Identifying Binary Realizations from Intervention Laws: Certificates, Recoding Obstructions, and a Bounded SPC-2/IIT Comparison

Can intervention-response laws identify a system’s binary causal decomposition when its coordinate labels are withheld? This paper develops a conditional identification method and examines its consequences for a frozen comparison between SPC-2 and Integrated Information Theory (IIT). For a completely distinguished finite state space, strictly positive response distributions that factor in an unknown binary coordinate chart identify that chart up to coordinate permutation and independent bit complementation when the uniformly centered log-probability matrix has full component rank. A projector onto its row space reconstructs Hamming adjacency. An explicit sufficient spectral-error certificate establishes stability under bounded perturbations. A synthetic five-register demonstration applies the method to an original implementation, a nonlocal recoding, and a signed-coordinate control. Each generator specifies independent-flip responses in its own modeled storage chart while withholding that chart from the estimator. Reconstruction recovers the respective binary structures. Combined with unchanged service and realization contracts, these structures recover the inherited SPC-2 two-versus-one assignments despite identical complete decoded service behavior. The paper also characterizes affine transport of product distributions, including deterministic and fair coordinates. For the frozen nonlocal recoding, no fully supported, exactly conjugate native product-kernel family approaches the deterministic endpoint, although sparse common stochastic families exist. An official PyPhi computation examines all matched states, both complete presets, and a frozen domain of spatial pair macrograins. Within that bounded search domain, the inherited positive-complex comparison persists. The response method identifies a candidate binary microchart; it does not select an intrinsic macrograin and is potentially applicable to different constitutive theories. The results are finite specializations and applications within established identifiable-component and causal-abstraction frameworks. The synthetic demonstration tests inference under a declared response model. It does not establish physical conformance of that model, derive primitive locality, or empirically validate consciousness attribution. AI-assisted checks are disclosed; independent human mathematical review remains outstanding. Author: Jeremy Rodgers, Independent ResearcherWebsite: https://everythingequation.comPaper 4 DOI: 10.5281/zenodo.23075828 Related works:Shadow Theory and Consciousness, Version 2: https://doi.org/10.5281/zenodo.22853774Paper 2 — Relational Boundaries and Awareness Localization: Robustness, Composition, and Identification Limits: https://doi.org/10.5281/zenodo.23075822Paper 3 — Learning Effective Interfaces from Opaque Stochastic Systems: Capacity, Selection, and Validation Limits: https://doi.org/10.5281/zenodo.23075824