Abstract 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 Sₛtruct. 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 Dquadᵉ, D_Ξℓ, 0, D_ηW, D_ΞW, and Dcomposeʸᵉ. Conditional Electron-Yukawa closure target: DKα = 0, Dquadᵉ = 0, D_Ξℓ, 0 = 0, D_ηW = 0, D_ΞW = 0, Dcomposeʸᵉ = 0 ⇒ Dᵧₑ = 0. A. 8 Proton-Side Mass Architecture Mₚ = ΛS · (λₚhysᵖ + Δᵣeadᵖ). 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 · (λₚhysᵖ + Δᵣeadᵖ) / 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 ΔKdarkᴼ = − BG† · CG⁻¹ · BG. 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.
CHRISTIAN BAGANZ (Wed,) studied this question.