the apparatus

The formal companion to the framework's prose. Thirteen sections; fifteen formal results; six measurable instruments; three intervention modes in a 2×2 causal taxonomy. The apparatus is the spine the prose mirrors at every load-bearing claim.

1. §0

   The spine

   The performative setup (P1)–(P5), the closure as fixed point of Φ, the three axiom groups

2. §1

   Foreclosure

   The closed loop drives the survival function at the reflective threshold to operational zero

   • Lemma 1 (foreclosure of the no-stimulus window)
   • Definition 1.1 (chrono-debt spectrum)
3. §2

   Reclamation as learnability

   Theorem 1' — sample-complexity divergence under closed-loop sampling

   • Theorem 1' (reclamation as sample-complexity divergence)
4. §3

   Auxiliary structural results

   Propositions 4'–7': dividual condition, metric superego, captured resonance, libidinal routing

   • Proposition 4' (dividual via posterior contraction)
   • Proposition 5' (metric superego)
   • Proposition 6 (somatic phase-locking, Kuramoto)
   • Proposition 7' (libidinal routing)
   • §3.5 Somatic optimization (Hawkes instrument)
5. §4

   Parametric continuity

   Lemma 2 — the technical scaffold for Mode A's parametric continuity argument

   • Lemma 2 (parametric continuity of regularized stable points)
6. §5

   Intervention modes

   Propositions 1'–3' on Modes A/B/C, the 2×2 site×source taxonomy, the Mode A scaffold's five features

7. §6

   Cohort gradient

   Proposition 8 — formal derivation of the developmental decay D_KL · e^(−2Kτ)

   • Proposition 8 (cohort gradient)
8. §7

   Performative do-calculus

   Theorem 9 — the unified sensitivity formula. Closure-amplification (I − D_θ Φ)^(−1)

   • Theorem 9 (causal sensitivity in performative systems)
9. §8

   Network coupling

   Theorem 10 + corollaries. Engagement cascade, percolation threshold, cohort graph-amplification

10. §9

    Rate-of-foreclosure refinement

    Theorem 11 — intermediate tail regimes between polynomial and exponential

    • Theorem 11 (stretched-exponential survival bound)
11. §10

    Crisis-boundary classification

    Theorem 12 — three generic bifurcation regimes at the contractivity boundary, plus the symmetry-protected pitchfork

    • Theorem 12 (crisis-boundary bifurcation classification)
12. §11

    Synthesis

    What the apparatus delivers; what it does not yet deliver

13. §12

    Empirical calibration

    Per-result calibration hooks; the reflexive measurement problem