CASE-110: Two members, one total: a coarse-graining by total output preserves an institution's performance, its survival and every prediction it makes, and still merges a state in which a member works within the agreed cap with one in which that member works far past it

The construction

A22: “Construct a small institutional system with two policies that preserve the same aggregate performance statistic but distribute opportunities or burdens differently among members. Demonstrate that the aggregate representation preserves the institutional performance judgment but not the relevant member-level evaluation.” A21 §16 gave the schema (J_inst(a) = J_inst(b) and J_i(a) ≠ J_i(b)). This is the smallest version.

• States. An institution has two members. A state is x = (w₁, w₂), the hours each works in a period, with w₁, w₂ ∈ {0, 1, …, 12}. Each hour produces one unit.
• The institution’s coarse description. C(x) = w₁ + w₂, total output.
• The institution’s judgments. Performance is adequate when output reaches the quota: J_inst(x) = [w₁ + w₂ ≥ 12]. The institution survives to the next period exactly when its performance is adequate: S(x) = J_inst(x).
• A member’s claim. The members agreed a cap of eight hours. Member 1’s claim is respected when w₁ ≤ 8: J₁(x) = [w₁ ≤ 8].
• Two policies. a sets x_a = (6, 6). b sets x_b = (10, 2).

What it shows

1. C preserves every institutional judgment. J_inst and S are functions of w₁ + w₂, so each is constant on C’s fibres and factors through C (CLAIM-350). Any prediction of the institution’s future that runs through its output runs through C. By stability, prediction and aggregate performance, C is a complete description of this institution.
2. C merges states the member’s claim distinguishes. C(x_a) = C(x_b) = 12, while J₁(x_a) = 1 and J₁(x_b) = 0. J₁ is not constant on the fibre C = 12, so by CLAIM-350 there is no j with J₁ = j∘C. No judgment made on total output can say whether member 1’s claim was respected.
3. The adequate description is fixed by the target. C′(x) = (w₁ + w₂, [w₁ ≤ 8], [w₂ ≤ 8]) preserves J_inst, S and both members’ claims. The identity preserves everything. So the failure is not that no coarse-graining works. It is that the one chosen because it serves the institution does not serve the member.
4. Nothing at the institution’s level registers the difference. Both policies meet the quota and the institution survives under both. The difference lies in a distinction the institutional description has discarded.

What it does not show

• Which policy is right. That member 1’s claim should prevail is a normative premise built into J₁, not shown. A21: “the evaluative target must not be justified solely by the coarse-graining that preserves it.” The construction does not justify J₁ either.
• That institutions in fact discard such distinctions, or how often.
• A22’s optional extension, a collective judgment rule that changes which evaluations acquire institutional authority. A22 marks it “an interpretive extension concerning institutional legitimacy”, and it is not built here. It is where CLAIM-271 and THEORY-127 would enter.
• Anything a richer institution could not repair. An institution whose survival depends on member 1’s continued membership, and member 1’s on the cap, would make J₁ bear on S. Then C would no longer be sufficient for survival. The construction depends on the claim being idle for the institution’s persistence, which is CLAIM-060’s case.