# Chapter 18: Completion, invariance, and empirical conservativity

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<a id="ch:completion"></a> 

<a id="section-18-1"></a>

## 18.1 The certified finite domain

<a id="section:completion:a-total-finite-construction"></a> A *certified finite realization* $R^\ast$ has finite physical component and state domains, rational physical instrument kernels, an explicit native time and route doctrine, executable finite recurrence witnesses, physical process provenance, and finitely represented native operating contracts for every candidate core. Each candidate's intrinsic carrier and common typed continuation grammar are generated before qualification. Its native instrument representation is closed, and its all-future predictive equivalence satisfies the checked congruence condition of [Assumption 16.4](/consciousness/monograph/endogenous-predictive-structure-across-horizons#ass:spc2-congruence). Nominated experience-facing report channels have rational physical kernels and satisfy the grounding criterion at every supplied channel context.

These are mathematical and physical input conditions. They do not assert that a candidate is conscious. A failure of closure, admissibility, or grounding makes the proposed realization uncertified for this theorem; it does not force a negative consciousness verdict. This distinguishes an incomplete model from a completed model that assigns no perspective. General hidden or history-dependent vessels may require a larger representation or another theorem.



**Theorem 18.1 (Constitutive completion of a certified finite realization).**

<a id="thm:completion"></a> Given a certified finite $R^\ast$ and A0–A3, the following assignments are determined on every supplied native time cell and along its supplied provenance history: 

1. (i) every physical component belongs to a unique maximal internally typed joint-influence SCC; qualification is a total predicate, and the qualifying components form a disjoint subject partition of their union;

2. (ii) core occurrences have a unique episode partition under the continuation law, with specified split, merger, and loss-of-qualification boundaries;

3. (iii) every admitted occurrence has a complete endogenous phenomenal structure and its actual point, unique up to the declared whole-structure gauge;

4. (iv) every nominated grounded report probability is induced by its physical channel, with no independently adjustable phenomenal decoder;

5. (v) the construction is covariant under the declared realization equivalences and independent of analyst-selected test subsets, forecast truncations, and observational windows that do not change $R^\ast$;

6. (vi) all finite presentations needed for these assignments, and all individual finite transcript probabilities, are exactly evaluable from the rational data. The claim does not include general exact optimization of an infinite-horizon contrast supremum.

 No further manifestation map is left unspecified after the realization, selection doctrine, and named constitutive laws have been fixed. 

 

**Proof.**

The finite physical comparisons determine the minimal joint-output dependence relation and its internally typed projection. Strongly connected components are a unique graph partition. The candidate construction produces the intrinsic carrier, grammar, and witness data from physical components without assuming qualification. Finite witness evaluation and the all-future equivalence calculation determine the qualification predicate; A1 supplies its experiential attribution.

For each candidate, finite rational observable-span closure decides equality of all native finite transcript laws. The nested finite quotients realize their inverse limit by [Proposition 16.2](/consciousness/monograph/endogenous-predictive-structure-across-horizons#prop:spc2-projective). Congruence is checked by finite sums over the resulting classes. The quotient-instrument theorem then supplies actual-class transitions and a finite presentation of every continuation law. A2 copies this complete structure and sends the realized intrinsic state to its quotient point. Two such copies are isomorphic through their identifications with the same physical object. This is the elementary structural-copy part of the proof; the well-defined physical construction is the substantive prerequisite.

The full core-provenance graph and the degree-one rule give the episode paths by [Proposition 17.5](/consciousness/monograph/the-shadow-psychophysical-constitution#prop:lineage). Channel grounding gives a unique report kernel on each realized predictive class at fixed external context. Transporting it through the A2 identification supplies the phenomenal expression with no new choice. The next theorem proves covariance. Restricting which data an analyst inspects leaves all these physical and constitutive data unchanged. Every operation on the finite presentation uses exact rational tests, finite graph algorithms, or finite linear algebra. Individual finite protocol probabilities are finite products and sums of the supplied instrument entries. 

□



The subject partition in (i) covers admitted core support, not all matter: components outside qualifying cores are not assigned additional subjects. A0 has no numerical role in these computations. It gives the resulting A1 assignments their proposed source-aspect meaning. The theorem establishes conditional determinacy, not the truth of the psychophysical identifications or a derivation of awareness from nonexperiential premises.

The same realization must support the whole continuation grammar. A collection of unrelated finite-resource models, one selected anew for each horizon, does not satisfy that condition. A finite bank may instead be included explicitly in the state and lead to an absorbing exhausted protocol type. That type is assessed on its own current return capabilities, not on a live type that is no longer physically available. Its all-finite transcript family remains well defined, while its physical life is finite. No infinite resource is obtained from the inverse-limit notation.



<a id="section-18-2"></a>

## 18.2 Redescription and operational presentation

<a id="section:completion:redescription-covariance"></a> A realization isomorphism transports physical states, component incidence and provenance, native cells, preparation and intervention domains, minimal joint-dependence witnesses, internal/external typing, candidate construction, native instruments and grammar, actual points, recurrence certificates, boundary contracts, and report channels. It is a bijection preserving this whole structure, not merely a permutation of the rows of one transition matrix.



**Theorem 18.2 (Realization covariance).**

<a id="thm:covariance"></a> Such an isomorphism carries qualifying cores, their all-finite endogenous objects, episode partitions, and grounded report laws to their corresponding objects. It preserves the assigned structure and point up to the declared gauge. 

 

**Proof.**

A transported intervention comparison has the same joint probabilities on corresponding output subsets, so minimal distinguishing subsets and their typed projected edges correspond. Graph components and physically executable witnesses therefore correspond. Native protocol trees transport with the same laws; equality at every finite horizon, and hence all-future equivalence, is preserved. Sums over corresponding predictive fibres preserve congruence and intertwine the quotient instruments. The provenance graph and its incoming and outgoing degrees are preserved, so the episode paths correspond. Report descent commutes with the same transport. A1–A3 consequently give the corresponding assignments. 

□



A modest extension concerns redundant *descriptions* of tests. Suppose two finite presentations implement exactly the same labeled native instruments, and their executable policies have mutually translating presentations with the same transcript laws, stopping rules, resource use, and admissible continuation. Identifying these presentations yields the same predictive equivalence and instrument object. Likewise duplicating a calibrated output label as a bijective encoding changes its presentation, not its content. These statements require actual equivalence of the specified native structure. Adding an unavailable control, inserting a new record, or changing physical timing is not a redundant presentation.

 

| Transformation | Scope of invariance |
| --- | --- |
| Coordinate or calibrated label change | Covariant when every affected physical and interpretive datum is transported. |
| Source-preserving realization isomorphism | Covariant by the theorem. |
| Redundant grammar presentation | Invariant when the same executable native instruments and continuations are represented. |
| Analyst test subset or forecast cutoff | Changes available evidence or an approximation, not the constitutional assignment. |
| State refinement or coarse-graining | Not generally invariant. Requires a separately verified equivalence preserving the joint graph, native quotient instruments, qualification and provenance. |
| Component regrouping or port retyping | Can change subjects; generally a different realization doctrine. |
| Native-time resampling | Can hide cycles, splits or gaps; generally changes the realization. Pure subdivisions preserve episodes only under the condition in [Section 17.4](/consciousness/monograph/the-shadow-psychophysical-constitution#sec:lineage-process). |
| Additional physical intervention grammar | Can change endogenous organization when genuinely implemented; analyst wish or apparatus readout alone does not add it. |

 



<a id="section-18-3"></a>

## 18.3 Independent composition

<a id="section:completion:independent-composition"></a> For two closed native systems with independent preparations, separate resources, product instruments, no internal cross-dependence, and a product grammar containing their separate tests, all-future equivalence of pairs is the product of the separate equivalences. Equal pairs of classes give equal laws by induction over any admitted common policy tree; separate tests distinguish unequal pairs. If each local quotient is congruent, summing the product instruments over pairs of classes proves product congruence. On a restricted jointly reachable domain only the corresponding realized subset is present.

A common adaptive controller can correlate transcript outputs by choosing one system's later input from the other's earlier record. The transcript distribution need not factor under such a policy. This does not invalidate the conditional product-instrument statement. If that controller becomes part of the internal organization or changes its interfaces, the graph and qualification must be recomputed. A mathematical product by itself creates no third subject beyond the qualifying cores.



<a id="section-18-4"></a>

## 18.4 Physical conservativity

<a id="section:completion:physical-conservativity"></a> 

**Theorem 18.3 (Conservativity of the aspect interpretation).**

<a id="thm:conservative"></a> Fix the complete physical realization, preparation, and all physical operation and report kernels. Adding A0–A3 and grounded phenomenal interpretations without changing those kernels leaves the probability of every finite adaptive physical transcript unchanged. 

 

**Proof.**

The initial law is the same. At a policy node the next action depends on the same recorded history and the conditional physical instrument is unchanged. Induction over the policy tree gives identical finite joint laws. The phenomenal variables add an interpretation of the realized organization, not an independent argument of the physical transition kernel. 

□



Grounding removes a free internal decoder, but it creates no physical prediction absent from the supplied realization. A rival accepting exactly the same complete physical transcript laws while denying their experiential interpretation cannot be excluded by those transcripts alone. This underdetermination includes primitive source awareness and any alternative that differs solely by transcript-inert qualitative claims.

Independent experience-facing relations can nevertheless constrain an application. A fixed proposed physical realization can fail to predict the observed records. A1's proposed boundary can conflict with an independently defended subject-individuation judgment. A2 can fail to match independently elicited phenomenal contrasts under a fixed calibration and held-out interventions. Such evidence bears on the realization and the psychophysical identification together; it does not turn a definitional copy into an independent observation of awareness.



<a id="section-18-5"></a>

## 18.5 An aspect and an independently variable force

<a id="section:completion:an-extra-awareness-variable-versus-an-aspect"></a> For comparison, let a complete physical model have state $(x,a)$ and kernel $T_u$. A reduced autonomous law on $x$ for every initial preparation exists exactly when 

$$

\sum_{a'}T_u((x,a),(x',a'))

$$

 is independent of $a$ for every $x,x',u$. Point-mass preparations prove necessity; conditioning proves sufficiency. This is ordinary controlled lumpability. If the condition fails, deleting $a$ leaves an incomplete physical description, regardless of the name assigned to it.

An aspect theory instead identifies experience with an aspect of the same physical process. Independently varying that aspect while fixing its complete realization is not an admitted intervention. The process remains physically causal; the theory does not add a second force. Conservativity establishes equality of the stipulated probabilities and does not, by itself, settle the ontology or every philosophical question about mental causation.