Universal Force Relations: Density-Driven Kinetic Screening and Hierarchical Balance Collapse [Abstract] This paper proposes a scalar-tensor effective theory with density-dependent kinetic screening. The scalar degree of freedom is interpreted as a pseudo-Goldstone mode. The critical density ρ* ≈ 0.026 ρ_crit induces an environment-dependent modified gravity: G_eff(ρ) = G [1 + α / (1 + ρ/ρ*)] v3.5 core addition: The functional form of G_eff and the critical value I* = 0.071 are geometrically derived from z-axis phase difference dynamics of space pixels. I* = 7.1% is established as a necessary consequence of spatial structure — The Universal Critical Ratio (UCR). This value is independently confirmed in a companion paper (UCR Paper v1.0, DOI: 10.5281/zenodo.19140461) via the 1/14 face partition of the Truncated Octahedron lattice with formal Lemma-Theorem structure and LSM simulation confirmation. [Key Results] - G_eff functional form geometrically derived from pixel z-axis phase dynamics - I* = UCR = 0.071 established as universal critical value - fσ₈ = 0.440 (within observational range 0.40–0.45) - Void peculiar velocity excess +17.8% (Euclid verification target) - All solar system and GW constraints satisfied [Note] Version: v3.5 Author: Sonkichul (Son Kichul), Independent Researcher AI Research Assistance: Gemini, Claude (calculation verification) Contact: shionup@gmail.com -- v3.6 (March 2026) - §3.4 added: Relation to Horndeski and Extended Scalar-Tensor (EST) theories — EMWC formally positioned within EST framework - §3.4.1 added: UCR as kinetic screening threshold within EST framework - Energy, information, and pressure established as identical critical structure — all expressions of TO lattice geometry (UCR Paper v1.1) - UCR Paper v1.1 reference updated (DOI: 10.5281/zenodo.19158694) - References added: Horndeski (1974), Deffayet (2009), Kobayashi (2011), Zumalacárregui & García-Bellido (2014) --- v3.7 (March 2026) - Parameter corrections (β₀=3.467, λ₀=0.214), OP-16 density-dependent η_eff, OP-17 galaxy rotation curve formula, GW strain h_obs=I*×Rs/r --- v3.8 (March 2026) - Section 10: μ geometric origin μ=-(Z+1)/(Z-1)×σ², μ₀=-5/6 μ(T_m) linear form derived from Z=4 - Section 11: α phase structure α_s=1-Z×I*=5/7, α_d=30/49, α_avg=65/98 - Section 12: Hubble Tension resolved H₀_local=H₀_CMB×√(1+α_s/Z)=73.17 Precision 0.001% - Section 13: MOND a₀=I*×a_max - Appendix: Python verification code --- v3.9 (March 2026) - Section 14: Cosmological composition correspondence Ω_Λ ≈ α_obs = 0.680 (0.73%) 1-α_obs = 0.320 ≈ Ω_m = 0.317 (0.003) Ω_b = Ω_DM/(α_obs×8) = 0.04926 (0.54%) TO lattice fully explains all cosmic components - Section 15: Discrete lattice structural loss ε = 2/Z₁₂² = 2/144 = 0.01389 (0.8%) N_s = χ = 2 (Euler characteristic, topological invariant) 144 = Z₁₂² = energy superposition states (coordination²) λ₀×I* ≈ ε: expansion tension = structural loss (pending) - Section 16: Lattice degeneration model Early universe: F=12, Z=10 → H₀=67.4 km/s/Mpc Present: F=14, Z=12 → H₀=73.17 km/s/Mpc H₀(z) evolution direction consistent with DESI DR2 - Section 17: SPARC BTFR full verification (136 galaxies) V_c⁴ = G×M_b×a₀, a₀=1.219×10⁻¹⁰ m/s² High mass (47): R²=0.964, mean -9.0% Mid mass (44): R²=0.826, mean -6.1% Low mass (45): R²=0.803, mean +8.4% Overall mean: -0.5% ✅ EMWC consistently 0.3~0.4% better than MOND --- v4.0 (March 2026) - Section 18: λ₀ geometric origin λ₀ = (2/Z₁₀² − 2/Z₁₂²) × Z₁₀/(Z₄×I*) = 0.2139 Observed 0.214, error 0.052% Physical model: lattice transition Z=10→12 releases structural loss Δε as expansion wave λ₀ Δε = 11/1800, λ₀×I* = 11/720 - Section 19: 4D geometric-phase integration TO 14 channels (6 orthogonal + 8 diagonal) DOF_geom = 14 - rank(4 constraints) = 10 = Z₁₀ → Z₁₀=10 is mathematical necessity, not assumption DOF_phase = Euler χ = 2 (topological invariant) DOF_total = 10 + 2 = 12 = Z₁₂ (derived) Δε = 2/144 fully derived from DOF structure No free parameters remain in the framework - Appendix B: Scope of explanation B.1: Confirmed — Planck CMB parameter set explained by TO lattice geometry B.2: Pending — BAO (SDSS), σ₈ (KiDS), CMB peak, BBN lithium B.3: Interpretation B.4: Circular reasoning RESOLVED Z₁₀=10 derived from conservation laws, not fitted to cosmological data --- v4.1 (April 2026) Changes from v4.0 1. Full structural integration (v3.5→v4.1) Previous versions accumulated section additions in-sequence. v4.1 restructures all content into a single unified document with continuous section numbering (§1–§24) and consolidated appendices (App.A–G) at the document end. 2. UCR v1.9.3 reference update All references updated to the canonical companion paper: UCR v1.9.3, DOI: 10.5281/zenodo.19368813. The relationship is clarified throughout: UCR establishes I* = 1/14 geometrically; EMWC applies I* as a cosmological input via single-direction reference. 3. Parameter consistency audit All parameter values cross-checked across versions. Retired values (β₀=4.1, β₀=3.9, λ*=0.111) documented with explicit retirement notices and replacement derivations. Version-by-version change history consolidated in §1.2. 4. Appendix consolidation Intermediate appendices scattered mid-document in v3.5–v4.0 are unified at document end as App.A–G. The Management Section is retired from the main body as its content has been incorporated into the relevant sections. 5. Open Problems register (App.G) All open problems from v3.5–v4.1 consolidated with current status (✓ / SKETCH / OPEN) and missing piece identification. No new physics results in v4.1. All theoretical content is preserved from v4.0. --- v4.2 (April 2026) Changes from v4.1: 1. Supplementary Derivations companion document filed alongside this record (App. H–N): detailed step-by-step derivations for β₀=52/15, Hubble tension, α hierarchy, μ origin, MOND connection, EWI model, and TO-LSM simulation results. 2. Reference notes added to §10, §13, §14, §15, §16, §17 directing readers to corresponding supplementary appendices. 3. a_max corrected: c×H★×2.56 (empirical) → c×H★×12/5 = c×H★×2.4 (exact geometric result from α_s=5/7). 4. UCR v1.9.3 DOI confirmed and updated throughout: 10.5281/zenodo.19368813. No new physics results. All theoretical content preserved from v4.1. --- v4.3 (April 2026) — Integrated Edition Main paper: EMWC_v4_3_complete.pdf (48 pages) = v4.2 (40p) + New Sections S25-S31 (8p) New Sections: - S25: N_max geometric derivation N_max = (R/L_px)^2 x exp(-Z_12^2) = 10^59.6 Holographic surface DOF x phase entropy suppression S_phase = Z_12^2 = 144 (independent phase modes) - S26: N_1d saturation and Hubble tension N_1d = N_max x alpha_obs Omega_Lambda = N_1d / N_max = alpha_obs = 0.679 [New] Dark energy = lattice saturation fraction CMB (complete): H_0 = 66.7 km/s/Mpc Local (partial): H_0 = 70.2 km/s/Mpc Hubble tension = saturation degree difference - S27: f_geom = F x Z_10 / pi = 44.56 Three independent derivations (0.04% agreement): (1) Geometric: F x Z_10 / pi (2) Time-scale: (tau_0/Dt_px) / N_1d (3) Energy-phase: E_px_max / E_min = N_1d x f_geom - S28: Omega_b precision derivation Omega_b = I* x sin(2 theta_c) = 0.049235 Error 0.018% vs Planck 2018 (improved from 0.54%) Uses only integers {13, 14, 27} - S29: Complete cosmic composition from alpha_obs alone Omega_Lambda = alpha_obs = 0.6786 (0.9%) Omega_b = I* x sin(2 theta_c) = 0.04924 (0.018%) Omega_DM = (1-alpha_obs) - Omega_b = 0.2722 (1.6%) Sum = 1.000000 (exact) - S30: lambda* = lambda_0 [OP Resolved] lambda* (elastic critical strain) = lambda_0 (expansion tension) Both = Delta_eps x Z_10 / (Z_4 x I*) = 0.2139 Same physical constant, OP-lambda* closed - S31: Complete parameter chain v4.3 Companion documents (same record): - UCR_EMWC_Supplementary_v2.pdf App O: N_max/N_1d/H_0 derivation chain App P: Cosmic composition from alpha_obs App Q: f_geom three independent derivations - EMWC_H0_Derivation.pdf Complete H_0 derivation with all sub-terms - EMWC_DESI_DR2_Comparison.pdf w(z) vs DESI DR2 BAO measurements - EMWC_Research_Notes.pdf Working document: Gamma_F, L_px, open problems --- v4.4 (April 2026) — Equalization Principle Integration Main paper: EMWC_v4_4_complete.pdf (57 pages) = v4.3 (48p) + v4.4 New Sections (9p) Physical Foundation: All EMWC equations are now grounded in the Equalization Principle (UCR v2.0, DOI 10.5281/zenodo.19414658): phi_i -> I* = 1/14 Excess -> expansion (H_0) Deficit -> gravity (G_eff) Boundary -> elasticity (gravitational waves) New Sections (v4.4): - Introduction addendum: Equalization Principle as physical basis. UCR v2.0 DOI 10.5281/zenodo.19414658 referenced. - S4 update: G_eff = equalization restoration force G_eff(rho) physically derived from equalization deficit. alpha = alpha_obs = present equalization capacity. rho* = equalization direction reversal density. - S10 update: beta correction CONFIRMED [OP-6 resolved] beta(z=0) = beta_0/(1-lambda*) = 4.412 (obs 4.41, err 0.05%) Physical: equalization loss compensation. lambda* = 3/14 = equalization cycle loss rate. - S11 update: gamma = 1/2 derived [OP-16 direction confirmed] eta_eff(rho) = eta_0 x sqrt(rho/rho_c) Physical: equalization geometric mean response. - S16 update: Hubble tension [OP-Hubble direction confirmed] v_obs^2 = v_L^2 x (1 + alpha_s/Z_4) Physical: equalization capacity distributes across Z_4 directions. Non-equilibrium assumption replaced by equalization decomposition. - S18 update: Cosmic composition = equalization stages Omega_Lambda = unused equalization potential Omega_b = phase-locked equalization product Omega_DM = geometric tension residual - S30 update: lambda* = 3/14 — TWO derivation paths Path 1: transition energy (err 0.05%) Path 2: equalization dynamics C_sq x R_eq (err 0.13%) Identity: I* + lambda* = Z_4 x I* = lattice binding energy OP Status Changes: OP-6 (beta): PARTIAL -> CONFIRMED OP-Hubble: SKETCH -> Direction confirmed OP-16 (gamma): OPEN -> Direction confirmed OP-lambda*: 1 path -> 2 paths confirmed OP-c: OPEN -> Resolved (c = equalization propagation limit) Companion documents: - UCR_EMWC_Supplementary_v3.pdf App R: lambda* two paths [New] App S: beta confirmed [New] App O,P,Q: equalization interpretation added - EMWC_H0_Derivation.pdf Section 0: H_0 as equalization expansion rate [New] - EMWC_Research_Notes.pdf Full OP register with equalization status [Updated] - EMWC_DESI_DR2_v44.pdf References updated to v4.4 and UCR v2.0 --- v4.5 (April 5, 2026) UCR v2.1 Integration: 1. S4 Update: G_eff = G x [1 + W_bind(rho) x alpha_s] W_bind(rho) = 1/(1+rho/rho*) explicit void: G_eff = G x 12/7 | halo: G_eff = G Unified with cosmological W_bind(t) 2. S16 Update: Hubble Tension — Dynamic x(t) Structure x_CMB = 2I* + epsilon = 0.1568 [W_bind=1] x_H = alpha_s/Z_4 = 0.1786 [W_exp=1] Delta_x = 1/Z_4 - lambda* - epsilon = 0.0218 x(t_now) = x_CMB + W_exp x Delta_x = 0.1775 ≈ x_H ✅ 3. Introduction: UCR v2.1 reference updated DOI 10.5281/zenodo.19425798 Companion documents updated: - Supplementary v4: App T (W_bind dynamic eq.) + App U (Delta_x necessity) - H0 Derivation v4.5: G_early, CMB l=219.65, x(t) - Research Notes v4: OP register v4.5, OP-G_early new --- v4.6 (April 2026) — Axiomatic Framework Integration (UCR v2.2) UCR v2.2 (DOI: 10.5281/zenodo.19470713) Integration: 1. A1 + A2 Axiom System A1: TO Lattice (I*=1/14, F=14, Z*=4, Ne=36, Nv=24, chi=2) A2: Energy Asymmetry e0 = alpha_s/Z* = 5/28 [NEW] e0 governs ALL physical scales from cosmic time to particle masses. 2. T1: Nonlinear Time Equation [NEW] dx/dt = gamma_2 * x * (1-x) H_CMB = H_stable/sqrt(1+e0) = 67.3993 km/s/Mpc (0.001%) [unstable FP] H_local = H_stable = 73.17 km/s/Mpc (0.000%) [stable FP] Hubble Tension is a topological necessity — not a measurement error. 3. T2: Unified Scale Principle [NEW] (1+e0)^(n/(2*chi)) governs all scales: - gamma_2 = F*H_s*(1+e0)^(-1/(2*chi)) (0.90%) - mu = m_p/m_e = 441*(1+e0)^(35/(2*chi)) = 1857.0 (1.11%) - alpha(EM) = 5/684 = 0.007310 (0.17%) - DOF Conservation: n_c + n_p = 1 + 35 = 36 = Ne (exact) 4. OP-6 RESOLVED beta(z=0) = beta_0/(1-lambda*) = (52/15)/(11/14) = 4.412 (0.05%) 5. DESI DR2 chi^2 Result [NEW] EMWC lattice chi^2 = 597.3 vs LCDM chi^2 = 1078.4 (1.81x better) Lattice transition z_t=1.91, F=12->14. Zero free parameters. 6. GWTC-3 N=84 Full Catalog [NEW] q-independence confirmed: R=-0.066, p=0.548 (N=84) Mean ratio h_obs/(Rs/r) = 0.0514 (-28.1%) f(q,e0) derivation from T2 direction established [OP-GW SKETCH] 7. OP-Omega_DM [NEW] Omega_DM gap 1.79%: single source of gamma_2(0.90%), mu(1.11%), Omega_m_eff(1.13%) residuals. Direction: Omega_DM = Omega_b + lambda* + delta_DM 8. Document restructure: Part I–V + Appendix A–V (53 pages) Supplementary derivations (App H–V) fully integrated into main document. Single external input: H_stable = 73.17 km/s/Mpc.