The Log-Harmonic Field Theory (LHFT)

Preface Log-Harmonic Field Theory The Log-Harmonic Field Theory (LHFT), developed by Christian Baganz in Potsdam, Germany, is a projective recovery framework for physics. It does not begin with a pre-given spacetime, a fixed particle inventory, or an observer standing outside the world. It begins by separating what can be made physical from what must remain beyond physical exhaustion. The deepest symbol of the framework, Sstruct, denotes the non-exhaustible structural presupposition of LHFT. It is not itself a physical object. It is boundary language, not a measured sector and not a derivational premise. Sstruct ∉ Pphys Therefore LHFT does not claim a physical derivation from Sstruct to the formal action layer S1L. Sstruct ↛phys S1L This boundary is essential. It prevents the theory from becoming an absolute ontology disguised as physics. The complete structural totality cannot be closed by a finite internal observer. What can be closed are finite, stable, observer-readable recovery windows. The formal theory begins with S1L. This object is not yet physics in the strict sense. It is the formal starting layer from which local variation data may be extracted. S1L → J4Φ*S1L → Kstruct The jet J4Φ*S1L is the formal local data layer. It collects local derivative information around an admissible background Φ*. It is not a physical phenomenon. It makes operator formation possible. From this local data layer, LHFT constructs the structural operator Kstruct. Only after projection, visible-hidden decomposition, Schur reduction, recovery mapping, and defect control can an observer-readable physical sector arise. S1L → J4Φ*S1L → Kstruct → KO → KeffO → XO Strict physics begins only at the level of a stable, measurable, defect-controlled recovery readout. XO = RO (KeffO), DO = 0 This is the central physical criterion of LHFT. A formal object is not physical merely because it is mathematically defined. A sector is physical only when its recovery map is specified and its defect functional vanishes. The Schur mechanism is the main visibility mechanism of the theory. A projected operator window is decomposed into a visible block and a hidden complement. The visible sector is not simply isolated; it is modified by the backaction of the hidden complement. KeffO = AO − BO†CO, adm−1BO This formula expresses the non-additive character of LHFT. What is visible is not merely what remains after exclusion. It is the stabilized visible readout after hidden-sector backaction has been included. The inverse CO, adm−1 is an admissible complement inverse. It is not generally identical to a Moore-Penrose pseudoinverse and it is not a substitute for gauge fixing. It must be defined on the physically admissible complement domain of the sector. CO, adm−1 ≠ CO+ The recovery maps of LHFT are sector-specific. There is no universal black-box map that turns every effective operator into physics. Geometry, quantum states, scalar fields, spinors, gauge structures, electromagnetic coupling, dark residuals, and geometry-matter co-recovery each require their own map and their own defect system. RO = RG, RΨ, RDirac, REM, Rgauge, RΔ,. . . Established physics is not rejected. Quantum mechanics, classical phase space, quantum field theory, special relativity, general relativity, Newtonian gravity, gauge theory, and the Standard Model are treated as recovery limits. Their empirical success becomes a constraint on the LHFT recovery maps. Dlimita = 0 ⇒ XaO → Xaknown The electromagnetic Alpha sector has the strongest current projective normal-form status in the LHFT program. It is treated as a finite observer-readable electromagnetic readout, not as an absolute closure of the whole structure. KαO = α−1 By contrast, the full Standard Model, the full QCD sector, gravitational coefficient recovery, dark-sector profiles, and cosmological amplitudes remain sector-specific proof programs unless explicitly closed by their own defect equations. The logarithmic coordinate u = ln (r/r0) remains important, but its status has changed. It is no longer treated as a deepest coordinate of the inaccessible totality. It is a recovery coordinate used after a scale, distance, or relational readout has already been stabilized. uO = ln (rO/r0) Earlier empirical log-scale searches are therefore treated as diagnostics, not as core confirmation. Unconfirmed SN1a log-shell results are not used as foundational support for the theory. diagnostic ≠ confirmation The present formulation of LHFT is organized into five cores: Core I defines the boundary, the formal starting layer, and the criterion for physical readability. Core II develops the jet, operator formation, projection, admissible complement inverse, and Schur reduction. Core III specifies sector-specific recovery maps and physical sector criteria. Core IV embeds known physics as recovery limits and states the limits of sector claims. Core V turns the architecture into a proof program based on minimal models and explicit defect realization. The most important open methodological demand is clear: the condition Da = 0 must be solved, not merely written down. LHFT must demonstrate the full pipeline first in minimal models, then in controlled field sectors, then in spinor and gauge sectors, and only later in QCD and the full Standard Model. Da = 0 must be solved, not assumed The purpose of this manuscript is therefore not to claim possession of the whole. Its purpose is to define a disciplined recovery architecture in which physical sectors become valid only through projection, Schur reduction, recovery mapping, and defect closure. The guiding principle is: projective closure ≠ absolute closure LHFT is limited to stable projective recovery. It does not claim absolute possession of the unreachable whole.