Collapse and Rebirth Why Structures Contract Only After Collapse: A Minimal Model of Maintenance Budget, Participation Base and Readings

Abstract The most common synthesis in comparative studies of rise and fall is that "internal causes set fragility, external causes set timing", so that the risk of collapse is roughly fragility times shock intensity. This synthesis can hardly be wrong, and for that very reason it excludes almost no outcome. This paper rewrites collapse as a failure of the maintenance relation and states a set of propositions in exclusionary form. First, a shock is not an independent factor multiplying fragility but an item of expenditure in the maintenance budget, proportional to the structural stock; the largest maintainable stock falls monotonically with the shock rate. When the budget is tight, alignment costs are cut first, which turns a depletion crisis into a channel-rupture crisis. Shocks also enter in a second way, as a one-off pulse. The larger the shock, the less the ordinary depth of a well decides who dies; width (the number of mutually independent units) and depth then enter the survival probability only as a product. Second, a U-shaped loss whose two arms are power functions has no fold, so "rigidity" by itself produces no cliff; cliffs come only from self-reinforcement. The paper isolates a source not previously isolated: rigid maintenance commitments spread over the participating units. It produces a saddle-node and hysteresis even in an activity equation without a synergy term. Third, the core of the paper is a two-dimensional fast–slow model and two extensions of it. In the basic model, if structure stops expanding only when participation falls below the fold, structure grows along the entire high-activity branch and contracts only in the low-activity well: collapse is the only route by which structure contracts, the last channel of error correction. "Tear down and start over" thereby acquires a precise meaning: it is not a law but the dynamical regime of the condition "structure contracts only after collapse". Adding a reading channel with lag lets structure contract ahead of the fold, and the cycle is replaced by an equilibrium; this requires the reading gain to exceed a closed-form threshold k_c and the reading lag to be shorter than a critical value of the same order as the time scale of structural accumulation (ω is the slow-to-fast time-scale ratio; at reading gain k = 0.2 the critical lag is about 1.2/ω, and it grows with k). Adding a share θ of structural income that does not depend on the participation base makes rebirth disappear at a closed-form critical value θ_c, and the system enters a state absent from the adaptive-cycle diagram, which this paper calls "hollow lock-in": the structure remains and even grows, while the participants are gone. On the way to θ_c the trough lengthens as 1/(θ_c − θ) while the boom shortens slightly, a precursor observable in advance. The result is a three-region phase diagram (adjustment, order–disorder cycle, hollow lock-in) whose three boundaries are all in closed form; all 30 numerical integrations fall in the regions the closed forms predict. Fourth, concentration goes together with lower distinguishability, a shared bias cannot detect itself, the identification delay of an organisation is set by the person least willing to admit error, and shared readings put a floor under synchronous failure; early warning must read the recovery rate, not the variance. Fifth, the conditions of rebirth fall into two groups: release conditions decide whether collapse can remove the burden, and reorganisation conditions decide whether the slow layers survive the removal. Seven outcomes follow (adjustment, rigidity trap, cycle, lock-in, rupture, absorption, replacement), each with an observable signature. The paper closes with a case-coding protocol, seventeen registered predictions and an execution ledger: the number of executions on external data is zero.