Log-Harmonic Field Theory (LHFT)

Log-Harmonic Field Theory By ‘log-harmonic’ we mean log-periodic signatures in u = ln r space (not the GFT definition f = h * overline(g)) (LHFT) is a structural framework that treats familiar 4D physics as an operational recovery of a deeper, scale-organized substrate. The central aim is conservative: not to replace standard physics in its successful domains, but to provide a coherent interface model in which special relativity, general relativity, and quantum field theory appear as stable limits while measurement-context effects and scale-structured residuals become natural consequences of the underlying geometry. LHFT is organized around a strict triad: Structure – State – Coupling. • Structure denotes the resolution-free substrate in which scale organization is most naturally expressed in a logarithmic radial coordinate (u = ln r). Discrete scale invariance (DSI) is treated as a structural symmetry principle, implying characteristic log-harmonic (log-periodic) residual patterns as a real-valued consequence of complex scaling modes. Intuitively, structure can be compared to a blueprint that contains only relations and proportions, but no absolute numbers or units. It specifies how things are related, not how large, heavy, or fast they are. Concrete values such as meters, seconds, or joules appear only after coupling to an observer and a measurement context. • State denotes the concrete structural configuration that evolves in a structural evolution parameter, carrying phases, distributions, and drift of structural degrees of freedom. Metaphorically, the state is like the current configuration of a blueprint being used in a construction process. The blueprint itself does not change (that is the structure), but the state tells us which parts of the plan are currently “active,” which sections are emphasized, and how the overall configuration shifts from one step to the next. It is still all about relations and proportions, not about concrete materials, sizes, or measurements. • Coupling denotes the physical embedding of an observer/apparatus. It determines which structural degrees of freedom are operationally accessible and how they appear as effective 4D observables. The “projection” language used in LHFT is not philosophical; it is the minimal operational interface required to define reproducible measurements from a resolution-free substrate (analogous to rendering a vector description into a finite, device-dependent raster readout). A crucial consequence of this view is endogeneity: there is no external or absolute viewpoint. Every observer is part of the structure, and every observation is necessarily made from within it. The framework explicitly separates invariant local kinematics from observer-dependent readout. In the recovery regime, Lorentz invariance and the constancy of c are preserved as stability requirements of the effective 4D description; apparent variations in measured speeds arise from projection geometry, finite bandwidth, time-windowing, and experimental context rather than from changes in the underlying light-cone structure. A second key element of LHFT is the use of logarithmic scale space. Instead of describing distances and scales in ordinary linear terms, LHFT uses a logarithmic representation. This choice is motivated by the empirical fact that many natural structures repeat across scales and exhibit approximate scale invariance. In logarithmic space, multiplicative scaling becomes additive, making such patterns simpler, more regular, and more stable to analyze. This shift is not a mathematical trick, but a natural coordinate choice for a scale-free structural description. This record contains the core LHFT documents as modular components: (i) the discrete-scale-invariant geometry and its stable representation in log-coordinate space, including analysis conventions; (ii) the drift concept as structural evolution and its observable phenomenology under coupling without spoiling recovery; (iii) the empirical program with preregistered (“freeze”) parameters, explicit null models, robust statistics, and auditable significance testing for log-harmonic signatures; and (iv) a structural reference backbone to keep definitions, notation, and consistency constraints aligned across the full manuscript set. Two-Layer (conceptual / ontological view): LHFT separates a structural substrate (“Structure–State” in the log-coordinate domain) from its recovered 4D description. A physical observer/apparatus is characterized by a coupling class, and the 4D world is the operational readout produced by that coupling. This view is best for explaining measurement-context effects, “projection,” and why standard 4D physics is a stable recovery regime. One-Layer (operational / working view): LHFT is written directly in the log-coordinate formulation as an effective single-layer theory: the same structural principles (DSI, log-harmonic modes, drift, recovery limits) are expressed in one mathematical arena, so calculations and data analyses can be done without continuously switching between “structure” and “4D.” This view is best for practical modeling, statistics, and empirical tests. Key point: Both are the same theory; the Two-Layer version makes the interface explicit, while the One-Layer version absorbs it into an operational formulation for computation and testing. Although LHFT is constructed to recover standard 4D physics in the appropriate limiting regimes (including the local validity of special and general relativity and the effective field-theoretic description of matter), its ontological starting point is intentionally far removed from the conventional spacetime-first worldview. In LHFT, spacetime is not assumed as fundamental but treated as an emergent, operationally recovered description arising from an underlying structural substrate and the coupling of an embedded observer/apparatus. For this reason, the present manuscript focuses on establishing full compatibility at the level of observable predictions and recovery behavior, while the final, fully “interface-complete” formulation—expressed in a way that is maximally seamless for the traditional spacetime language—remains an ongoing consolidation step of the program. The empirical “shell” program tests a fixed log-harmonic template in the logarithmic coordinate u = ln r, rather than fitting parameters after inspection. Shell centers are defined by a discrete scale-invariant ladder (u_n = u0 + n·lambda, equivalently r_n = r0·exp(n·lambda)), and each shell is evaluated with symmetric on-shell windows and equal-width off-shell control windows in u-space. Multiple statistics are used (shell count contrasts, magnitude contrasts for SDSS SN Ia, and fixed-frequency spectral projections at omega0 = 2π/lambda), and significance is estimated non-parametrically via explicit null ensembles (permutation and phase-randomized / synthetic-sample controls). In the SN Ia residual spectrum, the observed peak near the predicted template frequency is not reproduced by any of the reported null realizations, giving a conservative bound p < 10^-3 (finite-null resolution) and a Gaussian-equivalent scale quoted in the multi-sigma range (roughly ≥ 4.5σ as an orientation). Robustness checks (binning, smoothing, redshift slicing, and window variants) preserve the peak location and template alignment. If LHFT is presented explicitly as a mathematical construct that requires calibration, then its primary success criterion is internal coherence and recovery of baseline physics, not immediate empirical closure. This staging is standard in theoretical physics: it is a renormalization-group method and modern quantum-gravity program typically begin as mathematical frameworks. In this framing, empirical contact is treated as a calibration step rather than a validation shortcut. Parameters such as lambda are regarded as structural constants to be fixed by calibration, while log-harmonic templates are defined as candidate observables (targets for preregistered searches), not as claims of established detection. Likewise, the projection Pi_O is introduced as a mathematical map that links structural variables to an effective 4D description; it is not yet an operational measurement device. This removes premature “observer dependence” objections by keeping the discussion at the level of well-defined mappings and recovery requirements. Accordingly, LHFT should be evaluated against the appropriate comparison class: axiomatic field theories, pre-geometric models, and structural effective field theories. Direct competition with precision ΛCDM fitting or full Standard Model parameter derivations is not the initial goal; those belong to later stages once calibration, closure, and explicit recovery theorems are completed.