The question behind the coordinates

1 Question, scope, and relation to previous work

A complete behavioral description need not specify which changes are local to one physical storage degree of freedom. This distinction matters whenever a theory maps causal organization to a constitutive assignment. The central question here is not whether arbitrary state relabelings preserve a fixed model. It is whether additional evidence, short of a labeled netlist or a complete labeled overwrite algebra, can identify the decomposition used by that model.

SPC-2 is a candidate finite psychophysical constitution conditional on a certified realization. A1 admits recurrent organizations satisfying the native covering-return and predictive-nontriviality conditions; A2 assigns the complete endogenous predictive organization; A3 specifies nonbranching provenance continuation. Its awareness interpretation and constitutive axioms are not inferred from the calculations in this paper [1, 2]. Paper 3 concerns recovering effective stochastic interfaces from restricted data; it does not select a physical primitive decomposition [3].

FRD-1 supplied a finite logical realization doctrine and preserved both stable and ambiguous cases. FRD-2 then proved register-local assignment covariance and constructed a recoding obstruction [4, 5]. Those results already establish that signed-coordinate changes preserving the declared intervention algebra transport the assignment. Reproving this covariance would not answer the present identification question.

Our contribution is a finite identification-and-comparison construction. Response distributions replace supplied coordinate labels as chart evidence, under a declared product-response model. A projector reconstructs a binary cube and provides a quantitative sufficient stability condition. The executed application is a synthetic demonstration: each generator specifies independent flips in its own modeled storage chart, and that chart is withheld from the estimator. The reconstruction tests inference from the response laws, not whether nature realizes the chosen noise model. We then classify admissible common stochastic laws and execute a bounded official IIT grain comparison. The general strategy has substantial antecedents in conditional nonlinear independent-component analysis (ICA), finite-alphabet ICA, and causal abstraction [7, 8, 9, 10, 11, 12]; the result-level attribution is made explicit in Section 7. Physical conformance is a future obligation, not an outcome of this computation.

Three different equivalence questions.

A coordinate change transports all parts of one realization. A different storage implementation can instead have different elementary interventions while preserving decoded behavior. Selecting a description of one fixed device is a third problem. Our results do not identify these questions. In particular, a theory's different assignments to genuinely different causal implementations are not, by themselves, evidence of inconsistency.

Microchart identification is not macrograin selection.

A microchart is a bijective binary coordinate description within the response-model class. IIT's intrinsic-unit construction instead evaluates admissible units and grains on a supplied causal substrate [15]. The two tasks need not return the same kind of object. Independently justified response evidence could help establish the input microdescription for SPC-2, IIT, or another theory; the chart-identification method is not specific to SPC-2. A difference between such a chart and an IIT-selected macrograin would not, by itself, show that IIT selected physically incorrect units. Here the comparative question is only whether the inherited formal differences survive the stated grain-search domain.