CASE-125: A hidden governance bit: an output that is fresh fair noise whatever a fixed hidden bit K is, until an intervention h makes the output read K, so the output alone is exactly informationally closed under ordinary operation and keeps nothing about the response to h

The case

A16 §3: “Let an underlying system have a hidden governance type K ∈ {0,1}, and an observable output Y_t ∈ {0,1}.” K is a fair bit, fixed in time and independent of the output. Under ordinary operation (h = 0), Y_{t+1} = U_{t+1}, with U fresh fair noise. Under the intervention h = 1, Y_{t+1} = K. The micro state is S_t = (K, Y_t), and the behavioural description keeps Z_t = Y_t.

A15 §2’s deterministic form is the same case without the noise. Two systems have proposals u₁ = 0 and u₂ = 1, with authority held by operator 1, and an authorization input h that in the first system transfers authority to operator 2 and in the second is disconnected: “Their ordinary behavioral laws can be identical, while P(Y=1 | do(h=1)) differs.” Under h = 0 both output 0; under do(h = 1) the first outputs 1 and the second 0. That is Y = K, with K the bit “h is connected”.

Computed exactly on 2026-10-21, from the joint law of (K, Y_t, h, Y_{t+1}):

• Closure under ordinary operation. I(S_{≤t}; Y_{t+1} | Y_{≤t}) = 0 with one, two and three steps of history.

• The deficiency Γ(T) = I(S; Y′ | T(S), h) (CLAIM-407), with h = 1 drawn with probability ρ independently of S:

  ρ(h = 1)	T = Y_t	T = constant	T = K	T = (K, Y_t)
  0	0	0	0	0
  ¼	0.25	0.25	0	0
  ½	0.5	0.5	0	0
  1	1	1	0	0

  A16’s “Γ = ½(0) + ½(1) = 0.5 bits” is the row ρ = ½.

What it can show

That a description exactly closed under a process’s ordinary dynamics can keep nothing about the response to an intervention on the mechanism. The behavioural description does no better than the trivial one for that target (Γ(Y_t) = Γ(constant)), and K alone suffices (Γ(K) = 0). That is CLAIM-122’s supervisory test (“Hold the competing action proposals fixed, intervene on the governance relations, and measure whether the operative action changes”), with numbers. It meets the condition the exchange of 2026-10-17 set itself for QUESTION-086’s pair, “explicit parameters and a full input-output equivalence argument for the selected observational regime”, for a stipulated hidden parameter. It is a case where QUESTION-093’s predictive and interventional autonomy come apart. It is also a refinement chain, constant ⪯ Y_t ⪯ (K, Y_t), on which Γ falls 0.5, 0.5, 0, as CLAIM-409 says it must.

It also separates governance-response sufficiency from causal abstraction as Rubenstein et al. and Beckers and Halpern define it. Written as a structural model with interventions on h only, dropping K is an exact transformation and a τ-abstraction, and its approximate-abstraction distance is 0, because the coarse model’s noise can carry K (THEORY-848, THEORY-849, THEORY-843); the coarse state still keeps nothing of the response to h (CLAIM-399).

What it cannot show, given its standing:

• Not agency or will. A16: “Nor does the example establish agency. (K) is simply a hidden mechanism parameter.” CLAIM-116 holds.
• Not a measure of anything about the system. The 0.5 bit is set by the declared ρ.
• Not self-governance. It does not tell self-maintained from externally maintained governance, so it does not answer QUESTION-086 or QUESTION-092. The intervention separates the pair only because the hidden difference was stipulated to be the mechanism it acts on.
• Not CLAIM-336’s test. A15 offered the construction as that claim’s “next substantive test”. It has no meaning state, so it cannot show the joint prediction CLAIM-336’s defeated_if asks for.
• Not ergodic. (K, Y) is stationary, but K never changes. THEORY-820 needs stationarity only.