The escape routes and their boundaries

4 The escape routes and their boundaries

Status: Scope of the explicit countermodels.

Theorem 2.7 is an implication in a specified model class, not a census of all physical theories. Its examples establish three separate premise countermodels and one branching alternative outside its single-outcome domain. Multiple routes may be used together.

Branching.

The explicit state in (C1) has a complete initial ray and deterministic vector evolution. Its branch-weight table is $P_Q$, but assigning a unique actual outcome requires additional structure. The relative-state literature discusses the interpretation of these weights [30, 31, 32]. No such interpretation can be smuggled into the scalar outcome functions of (D). Accordingly (C1) refutes an unrestricted single-outcome ontology claim, not Lemma 2.2.

Fundamental chance.

Model (C2) keeps the initial state complete while outcomes are intrinsically stochastic. This suffices as a logical countermodel; a particular collapse dynamics is not required. The preparation-contextuality obstruction in Proposition 2.9 concerns the density matrix as a readout of preparation laws, not incompleteness of each pure-state ontic value [2, 12].

Measurement dependence.

Model (C3) preserves local deterministic responses while correlating source and settings. Specific independence hypotheses can have empirical consequences, but independence against an unrestricted omitted variable is not certified merely by observing random-looking settings. Brans-type constructions show the logical role of the premise [35]. No stronger impossibility of testing every specified setting mechanism is asserted.

Accessible nonlocality.

Model (C4) has no-signalling unconditioned marginals and signalling responses conditional on the readable source. This is an explicit counterexample to inferring (NS) for arbitrary side information from ordinary Bell marginals. The transition-set literature and Theorem 12.1 quantify mathematical nonequilibrium reweightings [13]; physically preparing them is a separate premise. The retained finite-pilot setting dependence in Theorem 11.7 is an already specified nonrelativistic constitutive prediction, not a demonstrated spacelike experiment.