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Adjacency Theory markAdjacency Theory

Phase 18 · joint incompatibility

The Joint Wall

AT-0143 → AT-0150. Phase 17 ranked the walls. This phase asks a harder question: what if there is no single wall? Individually reasonable requirements can be mutually incompatible, and then loosening any one of them changes nothing at all.

The failure mode this phase is built on

F₁ ≠ ∅, F₂ ≠ ∅, F₃ ≠ ∅, F₄ ≠ ∅ and F₁∩F₂∩F₃ = ∅ ⟹ no single relaxation reopens F₁∩F₂∩F₃∩F₄
An empty intersection does not imply a dominant blocker. It can be a property of the set as a whole.

Sometimes no single rule is the problem. The rules can be individually reasonable but mutually incompatible. Each one alone leaves plenty of room; together they leave none, and loosening any one of them still leaves none.

RECORDED TOY FEASIBILITY BENCHMARK, NOT PHYSICS. Seed 143, generic coupled toy system with 12 relational dimensions, 8 identity responses and 4 invariant responses. Among roughly 100,000 sampled positive candidate directions, no point satisfied progress ≥ 0.7, identity damage ≤ 1.0, invariant drift ≤ 0.8 and locality ≤ 4 simultaneously. The best minimax compromise still measured progress ≈ 0.416, identity damage ≈ 1.379, invariant drift ≈ 1.093, locality = 5, with a normalised worst violation of ≈ 0.405. The lesson is joint incompatibility, not a single dominant blocker.

Model controls

The joint wall

Running the feasibility sweep…

Running AT-0143 → AT-0150…