Energy Cohesion Model (ECM): An Energy-Balance Framework for Extracting Molecular Cohesion Energies, Cross–Interaction Energies, and Unknown Species Contributions in Gas Mixtures

ECM: Energy-Based Reconstruction of Binding Contributions in Unknown Gas Mixtures Abstract Most gas-mixture experiments reveal only two macroscopic quantities: the externally injected energy Ein and the outgoing energy Eout. All microscopic internal effects collapse into the measurable residual: Eres =Ein −Eout. This paper develops the Energy Composition Method (ECM), a general formalism for reconstructing or estimating intrinsic binding contributions using only this residual. Because internal structure is never directly observable, ECM classifies all possible information states (Case 0–4) and determines which parts of the composite residual are recoverable, partially identifiable, or fundamentally inseparable. The expanded formulation demonstrates how ECM extracts the maximum possible information allowed by physics, without any molecular-level access, spectroscopy, or species identification. 1 Introduction Determining binding contributions within a gas mixture is an indirect problem: microscopic interactions cannot be isolated experimentally. Instead they manifest only through the macroscopic energy difference Eres =Ein −Eout. However, this residual is a composite quantity originating from multiple internal sources: Eres =Ebind +Ecross +Eloss, where Ebind reflects intrinsic species-level binding, Ecross reflects cross-species influence, and Eloss captures equipment dissipation. 1 The central challenge is identifiability: What internal quantities leave distinguishable footprints on Eres? Under which conditions can binding be reconstructed, and when is only partial estimation possible? ECM establishes a complete classification (Cases 0–4) describing every logical information state and shows when reconstruction is exact, approximate, or non-unique. 2 Notation Symbol Meaning Ein Injected energy Eout Released energy Eres =Ein −Eout Energy residual Ebind Intrinsic binding contribution Ecross Cross-species influence Eloss Equipment dissipation xiMole fraction of species i BiBinding coefficient of species i nNumber of species in the mixture 3 Framework: Residual Decomposition The central structural decomposition is: Eres =Ebind +Ecross +Eloss. If composition is known, binding is a linear mixture: Ebind = n X i=1 xiBi. If composition is unknown, or if nis unknown, the problem becomes a mixture of parameter identification and inference. ECM provides systematic reconstruction rules for every information state. 2 4 Case-Based Reconstruction Framework This section provides a complete, structured reconstruction logic for all information states—from fully known systems to fully unknown mixture structures. ECM treats each Case as a distinct identifiability regime. The goal is always the same: Ebind from Eres =Ein −Eout. Because the observable dimension is only one scalar (Eres), the internal recoverability depends entirely on how much prior information is available. The five Cases below form the full ECM reconstruction chart. Case 0: Single-Species Mixture (Baseline Identifiable Case) This is the only regime in which internal binding can be recovered exactly. Ecross = 0. Residual is Eres =Ebind +Eloss. If the equipment loss is calibrated: Ebind =Eres −Eloss. Why this Case is special. There are no multiplicative unknowns and no structural ambiguity. No composition, no species interaction, no hidden dimensions. Thus, ECM recovers Ebind uniquely. Case 1: Multi-Species System with Known Composition When the composition vector xiis known, the binding term Ebind =X i xiBi becomes directly accessible in structure. Under this condition, the residual Eres =X i xiBi+Ecross +Eloss 3 still contains two uncontrolled components, but the knowledge of xisubstantially reduces the overall uncertainty. Because the composition is fixed, controlled variations in external conditions (e.g., pressure, temperature, or injection energy) introduce measurable changes that help constrain or separate the remaining terms. As a result, Case 1 provides a more stable and better-conditioned reconstruction setting compared to cases with unknown composition. Although the recovery is not fully exact in the strict mathematical sense, the structural information from xiallows the binding contribution to be estimated with moderate but dependable accuracy in typical experimental environments. Case 2: Species Count Known, Composition Unknown Unknowns: - xi(composition) - Bi(binding coefficients) Residual: E(k) res = n X i=1 xiB(k) i+E(k) cross +E(k) loss. Why this Case is difficult. The term PixiB(k) icontains multiplicative unknowns (xiBi). A single experiment cannot separate them. Requirement. ECM requires a set of Kindependent experimental conditions: - differing temperature, - differing input intensity, - differing process timing, which produces variation in B(k) i. Then the system becomes: Eres =A(x1, x2, ..., xn)T+ε with rank(A)≥nrequired. Outcome. Exact recovery is impossible, but ECM produces stable estimates for both xiand Bi. Case 3: Species Count Unknown Here even the dimensionality of the system is hidden. Residual set: {E(1) res , . . . , E(K) res } contains patterns caused by different species. Goal: infer n. ECM uses structural features: 1. Distinct response modes of Eres (temperature slopes, pressure curvature) 4 2. Singular value decay (elbow detection) 3. Stability across repeated conditions Once ˆnis inferred, the system reduces to Case 2 and the usual estimation applies. Outcome. Not exact, but structurally consistent estimation of xi,Bi, and ˆnbecomes possible. Case 4: Fully Unknown System No species count, no composition, no coefficients, no cross/loss models. Residual: Eest bind =Eres −b Ecross −b Eloss. Estimation tools used: - non-negativity constraints Ebind ≥0, xi≥0 - smoothing across conditions - physically reasonable monotonicity - consistency filters - minimizing residual mismatch Outcome. Case 4 does not allow exact recovery, but ECM still yields the closest possible estimate permitted by macroscopic information theory. Estimation Principle and Theoretical Justification Whenever the internal degrees of freedom exceed the dimension of the observable residual, exact reconstruction becomes mathematically impossible. This follows from the basic identifiability condition of linear algebra: rank(A)<number of unknowns ⇒no unique solution. In these regimes (Cases 2–4), ECM does not perform heuristic guessing. Instead, it applies a constrained inverse-problem formulation that selects the physically admissible solution consistent with all observations. The residual satisfies E(k) res =X i xiB(k) i+E(k) cross +E(k) loss. Thus the estimation step solves the constrained optimization problem: min xi,Bi, Ecross, Eloss X k E(k) res −X i xiB(k) i−E(k) cross −E(k) loss!2 , 5 subject to the physically required constraints: xi≥0,X i xi= 1, Ebind, Ecross, Eloss ≥0. These constraints ensure that ECM chooses solutions that are both mathematically consistent and physically meaningful. The method is therefore not a heuristic but a standard application of inverse-problem theory and constrained least-squares estimation. As a result, even when exact decomposition is impossible, ECM produces the closest consistent estimate permitted by macroscopic information, achieving theoretical soundness across all partial-information regimes. 5 Discussion It is important to emphasize that the estimation steps in ECM are not heuristic. They are dictated by the mathematical structure of the residual equation and follow standard principles from linear algebra, inverse problem theory, and constrained optimization. Whenever the number of internal unknowns exceeds the dimension of observable energy measurements, exact reconstruction becomes mathematically impossible. In such regimes, ECM performs the theoretically correct action: it selects the physically admissible solution that minimizes residual inconsistency under non-negativity and compositional constraints. Therefore, all estimation results are grounded in established identifiability theory rather than empirical guessing. ECM provides a complete classification of what can and cannot be recovered from energyonly measurements. Reconstruction hinges not on precision, but on structural identifiability. Even perfect measurement cannot reveal more information than the structure allows. Case 0 and Case 1 permit exact or near-exact recovery. Case 2–4 allow only partial recovery, depending on dimensionality. 6 Conclusion ECM shows that energy residuals encode significantly more information than previously recognized. The framework identifies precise reconstruction limits, clarifies when binding is recoverable, and provides estimation methods when it is not. 6 Future extensions include: temperature-dependent binding, nonlinear cross-interactions, and dynamic (time-resolved) reconstruction. 7 Numerical Examples (Appendix) To illustrate how ECM behaves under different structural assumptions, we provide three numerical examples. All examples use the same baseline definitions: Eres =Ein −Eout, Eres =Ebind +Ecross +Eloss. Each example demonstrates how ECM reconstructs or estimates the binding contribution depending on the available information (Cases 0–4). Example 1: Single-Species (Case 0) Let the experiment inject Ein = 12.0 J, Eout = 7.6 J. Residual: Eres = 12.0−7.6=4.4 J. If calibrated equipment loss is Eloss = 0.9 J, then the binding energy is uniquely recovered: Ebind =Eres −Eloss = 4.4−0.9=3.5 J. No uncertainty or estimation is required because the system contains no hidden degrees of freedom. Example 2: Two Species, Composition Known (Case 1) Let x1= 0.40, x2= 0.60. Observed residual under a given condition: Eres = 5.8 J. 7 Assume cross and loss contributions are partially identifiable: Ecross = 0.6, Eloss = 0.7. Then: Ebind =Eres −Ecross −Eloss = 5.8−0.6−0.7=4.5. Binding coefficients follow: x1B1+x2B2= 4.5. If an additional condition provides: x1B1+x2B2= 4.2, the two linear constraints bracket the true value. This is an example of ECM’s **structured estimation** approach. Example 3: Unknown Species Count (Case 3) Suppose three experiments yield: E(1) res = 4.1, E(2) res = 6.8, E(3) res = 5.3. Stacking residual responses and applying SVD: Eres =   4.1 6.8 5.3   ⇒σ= (10.63,0.41,0.02). The sharp spectral drop indicates an effective **rank of 1**, implying the mixture behaves as if dominated by a **single effective binding mode**, even if multiple species exist. ECM therefore reconstructs: b Ebind ≈Eres −(b Ecross +b Eloss) with rank-1 consistency constraints. This demonstrates how ECM extracts structure even in fully unknown systems. 8 8 Advantages of ECM ECM offers several advantages over conventional mixture-analysis approaches, especially in systems where microscopic or molecular-level information is inaccessible. 1. Works with Only Macroscopic Energies Most traditional mixture or binding-analysis methods require: •spectroscopy, •concentration measurement, •species identification, •direct mechanistic observability. ECM requires only (Ein, Eout) making it applicable even in highly opaque, unknown, or complex systems. 2. Handles Unknown Numbers of Species Many frameworks **assume** the number of species is known. ECM does not. It infers structure directly from: •variation of the residual, •rank patterns, •spectral curvature, •consistency across conditions. 3. Provides the Maximum Identifiable Information ECM never overclaims: it reconstructs only what is mathematically possible to extract from macroscopic energy. In underdetermined systems, it switches to: consistent estimation (structural constraints + non-negativity) instead of pretending to recover unobservable quantities. 9