The Principle of Nonuniformity :Three Structural Quantities, Two-Ledger Dynamics, and the Correct Classification of Aggregation Limits (V11)

Abstract

Abstract: The macroscopic phenomenological apparatus of open flow-through systems — an income-minus-expenditure master equation, a gradient-flow relaxation, and a quasi-potential landscape — is usually posed as a set of postulates. This paper assembles that apparatus into a diagnostic framework and draws its boundary of validity. The organizing proposition is retained as a first principle for open systems, the Principle of Nonuniformity: the state of an open system departs structurally from uniformity along both a cross-sectional and a temporal axis, a departure supplied continuously by work and paid for by non-negative internal entropy production. This version makes three retractions and five corrections relative to V10. All of them fall on load-bearing structure.Retraction one concerns the reference measure of the load-bearing variable. Earlier versions took the Kullback–Leibler divergence from an exponential baseline and converted it to energy units, calling the result an ordered free energy. For any system whose state-dependent coupling is positive, the stationary mark law is not exponential, so a quantity referenced to the exponential is not a rate function on any system this framework is about, and does not vanish on the system’s own stationary law. The correct construction splits one quantity into three, distinguished only by which measure sits in the reference slot. A structural stock U_str is measured from the passive baseline, the law to which the system relaxes when driving is withdrawn; this is what the master equation carries. A displacement U_LDP is the quasi-potential of the stationary distribution and enters the Kramers escape exponent. A structure reading U_exp is measured from the memoryless baseline; it locates the stationary law on the form spectrum and enters no equation. All three are dimensionless, energy units survive only at two explicitly marked absolute calibration points, and the ordered-free-energy symbol is retired.Retraction two concerns the potential. V10 listed a quadratic free energy, a Landau expansion, and a large-deviation logarithmic integral as three truncation levels of one object. They are not. The quartic is the antiderivative of the deterministic drift with its sign reversed; for polynomial drift it is exact rather than a Taylor truncation, and its quadratic coefficient is (c−B)/2, not c/2. The quasi-potential is the WKB potential of the jump process. The two share every critical point, but their curvatures at a fixed point differ by the exact factor 1/(Φ★·c) — 1.2747 against 2.108 at the maintaining state of the standard parameter set, and a factor 4.46 at the barrier — so the claim that they agree to second order does not hold. One identity falls out of the restatement: the critical margin equals the second derivative of the deterministic potential at the maintaining state.Retraction three concerns the fixed-point status of the fossil state. V10 stated that once the three expenditures are written multiplicatively, the zero of the structural stock becomes an unconditional exact fixed point. The drift there is the income term ε·Φ·η̂·W, which does not generally vanish, so that locus is a repelling line. The fixed-point status of the fossil state comes instead from the vanishing constant term in the recruitment rate: the activity equation carries an overall factor Φ, so Φ = 0 is an invariant manifold. The conclusion survives in cleaner form, and the absorbing conditions of the two coordinates merge into one.Correction one adds a sign branch to the master equation. Driving can push the stationary law to be more concentrated than the passive baseline or more homogeneous than it, and a divergence assigns a positive value to both directions alike. The sign branch is defined locally on the size side, as the sign of a difference of concentration readings, rather than through the Fano factor of the count distribution. The latter choice would make the definition of the central state variable depend on the falsifiable claim that the two coordinates share one sign, and the framework’s own equations supply a candidate counterexample region.Correction two supplies a single definition of the effective recovery rate. The margin formula in V10 used a coefficient that its own notation table never defined, and omitted the term responsible for bistability. The effective recovery rate is defined as the negated spectral abscissa of the linearized generator; under finite dimension, near-diagonality, and a fixed point it degenerates to the critical margin, whose closed form is Δ = γ + δ·f_shock − B·(1 − 2Φ★) + a₂·Φ★·(3Φ★ − 2). Each of the three conditions fails somewhere in the framework — under age structure, on limit cycles, and in spatially extended systems — and each failure is now labelled where it occurs.Correction three reattributes the screening length. V10 called the screening length and the tail index two properties of one coordinate. The mark law carries no spatial information, so that reading cannot stand. The correct form is a causal chain: spatial gradient surplus lets denser locations draw from further away, the effective multiplicative gain rises, the allocation exponent is pushed up, and the tail index falls. The screening length itself is a second reading of the same activity-field spectrum, ℓ = √(D/Δ), and the dissipation slot in that formula is a role variable identified per application.Correction four adds age structure, and with it the lowest-threshold prediction in the framework. Giving the stock one extra dimension of component age separates the two maintenance classes for the first time. An exogenous shock imposes a common rate shift on every age mode, so the second derivative of the logarithm of the recovery curve is exactly invariant under that shift. The falsifiable statement therefore reads: the log-recovery curve is convex for a high-turnover system and straight for a low-turnover one, and the test requires no control over the shock.Correction five redraws the line between exogenous and endogenous. V10 claimed that exogenously variable rates can only transcribe a tail that is already present. Mixing an exponential law over a Gamma-distributed rate gives a Lomax law: both the component and the mixing law are light-tailed, and the result is a genuine power law. The line that survives is drawn on response to work — an exogenously frozen departure has an identically vanishing derivative with respect to work and does not relax when work is withdrawn, and only endogenous state dependence makes the departure a function of work. Under the passive baseline this negative result becomes cleaner still: a departure produced by exogenous mixing is positive on the structure reading and identically zero on the stock.