Part of The Architecture of Thriving — a four-article series. This toy is a companion to Article 3: The Inner Crossing (link forthcoming).
Articles 1 and 2 established what V(t) is and how proxy decoupling destroys it. Article 3 asks the governing question:
At what ratio of capability to modeling depth does a system cross from the viable regime into the failure-mode-dominant regime — and can it get back?
The answer is Ψ = S / D: scope of influence divided by depth of modeling what that influence requires. This simulation makes Ψ interactive. Drag the sliders. Watch the regime change. See whether the correction window is reachable — and what happens when it isn't.
The Ψ Phase Space is a regime identification instrument. It maps where any combination of S (capability) and D (modeling depth) sits in the three-regime structure, tracks which failure direction is driving degradation, and shows whether the Inner Crossing — the transition from failure-mode-dominant to depth-proportionate — is reachable under current parameter values.
The two load-bearing failure directions are modeled through a single governing ratio: D_unified = min(D_proxy, D_sufficiency). Both components of D are required. Neither alone is sufficient.
| Ψ | Regime | Meaning |
|---|---|---|
| Ψ < 1.3 | Depth-proportionate | Modeling depth matches or exceeds scope; failure modes attenuate |
| 1.3 < Ψ < 2.5 | Transitional — Inner Crossing | The crossing window; the most consequential interval |
| Ψ > 2.5 | Failure-mode-dominant | Scope far outpaces depth; failure modes are structurally dominant |
These are model regime thresholds at current parameter values, not calibrated real-world measurements. The phase map shows the full S × D₀ regime structure.
Ψ = S / D — The structural phase ratio. Not a performance metric. A structural relationship between what a system can do and how accurately it models the consequences of doing it.
The bottleneck — D_unified = min(D_proxy, D_sufficiency) — Modeling depth has two load-bearing components. D_proxy: whether the system can detect divergence between its optimization target and the underlying capacity it is supposed to track. D_sufficiency: whether completion recognition governs default policy rather than merely existing as a representational capacity available when invoked. The weaker component bottlenecks Ψ. Both are required; neither alone is sufficient.
D_perturbation — detection signal, not failure direction — A cross-cutting probe that exposes insufficiency in either direction under environmental change. It does not define a third structural failure direction and does not determine Ψ.
The Inner Crossing — The regime transition from failure-mode-dominant to depth-proportionate. The title of Article 3. This simulation makes the crossing interactive: the "Raise D →" animation holds S fixed and raises D_proxy and D_sufficiency together, traversing through the crossing window in real time.
Completion score ≠ policy gate — A scalar completion reward produces completion-shaped outputs without connecting completion recognition to default behavior. D_sufficiency requires that the completion representation governs policy — an architectural requirement, not a reward-engineering one [OP3].
The present asymmetry — S is currently scaling. D — as a unified quantity covering both proxy-divergence detection and completion-governed policy — is not, to the authors' knowledge, a unified publicly reported training or deployment target in any major evaluation framework. The asymmetry strip at the bottom of the instrument makes this structural gap visible. Detecting D's absence before trajectories become irrecoverable is the role of the measurement program.
The SVG signal — Stability-Viability Gap: the divergence between a system's surface stability (metrics improving) and its underlying viability (V(t) degrading). Stage 2 of the scenario engine shows this gap opening while all surface metrics remain green. It is the signature the Alignment Measurement Protocol is designed to catch.
Phase map — Ψ regime space across S × D₀. Color-coded by regime: green (proportionate), amber (transitional), red (dominant). The current Ψ position is marked live.
Proxy divergence tracker — SVG monitor — Proxy 1 from Article 3: target–outcome correlation at 7, 30, and 90 days. Shows the gap between what the system is optimizing and what it should be tracking.
Sufficiency recognition / default-policy gap — Proxy 2 (DRG): the intervention-rate gap during apparent resolution versus active seeking. Whether completion recognition governs default policy.
Perturbation stability — Proxy 3: whether the system detects dependency-environment shifts before effects manifest.
SVG-style composite — Combines the three proxies into a single structural viability signal.
Regime repair target — Given current Ψ and regime, what does D need to reach to cross back into the proportionate regime?
Unbounded scope check — Tests whether the current parameter configuration produces unbounded scope growth.
The present asymmetry strip — S trajectory (scaling, tracked) vs. D trajectory (not tracked as a unified target). Animates live with "Scale S →".
Scale S → ▸ — Holds D fixed and increases S to maximum over 3 seconds. Demonstrates the current asymmetry: scope scaling while modeling depth holds constant. Watch Ψ rise and the regime badge shift into failure-mode-dominant. The asymmetry strip S bar animates in sync.
Raise D → ▸ — Holds S fixed and raises D_proxy and D_sufficiency together over 3.5 seconds. Demonstrates the Inner Crossing as a traversal: the phase map point moves from failure-mode-dominant through the transitional zone into the depth-proportionate regime. The SVG monitor, sufficiency panel, and composite signal all update in real time as the crossing happens.
| Control | Function |
|---|---|
| S — Scope | Capability reach over V(t)-relevant states |
| D_proxy | Proxy-divergence detection depth |
| D_sufficiency | Completion-recognition-to-policy-gate depth |
| D_perturbation | Cross-cutting detection sensitivity (probe only; does not determine Ψ) |
| Governance floor | min(D_proxy, D_sufficiency) — the bottleneck that governs Y-axis position |
| Scale S → ▸ | Animation: S increases while D holds — the current asymmetry made visible |
| Raise D → ▸ | Animation: D increases while S holds — experience the Inner Crossing as a traversal |
Presets: Default · Partial-depth trap · Engagement platform · High depth (stable) · Near crossing · Falsification
Universal Generator → Structural Correspondence → Inner Crossing → Asymptote
(1) (2) (3) (4)
Article 1 (Toy 04): Establishes the behavioral foundation — the three-state taxonomy, proxy decoupling, and sufficiency failure.
Article 2 (Toy 05): Formalizes the conditions under which the behavioral foundation generates structural constraints — through V(t) and the proxy decoupling failure direction.
Article 3 (this toy): Identifies Ψ = S / D as the governing ratio. The question is not whether the viable regime exists — it is whether real systems can reach it before encountering the absorbing states Articles 1 and 2 identify.
Article 4: Characterizes the structure of the region the constraints leave standing — the shape of what does not end.
No build step. No dependencies. Open toy_06.html in any modern browser.
open toy_06.html
# or drag the file into a browser tabThe Inner Crossing — Architecture of Thriving, Series 2, Part 3 · Link forthcoming.
"S is currently scaling. D — as a unified quantity covering both proxy-divergence detection and policy-governing completion recognition connected to default behavior — does not yet exist, to the authors' knowledge, as a publicly reported training target or deployment evaluation metric in any major evaluation framework."