Log-Harmonic Field Theory — Curated Formula Register

Log-Harmonic Field Theory — Curated Formula Register Title: LHFT Curated Formula Register: Equation Status Inventory, Recovery Architecture, and Defect-Gate Audit Register Overview This document is a curated formula register for Log-Harmonic Field Theory (LHFT). It collects the central equations, recovery chains, defect gates, and status-controlled theorem targets of the current LHFT program. Core purpose: The register is not merely a list of formulas. It is a status-controlled audit layer that separates definitions, derived relations, recovery statements, conditional normal forms, candidate structures, and open proof obligations. The register is designed to prevent overclaiming by making explicit whether a formula is a definition, a derived identity, a projected recovery statement, a normal-form closure, a candidate theorem target, or an open microscopic proof obligation. Core LHFT Architecture The formula register is organized around the LHFT validity and recovery chain: Sₛtruct ⇢ S₁Lᵛal → J⁴⏥*S₁L → Kₛtruct → (ΓX^𝒪, ΠX^𝒪, WX^𝒪, ZX^𝒪) → Kₑff, X^𝒪 → X_𝒪 In this chain: Sₛtruct denotes the open structural presupposition. S₁Lᵛal is the first validity-bearing mathematical access layer. J⁴⏥*S₁L is the finite local jet at the stationary carrier state. Kₛtruct is the structural Hessian operator. ΓX^𝒪 is the observer-sector coupling state. ΠX^𝒪 is the sectoral observer projection. WX^𝒪 is the admissible recovery window. ZX^𝒪 is the sectoral impedance or normalization. Kₑff, X^𝒪 is the Schur-effective sector operator. X_𝒪 is the observer-readable projected recovery output. Key Formula Pattern A central operator pattern of the register is the sectoral Schur readout: Kₑff, X^𝒪 = AX^𝒪 − (BX^𝒪) † (CX^𝒪) ⁻¹ BX^𝒪 This formula expresses the visible sector corrected by hidden-complement backreaction. The hidden block is not absent. It is suppressed from direct readout but remains dynamically effective through the Schur response. Compactly: hidden ≠ absentprojection ≠ destructionSchur reduction = objective readout lens Register Structure The register is organized into the following major parts: Part I — Master Architecture and Validity Boundary: theory object, validity chain, structural state, structural drift, observer clock, and scale coordinate. Part II — Projection, Hidden Complement, and Schur Readout: master projection, visible/hidden split, sectoral Schur complement, and recovery map. Part III — Closed and Audit-Relevant Internal Factors: Alpha anchor, R₅₀, ρ₅₀, ηW, ΞW, fₑ, eff², Ξ_ℓ, 0, yₑ, and quotient closures. Part IV — Electron, Weak, and Lepton Defect Gates: electron-Yukawa defect, near-node gate, deep-lepton witness gate, weak compatibility, and Koide diagnostic. Part V — Standard-Model Recovery Targets: gauge algebra, electroweak bridges, Yukawa matrices, CKM, PMNS, neutrino oscillation, and absolute mass split. Part VI — Strong, QCD, and Proton Sector: strong scale, recursion depth, QCD running, singlet projector, hadronic Schur operator, and proton mass architecture. Part VII — Energy, Mass, and Recovery Bridges: sectoral energy readout, EM phase energy, effective action quantum, recovery mass anchor, and relativistic energy relation. Part VIII — Geometry, DARK, and Cosmology: stress-energy recovery, geometry-matter bridge, DARK projection, DARK Schur response, and cosmological benchmark. Part IX — Measurement, Born Weights, Canon Rules, and Open Master Defects: Born gate, projector algebra, branch weights, transition probability, closure rules, no-fit rules, and global defects. Representative Formula Highlights Validity boundary: Sₛtruct ≠ S₁LᵛalKₛtruct = δ²S₁Lᵛal / δΦδΦ |⏥* Observer clock recovery: t_𝒪 = T_𝒪 (s) τ_𝒪 = dt_𝒪 / ds Log-scale coordinate: u^𝒪 = ln (r^𝒪 / r₀^𝒪) Alpha precision anchor: K_α = α⁻¹ Electron-Yukawa readout: yₑLHFT = fₑ, eff² √ (2π Ξ_ℓ, 0 / (K_α ηW ΞW) ) Weak role separation: ηW ≠ ΞWηW − ηW⁽⁰⁾ ∼ R₅₀²ΞW − 2/3 ∼ R₅₀ General closure rule: DX^𝒪 = 0 ⇒ X closed inside WX^𝒪DX^𝒪 = 0 ⇏ Sₛtruct absolutely closed Status-Control Function A central purpose of the document is to mark each equation according to its epistemic status. The register distinguishes: Definition: introduced as part of the formal LHFT language. Derived: follows from prior definitions or fixed normal forms. Recovery: belongs to an observer-readable projected physics regime. Normal-form closed: closed inside a specific normal-form architecture. Projective normal-form closed: closed inside a specified projection window. Candidate: structurally motivated but not yet microscopically forced. Open: explicitly registered as a proof obligation. Audit principle: A formula is not closed merely because it is elegant, useful, or numerically close. A theorem row requires a defect row. No-Fit Boundary The formula register explicitly protects LHFT from fitting-based overclaiming. Observed values may test a chain, but they must not define internal factors. Examples of no-fit rules include: yₑᵒbs ⇏ fₑ, eff²yₑᵒbs ⇏ Ξ_ℓ, 0sin²θₑff^ℓ, obs ⇏ ηWMₚᵒbs ⇏ ΛSnumerical agreement ≠ derivation This makes the register an audit document as much as a formula collection. Scientific Role The register is intended as a working reference layer for LHFT manuscripts, appendices, theorem notes, and empirical protocols. It provides a controlled equation inventory for checking whether a claim is definitional, derived, recovered, conditionally closed, candidate-level, or open. It is especially useful for: checking formula consistency across LHFT documents, tracking proof obligations, separating recovery statements from fundamental claims, identifying duplicated or historical formula variants, preventing numerical agreement from being misread as derivation, preparing future peer-review and publication audits. Claim Boundary This register records formula status, dependencies, recovery roles, and proof obligations within LHFT. It does not claim absolute closure of Sₛtruct. It does not treat normal-form closure as microscopic closure. It does not treat numerical agreement as derivation. Observable quantities listed here are to be read as projected recovery targets or defect-controlled normal forms unless explicitly marked otherwise. Final boundary: The register is a curated status-control document, not a complete microscopic proof of all listed sectoral recoveries. Suggested Citation Baganz, Christian. LHFT Curated Formula Register: Equation Status Inventory, Recovery Architecture, and Defect-Gate Audit Register. Log-Harmonic Field Theory (LHFT), 26. 06. 16-02 working register. Licensed under CC BY 4. 0.