When Does an Accounting Inequality Have Teeth? A landing criterion for the maintenance power bound in prebiotic chemistry, with a first empirical execution

Abstract

Abstract Background. Prebiotic chemistry has long lacked a common ledger. Synthesis experiments either reconstruct reaction pathways or investigate energetic conditions in isolation; over several decades the two have advanced separately, and the two standard strategies, “lower the loss by cold and dry conditions” and “raise the synthetic flux by activation chemistry,” have never been compared in magnitude within a single dimension. Gap. The four companion papers argued that what runs across levels is a lower bound rather than an extremum, supplied the maintenance dissipation floor for multiplicative open systems on the heavy-tailed branch, and treated that floor under self-reference and under truncation; but all four treat abstract systems, and a unification that never lands on a specific domain cannot be assessed for executability. Aim. This paper lands that line on prebiotic chemistry and protocells, writes the ledger out in full, and asks a question none of the four asked: under what conditions does the ledger have exclusionary power. It reconstructs no geological pathway and asserts no historically necessary step; what it gives is a necessary condition at the level of accounting. Method. On the theory side, each term of the master equation is identified with a physical carrier in this domain and judged item by item for measurability under guardrail G5. On the empirical side, published amino acid abundance spectra of carbonaceous chondrites are subjected to distribution identification and precision-factor scaling analysis, throughout under guardrail G1 (maximum likelihood plus Kolmogorov–Smirnov distance plus bootstrap; log-log regression prohibited) and guardrail G3 (stochastic entries report sample size, repetitions and seed). Five results. First, the master-equation ledger of this domain (the origin inequality) admits two readings: the one from the steady-state relation is an identity containing the conversion coefficient and is therefore unfalsifiable; to become a constraint it must be joined to a power floor independent of that coefficient. Second, joining the thermodynamic-uncertainty floor of the second paper yields a floor-ratio criterion: the exclusionary power of the origin inequality is set by the precision factor R ≡ (E[s])²/E[s²]. On the pure Pareto family, R splits by tail index into three bands: cutoff-independent for κ★ > 2; decaying as 2/ln b at κ★ = 2; decaying as b^−(2−κ★) for 1 < κ★ < 2; and in the untruncated limit every bound depending only on the first two moments vanishes for κ★ ≤ 2. It must be stated alongside that R is defined for any law with a finite second moment, the Pareto form being only a closed form, so “this system’s spectrum is not a power law” does not falsify the criterion. Third, this domain carries a structure no other in the series carries: income and shock share a physical carrier (ultraviolet and heat are simultaneously energy source and degradation source). Writing the overlap as Σ(W) = Σ₀ + cW^γ, the steady-state stock is single-peaked in input power if and only if γ > 1, with W★ = [Σ₀/(c(γ−1))]^(1/γ); at γ = 1 it rises monotonically to the saturation value Φη/c, and for γ < 1 it diverges as W^(1−γ). The interior optimum follows from no extremal principle but from an identifiable mechanism. Fourth, two zero-cost predictions are executed. On one CM2 chondrite (Murchison), an independent CM2 chondrite (Aguas Zarcas) and a fall-site soil control, distribution identification rejects the pure exponential and the pure Pareto on both meteorite arms (bootstrap goodness-of-fit p ≤ 0.002) and selects the truncated power law by AIC, in the predicted direction; but the truncated power law and the lognormal are not separable (Vuong p = 0.94 and 0.69), so the step that matters most, anchored versus unanchored, was not decided. The sample precision factor gives R̂ = 0.352 and 0.269, corresponding to effective tail indices of 2.24 and 2.17, on the side where the floor is non-trivial. Fifth, execution produced a new observable: the effective tail index recovered from R and that recovered from the concentration coordinate C must coincide on a true Pareto law, and differ here by 0.65 to 0.72; that difference is a measure of departure from Pareto requiring no tail fitting, registered as prediction P-L5. Limitations (to be read as prominently as the results). First, the irreversibility asymmetry A is not measurable on static abundance data; the floor computed here at A = 1 is an upper envelope at maximal irreversibility, not a measured floor. The floor scales as A²: at A = 0.5 it falls to one quarter, at A = 0.3 to about one eleventh. This paper therefore verifies only the factor R, and the statement “the origin inequality has teeth in this domain” currently holds only at the level of R. Second, at n = 23 the log-slope cannot separate κ★ = 1.5 from κ★ = 3.0; this execution delivers consistency, not discrimination. Third, meteorite spectra record parent-body aqueous alteration and are separated from any terrestrial candidate setting by transport, degradation and resynthesis. Path forward. The only setting in which the inequality can be fully confirmed or falsified is a continuous-flow experimental system (prediction P-L3), preceded by an over-identification test and a power scan; the four tiers of outstanding work, ordered by cost, are in Section 10.