridged by a minimal gradient-coupled excitation model (31878760). Three assumptions — bosonic statistics, gradient-coupled excitation energy E ∝ √g, and a single scale a₀ — reproduce the Bose–Einstein RAR μ(g) = 1/(exp(√(g/a₀)) − 1) with no free parameters. A lattice derivation shows the √g scaling is a structural consequence of harmonic restoring forces in a gravitational gradient (k_eff = g/l_g). This identifies a resolution pathway for KT3 (β = 0.29 → 0.50).

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:

ξ ≡ |∇Φ| / a₀ — Gradient regime coordinate. Measures the local gravitational acceleration relative to the MOND scale a₀. ξ > 1 corresponds to Newtonian (UV) regime; ξ < 1 corresponds to deep-MOND (IR) regime.

η ≡ μ_Λ Φ / |∇²Φ| — Screening regime coordinate. Measures the ratio of Λ-locked bulk screening to the local Laplacian source. η > 1 indicates screening-dominated (Newtonian recovery); η < 1 indicates IR-gradient-dominated (MOND-like emergence).

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.

Status	Meaning	Requirements
🟢 Completed (Data Likelihood Test)	Full quantitative comparison between EFC and ΛCDM using an established likelihood or statistical framework. This is a direct empirical test.	Real data; reproducible pipeline; documented parameters; Δχ² or Bayes factor
🟢 Completed (Consistency Check)	EFC predicts little or no deviation in the tested regime, and current data show no contradiction. EFC survives but does not gain positive support.	Near-null prediction by construction; observations within GR-consistent range
🟢 Completed (Structural Constraint)	Theoretical or geometric argument rules out certain operator classes or coupling types. Reduces model space but is not a likelihood comparison.	No parameter fitting; uses symmetry or configuration arguments
🟡 Phenomenological	Temporary parameterization motivated by EFC tested against data, but coupling not yet derived from field-level framework. Provides empirical insight but does not count as canonical validation.	No action-level derivation; explicitly labeled provisional
🟡 Framework-Level Diagnostic	Internal or cross-regime consistency evaluation of framework structure rather than direct observable comparison. Evaluates model architecture, not empirical fit.	Assesses coherence, parameter economy, or regime compatibility
🔵 Planned (Pipeline Ready)	Test with fully specified methodology and data source, awaiting execution. Actionable validation step.	Observable defined; dataset identified; statistical method known
🔵 Conceptual / Method Defined	Theoretical test formulated but requires new derivation, simulation, or pipeline development. Part of roadmap, not yet validation test.	Physical principle identified; observable link described
🔴 Discrepant	EFC predictions conflict with observations beyond statistical tolerance. Falsification point unless resolved by theoretical revision.	Quantitative comparison performed; disagreement exceeds uncertainty
⚪ Inactive / Not Dominant	EFC sector does not meaningfully contribute in this regime due to suppression or gating. Not failure, not success—regime not sensitive to tested coupling.	—

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.

Tier	Meaning	Typical evidence
T1 – Joint-likelihood	Full multi-probe joint likelihood comparison between EFC and ΛCDM with frozen parameters across all channels.	Δχ² or Bayes factor across ≥ 2 independent probes
T2 – Single-probe likelihood	Quantitative comparison with real observational data using a single probe or dataset. Direct empirical test, but not multi-channel.	Δχ² or goodness-of-fit on one dataset
T3 – Structural / analytical	Analytical proof, structural constraint, or self-consistent prediction. Reduces model space or establishes theoretical bounds but does not constitute direct data confrontation.	Sign lemma, symmetry argument, analytical bound
T4 – Framework diagnostic	Internal consistency evaluation, meta-model comparison, or cross-regime coherence check. Evaluates framework architecture, not empirical fit.	Coherence metric, parameter economy, regime compatibility

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

Parameter	Symbol	Role	Regime	Status
Late-time background coupling amplitude	β	Controls strength of EFC vacuum/background modification via H²(a)	L1–L2	Frozen after four-channel fit
Regime transition function	T(a)	Smooth activation function controlling when background coupling turns on	L0–L2	Form fixed; transition scale globally shared
Transition redshift	z_t	Characteristic redshift where T(a) activates	L1–L2	Shared across all late-time probes

Growth-Sector Parameters

Parameter	Symbol	Role	Regime	Status
Integrated regime transition strength	Δ_F	Measures cumulative deviation from GR growth across regimes	L1–L2	Physically bounded via Θ(ρ)
Effective gravitational modification	μ(k,z)	Perturbation-sector modification (only in Case B or structural diagnostics)	L1–L2	Inactive in baseline EFC-D; only Case B

Density-Saturation Mechanism

Parameter	Symbol	Role	Regime	Status
Density saturation function	Θ(ρ)	Suppresses EFC effects in high-density environments	L2–L3 (screening)	Form fixed; scale shared across all systems
Critical density scale	ρ_crit	Density where Θ(ρ) transitions toward suppression	L2–L3	Global constant

Lensing Phenomenology (Temporary Closures)

Parameter	Symbol	Role	Status
Lensing amplitude modifier	Σ(k,z)	Phenomenological shear amplitude boost (Case A only)	Temporary closure; not field-derived
Phenomenological lensing coupling	α_L2	Controls Σ² amplitude	Not a field-derived parameter; Case A only

⚠️ 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)

Parameter	Symbol	Role	Regime	Status
Gradient regime coordinate	ξ	\|∇Φ\| / a₀; classifies UV (ξ > 1) vs IR (ξ < 1) regime	All (L0–L3)	Dimensionless coordinate; not fitted
Screening regime coordinate	η	μ_Λ Φ / \|∇²Φ\|; classifies screening dominance	All (L0–L3)	Dimensionless coordinate; not fitted
Λ-locked screening scale	μ_Λ	IR screening mass derived from cosmological constant; Λ treated as bulk capacity, not free parameter	L2–L3	Fixed by Λ; not independently tunable
Graph operator prefactor	C	Structural renormalization constant from graph discretisation (C ≈ 2.32)	L2–L3	Structural; emerges from discretisation
Mass scaling exponent	β	IR mass-scaling exponent under Λ-locked screening (β ≈ 0.18)	L2–L3	Λ-locked; not independently fitted

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)

Observable / Test	L0	L1	L2	L3	Active Sector(s)
CMB TT power spectrum	⚪	—	—	—	BG (suppressed)
CMB TE/EE + lensing	⚪	⚪	—	—	BG, GRW (suppressed)
BBN expansion rate	⚪	—	—	—	BG (hard bound)
BAO isotropic	—	✅	—	—	BG
BAO anisotropic (D_A vs H)	—	✅	—	—	BG
SN Ia distances	—	✅	—	—	BG
RSD growth fσ₈(z)	—	✅	⚪	—	BG → GRW response
Full-shape P(k)	—	✅	⚪	—	GRW
Weak lensing (Case A)	—	⚪	✅	—	LEN (phenomenological)
Weak lensing (Case B)	—	⚪	✅	—	GRW + LEN
E_G statistic (lensing×RSD)	—	✅	⚪	—	BG + LEN
ISW cross-correlation	—	✅	—	—	BG → GRW (growth response only; no direct potential coupling)
Lyα high-z null test	⚪	⚪	—	—	BG (suppressed)
Galaxy rotation curves (SPARC)	—	—	✅	⚪	GRW + SCR
Cluster lensing geometry	—	—	✅	—	LEN + SCR
Cluster abundance N(M,z)	—	⚪	✅	—	GRW
Cluster entropy–structure	—	—	✅	⚪	SCR
Solar System PPN	—	—	⚪	✅	SCR
Binary pulsars	—	—	⚪	✅	SCR
Standard sirens (GW vs EM)	—	⚪	—	—	PROP
Perturbation-Sector Localization (v3.3)
CMB systematic localization	✅	✅	—	—	BG + GRW + LEN
μ–Σ valley	✅	⚪	—	—	GRW + LEN
Lensing barrier	✅	—	—	—	LEN
B0 fσ₈ bridge test	—	✅	—	—	GRW
Discrete Gravity Sector (Graph-Based Operator)
Newton recovery (KT1)	—	—	✅	✅	DGS (ξ ≫ 1)
Prefactor renormalization (KT2)	—	—	✅	—	DGS (ξ ~ 1)
Mass scaling / Λ-lock (KT3)	—	—	✅	—	DGS (ξ < 1, η < 1)
Superposition breaking (KT4)	—	—	✅	—	DGS (ξ ~ 1)
External field effect (KT5)	—	—	✅	—	DGS (ξ ~ 1)
Cosmological growth stability (CGS)	—	⚪	✅	—	DGS + BG (Λ-locked)

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)

Phenomenon	Primary dataset(s)	EFC-S	EFC-D	EFC-C	ΛCDM	Regime interpretation	Status	Evidence
Late-time background coupling architecture (β·T(a))	BAO (BOSS), RSD (BOSS), CMB lensing, SN Ia	⚪	✅	⚪	✅	Single-parameter background modification tested across four independent cosmological probes (BAO, RSD, SN Ia, CMB lensing). Poisson-sector coupling disfavoured by RSD; background channel produces consistent growth suppression via expansion history. v3.3 update: Systematic CMB localization (31368433) confirms background gate is observationally empty: α collapses to ≈0 under Planck+BAO joint fit (all combinations \|Δχ²\| < 2σ), with perfect α–H₀ degeneracy (corr = 0.975). CMB is blind to α; BAO breaks the degeneracy.	T1 Completed (multi-channel consistency)	31304980
Solar System / PPN / EP	Cassini / LLR / perihelion	⚪	⚪	⚪	✅	Density Saturation mechanism with source-smoothed effective density yields screening sufficient to pass Cassini, LLR, perihelion, and WEP bounds. Linearized field analysis shows PPN parameter γ = 1 at leading order, with residual deviations bounded far below current experimental sensitivity. Complementary result from Covariant EFT (31878334): exponential amplitude suppression A(r) ~ exp(−r/λ_S) provides an independent screening path for the minimally coupled scalar class. Full strong equivalence principle proof remains a target for action-level derivation.	T3 Quantitative compatibility supported (screened regime)	31244827 ● Cassini/Shapiro ● LLR bounds
Galaxy rotation curves	SPARC175 / MaNGA	✅	✅	⚪	✅	MOND-like regime emerges in IR gradient-dominated sector (ξ < 1, η < 1). Regime-dependent validity via EBE partition.	T2 Completed (Data Likelihood Test)	31047703, 31007248, 31045126, 31224538
Weak-lensing shear (phenomenological Case A)	KiDS-1000 Flinc	⚪	⚪	⚪	✅	Case A (lensing amplitude Σ²): Phenomenological test with P_EFC = P_ΛCDM × Σ(k,z)². Best-fit α_L2 = 0.10 yields Δ(−2 ln L) = −50.9 vs ΛCDM. Improvement localized to low-z (3.2× stronger than high-z), consistent with late-time activation. Critical finding: EFC (Case A) prefers lower S₈ = 0.685 vs 0.739, increasing Planck tension (3.6σ vs 2.3σ). Case A is phenomenological lensing boost, not S₈ tension solution. Case B (consistent MG with μ in growth equations) required for tension resolution test.	T2 Completed (phenomenological; Case A only)	31271917, 31188193
CMB lensing + gravitational slip	Planck lensing / E_G / RSD×lensing	⚪	⚪	⚪	✅	L0–L1 metric consistency: Φ vs Ψ potentials must satisfy derived slip relation η(k,z). Tests metric–structure coupling beyond shear-only constraints.	Planned – requires derived metric–entropy coupling	Method defined
Early galaxies (JWST)	COSMOS-Web z > 6–10	✅	✅	⚪	⚠️	EFC consistent with observed abundance in high-entropy regime; ΛCDM shows tension under standard assumptions (SF, dust, IMF interpretation-dependent)	T2 Completed (interpretation-dependent)	31059964
BAO transfer test (DESI → BOSS/eBOSS; no refit)	BOSS / eBOSS (transfer consistency check)	⚪	✅	⚪	✅	Out-of-sample transfer: late-time growth/geometry sector must remain consistent when carried across surveys without parameter re-tuning. Later confirmed by explicit four-channel background-coupling consistency test.	T2 Completed (transfer consistency)	31231522, 31304980
BAO + RSD joint fit (phenomenological α_L2)	BOSS DR12 consensus	⚪	⚪	⚪	✅	Superseded by four-channel architecture test showing consistent single-parameter background coupling (β = 0.08) across BAO, RSD, SN Ia, and CMB lensing.	T2 Completed (phenomenological test)	31243828, 31304980
MVP-G1 Growth: fσ₈ Leave-One-Out robustness (N2a)	fσ₈ extended (7 pts, z=0.02–0.85) + BAO(14)	⚪	✅	⚪	✅	T7 Leave-One-Out robustness (fσ₈, N2a-mode: r_d=147.09, σ₈~N(0.81,0.02)): 7/7 LOO runs pass (\|α\|/σ ≥ 1.7 AND ΔAIC ≤ 0). α-estimate extremely stable: α = −1.00 ± 0.46 (2.20σ), LOO-range [−1.11, −0.88], spread only 0.23. No single fσ₈ measurement drives the hint; lowest significance at z=0.02 drop (1.84σ), highest at z=0.15 drop (2.42σ). Correlations stable across LOO: corr(α,Ω_m) ≈ +0.59, corr(α,σ₈) ≈ −0.26. Full MVP-G1 control suite passed: N1 r_d-independence, N2 σ₈-prior strengthening, T7 LOO robustness (7/7). Conclusion: Robust, distributed ~2σ preference for α<0 in EFC growth channel; signal is structural, not artefactual.	T2 Completed (robust hint; ~2σ)	31332730
Regime transition metric (Δ_F)	DESI + Fugaku-class N-body + SPARC	⚪	✅	⚪	⚪	Growth-sector constraint: Integrated transition strength across L0–L1–L2 regimes. Measures effective gravitational enhancement rather than a background density parameter. Now physically bounded by Density Saturation Θ(ρ) in high-density regimes.	T3 Completed (Consistency Check)	31144030
Regime response surface R(k,S,ρ)	DES Y6 regime structure (multi-probe L2)	⚪	⚪	⚪	⚪	Framework linking μ(a) and μ(k,z) through structural state S; defines global falsification conditions across L2 probes. Extended to R(k,S,ρ) with explicit density axis enabling dwarf–cluster discrimination.	Framework defined – quantitative tests pending	31211437
Cluster merger geometry	Cluster lensing & mass–gas offsets	⚪	✅	⚪	⚪	Strong-field L2 geometric constraint; disfavours local coupling classes and imposes structural requirements on admissible operators	T3 Completed (structural exclusion test)	31173850
Cluster shock-front lensing asymmetry	JWST κ maps + Chandra k_T, n_e (Bullet, Abell 2146)	⚪	⚪	⚪	⚪	L2 geometric symmetry test. Pre-registered directional residual estimator (A_sig) measures whether lensing convergence residuals show shock-locked asymmetry after density-based mass model subtraction. Discriminates isotropic metric theories from entropy- or structure-coupled anisotropic lensing responses.	Pre-registered – data analysis pending	Methodology locked (2026)
Cluster core entropy–structure coupling	ACCEPT / TNG-Cluster	⚪	⚪	⚪	❌	Pre-registered rank correlation test between entropy profile slope (α) and core thermodynamics (K₀, t_cool). Under matched Cavagnolo et al. (2009) definitions, TNG-Cluster shows strong negative correlation (ρ ≈ −0.4 to −0.6), opposite to positive correlation in ACCEPT observations. Mass-controlled partial correlations confirm sign flip. Diagnostic mismatch points to deficiencies in simulated ICM thermodynamic regulation.	T4 Completed (diagnostic mismatch)	31286368
Cluster abundance / mass function N(M,z)	eROSITA / SPT / Planck SZ	⚪	⚪	⚪	✅	L2–L3 integrated growth test: Mass function evolution N(M,z) discriminates between EFC and ΛCDM structure formation. Tests whether geometry consistency extends to actual halo abundance.	Planned – methodology defined	Method defined
CMB power spectrum	Planck 2018 (TT)	⚪	⚪	⚪	✅	ΛCDM preferred in L0; EFC dynamically suppressed. EFC terms expected to vanish at recombination (Θ(ρ→0)); explicit Boltzmann-level validation pending.	GR recovery expected (Boltzmann validation pending)	31095466
ISW cross-correlation	Planck × DESI DR1 tracers	⚪	✅	⚪	✅	L1 linear perturbations: Under fully self-consistent growth-channel implementation (ODE-based D(a), no potential μ′ term), EFC predicts a uniform late-time ISW amplitude suppression A_ISW ≈ 0.89 ± 0.03 across tracer bins. No strong tomographic gradient survives internal consistency audit. Definitive discriminator: Measured A_ISW must deviate from unity by ≈10% in a tracer-independent manner. A statistically significant detection of A_ISW ≈ 1.0 ± 0.02 would disfavor the background-driven EFC channel.	T3 Completed (self-consistent prediction; awaiting observational confrontation)	31301953, 31329082
High-z null regime consistency	DESI Lyα P1D (z ≈ 3)	⚪	✅	⚪	✅	Same coupling T(z) predicts near-zero modification at high redshift; data consistent with null detection, as required by the same transition function T(z) used in the four-channel background test.	T2 Completed (null-prediction consistency check)	31304995
EFCLASS Sign Structure: background channel σ₈ exclusion	CLASS v3.3.4 (analytical + numerical)	✅	⚪	⚪	⚪	Sign Lemma: ΔE²(z) = A[g(z) − g(0)] ≤ 0 for all z > 0 under E(0)=1 normalisation. Numerical verification < 0.3% agreement at 9 redshift points. Growth enhancement +0.5% to +0.76% in fσ₈ at BOSS DR12 redshifts. Implication: Background-level EFC cannot suppress σ₈; perturbation-level μ(a) < 1 required. 77 lines of C code across background.h, background.c, input.c.	T3 Completed (structural proof)	31333414
Perturbation-Level σ₈ Suppression via μ(a) < 1: WP1a reference model	BOSS DR12 RSD (3 pts, full covariance)	✅	⚪	⚪	⚪	Universal factor-2: Δσ₈(n=2)/Δσ₈(n=6) = 2.00 ± 0.01. WP1a reference model: A=0, B=0.187, n=2, z_t=1.01; μ₀=0.85, σ₈=0.773, S₈=0.790. S₈ gap closed: 73%. Planck status: ~1.5σ from μ₀=1. Structural ceiling: Scale-free μ(a) closes at most ~43% within Planck 1σ.	T3 Completed (structural constraint)	31333600
CMB systematic localization	Planck 2018 TT/TE/EE + BAO (BOSS/eBOSS)	✅	✅	⚪	✅	Systematic localization of EFC effects across CMB+BAO parameter space. Background gate formally empty: α collapses to ≈0 under all Planck+BAO combinations (\|Δχ²\| < 2σ). Perfect α–H₀ degeneracy (corr = 0.975) confirms CMB blindness to α; BAO breaks it. Stronger than four-channel consistency: observationally demonstrates background sector is empty, not merely consistent.	T2 Completed (CMB systematic localization)	31368433
Perturbation-sector μ–Σ valley	Planck 2018 + BAO + fσ₈ (grid scan)	✅	⚪	⚪	⚪	Narrow survival zone mapped: μ ∈ [0.93, 0.96], Σ ∈ [1.03, 1.07]. Best-fit at (μ = 0.94, Σ = 1.05) with Δχ² = −0.45 (free cosmological parameters). Gravitational slip η ≠ 1 is structurally necessary. Implies EFC perturbation sector must produce both growth suppression (μ < 1) and lensing enhancement (Σ > 1) simultaneously.	T3 Completed (structural constraint)	31368433
Lensing barrier: pure μ < 1 excluded	Planck CMB lensing (C_ℓ^φφ)	✅	⚪	⚪	⚪	Pure μ < 1 with Σ = 1 is excluded by CMB lensing: Δχ² = +19 at μ = 0.93. Lensing power requires Σ > 1 to compensate growth suppression from μ < 1. This is a hard structural barrier: any EFC perturbation-sector model must modify both Poisson and lensing potentials.	T3 Completed (structural constraint)	31368433
A_L degeneracy diagnostic	Planck 2018 CMB (TT/TE/EE + lensing)	✅	⚪	⚪	⚪	Σ > 1 reproduces the A_L > 1 anomaly identically (cosine similarity = 0.93 between C_ℓ residuals). CMB alone cannot distinguish physical metric modification (Σ > 1) from phenomenological A_L > 1. External probes (fσ₈, galaxy lensing) required to break degeneracy.	T3 Completed (diagnostic)	31368433
B0 fσ₈ bridge test — sign constraint μ < 1	fσ₈ compilation (7 pts, z = 0.38–0.85)	✅	⚪	⚪	⚪	fσ₈ independently confirms μ < 1 sign: Δχ² = −5.22 (> 2σ) improvement via F3 tanh filter at k ≈ 0.05–0.1 h/Mpc. μ > 1 ruled out with Δχ² > +400. Original GRAV-sector screening scale (k_Λ = 0.0014 h/Mpc) gives Δχ² = −0.01 (no effect), confirming the modification must operate at much larger k than the discrete gravity sector provides. This is a CMB-independent sign constraint that breaks the A_L degeneracy: μ < 1 is required by growth data regardless of lensing interpretation.	T2 Completed (sign constraint)	31368433
GRAV→(μ,Σ) structural gap — CLOSED	EFC Relativistic Action (variational derivation)	⚪	✅	⚪	⚪	Gap closed by EFC Relativistic Action (31876324). Single variational action produces: μ < 1 via entropy-stiffness mechanism (K(ρ) divergence opposes collapse), η > 1 via flow-anisotropy mechanism (Lagrange multiplier generates tensorial stress absent in standard scalar-tensor theories), Σ > 1 via slip compensation. Typical values: μ ≈ 0.94, Σ ≈ 1.05, η ≈ 1.10 — within the survival valley. c_T = c exactly (GW170817 passed). Automatic GR screening via K(ρ) → ∞. Six action-level falsification conditions (FA1–FA6) supersede phenomenological F7.	T3 Completed (structural gap closed)	31876324
Structural coherence across regimes (SCE evaluation)	KiDS, BAO, RSD, Lyα, CMB lensing	⚪	⚪	⚪	⚪	Framework-level diagnostic (not empirical observable): EFC shows higher layer consistency and lower model plasticity; ΛCDM retains higher compression	T4 Completed (meta-model evaluation)	31305550
BAO covariance-aware consistency test	BOSS DR12 consensus (6×6 cov) + cosmic chronometers	⚪	✅	⚪	✅	Fixed-parameter transfer test using official Cobaya likelihood with full covariance matrix. Δχ² = −2.4 (BAO) + −0.6 (CC) = −3.0 total. No validated geometric channel penalises EFC relative to ΛCDM.	T2 Completed (consistency test)	31314922
H₀ tension	SH0ES / JWST	⚪	✅	⚪	❌	Local–global inference separation; EFC survives but does not resolve the tension	T2 Completed (Consistency Check)	31026151
Dark-energy evolution w(z)	DESI DR2 (BAO)	⚪	⚪	⚪	⚠️	Geometric BAO-only phenomenology; full dynamical multi-probe test pending	T2 BAO phenomenology completed	31127380
Small-scale structure	Rubin LSST / Euclid	⚪	⚪	⚪	⚪	Non-linear discriminators	Planned	Metric spec pending
Linear & quasi-linear matter power spectrum shape P(k,z)	BOSS full-shape, eBOSS QSO, DESI full-shape	⚪	⚪	⚪	✅	Tests scale-dependent growth and turnover structure under background-driven growth suppression. Shape consistency required if Poisson sector remains GR while H(a) is modified.	Planned – full-shape likelihood implementation required	Method defined
CMB polarization & lensing internal consistency (TE/EE/φφ)	Planck 2018 TE/EE + CMB lensing	⚪	⚪	⚪	✅	Linear-regime metric and potential evolution constraint. Confirms that regime suppression fully removes EFC effects during recombination and linear post-recombination evolution.	Planned – Boltzmann-level test required	Method defined
Multi-epoch RSD growth trajectory	6dF, SDSS MGS, BOSS, eBOSS, DESI ELG	⚪	⚪	⚪	✅	Tests redshift continuity of growth suppression across L1→L2 transition. β·T(a) must produce a smooth, monotonic deviation pattern with no retuning.	Planned – cross-survey joint trajectory fit	Method defined
E_G gravitational slip consistency (lensing × RSD)	DESI + KiDS/DES lensing	⚪	⚪	⚪	✅	Direct observable linking geometry and growth. With Poisson sector fixed (μ=1), EFC predicts specific redshift-dependent E_G evolution driven solely by expansion history.	Planned – cross-probe estimator implementation	Method defined
BAO anisotropic split (D_A vs H(z))	BOSS, eBOSS, DESI BAO anisotropic fits	⚪	⚪	⚪	✅	Separately constrains transverse and radial expansion. Background coupling must shift D_A and H consistently; forbids geometry retuning.	Planned – anisotropic likelihood test	Method defined
Big Bang Nucleosynthesis expansion-rate bound	Primordial D/H, He⁴ abundance	⚪	⚪	⚪	✅	L0 regime hard constraint. Early-time suppression must ensure ΔH/H ≪ 1 at MeV temperatures. Equivalent to ΔN_eff bound.	Planned – early-time suppression consistency check	Method defined
Galaxy bias evolution consistency	BOSS, DESI clustering + lensing cross-correlation	⚪	⚪	⚪	✅	Ensures that modified growth history does not produce inconsistent galaxy–matter bias evolution. Required for internal consistency of clustering and lensing in L2.	Planned – bias-growth joint modeling	Method defined
Global parameter-lock cross-probe test	BAO + RSD + lensing + ISW + Lyα	⚪	⚪	⚪	⚪	Framework-level stress test: a single β and transition function T(a) must fit all late-time probes without retuning. Eliminates hidden model plasticity.	Planned – full joint likelihood with frozen parameters	Method defined

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.

Test ID	Test	Regime (ξ, η)	Result	Status
KT1	Newton Recovery	ξ ≫ 1, η ≫ 1	Newtonian limit recovered in high-acceleration regime; discrete operator reduces to standard Poisson equation	PASS
KT2	Prefactor Renormalization	ξ ~ 1 (transition)	C ≈ 2.32 (structural constant; not a free parameter)	PASS (structural)
KT3	Mass Scaling	ξ < 1, η < 1	Λ-locked; β ≈ 0.18 (classical); β = 0.50 predicted by statistical occupation (31878760)	PASS (Λ-locked) — resolution pathway identified
KT4	Superposition Breaking	ξ ~ 1 (multi-source)	Non-linear superposition correctly handled by graph operator	PASS
KT5	External Field Effect	ξ ~ 1 (composite field)	EFE signature present; amplitude partially reproduced	PARTIAL PASS
CGS	Cosmological Growth Stability	ξ variable, η < 1	Growth evolution survives under Λ-locked IR screening; unstable without Λ-lock	SURVIVES (Λ-locked only)

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.

Test	Dataset	Result	Interpretation
BAO+RSD joint fit (α_L2)	BOSS DR12 consensus	α_L2 = 0.040 ± 0.024 (1.7σ)	Mild preference for positive coupling; consistent with but not conclusive for EFC
Weak-lensing shear (Case A)	KiDS-1000 Flinc ξ±	α_L2 = 0.10 ± 0.01; Δ(−2 ln L) = −50.9 S₈ = 0.685 (EFC) vs 0.739 (ΛCDM)	Strong fit improvement, but EFC prefers lower S₈, increasing Planck tension (3.6σ vs 2.3σ). Case A is phenomenological lensing boost, not S₈ tension solution.
MVP-G1 Growth: fσ₈ LOO robustness (N2a)	fσ₈ extended (7 pts) + BAO(14)	α = −1.00 ± 0.46 (2.20σ); ΔAIC = −2.91 LOO 7/7 pass; \|α\|/σ range [1.84, 2.42]	Robust ~2σ preference for α<0 in EFC Hubble friction channel. Signal distributed across all 7 growth data points — not driven by any single measurement. Full MVP-G1 control suite: N1 r_d-independence, N2 σ₈-prior strengthening, T7 LOO (7/7). [10.6084/m9.figshare.31332730]
Solar System screening	Cassini / LLR / perihelion	Compatible (γ → 1)	Quantitative compatibility via Density Saturation; complementary exponential amplitude suppression from Covariant EFT (31878334); full SEP proof pending
Cluster entropy–structure (diagnostic)	ACCEPT / TNG-Cluster	Sign flip (ρ_sim < 0, ρ_obs > 0)	Sim–obs mismatch in ICM thermodynamic regulation; motivates entropy-flow physics
ISW cancellation metric 𝒞 (DEPRECATED)	Planck × DESI DR1 tracer windows (LRGz0, LRGz1, LRG+ELG, ELG)	Previously: 𝒞 = 0.59–0.96 (monotonic); A_ISW = 0.29–0.56	The previously defined cancellation metric 𝒞 relied on mixed-channel implementation and is no longer considered a canonical observable under growth-driven EFC. Superseded by uniform A_ISW ≈ 0.89 ± 0.03 under self-consistent ODE-based growth-channel (31329082).

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.

Survey	S₈ Value	Status	Note
KiDS-1000 (original)	0.759 ± 0.024	DEPRECATED	Superseded by Legacy re-analysis
KiDS Legacy	0.815 ± 0.016	CURRENT	Updated shear calibration + photo-z
HSC Y3	0.805 ± 0.022	CURRENT	Independent ground-based confirmation
DES Y3 (original)	0.776 ± 0.017	DEPRECATED	Pending Y6 re-analysis
Stage-III Combined	0.813 ± 0.012	CURRENT	Weighted mean of CURRENT values
Planck 2018	0.834 ± 0.016	CMB reference	~1.3σ from Stage-III combined

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)

Constraint	Scale / Regime	Effect on EFC
Density Saturation Θ(ρ) (v1.3)	All regimes (L0–L3)	Enforces automatic GR recovery in high-density environments (Solar System, stellar interiors); physically bounds Δ_F; provides derivation path for weak-lensing coupling; enables R(k,S,ρ) extension with explicit density discrimination
Cluster merger geometry (31173850)	L2 non-linear	Constrains local entropy-gradient couplings; requires operator classes with component sensitivity and geometric smoothing to remain viable
Weak-lensing phenomenological closure (Postulate A)	L1 linear perturbations	Documents the current absence of a field-derived lensing coupling; introduces temporary phenomenological closure pending action-level derivation. Density Saturation (v1.3) provides a candidate path toward physical derivation.
Case A S₈ direction (v1.4)	L1–L2 lensing	Phenomenological lensing amplitude modification (Σ²) drives S₈ lower, increasing CMB tension. Case A cannot resolve S₈ discrepancy; Case B (consistent MG) required for tension resolution test.
Core Lock consistency enforcement (31223503)	All regimes (method layer)	Prevents parameter drift and cross-regime leakage; enforces frozen-parameter boundaries and explicit translation rules across L0–L3
EFCLASS Sign Structure: background channel σ₈ exclusion (31333414)	L1 background (Friedmann equation)	Analytical proof that ΔE²(z) ≤ 0 for all z > 0: additive gate modification reduces Hubble friction, enhancing growth. Background channel structurally excluded for σ₈ suppression. Verified in CLASS v3.3.4.
Perturbation-Level σ₈ Suppression via μ(a) < 1 (31333600)	L1 perturbations (growth ODE)	WP1a reference model: μ₀=0.85, σ₈=0.773, S₈=0.790 (73% gap closure). Universal factor-2 from gate broadening. Structural ceiling: scale-free μ(a) closes at most ~43% within Planck 1σ.
Lensing barrier: pure μ < 1 excluded (31368433)	L0–L1 perturbations (CMB lensing)	Pure μ < 1 (Σ = 1) excluded at Δχ² = +19. CMB lensing power demands Σ > 1 compensation. Any EFC perturbation model must modify both Poisson and lensing potentials simultaneously.
μ–Σ survival valley (31368433)	L1 perturbations (CMB + BAO + fσ₈)	Narrow survival zone: μ ∈ [0.93, 0.96], Σ ∈ [1.03, 1.07]. Gravitational slip η ≠ 1 structurally required. Target matched by EFC Relativistic Action (31876324): μ ≈ 0.94, Σ ≈ 1.05, η ≈ 1.10.
GRAV→(μ,Σ) structural gap — CLOSED (31876324)	DGS → L1 perturbations (cross-regime)	Gap identified in v3.3 (Grid-AQUAL gives μ > 1, η = 1, wrong scale). Closed in v3.4 by EFC Relativistic Action: variational derivation from F(φ)R + K(ρ) + flow constraint gives μ < 1, η > 1, Σ > 1 within survival valley. c_T = c exactly. Action-level falsification conditions (FA1–FA6) now govern the perturbation sector.
Microphysical gap — BRIDGED (31878334 → 31878760)	DGS / EFT (galactic regime)	Gap identified in v3.5: classical field equation produces correction increasing with acceleration (wrong direction for RAR). Bridged in v3.6 by minimal gradient-coupled excitation model (31878760): three assumptions (bosonic statistics, gradient-coupled energy E ∝ √g, single scale a₀) reproduce the BE form μ(g) = 1/(exp(√(g/a₀)) − 1) with no free parameters. Lattice derivation: k_eff = g/l_g → ω ∝ √g. Kinematic bridge only; full grid Hamiltonian and QFT derivation remain outstanding. KT3 resolution pathway: statistical occupation should restore β = 0.50.

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.

#	Falsification Condition	Sector	Consequence if observed
F1	Measured A_ISW = 1.00 ± 0.02 (tracer-independent) on DESI DR1 × Planck	BG (L1)	Rules out background-driven EFC growth channel; β·T(a) coupling disfavoured
F2	Multi-epoch RSD trajectory shows no statistically significant deviation from ΛCDM across z = 0.02–1.5	GRW (L1–L2)	Eliminates growth-channel modification; EFC must reduce to pure background cosmology
F3	Cluster lensing asymmetry absent at pre-registered significance level on JWST κ maps (A_sig consistent with zero)	LEN + SCR (L2)	Disfavours entropy- or structure-coupled anisotropic lensing response
F4	Full-shape P(k) inconsistent with background-only growth modification at > 3σ	GRW (L1–L2)	Scale-dependent growth pattern requires Poisson-sector modification not present in baseline EFC
F5	Discrete gravity operator fails to reproduce observed rotation curve diversity across morphological types (SPARC + MaNGA joint likelihood)	DGS (L2–L3)	Graph-AQUAL operator structurally insufficient; discrete sector requires revision or abandonment
F6	Λ-locked screening produces cosmological growth instability that cannot be resolved without introducing free parameters	DGS + BG	Λ as bulk capacity hypothesis fails; screening mechanism requires replacement
F7 (theoretical) — PASSED	Relativistic EFC perturbation theory produces η = 1 (no gravitational slip) in the cosmological regime, failing to generate the required μ–Σ split with μ < 1 and Σ > 1	GRW + LEN (L1)	PASSED (v3.4): EFC Relativistic Action (31876324) derives η = 1 + 2(ε_F + ε_λ + ε_λ,resp) ≠ 1 from three independent slip sources (scalar-tensor, flow-constraint, response-field). The action produces μ ≈ 0.94, Σ ≈ 1.05, η ≈ 1.10 — matching the survival valley. Superseded by action-level conditions FA1–FA6.

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.

Falsified Element	Failure Mode	Resolution	Status
EFC v1 Scalar Index Model	Original scalar-index parametrisation produced monotonic radial boost incompatible with observed rotation curve diversity. Failed to reproduce flat/declining outer profiles.	Abandoned in favour of regime-dependent EBE framework with explicit density saturation Θ(ρ). Monotonic Radial Boost Problem resolved by structural revision.	FALSIFIED → Superseded
EFC-R on SPARC (direct application)	Direct application of EFC-R coupling to full SPARC sample produced systematic residuals in Regime 2–3 galaxies. Only Regime 1 (low-acceleration) showed acceptable fits.	Led to regime partitioning (EBE framework) and recognition that single-regime coupling cannot span full acceleration range. Multi-regime structure now core architectural feature.	PARTIAL FAIL → Resolved by regime separation
ISW cancellation metric C (v1.6)	Original tomographic cancellation prediction relied on mixed-channel implementation producing spurious tomographic gradient.	Superseded by ISW Consistency Audit v1.1. Self-consistent ODE-based growth channel produces uniform A_ISW ≈ 0.89 ± 0.03. Original C-metric deprecated.	DEPRECATED → Revised (v2.1)
Poisson-sector coupling (direct μ boost)	Early EFC assumption that Poisson-sector μ > 1 could enhance growth was tested against RSD data. Found to be disfavoured; background channel preferred.	Led to structural separation of background (H²) and perturbation (μ) channels. Current architecture: background channel primary, perturbation channel requires μ < 1 for σ₈ suppression.	DISFAVOURED → Architecture revised
Naïve GRAV→cosmological (μ,Σ) extrapolation (v3.3)	Grid-AQUAL operator directly extrapolated to cosmological perturbation sector produces μ > 1 (wrong sign for σ₈ suppression), η = 1 (no gravitational slip), and screening at galactic rather than cosmological scales.	Formally documents that discrete gravity operator cannot be naïvely mapped to cosmological perturbation parameters. Relativistic perturbation theory is required as an explicit derivation step. Added as pipeline-critical item.	FALSIFIED → Relativistic derivation required (31368433); RESOLVED by EFC Relativistic Action (31876324)

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

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)

6. Recent Updates

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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