The construction
A4’s three properties: “D — Dynamical separation: A macrovariable evolves much more slowly than constituent activity”, “E — Endogenous maintenance: Constituents respond to a shared variable and, through their behavior, regenerate it”, and “H — Historical transmission: A later organization inherits an operative structure from an earlier one through a causal process.”
Arm A, slow without uptake. State (Z, U₁, U₂) ∈ {0, 1}³. Z_{t+1} = Z_t ⊕ B_t with B_t ~ Bernoulli(0.01); U₁ and U₂ are resampled independently at each step. The Z chain has transition matrix [[0.99, 0.01], [0.01, 0.99]], with eigenvalues 1 and 1 − 2(0.01) = 0.98. The resampling of (U₁, U₂) is a rank-one matrix, with eigenvalues 1, 0, 0, 0. The joint chain is their Kronecker product, with eigenvalues 1, 0.98 and six zeros, as A4 says. Dependence on Z’s initial state decays as 0.98^t (0.904 at t = 5, 0.133 at t = 100); the members forget in one step. A4: “Strong timescale separation exists, but constituent agents never read or regenerate Z. It is a persistent statistic”.
Arm B, regenerated without slowness. Two members read a shared register Z_t and act: A_i = Z_t ⊕ N_i, N_i ~ Bernoulli(0.4). Member 1’s action writes the next register, Z_{t+1} = A₁. So P(Z_{t+1} = Z_t) = 0.6 and the transition matrix is [[0.6, 0.4], [0.4, 0.6]], with eigenvalues 1 and 0.2. Under do(Z_t = 0), P(Z_{t+1} = 1) = 0.4; under do(Z_t = 1), 0.6. The members’ reading of Z makes a difference to its next value, and the register forgets as 0.2^t, on the members’ own timescale. A4: “A shared variable can be causally regenerated by members without exhibiting strong temporal persistence.”
Arm C, transmitted without continuous operation. Episode A runs with rule θ_A, which is written to an archive. Episode B, which never runs at the same time as A, reads it and executes b_B = θ_B = θ_A. Intervening on θ_A changes B’s behaviour. Neither episode needs ongoing feedback or a slow internal state.
Every eigenvalue, the decay values in A4’s chart and the interventional probabilities were rechecked by hand and numerically for this filing. A3 said it checked them “using exact rational arithmetic”, in a bundle that never reached the repository.
What it can show
- Arm A: slowness is not sufficient for endogeneity. That is already CLAIM-327’s “timescale separation (τ_Z ≫ τ_X, not sufficient)” and CLAIM-331’s “slowness is a mark of a level and not its definition”. The arm makes it a construction, and a trivial one: Z does not depend on the members at all.
- Arm B: slowness is not necessary for endogeneity. This is the converse, which the record did not have. It holds within A4’s limit, kept verbatim: “This demonstrates only weak endogenous maintenance. If maintenance is defined to require robust long-term persistence, this counterexample no longer establishes the separation.” A3 put the next test the same way: whether the separation “survives stronger definitions of maintenance”.
- Arm C: transmission without running. It is CLAIM-287’s dormant coalition, the capacity to reinstate an organization rather than its continuous running, and CLAIM-329’s “a lapse in its coherence need not end it”. A3’s refinement: “Dynamical separation and endogenous maintenance are properties of processes. Historical continuity is a relation between processes at different times.”
What it cannot show
- Identity. A4: “causal transmission alone does not settle whether A and B are numerically the same collective. That requires an additional individuation criterion.” CLAIM-287 says continuity of this kind can branch, so it is not identity.
- A general theorem. A4: the constructions “do not establish a model-independent theorem about collective identity.” They separate the properties under the operational definitions chosen here, D as a second eigenvalue, E as a nonzero intervention effect of the members’ reading.
- That members of arm B are agents, or that Z is a level. Member 2 reads Z and its action changes nothing, and nothing in arm B integrates the members (CLAIM-327’s “Not that the variable individuates a collective”).
Note of 2026-10-27: Flack’s stipulation, and arm B
Flack 2017 (NOTE-1216) agrees with arm A: a slow variable no one tunes to gives no downward causation. Arm B, read and rewritten but fast, runs against her stipulation that the variable be slower than the behaviour producing it (§3), so arm B contradicts a stipulation, not a finding. Her reason for the stipulation is that tuning to an estimate needs a stable target, and that reason does not reach arm B, whose members read the variable directly with no estimation step. Arm C is not addressed.