Synthesis of the LHFT Energy/Mass Program

This note presents a compact, status-controlled summary of a central finite-depth readout chain in Log-Harmonic Field Theory (LHFT). The guiding idea is that observable physics is not treated as the fundamental layer itself, but as a projected recovery regime produced by structure, state, coupling, projection, and Schur-complement reduction. In this reading, the usual linear scale description in r is replaced by the logarithmic access coordinate u = ln(r / r₀). Stable physical observables are therefore interpreted as finite projective Schur readouts across logarithmic structural depth. structure → state → coupling → projection → recovery readout A.1 Alpha Status and Closure Boundary The electromagnetic precision anchor is the finite Alpha normal form. In the present LHFT status, the finite readout of α is treated as [Projective normal-form closed], while the direct microscopic forcing from S₁L remains [Open]. This distinction is essential: projective normal-form closure is not identical to absolute microscopic closure of S_struct. Kα = α₅₀⁻¹ [Projective normal-form closed] S₁L ⇒ Dα = 0 [Open as direct microscopic forcing] A.2 Finite Alpha Normal Form The finite LHFT Alpha normal form is: α₅₀⁻¹ = 4π³ + M₂(50) / 16 − (7/16)ρ₅₀ − (1/16)ρ₅₀² + (2/3)ρ₅₀³. ρ₅₀ = (23/110) · √(M₂(50) / M₄(50)). The finite moment functions are: M₂(N) = (N² − 1) / 12 M₄(N) = ((N² − 1)(3N² − 7)) / 240 A.3 Numerical Evaluation of Kα For N = 50, the moment values become: M₂(50) = (50² − 1) / 12 = 208.25 M₄(50) = ((50² − 1)(3 · 50² − 7)) / 240 = 78020.8625 ρ₅₀ = (23/110) · √(208.25 / 78020.8625) = 0.0108024504370528... Kα = α₅₀⁻¹ = 137.03599919620436... α₅₀ = 1 / Kα = 0.00729735256330877... Thus, the electromagnetic anchor used below is not inserted as a measured parameter. It is taken from the finite LHFT Alpha normal form. A.4 Electron-Yukawa Master Formula The conditional LHFT Electron-Yukawa readout is: yₑᴸᴴᶠᵀ = fₑ,eff² · √[ (2π · Ξℓ,0) / (Kα · ηW · ΞW) ]. In this formula, Kα is the closed electromagnetic impedance anchor, fₑ,eff² is the projected electron near-node suppression, Ξℓ,0 is the deep charged-lepton accessibility factor, ηW is the weak projection-share normalization, and ΞW is the charged weak accessibility factor. A.5 Electron-Yukawa Factor Set Kα = α₅₀⁻¹ = 137.03599919620436... R₅₀ = √[20 / (3N∗² − 7)], with N∗ = 50. fₑ,eff² = 15 / (3N∗² − 7) = 15 / 7493. Ξℓ,0 = 20 / [3(4cF N∗)² − 7], with cF = 5. ηW = (1/cF) · [1 + (16/3)R₅₀²]. ΞW = (2/3) · (1 + R₅₀). A.6 Numerical Evaluation of the Electron-Yukawa Factors R₅₀ = √(20 / 7493) = 0.0516638933946005... fₑ,eff² = 15 / 7493 = 0.00200186841051648... Ξℓ,0 = 20 / [3(4 · 5 · 50)² − 7] = 20 / (3 · 1000² − 7) = 6.66668222225852 × 10⁻⁶. ηW = (1/5) · [1 + (16/3)(20/7493)] = 0.202847101739401... ΞW = (2/3) · [1 + √(20/7493)] = 0.701109262263067... A.7 Fully Expanded Active Electron-Yukawa Formula Substituting the active finite-depth factors gives: yₑᴸᴴᶠᵀ = (15/7493) · √{ [2π · (20 / (3 · 1000² − 7))] / [Kα · ((1/5) · (1 + (16/3)(20/7493))) · ((2/3) · (1 + √(20/7493)))] }. yₑᴸᴴᶠᵀ = 2.93484890024607 × 10⁻⁶ calculation The resulting Electron-Yukawa readout is therefore a conditional LHFT value. It uses the projectively closed Alpha anchor, but still depends on the independent closure of the non-Alpha gates D_quadᵉ, D_Ξℓ,0, D_ηW, D_ΞW, and D_composeʸᵉ. Conditional Electron-Yukawa closure target: D_Kα = 0, D_quadᵉ = 0, D_Ξℓ,0 = 0, D_ηW = 0, D_ΞW = 0, D_composeʸᵉ = 0 ⇒ D_yₑ = 0. A.8 Proton-Side Mass Architecture Mₚ = ΛS · (λ_physᵖ + Δ_readᵖ). The proton-side formula is not treated as closed here. It states the intended strong-sector architecture: a strong recovery scale ΛS multiplied by a dimensionless proton readout plus a residual readout term. Its closure requires independent strong-sector and hadronic Schur gates. A.9 Proton/Electron Ratio Mₚ / mₑ = [√2 · ΛS · (λ_physᵖ + Δ_readᵖ)] / [yₑ · vH]. This expression is a cross-sector theorem target. It may only become a prediction if the electron chain, the strong-sector mass readout, the Higgs recovery scale, and the common projection window are independently closed. A.10 Dark-Sector Schur Response ΔK_darkᴼ = − B_G† · C_G⁻¹ · B_G. The dark-sector expression is a candidate gravitational Schur response of the hidden or electromagnetically suppressed complement. It is not a fitted dark-matter term and is not claimed as closed in this abstract. Status Statement numerical agreement ≠ derivation projective normal-form closure ≠ absolute microscopic closure The present contribution therefore reports a closed finite Alpha readout, a conditional Electron-Yukawa readout built on that anchor, and a structured proof program for the remaining non-Alpha gates. The proton and dark-sector formulas are included only as architectural continuations of the same projective Schur-readout principle.