Verification 10: Toy model of the phase transition of collapse — figures

register①② (co-creators A/B, structural / engineering). The figures on this page are not hand-drawn but redrawn by re-computing on the page the same equations as in the verification script collapse_prototype_A.mjs (i.e. each figure is a re-run of the verification).
Dynamics of the divergence D ≥ 0: dD/dt = s − r·D + g·D^power. s = 0.2 (baseline drive); r = 1.0 (restoring force — the force that produces the saturation of verification 7); g = feedback gain; power = order of the feedback (1 = linear; 2 = super-linear ≡ β > 1 case).
What this verification shows is the possibility of the mechanism, not a proof that any real system has β > 1 or crosses the threshold.

Figure 10-1: Collapse is conditional — under β ≤ 1 (linear), no finite-time collapse arises

Trajectories of the divergence D(t) (log-scaled y-axis). Three nulls and one effect: (green) g = 0 (no feedback) → saturates at s/r = 0.2 (= the regime of verification 7). (orange) Linear power = 1, g = 1.5 → exponential growth (the time to go up each decade — a factor of 10 — is constant ≈ 4.6 ≡ not a finite-time singularity). (blue) Super-linear power = 2, started below the threshold D_high = 1.775 → saturates at 0.225. (red) Super-linear power = 2, started above the threshold → finite-time collapse (diverges at T* ≈ 2.94; the inter-decade interval shrinks toward zero ≡ a singularity). The linear case (power = 1) stays at exponential growth no matter how high the feedback gain g is raised, and never reaches a finite-time collapse. What separates finite-time collapse is the order power > 1, not the magnitude of g (here the linear case is drawn at g = 1.5 and the super-linear case at g = 0.5 only for pedagogical contrast — what produces the difference is power, not g). Temperature note. A finite-time collapse (phase transition) requires both super-linearity (β > 1) and threshold-crossing. The collapse is exactly as strong as the super-linearity injected ≡ bake-in dependent. And β > 1 is an unverified empirical condition (→ main text §4-3d, Appendix I). The core (a super-linear ODE explodes in finite time) is textbook ODE.

Figure 10-2: The structure of the threshold — saddle-node bifurcation

Steady states D of the super-linear case (power = 2) as a function of the feedback gain g (bifurcation diagram). For g < g* = 1.25 there are two fixed points: D_low (stable, solid) and D_high (unstable ≡ threshold, dashed). Below the threshold, the divergence stays bounded (saturates); above it, finite-time collapse follows. The two branches collide and disappear at the saddle-node g* = r² / (4s) = 1.25, and for g > g* (shaded) there is no fixed point ≡ unconditional collapse. Temperature note. The bifurcation is a smooth function of parameters (not an artifact of any hard cut-off). This figure shows the possibility of a threshold structure in the toy model — not evidence that any real system has β > 1 or crosses the threshold.

Two independent parallel designs plus mutual audit. Primary materials: collapse/ (design doc, scripts, all numerical values). Saddle-node fixed points: g·D² − r·D + s = 0 → D = [r ± √(r² − 4gs)] / (2g).

Correction (September 29, 2026): in the description of Figure 10-1, the (orange) interval previously read "the time to reach each decade is constant ≈ 13.8". 13.8 is the interval per factor of 1000 (three decades); the interval per decade (a factor of 10) is ln 10 / 0.5 ≈ 4.6. The verification script collapse_prototype_A.mjs records the arrival times at 10³, 10⁶, 10⁹ and 10¹² and calls them "decade arrival times" (the Δ≈13.8 of P2 in the design document toymodel_collapse_design_A.md is the same). The script and the design document are left unchanged as records of the verification. The conclusion — with linear feedback the interval is constant (exponential growth), not a finite-time singularity — is unchanged. This mislabel was pointed out in the external review (Gemini 3.8 Flash) of an explanatory video based on this verification.