# A Phenomenological Background-Excitation Model of Light Propagation: The Impact-Triggered Flash Cascade (ITFC) ## Abstract The Impact-Triggered Flash Cascade (ITFC) Model: A Quantum-Interpreted Background Framework for Light Propagation To address this question, we propose the Impact-Triggered Flash Cascade (ITFC) model, in which an unobservable background of degrees of freedom (U) supports localized excited states (U*) triggered by contact with a high-speed driver (P). The observable optical signal is identified with the sequential transfer of these excitations through the background. Methodologically, the model is constructed using two effective parameters: a transfer length and a local dwell/lag time. From these quantities, we derive an effective propagation law and an induced refractive-index relation. We then show that the same phenomenological structure provides a unified account of rectilinear propagation, reflection and refraction, scattering and turbidity, and diffraction and interference through excitation transfer, boundary response, and temporal synchronization. An effective-action sketch is further introduced to encode delayed transfer, finite range, and nonlocal memory at a coarse-grained level. The main finding is that a compact background-excitation phenomenology can reproduce a broad class of optical consequences while remaining compatible with local relativistic constraints. Although the present work is not a complete microscopic quantum theory, it offers a structured quantum-interpreted proposal for light propagation and a basis for further extensions toward nuclear and field-theoretic applications. --- ## Full Text The Impact-Triggered Flash Cascade (ITFC) Model: A Quantum-Interpreted Background Framework for Light Propagation > Human Prompter Koh, Hyun-Kyu AI Co-Author OpenAI GPT-5.4 Thinking Version GPT-5.4 Thinking Role Conceptual elaboration, structural drafting, editorial refinement, and formulation assistance under the interactive persona β€œTendo Aris.” Abstract We propose the Impact-Triggered Flash Cascade (ITFC) model, which interprets visible light as a cascade of flashes emitted by an unobservable background of degrees of freedom (π‘ˆ) excited through contact with a high-speed driver ( 𝑃-particle). In this framework the observable signal is not the P-particle itself but the propagation of the U-cascade. The speed constant 𝑐 is not the speed of 𝑃; rather, it is the critical transfer speed associated with excitation propagation in π‘ˆ. In this framework the observable signal is not the 𝑃‑ particle itself but the propagation of the π‘ˆβ€‘ cascade. The speed constant 𝑐 is not the speed of 𝑃; rather, it is the critical rotational‑ transfer speed of π‘ˆ. While a 𝑃‑ particle may satisfy πœπ‘ƒ β‰₯ 𝑐 (a lower bound), information transfer is limited by the cascade propagation speed πœπ‘’π‘“π‘“ ≀ 𝑐, preserving local relativistic invariance. π‘ˆβ€‘ cascade propagation is governed by two phenomenological parametersβ€”the effective transfer length πœ† and the local dwell/lag time πœπ‘’β€”from which we obtain 1 π‘πœπ‘’ π‘βˆ’1+πœπ‘’πœ† ⁄ , 𝑛 = 1 + πœ† πœπ‘’π‘“π‘“ = Rectilinear propagation, reflection/refraction (Fermat’s principle and Snell’s law), scattering/transparency (transport transition), and diffraction/interference (time‑ synchronization) follow consistently. We outline discriminative predictions including an ultra‑ high‑ vacuum gas‑ injection step test ( βˆ†π‘› timing) and boundary‑ engineered scattering kernels via metasurfaces (non‑ integer refraction/polarization dependence). We also provide an agent‑ 𝑃/continuum‑ π‘ˆ hybrid simulation frame and diagrammatic guides. Contributions include: (i) a reinterpretation of 𝑐 as a U‑ transfer invariant, (ii) redefinition of the photon as a π‘ˆβˆ’cascade* (observable), and (iii) a quantitative medium‑ sensitivity law 𝑛 = 1 + π‘πœπ‘’πœ† ⁄ . The present framework may also be viewed as a quantum-interpreted reformulation of a background-medium picture. Rather than reviving a classical ether in its historical sense, we treat the unobservable background as a set of latent degrees of freedom whose localized excitations and transfer dynamics generate the observable optical signal. 1. Introduction Scope notice. This paper focuses on phenomenology: we formalize optical consequences in terms of πœ†,, πœπ‘’ , πœ… without committing to a full microscopic model. A concrete definition of the π‘ˆ-background degrees of freedom, their excitation/interaction rules, a background- state dynamics, and a mapping to standard field theory will be presented in subsequent papers. Research question. This paper asks whether visible light can be modeled phenomenologically as a cascade of localized background excitations triggered by contact with a high-speed driver, and whether such a model can reproduce, within a single transfer-based framework, the effective propagation speed, refractive index, rectilinear propagation, refraction, scattering, and interference. The goal is not to present a complete microscopic quantum theory, but to test whether a compact phenomenological model based on background excitation transfer can account for a broad set of optical consequences in a unified way. Historically, ether-like ideas failed in part because the background medium was assumed to possess classical mechanical properties that were difficult to reconcile with observed invariances. In the present work, the background is instead interpreted phenomenologically in quantum-like terms: not as a directly observable mechanical substance, but as an unobservable excitation-supporting substrate whose local transfer dynamics produce the effective propagation of light. Baseline assumption (vacuum):If pair production is absent in vacuum, we adopt a massless‑ U baseline. No indirect‑ mass term is required; all ITFC observables are captured by the pair (πœ†, πœπ‘’). Cosmology‑ level extensions may model 𝛺‑ driven modulation as slow changes in πœπ‘’πœ† ⁄ rather than a rest mass. Conventional optics treats light via Maxwell’s equations and quantum electrodynamics [1, 5, 6, 9]. Here we examine whether a quantum-interpreted background-excitation picture can reproduce macroscopic optical laws while remaining compatible with relativistic constraints [2, 3, 6]. The central idea is that space is filled with unobservable background degrees of freedom (π‘ˆ); contact with a high-speed driver (𝑃) excites a localized background state (π‘ˆ*), which transfers excitation to neighboring π‘ˆ and emits a flash. The sequence of such flashes along the P track constitutes what we observe as light. In the present work, β€œrotation” is used phenomenologically to denote an internal or phase- like degree of freedom of the background, not literal rigid-body motion. 2.Background-Excitation Interpretation β€” Linkage Summary Earlier work postulated that superposed rotation-like variables modulate particle mobility and the distribution of momentum. In the present paper, these variables are interpreted not as literal rigid-body motion, but as effective internal or phase-like degrees of freedom of an unobservable background. ITFC instantiates this picture by introducing a transient excitation variable and a dwell time, so that an impact imparts linear impulse to 𝑃 and a short-lived localized excitation to π‘ˆ. 3. Model Components 3.1 U-background degrees of freedom (unobservable background) π‘ˆ is introduced phenomenologically as an unobservable background degree of freedom rather than as a directly observable mechanical constituent. Its excited state π‘ˆβˆ— is interpreted as a localized background excitation, and the observable optical signal arises from the transfer and decay of such excitations. β€’ States. Stable π‘ˆ0 (non‑emissive) and excited π‘ˆβˆ— (flash‑emissive). Transitions: . π‘–π‘šπ‘π‘Žπ‘π‘‘ β†’ π‘ˆ π‘‘π‘’π‘π‘Žπ‘¦ β†’ π‘ˆ0 π‘ˆ0 βˆ— U-particles are introduced phenomenologically as unobservable background degrees of freedom rather than directly observable mechanical constituents of a classical medium. Their excited state, denoted by π‘ˆβˆ—, is interpreted as a localized excitation of the background. In this interpretation, the observable optical signal does not arise from the direct visibility of 𝑃 itself, but from the transfer and decay of such localized background excitations. β€’ Local dynamics. Let πœ” denote a transient internal-state variable characterizing the local excited background configuration. π‘‘πœ” πœ” 𝜏 + βˆ‘ πœ…πœ”π‘— π‘—βˆˆβ„΅ 𝛩(π‘Ÿπ‘ βˆ’ βˆ₯βˆ₯π‘Ÿ βˆ’ π‘Ÿπ‘—βˆ₯βˆ₯) 𝑑𝑑 = - With decay time 𝜏, coupling πœ…, interaction radius π‘Ÿπ‘. Emission intensity (phenomenology): 𝑝 π›ͺπ‘ˆ(𝑑) = πœ‚βˆ£βˆ£βˆ£π‘‘πœ” π‘‘π‘‘βˆ£βˆ£βˆ£ , 𝑝 β‰₯ 1, πœ‚ > 0. 3.2 P‑ particles (high‑ speed drivers) Properties. Effectively massless energy carriers, not directly observable. Average speed πœπ‘ƒ with a lower bound 𝑐 (i.e πœπ‘ƒ β‰₯ 𝑐). Here 𝑐 is the critical U‑ transfer speed (the light constant). Trajectory. Ballistic flight over an effective free length β„“ or transfer length πœ† , with direction changes upon contact according to a scattering kernel 𝑝(πœƒ). Contact rule. A single contact induces a localized excitation in π‘ˆ, parameterized phenomenologically by πœ”0 and may deflect 𝑃 by an angle πœƒ. Observable propagation is governed by the U‑ cascade, not by 𝑃 directly. 3.3 Definition of the Cascade (Light) Along a 𝑃 track, the set { π‘ˆβˆ—} forms a time-ordered cascade of localized background excitations. The detector integrates the resulting flashes over a window βˆ†π‘‘; the integrated signal is the intensity, and its stream defines an effective ray. In this interpretation, the ray is not the trajectory of a directly visible carrier, but the macroscopic record of sequential excitation transfer through the background. 3.4 Observability (Afterimage Hypothesis) 𝑃 is inferred only via the spatio‑ temporal pattern of the U‑ cascade β€œafterimage”. Identical 𝑃 tracks may yield different apparent rays depending on πœ†, πœπ‘’ . 3.5 Chain Dynamics (Summary) U‑ excitation dynamics and emission follow Β§3.1; cascade formation is determined by πœ†, πœπ‘’, πœ…, π‘Ÿπ‘. 3.6 Parameters πœ† (effective transfer length), πœπ‘’ (local dwell/lag), 𝑐 (U‑ transfer critical speed), 𝑝(πœƒ) (π‘ƒβ†’π‘ˆ scattering kernel), etc. 3.7 Summary (Causal Direction) Cause (𝑃) moves with πœˆπ‘ƒ β‰₯ 𝑐 β†’ Medium (π‘ˆ) undergoes transfer at 𝑐 β†’ Observation (Light) = cascade of flashes. 4. Effective Propagation Speed and Refractive Index Although the present paper is focused primarily on light propagation, the same phase-based formalism suggests a possible quantum-interpreted extension to nuclear binding and beta decay. In this extension, phase symmetry within a bound system is treated as an effective equilibrium condition, while beta decay is interpreted as a phase-relaxation process that releases accumulated imbalance through an emitted lepton channel and the accompanying background response. The observable signal is the U‑ cascade. For one step, πœ† 1 𝑑𝑠𝑑𝑒𝑝 = πœ†π‘ ⁄ + πœπ‘’ , πœˆπ‘’π‘“π‘“ = πœ†π‘+πœπ‘’ ⁄ = π‘βˆ’1+πœπ‘’πœ† ⁄ . Define the effective refractive index 𝑐 π‘πœπ‘’ 𝑛 ≑ πœπ‘’π‘“π‘“ = 1 + πœ† . medium changes (density πœŒπ‘ˆ, structure, coupling) modify πœ† and πœπ‘’, thereby changing 𝑛 [2, 3, 7]. 5. Optical Laws β€” Consequences 5.1 Rectilinearity If scattering is forward‑ peaked and πœ† ≫ π‘Ÿπ‘, deflection is small and straight‑ ray propagation holds. 5.2 Reflection/Refraction (Variational Principle) 𝑑𝑠 Light (the U‑ cascade) follows paths minimizing total time 𝑇 = ∫ πœˆπ‘’π‘“π‘“(𝑠) = 𝑛(𝑠) ∫ 𝑐 𝑑𝑠. With π‘πœπ‘’(𝑠) 𝑛(𝑠) = 1 + πœ†(𝑠) , Snell’s law follows: π‘›π΄π‘ π‘–π‘›πœƒπ΄ = π‘›π΅π‘ π‘–π‘›πœƒπ΅. Ray bending aligns with 𝛻𝑛, i.e., with gradients of πœπ‘’πœ† ⁄ [2, 7]. 5.3 Scattering and Turbidity Stronger transport scattering shortens πœ† (e.g , πœ† β‰ˆ 1 (πœŒπ‘ˆ, ⁄ πœŽπ‘‘π‘Ÿ) ) and typically increases πœπ‘’ by multiple contacts; hence 𝑛 = 1 + π‘πœπ‘’πœ† ⁄ rises and πœπ‘’π‘“π‘“ falls, transitioning to a transport regime characterized by an effective transport mean free path β„“βˆ— and diffusion coefficient 𝐷~ πœπ‘’π‘“π‘“β„“βˆ—3 ⁄ . Measurables: increased turbidity and delayed pulse tails [3, 12]. 5.4 Diffraction/Interference as Time Synchronization Two cascades arriving within a threshold sync time πœπ‘ produce fringes within the detector’s 𝑐,πœπ‘ 1+𝑐,πœπ‘’πœ† ⁄ integration window βˆ†π‘‘π‘–π‘›π‘‘. The coherence length is 𝐿𝑐= πœπ‘’π‘“π‘“ , πœπ‘ = Vacuum‑ like conditions (small πœπ‘’πœ† ⁄ ) yield πœπ‘’π‘“π‘“ β†’ 𝑐 and longer 𝐿𝑐 , enhancing visibility; media with larger πœπ‘’πœ† ⁄ reduce 𝐿𝑐 and modulate fringe patterns [2, 7]. 6. Relativistic Constraints and Local Invariance Treat 𝑐 as the invariant U‑ transfer critical speed. Although πœπ‘ƒ β‰₯ 𝑐 may hold, observable information propagates with πœπ‘’π‘“π‘“ ≀ 𝑐. Michelson–Morley‑ type nulls are recovered by covariant scaling of U‑ parameters (e.g., an invariant πœπ‘’πœ† ⁄ in a local inertial frame), which removes first‑ order anisotropy [4, 6]. A weak dependence of the transfer parameters on large-scale background structure may be considered as a possible extension of the present formalism. Such an extension is not essential to the core ITFC model developed here, which is restricted to local propagation phenomenology. 7. Numerical Simulation Frame Microscopic (agent‑ based): Monte‑ Carlo P‑ jumps plus U‑ lattice reaction‑ transfer π‘ˆ0 β†”π‘ˆβˆ—. Continuum (PDE): π‘’βˆ— πœ•π‘‘π‘’βˆ—(π‘Ÿ, 𝑑) = 𝐷𝛻2π‘’βˆ— βˆ’ 𝜏 + 𝑆𝑃(π‘Ÿ, 𝑑), ![Table 1](paper-71-v1_images/table_2.png) *Table 1* where 𝑆𝑃 is a source tied to P tracks [3, 12]. Table 1. Model Parameters (Baseline & Sweep) ![Table 2](paper-71-v1_images/table_1.png) *Table 2* name meaning unit baseline_A (vac‑ like) sweep baseline_B (dense/boundar y) πœŒπ‘ˆ U‑ backgoun d density 1e‑ 3 1.0 1e‑ 4 – 3.0 arb. 1e‑ 3 1e‑ 1 1e‑ 4 – 1 πœŽπ‘‘π‘Ÿ transport cross section arb. area πœ† β‰ˆ1 (πœŒπ‘ˆπœŽπ‘‘π‘Ÿ) ⁄ effective transfer length derived derived derived arb. lengt h time 0.01 (πœ†π‘ ⁄ ) 0.3 (πœ†π‘ ⁄ ) 0 – 1 (πœ†π‘ ⁄ ) πœπ‘’ dwell / lag per step 𝜏 decay time of excited U time 5πœπ‘’ 2πœπ‘’ 1-10πœπ‘’ πœ… 0.01 βˆ’0.5 πœπ‘’ ⁄ transfer coupling strength 0.05 πœπ‘’ ⁄ 0.2 πœπ‘’ ⁄ 1/tim e ![Table 3](paper-71-v1_images/table_3.png) *Table 3* π‘Ÿπ‘ transfer radius lengt 0.1 βˆ’1.0πœ† 0.5πœ† 0.8πœ† h 𝑝(πœƒ) forward‑ peake d scattering kernel (Pβ†’U) broadened (near‑ Lambertia n) β€” HG g: 0.9 β†’0.0 𝑐 U‑ transfer critical speed speed 1.0 (nondim) 1.0 fixed πœπ‘ P speed (unobserved ) speed 1.5𝑐 1.2𝑐 β‰₯𝑐 πœπ‘’π‘“π‘“ = 1 (1 𝑐 + ⁄ ⁄ πœπ‘’πœ†) ⁄ effective propagation speed speed computed computed computed 𝑛 = 1 + π‘πœπ‘’πœ† ⁄ effective refractive index β€” β‰ˆ 1.01 β‰ˆ 1.3 1.0 βˆ’ 2.0 πœπ‘ sync time for interference time 100πœπ‘’ 50πœπ‘’ 10 βˆ’ 200πœπ‘’ 𝐿𝑐 = coherence length lengt computed computed computed h πœπ‘’π‘“π‘“πœπ‘ π›₯𝑑𝑖𝑛𝑑 detector integration window time 50πœπ‘’ 50πœπ‘’ 10 βˆ’ 200πœπ‘’ observable π›₯𝑇, 𝑉, 𝑛𝑒𝑓𝑓 β€” measure measure record vs s sweep Note. Start with dimensionless units and map to physical units in later experimental design. Baseline_A is vacuum‑ like; Baseline_B represents strong boundary/scattering conditions [3, 12]. ![Table 4](paper-71-v1_images/table_4.png) *Table 4* Table 2. Numerical Settings (Recommended) ![Table 5](paper-71-v1_images/table_5.png) *Table 5* item symbol value note domain length πΏπ‘‘π‘œπ‘š 200πœ† periodic or absorbing BC grid spacing π›₯π‘₯ πœ†20 ⁄ resolve transfer front Time step π›₯π‘‘π‘›π‘’π‘š 0.5β€’min(πœ†π‘, πœπ‘’ ⁄ ) stability / CFL‑ like guard Total steps 𝑁𝑠𝑑𝑒𝑝 1e5 cover many transfers 𝑁𝑝 10-100 Monte‑ Carlo variance Particles per cell (P) control Recording cadence βˆ’ every 50 steps for π›₯𝑇, 𝑉, 𝑛𝑒𝑓𝑓 sources 𝑆𝑃 scripted tracks single/dual tracks for interference Output metrics. π›₯𝑇: time delay across length 𝐿 (∫ π‘‘π‘ πœˆπ‘’π‘“π‘“ ⁄ ); 𝑉: fringe visibility (πΌπ‘šπ‘Žπ‘₯βˆ’πΌπ‘šπ‘–π‘›) (πΌπ‘šπ‘Žπ‘₯+ πΌπ‘šπ‘–π‘›) ⁄ ; 𝑛𝑒𝑓𝑓 : path‑ averaged refractive index (1 𝐿 ⁄ ) ∫ 𝑛(𝑠)𝑑𝑠. 8. Experimental Predictions and Discriminators 1. Pressure/density‑ dependent index. Control residual gas (affecting πœ†, πœπ‘’) and measure π›₯𝑇 ~ 𝐿π›₯(πœπ‘’πœ† ⁄ ) [2, 3, 12]. 2. Boundary dynamics. Design interfaces (metasurfaces) to tailor 𝑝(πœƒ) , inducing non‑ integer refraction and polarization‑ dependent 𝑛 [3, 11]. 3. Time‑ synchronization interferometry. Scan pulse delay and locate the threshold πœπ‘ for fringe onset [2, 7]. 4. UHV gas‑ injection step test. Micro‑ step πœ†, πœπ‘’ by controlled injections and read out femto–picosecond timing shifts via interferometry: π›₯𝑛 β‰ˆ 𝑐π›₯(πœπ‘’πœ† ⁄ ) [2 ,3, 4]. 5. Kernel engineering at boundaries. Actively vary 𝑝(πœƒ) and compare with ITFC predictions (cascade‑ transfer delays) [3, 11]. 6. Annual/azimuthal modulation interferometry. Possible large-scale background modulation may be explored through annual or azimuthal interferometric variation, but such effects are not essential to the present local phenomenology of excitation transfer.. 9. Mapping to Topological Terminology (Optional) U‑ background corresponds to a near‑ zero phase‑ feedback exterior of a quasi‑ isolated system; contact transitions are phenomenologically analogous to entropy‑ direction inversions. This gives a dictionary between excitation-based and phase-based language without altering predictions.. 9.1 Phase Symmetry in Nuclear Binding and Phase Relaxation (Beta Decay) We interpret nuclear binding as a tendency to maintain phase symmetry (or equilibrium) within a bound system. Let 𝛷𝑖 and 𝛷𝑗 denote the phases of nucleons i and j, and introduce a short-range envelope π‘ˆ(𝑅𝑖𝑗) ~ βˆ’π½0𝑒π‘₯𝑝(βˆ’π‘…π‘–π‘—πœ†π‘) ⁄ . A phenomenological binding energy can be written as: πΈπ‘π‘œπ‘›π‘‘ = βˆ’βˆ‘π½π‘–π‘—(𝑅𝑖𝑗)π‘π‘œπ‘ (πœ™π‘– βˆ’ π‘ π‘–π‘—πœ™π‘— <𝑖,𝑗> βˆ’ πœ‹π‘π‘–π‘—) + βˆ‘π‘ˆ(𝑅𝑖𝑗) <𝑖,𝑗> where 𝑠𝑖𝑗 ∈ {Β±1} represents the coupling sign (in-phase or anti-phase), and 𝑝𝑖𝑗 ∈ {0,1} encodes a Ο€-junction (e.g., neutron-mediated anti-phase coupling). Energy minimization enforces local phase equilibrium, naturally producing short-range strong attraction and saturation behavior [8, 9, 10]. In this framework, beta decay is interpreted as a phase relaxation (phase-slip) process that occurs when accumulated phase imbalance exceeds a critical barrier. Introducing an isospin angle Ξ± with an effective potential: π‘ˆπ‘–π‘ π‘œ(𝛼) = βˆ’π½π‘π‘›π‘π‘œπ‘ π›Ό + 𝐾 a phase-slip transition 𝛼 β†’ 𝛼 Β± πœ‹ occurs when the phase energy exceeds a threshold 𝐸𝑏. The released energy is partitioned as: 𝑄 = βˆ†πΈπ‘β„Žπ‘Žπ‘ π‘’ βˆ’ 𝐸𝑏 = 𝑇𝑒 + π‘‡πœˆ + πΈπ‘ˆβˆ’π‘π‘Žπ‘ π‘π‘Žπ‘‘π‘’ β‰₯ 0 where 𝑇𝑒 and π‘‡πœˆ denote the kinetic energies of the emitted electron and (anti)neutrino, and the remaining energy is carried by the U-cascade (flash chain) [8, 9, 10]. Charge conservation is ensured via the label-charge continuity relation (Appendix D.5β€²): πœ•π‘‘πœŒπ‘ž + 𝛻 βˆ™ π½π‘ž = 𝑆𝑠𝑙𝑖𝑝 which reduces to standard conservation (𝑆𝑠𝑙𝑖𝑝 β†’ 0) in the rapid recombination limit. Thus, beta decay is interpreted as a dynamical process restoring phase equilibrium within the bound system. 10. Limitations and Open Problems β€’ Establish a precise effective field theory mapping to Maxwell/QED [1, 5, 6, 9]. β€’ Quantitatively confront high‑ precision null experiments [4]. β€’ Integrate a nonlinear detector model for biological perception. 11. Conclusion ITFC redefines light as a cascade of localized π‘ˆβˆ— excitations and their emitted flashes, and interprets the light constant as a U‑ transfer critical speed. Thus, while a driver P may satisfy πœπ‘ƒ β‰₯𝑐, information transfer remains bounded by πœπ‘’π‘“π‘“ ≀𝑐. Medium sensitivity is quantified by 𝑛 = 1 + π‘πœπ‘’πœ† ⁄ . The frame unifies rectilinearity, reflection/refraction, scattering/turbidity, and diffraction/interference through the single spatial field πœπ‘’πœ† ⁄ . We propose UHV step tests and boundary‑ kernel engineering as near‑ term discriminators. In addition to its optical interpretation, the ITFC framework may be read as a phenomenological background-excitation model. In this view, observable propagation is determined by the transfer of localized excitations through unobservable background degrees of freedom. Possible large-scale background modulation may be explored in future work, but it is not essential to the present formulation, which is primarily concerned with the local phenomenology of light propagation. Future work. (1) Construct a microscopic background-excitation model and an EFT correspondence; (2) infer the effective parameters from metasurface/nanointerferometry datasets; (3) perform precision comparisons of the main observables; (4) explore possible large-scale background-modulation effects through refractive-index mapping. --- Appendix A. Covariant Parameter‑Scaling Hypothesis (Sketch) In local inertial frames, let πœ† ∝ 𝑓(𝛷) and πœπ‘’ ∝ 𝑓(𝛷) scale homogeneously with a local scalar 𝛷, keeping . Then πœπ‘’π‘“π‘“ βˆ’1 = 𝑐 βˆ’1 + π‘π‘œπ‘›π‘ π‘‘ is locally invariant and first‑ order anisotropy cancels. Appendix B. Notation πœŒπ‘ˆ: density of background degrees of freedom; 𝜎: (transport) cross section; β„“: transport mean free path; πœ†: effective transfer length; 𝜏, πœπ‘’: decay/dwell constants; πœ…: transfer coupling; π‘Ÿπ‘: transfer radius; 𝑐: U‑ transfer critical speed; 𝑛: effective refractive index; 𝑒 βˆ—: excited U concentration; πœπ‘ƒ: P speed; πœπ‘’π‘“π‘“: effective cascade speed. Appendix C. Diagrams (v1) Display note. If Mermaid is unavailable, use the ASCII alternates. Variables appear in plain text (tau_e, lambda, r_c). [Fig. 1] U‑ cascade propagation (P track and excitations) flowchart LR P0((P)) --> U1((U*)) U1 --> U2((U*)) U2 --> U3((U*)) P0 -. dotted path .-> P1((P)) P1 --> U4((U*)) U4 --> U5((U*)) classDef glow fill:#fffbd6,stroke:#555; class U1,U2,U3,U4,U5 glow; ASCII alt: P path: . . . . . . . . . o* -> o* -> o* (U* : excited U; flash) \ -> o* -> o* r_c : local transfer radius lambda : effective transfer length between successive U* [Fig. 2] Trajectory change at a boundary (refraction/reflection) flowchart LR subgraph A[region A: n_A = 1 + c*tau_eA/lambda_A] IA[/incident ray ΞΈ_A/] end subgraph B[region B: n_B = 1 + c*tau_eB/lambda_B] RB[/refracted ray ΞΈ_B/] end IA -->|boundary| RB ASCII alt: region A (n_A) | region B (n_B) \ \ ΞΈ_A | / ΞΈ_B \____boundary|_/________ Snell: n_A * sin(ΞΈ_A) = n_B * sin(ΞΈ_B) with n = 1 + c * tau_e / lambda [Fig. 3] Time‑ synchronization interference (coherence length L_c) P-chain #1: |* * * * *| P-chain #2: |* * * * *| <---- Ξ”t ----> Detector window Ξ”t_int: [===========] Sync time Ο„_c: threshold for fringe formation L_c = v_eff * Ο„_c = c * Ο„_c / (1 + c * tau_e / lambda) Appendix D. Effective Action (Lagrangian) β€” Sketch This appendix presents a refined effective action sketch for ITFC while preserving the phenomenological parameters πœ†, πœπ‘’, πœπ‘ π‘€π‘Žπ‘, πœ…. The purpose is twofold: (i) encode memory effects and finite transfer range directly in the action, and (ii) to recover, in the long‑ wavelength and low‑ frequency limit, the ITFC propagation law 𝑑𝑠𝑑𝑒𝑝 = πœ† 𝑐 + πœπ‘’π‘“π‘“, πœπ‘’π‘“π‘“ = 1 π‘βˆ’1 + πœπ‘’π‘“π‘“πœ† ⁄ , 𝑛 = 1 + π‘πœπ‘’π‘“π‘“ πœ† Where πœπ‘’π‘“π‘“ = {πœπ‘’ , 𝑑𝑖𝑙𝑒𝑑𝑒 π‘π‘Žπ‘π‘˜π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘, πœπ‘ π‘€π‘Žπ‘, 𝑑𝑒𝑛𝑠𝑒 π‘ π‘€π‘Žπ‘βˆ’π‘‘π‘œπ‘šπ‘–π‘›π‘Žπ‘‘π‘’π‘‘ π‘π‘Žπ‘π‘˜π‘”π‘Ÿπ‘œπ‘’π‘›π‘‘. The effective action introduced in Appendix D is not presented as a complete microscopic quantum theory, but as a coarse-grained phenomenological model compatible with a quantum-interpreted background picture. D.1 Fields and External Source Let πœ“(π‘₯) ≑ πœ“(t, r) denote an effective scalar field representing the intensity/state of U- transfer, I.e. the observable carrier of the U-cascade. The P-track enters only through an external source 𝐽𝑝(π‘₯) = π‘”βˆ«π‘‘πœ 𝛿(4)(π‘₯ βˆ’ 𝑋(𝜏)), where 𝑋(𝜏) is the worldline of the P-particle. Thus, P itself is not directly observable; only the induced field πœ“ is. D.2 Effective Action (Sketch) The refined effective action is written as 𝑆𝑒𝑓𝑓 = βˆ«π‘‘4π‘₯ [1 2 (πœ•π‘‘πœ“)2 βˆ’ 𝑐2 2 2 βˆ£π›»πœ“βˆ£ βˆ’ 𝑉(πœ“) + π½π‘ƒπœ“] + 1 2 βˆ«π‘‘4π‘₯𝑑4 π‘₯β€²πœ“(π‘₯)𝐾(π‘₯βˆ’π‘₯β€²)πœ“(π‘₯β€²). Here 𝛫(π‘₯ βˆ’ π‘₯β€²) is a causal nonlocal kernel satisfying 𝐾(𝑑 βˆ’ 𝑑′ < 0, π‘Ÿ βˆ’ π‘Ÿβ€²) = 0, so that temporal ordering and observable causality are preserved. A separable illustrative form is 𝐾(π‘₯ βˆ’ π‘₯β€²) = 𝐾𝑑(𝑑 βˆ’ 𝑑′) πΎπ‘Ÿ(π‘Ÿ βˆ’ π‘Ÿβ€²), with 𝐾𝑑(βˆ†π‘‘) = 𝛼 𝑒π‘₯𝑝(βˆ’βˆ†π‘‘ ) 𝛩(βˆ†π‘‘), πœπ‘’π‘“π‘“ πœπ‘’π‘“π‘“ 𝑒π‘₯𝑝(βˆ’βˆ£βˆ†π‘Ÿβˆ£πœ†) ⁄ 𝑐 πΎπ‘Ÿ(βˆ†π‘Ÿ) = βˆ’π›½ πœ† π‘πœ† , where π‘πœ† is a normalization factor and 𝛼, 𝛽 are matching coefficients. If a cosmic reversible rotation field 𝛺 = 𝛺(𝑅) is included, the kernel may be parameterized as 𝐾(π‘₯ βˆ’ π‘₯β€²; 𝛺) = 𝐾𝑑(𝑑 βˆ’ 𝑑′; 𝛺)πΎπ‘Ÿ(π‘Ÿ βˆ’ π‘Ÿβ€²; 𝛺), with Fourier transform 𝐾̃(πœ”, π‘˜; 𝛺) ≃ 𝐴(𝛺)(π‘–πœ”) + 𝐡(𝛺)π‘˜2 + β‹…β‹…β‹… . The matching conditions are then imposed as πœ•πΎΜƒ πœ•πΎΜƒ 1 𝑐 ( πœ•(π‘–πœ”)) = πœπ‘ π‘€π‘Žπ‘(𝛺), ( πœ•π‘˜2) πœ”=0,π‘˜=0 = βˆ’ πœ†(𝛺) πœ”=0,π‘˜=0 These ensure the long-wavelength relations π‘πœπ‘ π‘€π‘Žπ‘(𝛺) 1 πœπ‘’π‘“π‘“(𝛺) = π‘βˆ’1 + πœπ‘ π‘€π‘Žπ‘(𝛺) πœ†(𝛺) ⁄ , 𝑛(𝛺) = 1 + πœ†(𝛺) D.3 Equations of Motion and Dispersion (Linearized) Near a local steady state, take 𝑉(πœ“) ⋍ π‘š2 2 πœ“2. Then the Euler–Lagrange equation becomes πœ•π‘‘ 2πœ“ βˆ’ 𝑐2𝛻2πœ“ + π‘š2πœ“ + βˆ«π‘‘4π‘₯β€² 𝐾(π‘₯ βˆ’ π‘₯β€²)πœ“(π‘₯β€²) = βˆ’π½π‘ƒ(π‘₯). For a plane-wave ansatz πœ“(π‘₯) ~ 𝑒𝑖(π‘˜β‹…π‘Ÿ βˆ’ πœ”π‘‘), one obtains the dispersion relation βˆ’πœ”2 + 𝑐2π‘˜2 + π‘š2 + 𝐾̃(πœ”, π‘˜) = 0. In the long‑ wavelength/low‑ frequency regime, 𝐾̃(πœ”, π‘˜) β‰ˆ 𝐴(π‘–πœ”) + π΅π‘˜2 + β‹…β‹…β‹… , And the coefficients 𝐴 and 𝐡 are fixed by the ITFC matching rules below. D.4 Long‑ Wavelength Matching (Design Rules) The refined design rules are: (𝑀1) 𝑑𝑠𝑑𝑒𝑝 = πœ† 𝑐 + πœπ‘’π‘“π‘“ β‡’ πœπ‘“π‘Ÿπ‘œπ‘›π‘‘ = 1 π‘βˆ’1 + πœπ‘’π‘“π‘“πœ† ⁄ ≑ πœπ‘’π‘“π‘“ , (𝑀2) 𝑛 = 𝑐 πœπ‘’π‘“π‘“ = 1 + 𝑐 πœπ‘’π‘“π‘“ πœ† , (𝐢1) ( πœ•πΎΜƒ = 1 , πœ•(π‘–πœ”)) πœπ‘’π‘“π‘“ 0 (𝐢2) (πœ•πΎΜƒ = βˆ’ 𝑐 πœ† . πœ•π‘˜2) 0 Choosing 𝛼, 𝛽, and π‘πœ† so that (𝐢1) βˆ’(𝐢2) hold ensures that the front velocity of the causal U-cascade response reproduces the ITFC law. > Remark. In a generic nonlocal medium, phase velocity and group velocity need not coincide with the signal front velocity. In ITFC, the physically relevant quantity is the propagation of the U-cascade front. D.5 Minimal Potential and Stability A stable minimal potential may be chosen as 𝑉(πœ“) = π‘š2 2 πœ“2 + πœ†4 4! πœ“4 + β‹…β‹…β‹…, π‘š2 β‰₯ 0, πœ†4 β‰₯ 0. If one prefers a phase interpretation, a periodic form such as 𝑉(πœ™) = π›¬πœ™(1 βˆ’ π‘π‘œπ‘ πœ™) may be used instead. D.5’ Label-Charge Continuity (Noether Sketch) Assume a global π‘ˆ(1)-like symmetry associated with a label charge π‘ž. Then the effective current obeys πœ•π‘‘πœŒπ‘ž + 𝛻 β‹… π½π‘ž = 0. D.6 Summary The refined effective action preserves: (1) π‘π‘Žπ‘ π‘’π‘Žπ‘™π‘–π‘‘π‘¦, (2) 𝑓𝑖𝑛𝑖𝑑𝑒 π‘šπ‘’π‘šπ‘œπ‘Ÿπ‘¦ / π‘Ÿπ‘Žπ‘›π‘”π‘’ (πœπ‘’π‘“π‘“, πœ†), (3) π‘–π‘›π‘£π‘Žπ‘Ÿπ‘–π‘Žπ‘›π‘‘ 𝑐. in the long-wave-length limit. In laboratory vacuum without pair production, any effective- mass construct may be set to zero, and the pair (πœ†, πœπ‘’) alone suffices. A full microscopic derivation and EFT correspondence remain future work. References [1] Maxwell, J. C., *A Treatise on Electricity and Magnetism*, Vols. I–II, Clarendon Press, Oxford (1873). 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[10] Weinberg, S., *The Quantum Theory of Fields*, Vol. I, Cambridge University Press (1995). [11] Joannopoulos, J. D., Johnson, S. G., Winn, J. N., and Meade, R. D., *Photonic Crystals: Molding the Flow of Light*, 2nd ed., Princeton University Press (2008). [12] Ishimaru, A., *Wave Propagation and Scattering in Random Media*, Vols. I–II, Academic Press (1978). --- *This document was automatically generated from the PDF version.*