Energy-Based Reconstruction of Internal Structural Contributions (ECM)

Energy-Based Reconstruction of Internal Structural Contributions (ECM) Abstract This work develops a generalized formulation of the Energy Contribution Method (ECM), a framework that reconstructs internal structural energy effects of an unknown system using only externally measurable energy differences. Instead of assuming specific microscopic mechanisms (such as chemical bonding), ECM defines a universal internal contribution Eeff representing the system’s intrinsic structural response to injected energy. By observing changes across experimental conditions, ECM estimates both Eeff and the remaining non-intrinsic components. The approach is fully macroscopic, model-independent, and applicable to gases, liquids, solids, plasmas, composites, or any medium where internal interactions leave a repeatable energy footprint. A full case classification (0–4) describes identifiability limits under different information levels, and numerical examples illustrate each regime. 1 Introduction In many physical systems, internal processes are not directly observable. Experimental access is often limited to boundary-level quantities such as an injected energy Ein and a measured outgoing energy Eout. All micro1 scopic events—conversion, storage, dissipation, structural reconfiguration— are compressed into the residual Eres =Ein −Eout.(1) Traditional analyses attempt to decompose this residual into specific components (e.g. bonding, friction, coupling). However, such a decomposition usually requires strong microscopic assumptions that are not justified for many systems (especially gases, plasmas, or complex heterogeneous media). ECM adopts a different viewpoint. Rather than asking which microscopic mechanism created the residual, it defines an intrinsic structural contribution that captures how the system, as a whole, consistently modifies the energy that passes through it. This intrinsic part is denoted by Eeff and is separated from extrinsic, setup-dependent effects Eext. The central questions are: 1. How can Eeff be estimated from macroscopic measurements alone? 2. Under what conditions is such an estimate unique, approximate, or only partially identifiable? ECM as a black-box structural method We treat the system as a black box: energy is injected, interacts with internal structure, and emerges. The role of ECM is to infer from repeated measurements which part of Eres is intrinsic to the system and which part is due to external or incidental effects. 2 2 Basic ECM Formulation We decompose the residual as Eres =Eeff +Eext,(2) where: •Eeff:intrinsic structural contribution (system-specific), •Eext:extrinsic contribution, including: –equipment loss Eloss, –cross-effects Ecross, –condition-dependent deviations Econd. Thus Eext =Eloss +Ecross +Econd.(3) Conceptually, Eeff is the part of the residual that: 1. is repeatable across identical experimental runs, 2. remains stable under moderate scaling of Ein, 3. originates from the system’s internal structure, not from external devices. 3 Extraction Principles Suppose we perform Nexperiments indexed by k, possibly with varying conditions (intensity, frequency, geometry). For each: E(k) res =Eeff +E(k) ext.(4) 3 3.1 Averaging and baseline estimation If E(k) ext fluctuates around a mean value Eavg ext , then: b Eeff =1 N N X k=1 E(k) res −Eavg ext .(5) When Eavg ext is calibrated or negligible, the sample mean of E(k) res becomes a direct estimator of Eeff. 3.2 Intensity sweep Varying the input energy Ein allows us to separate contributions that scale with intensity from those that do not. If we denote intensity by Iand model Eext(I) = aI +bI2+c, (6) then regression on (I, Eres) pairs can estimate (a, b, c) and isolate Eeff as the remaining constant component. 3.3 Pattern extraction via SVD For more complex data (multiple conditions, frequencies, geometries), we can form a matrix Rwhose entries are residuals: Rkℓ =E(k,ℓ) res ,(7) with kindexing macroscopic conditions and ℓindexing control parameters. Applying singular value decomposition (SVD): R=UΣV⊤,(8) 4 reveals how many dominant internal modes are required to represent the data and how much of the structure can be ascribed to a low-dimensional intrinsic response. Figure 1 illustrates a generic singular value spectrum. i σiσ1 σ2 noise level Figure 1: Schematic singular value spectrum. Dominant modes lie above the noise floor. 4 Case Classification with Numerical Examples We classify all possible information regimes into five cases (0–4) and illustrate each with a concrete numerical example. Case 0: Single internal mode, fully calibrated losses This is the most favorable regime: a single intrinsic mode and perfectly calibrated external loss. We have Eres =Eeff +Eloss.(9) Numerical example. Assume Ein = 50 J, Eout = 33 J, 5 so Eres = 50 −33 = 17 J. If the equipment loss is known to be Eloss = 2 J, then Eeff =Eres −Eloss = 17 −2 = 15 J. Here Eeff is determined exactly. Case 1: Known intrinsic structure, partially known external variation We assume that the intrinsic contribution Eeff is fixed, but external losses vary slightly across conditions. Then E(k) res =Eeff +E(k) loss,(10) with E(k) loss partially known. Numerical example. Consider two conditions A and B: Condition Ein Eout Eloss A 40 25 1.0 B 40 23.6 1.4 Residuals: EA res = 40 −25 = 15, EB res = 40 −23.6 = 16.4. Thus EA eff = 15 −1.0 = 14, EB eff = 16.4−1.4 = 15. 6 A simple average yields b Eeff = 14.5 J. This case illustrates partial identifiability with small uncertainty. Case 2: Unknown structure values, known internal mode count Suppose we know that there are nintrinsic modes but not their strengths. We write E(k) eff = n X i=1 αiu(k) i,(11) where u(k) idescribes how mode iresponds under condition k. Numerical example. Assume n= 2 intrinsic modes and three conditions A, B, C. Measured residuals and losses: Cond. Eres Eloss Eres −Eloss A 20 2 18 B 26 3 23 C 32 4 28 Model: 18 = α1+α2, 23 = 2α1+α2, 28 = α1+ 2α2. Solving: from the first two equations, 5=α1⇒α1= 5, then 18 = 5 + α2⇒α2= 13. 7 ECM here finds unique mode strengths (α1, α2) by using multiple conditions. Case 3: Unknown mode count, structure estimated by SVD In this regime we do not even know how many intrinsic modes exist. We construct a residual matrix and apply SVD to estimate the number of effective modes. Numerical example. Take a 3 ×3 residual matrix R=   15 16 17 22 24 26 29 32 35   . A numerical SVD yields singular values σ1≈48.3, σ2≈2.1, σ3≈0.4. The large gap between σ2and σ3suggests that two intrinsic modes dominate, while the third is near noise level. Thus, ECM infers an internal dimensionality of approximately 2, without direct microscopic access. Case 4: Fully unknown system (closest consistent estimate) Here we only have residual data and no reliable knowledge of losses, modes, or external patterns. ECM then seeks the closest consistent intrinsic value that minimizes residual variation: b Eeff = arg min EX kE(k) res −E.(12) 8 The minimizer is the median of {E(k) res }. Numerical example. Given four measurements: E(k) res ∈ {15,16,17,19}, the median is b Eeff = 16.5 J. No exact reconstruction is possible, but ECM still provides the most stable intrinsic estimate consistent with the data. 5 Summary of Numerical Behavior Table 1 summarizes the numerical examples for Cases 0–4. Case Info level Example outcome Identifiability 0 Single mode, calibrated loss Eeff = 15 J exact 1 Known structure, variable loss Eeff ≈14.5 J high 2 Known mode count, unknown values (α1, α2) = (5,13) exact (w/ data) 3 Unknown mode count modes ≈2 structural only 4 Fully unknown b Eeff = 16.5 J closest estimate Table 1: Summary of ECM behavior in Cases 0–4. 6 Discussion: Advantages and Limits ECM does not attempt to reconstruct every microscopic contribution separately. Instead, it focuses on what can be robustly recovered from boundary measurements: •an intrinsic, structurally repeatable component Eeff, 9