Worked Research Cases
How the public stack moves from concepts to recomputable boundary behavior
R1 — Threshold Release Gate
Setup: a 3-of-5 release gate with shared-root dependence.
Research purpose: test whether nominally separate authority domains remain robustly independent when a common root or common dependency can collapse that independence.
Boundary idea: counting signers is not sufficient if several signers are controlled through the same underlying authority path. PISF therefore distinguishes nominal threshold structure from a robust threshold such as κ̲_E.
R2 — Time-Bounded Emergency Refusal
Setup: intervention latency is decomposed into detection, decision, authorization, enforcement, and verification.
Research purpose: determine whether refusal can complete before the last safe intervention point, rather than merely whether a stop procedure exists.
Connection to L30-RET: this case directly connects to Temporal Sufficiency. A refusal that cannot complete in the remaining safe interval is not made effective simply by formal authority.
R3 — Fallback-Capacity Dynamics
Setup: a finite-horizon model h(t+1)=h(t)-αa(t)+βm(t).
Research purpose: study how automation/dependency can reduce fallback capacity while maintenance or investment can preserve it, and how the result changes with the time horizon.
Boundary idea: no dramatic failure event is required for revisability to erode. Ordinary adoption and dependency can gradually move a system outside the human-revisable region.
R4 — Viable but path-locked
Setup: a four-state toy graph in which two states remain inside the human-revisable viability region for the whole horizon, but preserve different sets of materially distinct future terminal regions.
Research purpose: distinguish remaining viable from preserving meaningful future optionality.
Boundary idea: a state can remain viable while becoming path-locked. PISF v1.2 therefore uses the optionality object ΩδH to distinguish a state preserving at least two materially distinct terminal regions from one preserving only a single such region.
Result in the toy case: both states remain in K_R^H, while ΩδH(s0) ≥ 2 and ΩδH(s1) = 1. Viability alone therefore does not establish preservation of materially distinct future choices.
What to recompute
| Case | Object | Question |
|---|---|---|
| R1 | Nominal vs robust authority threshold | Does common-root dependence reduce effective independence? |
| R2 | Intervention/refusal margin | Can the full refusal process complete before the last safe intervention point? |
| R3 | Finite-horizon revisability | Does fallback capacity remain inside a human-revisable region under automation and maintenance dynamics? |
| R4 | Viability + optionality | Does the state preserve more than one materially distinct future path, rather than remaining merely viable but path-locked? |
What these examples are for
The point is not to prove that real systems are safe. The cases make definitions executable, procedures reproducible, parameter boundaries independently recomputable, and boundary behavior inspectable.