Exact completeness and predictive-identity results

Appendix A. Content summaries and same-occurrence completeness

An inventory can preserve some content relations while losing which occurrence supplied them. The finite result makes that loss exact.

Let $V$ be a finite comparison universe of nominated phenomenal constituent tokens. Let $C(e)\subseteq V$ be the nonempty nominated contents of occurrence $e$. This representation abstracts from the organization and modes in which those constituents occur. Define the unindexed existential summary by

$$\mathcal J(K)\ \Longleftrightarrow\ \exists e\;K\subseteq C(e), \qquad \varnothing\ne K\subseteq V. \tag{6}$$

For $\mathcal F=\{C(e)\}$, its true queries form the downward closure $\mathcal J=\downarrow\mathcal F$ on nonempty subsets. A true query tells us that some occurrence contains all the queried constituents. It does not name the occurrence or give its multiplicity.

Proposition A.1: finite inverse family

If $V$ is finite, the content-set families inducing exactly $\mathcal J$ are precisely the families $\mathcal G$ satisfying

$$\operatorname{Max}(\mathcal J)\subseteq\mathcal G\subseteq\mathcal J. \tag{7}$$

Their number is

$$2^{|\mathcal J|-|\operatorname{Max}(\mathcal J)|}.$$

Proof. Every member of $\mathcal G$ witnesses itself, so it belongs to $\mathcal J$. Each maximal $A\in\mathcal J$ must lie inside some $B\in\mathcal G$. Maximality forces $B=A$, so every maximal member must be included. Conversely, including every maximal member recovers all of $\mathcal J$, because each member of a finite family lies below a maximal member. Restricting included sets to members of $\mathcal J$ introduces nothing outside it. Each nonmaximal member can independently be included or omitted, giving the displayed number.

For maximal witnessed contents $\{a,b,c\}$ and $\{c,d\}$, there are nine nonempty true queries and two maximal members. There are therefore $2^{9-2}=128$ compatible content-set families. Overlap at $c$ provides no witness for $\{a,b,c,d\}$. The summary also discards multiplicity when distinct occurrences have the same nominated content set. Organized realization of a field therefore supplies information that this existential summary omits.

For a fixed independently identified occurrence, define $\mathcal J_e(K)$ by $K\subseteq C(e)$. Indexing the occurrence permits a stronger, exact relative certificate.

Proposition A.2: same-occurrence extent

If $\mathcal J_e(B)$ holds and $\neg\mathcal J_e(B\cup\{x\})$ holds for every $x\in V\setminus B$, then $C(e)=B$ relative to $V$.

Proof. The positive premise gives $B\subseteq C(e)$. Any $x\in C(e)\setminus B$ would make $B\cup\{x\}\subseteq C(e)$, contradicting its negative premise. No constituent of $C(e)$ lies outside $B$, so the sets are equal within the comparison universe.

The negative premises require actual warrant. Failure to report a constituent is not their logical equivalent. The certificate concerns the contents of an already fixed occurrence. The enacted account supplies law-relative constitution through the complete unit realization and current incorporation. Neither method replaces relational organization with an unstructured item list.

Appendix B. Compatibility with complete predictive identity

Fix an exact intrinsic carrier $X$ and its full native contract. Let

$$q_\infty:X\longrightarrow Z_\infty$$

be its all-finite-futures predictive quotient, carrying the required labelled operations, records and successor structure.

Proposition B.1: full-A2 compatibility

If full A2 applies to two nominations of the same complete phenomenal organization with the same actual phenomenal point, their complete labelled pointed predictive objects are isomorphic.

If $f:X\to Y$ is asserted to be a fixed phenomenal property under A2 and the enacted correspondence, $f$ must be constant on $q_\infty$ fibres. Equivalently, a function $\bar f$ exists on the attained predictive classes such that

$$f=\bar f\circ q_\infty.$$

Proof. Compose one structural identification with the common phenomenal organization and the inverse of the other. The composition is the required pointed labelled isomorphism. Equal predictive classes receive the same assigned pointed phenomenal organization. Every fixed property of that organization therefore agrees on each fibre. Conversely, fibre constancy makes the definition

$$\bar f\bigl(q_\infty(x)\bigr)=f(x)$$

well-defined on the attained classes, yielding the factorization.

The last statement is a general factorization criterion. Suppose a payload is $v:W\to V$ and the target is $F:W\to P$. Showing that a contextual relation $R$ fails to factor through $v$ does not show that $F$ also fails to factor through it. On binary pairs, take

$$R(a,b)=(a,b),\qquad v(a,b)=a,\qquad F(a,b)=a.$$

The payload omits context $b$ while preserving every $F$ distinction. The relevant obstruction is an equal-$v$, different-$F$ pair. In an A2 application, the compatibility demand applies to properties actually asserted to be phenomenal under the correspondence. Naming an arbitrary operational flag a phenomenal predicate supplies no such assertion's warrant.

References

The references below retain the sources and editions identified in the paper. The original publication is Whole-Field Constitution as Enacted Organization, DOI 10.5281/zenodo.23608069, Jeremy Rodgers, Independent Researcher, 23 October 2026.

1. Tim Bayne and David J. Chalmers. “What is the unity of consciousness?” In Axel Cleeremans, editor, The Unity of Consciousness: Binding, Integration, and Dissociation, pp. 23–58. Oxford University Press, 2003. Author text.
2. Jeremy Rodgers (2026a). Constituted Agency: Self-Authorship, Local Control, and Conscious Participation. Author manuscript dated 22 October 2026. DOI 10.5281/zenodo.23581587. Complete web treatment.
3. Jeremy Rodgers (2026b). Block Universe and Shadow Theory: Source, History, and Enacted Temporal Reality. Version 1.0, 23 October 2026. DOI 10.5281/zenodo.23604713. Complete web treatment.
4. Jeremy Rodgers (2026c). Shadow Theory and Consciousness: Awareness, Perspectival Realization, and the Source-to-Experience Problem. SPC-2, version 2 publication edition, 20 September 2026, Chapters 16–17. DOI 10.5281/zenodo.22853774. Consciousness programme.
5. Jeremy Rodgers (2026d). Experience as Enacted Scaffolding: A Relational Constitutive Account within Shadow Theory. Unpublished manuscript, 23 October 2026. The paper cites its identity and unit rule at Sections 7.4–7.7; no published DOI is asserted.
6. Jeremy Rodgers (2026e). Experienced Temporal Passage: Enacted Scaffolding and the Organization of Succession. Author manuscript dated 23 October 2026. DOI 10.5281/zenodo.23591626. Complete web treatment.