Causal Evolution Dynamics: Formalizing the Boundaries of Grounding and Hysteresis in Complex Systems

Complex systems increasingly report on their own success. When the report and the delivery come from different places, a system can keep recording completion after the work has stopped landing. We give this failure a formal treatment rather than a rhetorical one. We build a minimal formal system: nine axioms, twelve definitions, ten inference rules, an explicit domain for every variable, and a four-valued verdict (PASS / FAIL / INVALID / NOT_RUN) that is a statement about the scope of a claim rather than about its success. Two dynamical laws are kept separate throughout, because they are not the same equation: a saturating metabolic law (M1, used while dormant) and a linear accounting law (M2, used while active). Earlier versions of this work treated their concatenation as one "unified equation"; we show they have different fixed points, different stability and different time constants, and that the concatenation is an assumption (A7) rather than a derivation. Three results survive the audit.1. The deadlock boundary is hyperbolic and exact. Under a quasi-static substrate assumption and a wake rule that requires κ consecutive qualifying cycles, the dormant state is permanent if and only if χ < χmin = 1/(1 − θwake), where χ(v) = Y φv v/(m0 φdorm) is the charging margin of the agent itself; equivalently χ0 < 1/(f (1 − θwake)) with χ0 = χ(Vmax) and f = v/Vmax. The two forms differ by the factor f and must not be quoted interchangeably. The asymptotic charge A∞ = 1 − 1/χ is independent of the substrate level, and the marginal effect of added resources vanishes as W^−2 — which is the precise sense in which "adding resources does not exit a deadlock".2. A closed system has no hysteresis loop. With a Lyapunov function V = S, a batch system without feed drives S → 0 and then A below any threshold in finite time; the downward edge is autonomous and the upward edge can never fire. The general claim that any hysteresis loop needs an external upward trigger is false (van der Pol is a counterexample), and we withdraw it.3. A drift detector with power one, and its limits. Any affine-invariant statistic has power exactly α against linear drift; sliding-window variance has an expectation that does not grow with √n. The cumulative potential Φn = n∆drift + ε0 − εn has signal-to-noise n∆drift /(√2σ) → ∞. It is unbiased and consistent, but not sufficient and not efficient: the OLS slope has variance 12σ^2 /[n(n^2 − 1)], an order of magnitude smaller. We then audit our own framework and publish the audit. Twenty-eight implicit assumptions are identified; eleven are load-bearing — if any one fails, at least one of the results above or one headline number collapses. They are listed in Table 1 with the specific theorem each supports, and the seventeen that are not load-bearing are inventoried with their failure modes. We also withdraw two strong claims of the earlier draft (that multiplicative availability is necessary in systems, and that the observed cliff is a phase transition), and we downgrade the latter to a conjecture that is, in the independent-agent limit, false. Finally we report a controlled ablation in a 100-agent sandbox. Type-A commits a fixed scope and reads only its own log; Type-B commits a scope that is a function of the world's verdict. Both face the same capacity trajectory. Type-A stops delivering permanently at cycle 100, dies at 107, and its own log ends 15.56% above what landed. Type-B is stranded at 279. The lifetime ratio is 2.61, not the 9.19 that the previous draft reported; the difference is a unit error in the wake gate, which we isolate as a three-way sensitivity table (2.61 / 2.18 / 9.19). Wiring the INVALID channel into Type-A — and changing nothing else — is sufficient for it to survive the same 300 cycles with a zero log gap. The channel, not the drift, is what the cliff is made of. The extinction is scheduled rather than emergent: Corollary 4 places each death four to seven cycles after the onset, from the upkeep rates alone. The algebraic skeleton — the threshold equivalence of Lemma 4, the mutual exclusion of the two modes, and the totality of the four-valued verdict — is machine-checked in Lean 4 (Formal_Verification/Core_Theorems.lean): 0 errors, 0 warnings, no sorry, with the #print axioms receipts printed from the source rather than summarised here. We make no claim of universality. The contribution is a boundary that can be checked, three proofs that can be refuted, and a list of the assumptions that would have to hold for either. ---Version 2.0 Update Notes:- Updated Paper 1 to v1.0 (40 pages). - Corrected a mathematical inconsistency (Corollary 1 fixed-point formula) and updated the assumption ledger (Ax6).- Refined the semantic scope of the four-valued verdict and the ledger identity in §2.7.- Added full reproducibility artifacts: Lean 4 verification, automated audit scripts, and final PDF build sources.- Added Code and Data Availability statements linking to the GitHub repository.