# Section 1: Introduction

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## 1 Introduction

<a id="sec:introduction"></a> An explicit theory of awareness localization faces two different obligations. It must specify which physical organizations its law selects, and it must justify interpreting those organizations as experiential perspectives. Mathematical determinacy of the first assignment does not discharge the second obligation. Conversely, treating awareness as fundamental does not determine where a local perspective begins or ends.

*Shadow Theory and Consciousness* introduces the Shadow Psychophysical Constitution, SPC-2, to make these commitments explicit <a id="citation-1"></a>[[23](/consciousness/research/paper-2/references#bib-RodgersSPC2)]. Its source-awareness postulate A0 is an ontological interpretation, while A1 supplies perspective admission, A2 supplies phenomenal relational organization, and A3 supplies episode continuation. The constitution is evaluated only after a physical realization and a selection doctrine have been supplied. It therefore already has a localization bridge; the present paper is not an attempt to supply an omitted first bridge. Its subject is the robustness and evidential status of the functional boundary on which that bridge operates.

The audit begins with a specific vulnerability. The published dependence construction retains a joint target only when no proper subset distinguishes a fixed source contrast. This excludes irrelevant target padding and detects correlation-carried information. Yet exact minimality can be lost when a previously uninformative singleton acquires an arbitrarily small effect. The projected edge then disappears although the joint experiment changes only slightly. The problem is not that joint information is unreal. It is that the combinatorial rule can cease to represent it for a reason unrelated to the magnitude of the surviving information.

We replace this information diagnostic by a reconstruction question: after part of a joint observation is removed, can one source-independent stochastic kernel reproduce the complete family of response laws? The relevant tool is directed statistical deficiency, with established roots in comparison of experiments and approximate sufficiency <a id="citation-2"></a>[[6](/consciousness/research/paper-2/references#bib-Blackwell1953), [16](/consciousness/research/paper-2/references#bib-LeCam1964), [24](/consciousness/research/paper-2/references#bib-vanRooyenWilliamson2014)]. The contribution is not a new information measure. It is an exact application to the boundary obstruction and a disciplined integration with native execution and resource-constrained simulation.

That integration matters because three statements are inequivalent: information is present jointly; a physical schedule returns an intervened distinction; and no permitted disconnected implementation reproduces its recorded behavior. A two-stage encoder–decoder can satisfy the first two statements while having zero cut-simulation cost under a specified shared-randomness and local-memory contract. We consequently retain a family of quantitative, schedule-specific candidates rather than forcing a single subject partition.

The next question is compositional. A lower relational organization can serve as a higher-level component only if its effective interface retains the distinctions that the admitted exterior can subsequently use. For finite joint instruments, a strong marked quotient gives an exact construction. We prove causal substitution and minimization-compatible composition, with explicit restrictions on timing, initial correlations, shared events, and resource ownership. We also show why fidelity of the target process alone does not preserve its distance from a class of cut simulators. The comparison class must be transported too.

Finally, the source/readout issue is expressed at the level of complete experiments. An attribute is determined by their exact laws if and only if it is constant on the corresponding operational-equivalence classes. This criterion neither determines the intended phenomenal meaning of a label nor turns exact-law identification into uniform finite-data certification. A separate conservative-extension argument identifies the premise needed to connect functional structure to a local phenomenal predicate. Its scope is logical: it does not refute a theory that already supplies such a connection.

The resulting position is deliberately asymmetric. The original A1 remains a coherent conditional constitutive rule, but part of its candidate construction is vulnerable to perturbation and finite certification. The replacement analysis is more robust without being a uniquely selected replacement admission law. This distinction lets the paper retain the philosophical direction of SPC-2 while making its mathematical commitments available to readers who do not accept A0.



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#### Contributions and provenance.

 The principal results are the masking obstruction and exact reconstruction optimum, the separation of information, execution and cut simulation, and the joint conditions for compositional transfer of the resulting diagnostics. Statistical sufficiency, elementary metric inequalities, partition refinement and operational quotient factorization are established tools or direct constructions; their application here is stated without historical-priority claims. All general results below have analytical proofs under explicit hypotheses. Finite arithmetic checks are supplementary, and independent external proof review remains outstanding.



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#### Organization.

 [Section 2](/consciousness/research/paper-2/the-inherited-realization-and-finite-operational-setting#sec:setting), [Section 3](/consciousness/research/paper-2/minimal-target-masking#sec:masking) reconstruct the relevant inherited setting and the masking failure. [Section 4](/consciousness/research/paper-2/robust-reconstruction-of-joint-experiments#sec:reconstruction), [Section 5](/consciousness/research/paper-2/native-execution-and-resource-relative-severability#sec:execution) develop the replacement diagnostics. [Section 6](/consciousness/research/paper-2/relational-encapsulation-through-a-causal-interface#sec:composition), [Section 7](/consciousness/research/paper-2/approximation-and-preservation-of-comparison-meaning#sec:approximation) establish exact and approximate encapsulation, including the comparison-class obligation. [Section 8](/consciousness/research/paper-2/operational-identification-and-its-ceiling#sec:identification), [Section 9](/consciousness/research/paper-2/what-a-phenomenal-bridge-adds#sec:bridge) separate operational identification from phenomenal interpretation. [Section 10](/consciousness/research/paper-2/consequences-for-the-spc-2-boundary#sec:consequences), [Section 11](/consciousness/research/paper-2/limitations-and-verification-status#sec:limits) state the implications and limitations for SPC-2. The appendices supply calculation, composition and provenance details, including selected relational-state constructions rather than the full history of the research programme.