Three Densities: Proposition 4 — Hysteresis

The original essay (Three Densities) separated platform density into three layers and showed that conversion between them is non-conservative. The state classification there is static: it maps a platform at a single moment. This note introduces time. Proposition 4 (Density Hysteresis) D3 responds to a decline in D1 with delay. When D1 is high, participants form context-sharing networks: mutual recognition, role awareness, shared reference points. These do not collapse the moment D1 drops. They are cumulative structures. Proposition 2 established that a rise in D1 does not guarantee a rise in D3. The same non-conservation operates in reverse. D1 falls; D3 stays above threshold — for a while. This delay is density hysteresis. The analogy to physical hysteresis is exact. A ferromagnet retains magnetization after the external field is removed. D3 persists as residual context-sharing after D1, the external input, has diminished. Corollary 4.1 (Contracting Scene) The original essay defined four states: Death, False Growth, Comfortable Stagnation, Scene. Proposition 4 demands a fifth. Contracting Scene — D1 is declining. D3 remains above threshold but is falling. The space is alive. On borrowed time. When the hysteresis runs out, D3 crosses below threshold. A Contracting Scene is a transit state toward Death. Not irreversible: if D1 rises again, D3 decline can halt. But D3 does not recover on its own without renewed D1 input. Corollary 4.2 (Performative Proof) A declaration that a space is dying sometimes generates massive cross-context engagement. Participants from unrelated contexts converge on a single topic and produce context-aware responses. This happens. The phenomenon is observational evidence that D3 > threshold. The death declaration produces proof of life — a performative contradiction. This does not refute the decline in D1. It confirms hysteresis. A Contracting Scene is exactly the state that generates performative contradictions: the numbers are falling, but the space still responds. That it still responds does not guarantee it will keep responding. Open Question 2 What determines the duration of hysteresis? Three candidates. a. Peak D1. Higher peaks accumulate more context-sharing. More residual. Longer delay. b. Duration at peak. A brief spike builds shallow context-sharing. A sustained high-density period builds deep context-sharing. Depth, not height, may govern persistence. c. Internal structure of D3. Of the three components — identity, role, sustained anticipation — sustained anticipation depends on post flow and is the most fragile. Role awareness depends on recognition from others and is more durable. Identity is rooted in self-concept and is the most persistent — but when belonging to a space drops out of self-concept, collapse is irreversible. Verification methods for these candidates remain undefined.