Governance delay links observed resource decline to deployed effort. A stock–memory– effort model compares mobilising effort escalation with protective quota tracking, separating deployment latency from periodic review. For four registered mobilising vectors, an analytic characteristic-root count and interval crossing-order checks classify equilibrium stability on the entire positive delay axis. Candidate A has two stable delay windows, not only the interval between the fundamental Hopf crossings. Classical nonlinear Hopf hypotheses are verified at the declared crossings; these local results do not classify all nonlinear attractors. For the protective law, a nonlinear asymptotic-gain theorem identifies the complete positive-equilibrium basin within the physical state space. It covers every fixed finite delay and every positive fixed cadence under native signal-hold and exact nonlinear quota review. A sufficient gain is below 0.475 at the common point and below 0.687 throughout a declared parameter box. The excluded extinction and saturated-effort faces remain invariant; Euler and mixed-channel laws do not inherit the result. Periodic continuation and capture checks connect the upper stable family. Odd-grid fold refinements and rigorous infinite-Fourier orbit and kernel residual bounds advance con- tinuum validation, without substituting for the missing bordered inverse and contraction proof. Retrospective North Sea herring tests add quantitative field challenges: the pro- tective response narrowly beats AR(1), but the ecological and recursive annual-surrogate stock predictions fail baseline comparisons. These mixed outcomes distinguish conditional response skill from joint dynamic validation of the autonomous delay model. Response law, latency and cadence are mathematical design coordinates whose field identification requires direct institutional measurements and an adequate ecological observation model.

Depletion indicators can carry similar units while built to inform distinct questions. Reserve- life ratios, trend-persistence metrics, and gross-removal pressure scales are widely interpreted as a literal “time to depletion,” despite measuring distinct physical or statistical quantities. Compounding this ambiguity, compensatory scalar aggregation routinely masks critical physical deficits behind positive aggregates. In this paper, we resolve these confusions by constructing a typed stock–flow accounting ledger. By tracking conserved substance classes (moieties) independently, conservation laws remain typed: under the declared type discipline, biomass, financial flows, and biodiversity metrics do not form one typed scalar conservation readout. A cross-type numerical sum may still be computed, but it is an additional declared readout—not a conservation law of the typed ledger. Mass conservation follows directly from the incidence structure of the compartment–flux network, nonnegativity is guaranteed by donor limitation (primitive outflows vanish whenever their source compartment empties), and ecosystem services are treated strictly as observable readouts rather than conserved mass. We formalize three certification layers: an exact flux-identifiability criterion for what an observed readout trajectory can and cannot determine about unobserved primitive fluxes, the conservation-law projection of an already balanced trajectory, and an envelope theorem providing verifiable flux-bounding certificates. The closure calculus is state- and boundary- conditioned, and it separates material not yet returned by a declared use deadline from ma- terial never returned: the first is a recoverable in-transit or pipeline requirement, while only the second liquidates the available-plus-recoverable stock. For the closed finite-donor ledger, we establish the natural-block mass identity, orthant invariance, the absence of interior rest points under positive extraction effort, a characterization of the vanishing-extraction rest set (extinction, carrying capacity, and a frozen-biomass face), and extraction integrability over fi- nite budgets. Furthermore, we unpack “depletion time” into three mutually non-interchangeable concepts: gross turnover intensity, a frozen-rate local ratio, and a scenario-conditioned hitting time equipped with uniform-drift bounds. We classify three widely cited public-data indicators within this taxonomy: the G3P ground- water anomaly index is shown to be a record-relative statistical metric rather than a physical stock ratio; the USGS phosphate reserve-life ratio is an arithmetic convention of an economic class rather than an exhaustion forecast; and fisheries removals-only timescales represent gross pressure scales rather than demographic depletion diagnostics. We prove that no nonnegative linear weighting can universally certify componentwise nonnegativity on arbitrary balance do- mains, and we establish five double-counting rules to prevent the emergence of phantom mass. Finally, we formulate first-passage semantics on declared stochastic surrogates against record- relative barriers alongside explicit non-claims. The argument separates two questions often compressed into “weak versus strong sustainability”: whether material cycles close at demanded throughput—a state-conditioned dynamical feasibility question—and whether an assessment is scalar-compensatory or componentwise—a certification-semantics question. A stationary bridge between them is proved, but the two distinctions are not identified.

Composite indices can certify transitions under weight-adaptive or common-plan rules. We formulate the comparison as a robust finite-horizon assessment with path-wise constraints and declared distur-bances. The full-cone aggregate is lossless at a fixed trajectory, yet the plan/weight quantifiers need not commute. For finite menus in the stated margin-first class, the weight-adaptive accepted set is conv(D) + Rⁿ₊ and the common-plan set is the union of single-plan acceptance orthants; an abstract convexification of required margins closes that gap, while physical implementation requires a separate control assumption. An exact rational witness has a nonempty gap interior: at its canonical point both normalized margins are 6/5, so the equal-weight margin sum s₁ + s₂ = 12/5 exceeds the witness’s licensing threshold 2; each weighting licenses at least one primitive plan, though no single primitive plan meets both typed floors. A closed-form reserve threshold separates rescuable from impossible states. Under the declared linear-flow blend, fractional control closes the action-indexed gap, while the specified shared-shock blend does not restore feasibility at the canonical point; the complete-excursion result is conditional on its union-of-tubes assumption. On the witness-defined diagonal slice, every positive-elasticity member of the stated CES family falsely certifies an exact interval; the Leontief limit is typed-safe under the declared collapse convention. The shared-shock gap survives the exact 1/2-cut, while a distinct replace reading with identical plan- and weight-independent dips collapses the distinction (SI S4). Northern-cod values supply a historical unit comparison only; the benchmark is not a fitted fishery model.

We develop an exact finite-dimensional theory of the replica mechanics underlying measurement-induced entanglement transitions in monitored quantum circuits, separating three structures that are often conflated: linear outcome-summed replica channels, normalized conditional trajectories, and asymptotic large deviations. For fixed-basis projective measurements we determine the minimal invariant unital algebra of the one-wire feature space and prove that independently Haar-averaged two-site gates on a perfect matching close the \( n \)-replica permutation space exactly for every replica number, with an explicit Weingarten bond channel. The resulting one-period transfer operators have real, positive, semisimple nonzero spectra that compress to bond-label space; at \( n=2 \) the model is exactly the anisotropic triangular-lattice Ising model of Bao, Choi and Altman, with a closed-form critical point and a closed free-fermion spectrum. Carrying the same computation through \( n=3,4,5 \) locates the annealed critical points \( p_c^{(2)}=0.2338 \) <\( p_c^{(3)}=0.305(3) \)<\( p_c^{(4)}\approx0.383 \)<\( p_c^{(5)}\approx0.47 \) with the Potts-class exponents at \( n=3,4 \), and supports, at the three-size level, the first-order prediction of the $q=n$ -Potts identification at $n=5$ : an exact three-size test ( $L=4,6,8$ ) on a symmetry-reduced orbit basis of dimension 3,865 over the $2.1\times {10}^8$ bond-label space shows the two-phase gap closing faster than the continuous baseline at every size, with the scaled gap falling below the continuous envelope at the matched sizes and dual effective closing rates ordered consistently with the continuous, marginal and first-order behaviours at matched sizes---a directional trend, with the registered conditional that completes it: if the closing is genuinely exponential, a fourth size ( $L=10$ ) will show the effective closing rate stabilizing at a positive interface tension, and if the asymptotic is a slowly decaying power law, that rate will continue to decay---and at local dimension $d=3$ the same three-size test keeps the gap level below the $d=2$ values at every size. We further prove a no-freezing theorem for finite-reachable record processes, establish the worst-case complexity of conditional-trajectory decision, derive the exact replica interpolation identity with its measurable self-averaging criterion, and give the first numerical quenched record scaled cumulant generating functions for Haar circuits, which exhibit genuine record multifractality; the Clifford record-count disorder statistics are closed exactly through a verified conjugation 3-design. dummy