Organisms, Viruses, and Computers: Closure, Operadic Autopoiesis, and Thermodynamic Viability

Author: Anonymous

Source: https://abstract-organisms.grok.me/

An (∞)-Topos-Theoretic Separation of Organisms, Viruses, and Computers

Abstract

We formulate a categorical distinction between living organisms, viruses, and computational machines inside an $(\infty)$-topos $\mathcal{E}$. The distinction is not identified with computational behavior, nor with material substrate alone. A physical system carries typed components, internally generated processes, required organizational resources, and interfaces to an environment.

For each organizational mode $a$, a system $X$ has a demand morphism and an endogenous production morphism $$d_a^X \colon D_a(X) \longrightarrow R_a(X), \qquad p_a^X \colon P_a(X) \longrightarrow R_a(X).$$ The system is closed in mode $a$ when $\operatorname{im}(d_a^X) \le \operatorname{im}(p_a^X)$ in the subobject lattice of $R_a(X)$. Everything required as an organizational enabling condition is then generated by processes internal to the system. Matter and energy may still enter through explicitly external channels.

This yields three notions: autonomous closure, host-relative closure, and externally maintained machinery. A cellular organism is modeled as organizationally closed over an appropriate signature. A virus is not autonomously closed, but becomes closed after supplementation by host processes. An ordinary computer is not closed for fabrication, repair, and physical reproduction, though it may be closed for a restricted signature of computational operations.

These closure properties are homotopy invariant, preserved by equivalences and by geometric inverse-image functors, and satisfy descent. They are therefore genuine obstructions to equivalence. A material-support invariant is defined separately and shown to be logically independent of organizational closure. The analogy between a virus and a program arises only after a non-conservative forgetful functor erases structure needed to compare either of them with a cell.

1. Ambient (∞)-topos

Let $\mathcal{E}$ be an $(\infty)$-topos. One may think of $\mathcal{E}$ as a category of homotopy-coherent sheaves of physical states, processes, or observations, over a site of spacetime regions, experimental contexts, or interaction contexts.

An $(\infty)$-topos, rather than an ordinary category, supplies the following.

• arbitrary higher homotopies between realizations
• descent and locality
• an internal intuitionistic higher-order logic
• effective epimorphism–monomorphism factorizations
• a well-behaved theory of subobjects and images

The properties used below are the standard ones of $(\infty)$-topoi in the sense of Higher Topos Theory.

Definition 1.1 Subobject

For $Y \in \mathcal{E}$, a subobject of $Y$ is a $(-1)$-truncated morphism $A \hookrightarrow Y$. Equivalence classes of such morphisms form a partially ordered set $\operatorname{Sub}_{\mathcal{E}}(Y)$. The order is $A \le B$ if and only if $A \hookrightarrow Y$ factors through $B \hookrightarrow Y$. Internally, $A \le B$ means that every element satisfying $A$ also satisfies $B$.

Definition 1.2 Image

Every morphism $f \colon X \to Y$ admits a canonical factorization $$X \twoheadrightarrow \operatorname{im}(f) \hookrightarrow Y,$$ in which the first factor is $(-1)$-connected — equivalently, an effective epimorphism — and the second is $(-1)$-truncated. This is the $(-1)$-connected/$(-1)$-truncated factorization system of an $(\infty)$-topos. Hence $\operatorname{im}(f) \in \operatorname{Sub}(Y)$ is well defined up to a contractible space of choices.

2. Typed physical systems

We first encode material composition. Let $K \in \mathcal{E}$ be an object of physical component types. In a sufficiently concrete model, points of $K$ might include protein, lipid, DNA, silicon, copper, transistor, ribosome, and so on. No discreteness assumption is required.

Definition 2.1 Typed system

A $K$-typed physical system $X$ consists in part of an object $C_X \in \mathcal{E}$ of components together with a typing morphism $\tau_X \colon C_X \to K$. Its material support is the image $$\operatorname{Supp}(X) := \operatorname{im}(\tau_X) \hookrightarrow K.$$ Thus $\operatorname{Supp}(X)$ is the subobject of component-types actually instantiated by $X$.

Definition 2.2 Type-preserving equivalence

An equivalence $X \simeq Y$ of $K$-typed systems is required to contain an equivalence $f \colon C_X \xrightarrow{\sim} C_Y$ and a homotopy-commutative square $$\begin{CD} C_X @>{f}>> C_Y \\ @V{\tau_X}VV @VV{\tau_Y}V \\ K @= K, \end{CD}$$ together with the analogous coherent equivalences on the remaining structure. Commutativity says $\tau_X \simeq \tau_Y \circ f$.

3. The substrate invariant

Theorem 3.1 Material-support invariance

If $X \simeq Y$ as $K$-typed physical systems, then $\operatorname{Supp}(X) \simeq \operatorname{Supp}(Y)$ in $\operatorname{Sub}(K)$.

Proof.

The equivalence supplies $f \colon C_X \xrightarrow{\sim} C_Y$ with $\tau_X \simeq \tau_Y \circ f$. Homotopic morphisms determine the same image subobject: image depends on the morphism only through its class in the homotopy category, via the canonical $(-1)$-connected/$(-1)$-truncated factorization. Precomposition with an equivalence does not change that image, so $$\operatorname{im}(\tau_X) \simeq \operatorname{im}(\tau_Y \circ f) \simeq \operatorname{im}(\tau_Y).$$ Hence $\operatorname{Supp}(X) \simeq \operatorname{Supp}(Y)$. The general homotopy statement is Lemma 6.1.

Corollary 3.2 Substrate obstruction

If $\operatorname{Supp}(X) \not\simeq \operatorname{Supp}(Y)$, there is no equivalence $X \simeq Y$ in any category whose equivalences preserve the physical typing $\tau \to K$.

Differences of physical composition are therefore a genuine obstruction, provided composition is among the structure being preserved. A category that deliberately forgets $\tau$ cannot, by construction, detect the distinction.

4. Organizational signatures

Material composition is not the only relevant invariant. Let $\Sigma$ be a collection of organizational modes. Typical elements might be $$\Sigma = \{ \mathrm{boundary},\ \mathrm{catalysis},\ \mathrm{repair},\ \mathrm{translation},\ \mathrm{replication},\ \mathrm{regulation},\ \ldots \}.$$ The claim is not that every organism instantiates exactly this list. The signature $\Sigma$ records which structural capacities a given theory treats as organizationally essential.

Generic matter and free energy need not belong to $\Sigma$. Living systems are thermodynamically open. The closure defined here is not closure to matter and energy. It is closer to Rosen’s closure to efficient causation, and to the organizational closure of Maturana and Varela. Maturana characterized autopoietic organization as a network of component-producing processes that recursively regenerates the network and its boundary. Rosen’s relational biology likewise separates material flows from internally regenerated organizational causes.

5. Organizational resource data

Definition 5.1 Σ-system

A $\Sigma$-system $X$ consists, for each $a \in \Sigma$, of an organizational-resource object $R_a(X)$, a demand morphism $$d_a^X \colon D_a(X) \longrightarrow R_a(X),$$ and an endogenous-production morphism $$p_a^X \colon P_a(X) \longrightarrow R_a(X).$$ Here $D_a(X)$ enumerates organizational enablers required by $X$, and $P_a(X)$ enumerates organizational enablers produced by processes belonging to $X$. Write $$\operatorname{Req}_a(X) := \operatorname{im}(d_a^X), \qquad \operatorname{End}_a(X) := \operatorname{im}(p_a^X).$$ Both are subobjects of $R_a(X)$.

Definition 5.2 Closure in a mode

The system $X$ is closed in mode $a$ when $\operatorname{Req}_a(X) \le \operatorname{End}_a(X)$. Equivalently, the demand image factors through the production image: $$\operatorname{Req}_a(X) \longrightarrow \operatorname{End}_a(X) \hookrightarrow R_a(X).$$ Write $\mathrm{Cl}_a(X)$ for this proposition.

Definition 5.3 Σ-closure

The system is $\Sigma$-closed when it is closed in every mode: $$\mathrm{Cl}_{\Sigma}(X) \iff \bigwedge_{a \in \Sigma} \mathrm{Cl}_a(X) \iff \bigwedge_{a \in \Sigma} \bigl( \operatorname{Req}_a(X) \le \operatorname{End}_a(X) \bigr).$$ This is an internal proposition of the theory.

6. Homotopy invariance

Lemma 6.1 Homotopic maps have the same image

If $d \simeq d' \colon D \to R$, then $\operatorname{im}(d) \simeq \operatorname{im}(d')$. The same holds for production maps.

Proof.

Image is determined by the canonical $(-1)$-connected/$(-1)$-truncated factorization of the morphism represented in the homotopy category. Homotopic morphisms define equivalent objects of the arrow $(\infty)$-category $\operatorname{Fun}(\Delta^1, \mathcal{E})$, and therefore equivalent image factorizations. Their image subobjects coincide.

Corollary 6.2 Closure is not presentation-dependent

The truth value of $\mathrm{Cl}_a(X)$ is invariant under all coherent higher homotopies in the presentation of $X$. The closure invariant is $(\infty)$-categorical.

7. Closure is preserved by equivalence

Theorem 7.1 Organizational closure is invariant

Let $X \simeq Y$ be an equivalence of $\Sigma$-systems preserving the organizational-resource diagrams. Then $\mathrm{Cl}_a(X)$ if and only if $\mathrm{Cl}_a(Y)$, for every $a \in \Sigma$. Consequently $\mathrm{Cl}_{\Sigma}(X)$ if and only if $\mathrm{Cl}_{\Sigma}(Y)$.

Proof.

A structure-preserving equivalence supplies $R_a(X) \simeq R_a(Y)$, $D_a(X) \simeq D_a(Y)$, and $P_a(X) \simeq P_a(Y)$, under which the demand and production diagrams agree up to coherent homotopy. By Lemma 6.1 the image subobjects correspond: $$\operatorname{Req}_a(X) \simeq \operatorname{Req}_a(Y), \qquad \operatorname{End}_a(X) \simeq \operatorname{End}_a(Y).$$ An equivalence induces an isomorphism of the corresponding subobject posets and preserves their order. Therefore $\operatorname{Req}_a(X) \le \operatorname{End}_a(X)$ if and only if $\operatorname{Req}_a(Y) \le \operatorname{End}_a(Y)$.

Corollary 7.2 Organizational obstruction

If $\mathrm{Cl}_a(X)$ and $\neg \mathrm{Cl}_a(Y)$ for some mode $a$, then $X \not\simeq Y$ as $\Sigma$-systems.

This is the basic separation theorem. Equivalence cannot cross a difference of closure.

8. External supplementation

A virus is not handled by merely changing the context in which it is observed. The host contributes machinery. Base change and supplementation are different operations.

Suppose an external system $H$ supplies additional $a$-resources along $q_a^H \colon Q_a(H) \to R_a(X)$. Form the combined supply $$P_a(X) \sqcup Q_a(H) \xrightarrow{\,[p_a^X,\, q_a^H]\,} R_a(X).$$

Definition 8.1 Relative closure

The system $X$ is closed in mode $a$ relative to $H$ when $$\operatorname{Req}_a(X) \le \operatorname{im}\bigl([p_a^X, q_a^H]\bigr).$$ Write $\mathrm{Cl}_a(X \mid H)$.

Definition 8.2 Strict external dependence

The system $X$ is strictly dependent on $H$ in mode $a$ when $\neg \mathrm{Cl}_a(X)$ but $\mathrm{Cl}_a(X \mid H)$. Thus $X$ cannot generate all of its required organizational resources, while the coupled system can.

9. Image of combined production

Lemma 9.1 Image of a copairing

Let $p \colon P \to R$ and $q \colon Q \to R$. Then $$\operatorname{im}([p, q]) = \operatorname{im}(p) \vee \operatorname{im}(q)$$ in $\operatorname{Sub}(R)$.

Proof.

Both $p$ and $q$ factor through $\operatorname{im}([p, q])$, so $\operatorname{im}(p) \le \operatorname{im}([p, q])$ and $\operatorname{im}(q) \le \operatorname{im}([p, q])$. Hence their join is contained in the image of the copairing.

Conversely, $[p, q]$ factors through every subobject through which both $p$ and $q$ factor, and in particular through $\operatorname{im}(p) \vee \operatorname{im}(q)$. By minimality of the image, $\operatorname{im}([p, q]) \le \operatorname{im}(p) \vee \operatorname{im}(q)$.

Relative closure may therefore be rewritten $$\operatorname{Req}_a(X) \le \operatorname{End}_a(X) \vee \operatorname{Host}_a(H),$$ where $\operatorname{Host}_a(H) := \operatorname{im}(q_a^H)$.

10. Base change and descent

Let $f^* \colon \mathcal{E} \to \mathcal{F}$ be the inverse-image functor of a geometric morphism of $(\infty)$-topoi. It preserves finite limits and all small colimits.

Lemma 10.1 Inverse image preserves factorization

The functor $f^*$ preserves effective epimorphisms, monomorphisms, and image factorizations.

Proof.

Effective epimorphisms are the morphisms whose Čech nerves are colimit diagrams. Since $f^*$ preserves colimits and finite limits, it sends Čech nerves to Čech nerves, and effective epimorphisms to effective epimorphisms. Left exactness preserves $(-1)$-truncated morphisms, hence monomorphisms. Applying $f^*$ to an effective-epimorphism/monomorphism factorization therefore yields the image factorization after base change.

Theorem 10.2 Geometric invariance

If $\mathrm{Cl}_a(X)$ holds in $\mathcal{E}$, then $\mathrm{Cl}_a(f^* X)$ holds in $\mathcal{F}$.

Proof.

From $\operatorname{Req}_a(X) \le \operatorname{End}_a(X)$, apply $f^*$. Lemma 10.1 gives $$f^*\operatorname{Req}_a(X) \simeq \operatorname{Req}_a(f^* X)$$ and the analogous identification for endogenous production. Since $f^*$ preserves factorization through monomorphisms, $$\operatorname{Req}_a(f^* X) \le \operatorname{End}_a(f^* X).$$

11. Closure satisfies descent

Let $u \colon U \twoheadrightarrow B$ be an effective epimorphism. Pullback induces $u^* \colon \operatorname{Sub}(B) \to \operatorname{Sub}(U)$. Along an effective epimorphism this map is conservative on subobjects: it reflects order.

Theorem 11.1 Local-to-global closure

Suppose a system over $B$ becomes organizationally closed after pullback along an effective epimorphism $u \colon U \twoheadrightarrow B$. Then it was already organizationally closed over $B$.

Proof.

The hypothesis is $u^*\operatorname{Req}_a(X) \le u^*\operatorname{End}_a(X)$. Pullback along an effective epimorphism reflects the order of subobjects, so $\operatorname{Req}_a(X) \le \operatorname{End}_a(X)$. Thus $\mathrm{Cl}_a(X)$. Applying this to every $a \in \Sigma$ yields $\mathrm{Cl}_{\Sigma}(X)$.

Corollary 11.2 A host is not a local context

Suppose $\neg \mathrm{Cl}_a(V)$. There is no effective-epimorphic observational cover $U \twoheadrightarrow 1$ such that mere restriction to $U$ makes $V$ closed. If $\mathrm{Cl}_a(u^* V)$ held, Theorem 11.1 would give $\mathrm{Cl}_a(V)$.

Host dependence cannot be represented by localization or by a change of observational context. The host must contribute additional organizational morphisms $q_a^H \colon Q_a(H) \to R_a(V)$. Restriction and coupling are different.

12. Axiomatized cells

Let $\Sigma_{\mathrm{org}}$ contain modes such as boundary, catalysis, repair, biosynthesis, and replication.

Definition 12.1 Autopoietic Σ-object

An autopoietic $\Sigma$-object is a system $B$ with $\mathrm{Cl}_{\Sigma_{\mathrm{org}}}(B)$. External material and energy maps $M_{\mathrm{ext}} \to B$ are explicitly permitted. The condition says only that the organizational machinery named by $\Sigma_{\mathrm{org}}$ is internally regenerated.

This is not thermodynamic closure. It formalizes the core of autopoiesis and of closure to efficient causation. A single-celled organism is then modeled as an object $$B \in \mathbf{Aut}_{\Sigma}(\mathcal{E}) \subseteq \mathbf{Sys}_{\Sigma}(\mathcal{E}).$$ That is an empirical modeling hypothesis, not a theorem of pure mathematics.

13. Axiomatized viruses

Viral replication depends on host-cell machinery: on host metabolism, translation, membranes, and biosynthetic resources. Let $V$ be a viral system and $H$ a host.

Definition 13.1 Host-dependent replicator

The virus $V$ is a host-dependent replicator if for some $a \in \Sigma_{\mathrm{rep}}$, $$\neg \mathrm{Cl}_a(V) \qquad \text{and} \qquad \mathrm{Cl}_a(V \mid H).$$ Equivalently, $\operatorname{Req}_a(V) \nleq \operatorname{End}_a(V)$, while $$\operatorname{Req}_a(V) \le \operatorname{End}_a(V) \vee \operatorname{Host}_a(H).$$

The viral replication cycle closes only in the coupled virus–host system.

14. Axiomatized conventional computers

Let $C$ be an ordinary contemporary computer. Several signatures are available. For the restricted signature of instruction execution, one may have $\mathrm{Cl}_{\Sigma_{\mathrm{exec}}}(C)$: a running machine can carry out many of the operations required for continued execution using its own hardware.

Enlarge the signature to repair, chip fabrication, component replacement, assembly, and physical reproduction. For an ordinary computer, $\neg \mathrm{Cl}_{\Sigma_{\mathrm{phys}}}(C)$. There is instead an industrial environment $I$ such that, for the relevant modes, $$\operatorname{Req}_a(C) \le \operatorname{End}_a(C) \vee \operatorname{Ind}_a(I),$$ so that $\mathrm{Cl}_a(C \mid I)$ may hold while $\mathrm{Cl}_a(C)$ does not.

The causal picture is then approximately $I \longrightarrow C$, where $I$ contains semiconductor fabrication, mining, refining, electricity generation, human maintenance, transport, and manufacturing infrastructure.

15. Cell/computer separation

Theorem 15.1 Cell and computer

Let $B$ be an axiomatized cellular organism and $C$ an ordinary computer. Suppose some mode $a$ satisfies $\mathrm{Cl}_a(B)$ and $\neg \mathrm{Cl}_a(C)$. Then $B \not\simeq C$ as $\Sigma$-systems.

Proof.

An equivalence $B \simeq C$ would imply $\mathrm{Cl}_a(B)$ if and only if $\mathrm{Cl}_a(C)$, by Theorem 7.1.

16. Virus/cell separation

Theorem 16.1 Virus and cell

Suppose $B$ is autonomous in a mode $a$, so $\mathrm{Cl}_a(B)$, while a virus $V$ is strictly host-dependent in that mode: $\neg \mathrm{Cl}_a(V)$, though $\mathrm{Cl}_a(V \mid H)$ for some host. Then $V \not\simeq B$.

Proof.

Immediate from Theorem 7.1. Relative closure of $V$ with $H$ is a property of a different structured system, not of $V$ alone.

A virus may be biologically adjacent to life without inhabiting the same full subcategory as autonomously organized cells.

17. The virus/computer analogy

There is nevertheless a legitimate abstraction under which viruses and programs resemble one another. Let $\mathbf{Sys}_{\Sigma}$ be the category of structured physical systems, and let $$U \colon \mathbf{Sys}_{\Sigma} \longrightarrow \mathbf{Dep}$$ be the forgetful functor that remembers only a replicator or executable specification, a required host, and successful production after host supplementation. It forgets material typing, biochemical organization, evolutionary lineage, boundary production, thermodynamic organization, and most of the physical causal structure.

Under $U$ one may have $U(V) \simeq U(P)$ for a virus $V$ and a program $P$. Both have the shape “replicator plus host yields copies”: $$V + H \longrightarrow V^{\sqcup n}, \qquad P + M \longrightarrow P^{\sqcup n}.$$

18. Non-conservativity of the analogy

Proposition 18.1 The dependency functor need not be conservative

The functor $U \colon \mathbf{Sys}_{\Sigma} \to \mathbf{Dep}$ need not reflect equivalences. That is, $U(X) \simeq U(Y)$ does not imply $X \simeq Y$.

Proof.

Choose $X$ and $Y$ with $U(X) \simeq U(Y)$ but $\operatorname{Supp}(X) \not\simeq \operatorname{Supp}(Y)$. Theorem 3.1 forbids $X \simeq Y$. Since $U$ forgets material support, it may still identify their dependency skeletons.

Corollary 18.2 Analogy is not equivalence

The statement that viruses are like computer programs can be correct after a sufficiently coarse forgetful functor, without entailing that a virus is equivalent to a computer as a physical system, and still less that a computer is equivalent to a cell.

19. The original argument, categorically

Let $B$ be a bacterium, $V$ a virus, and $C$ a computer. At the coarse dependency level one may have $U(V) \simeq U(C)$. In the richer physical category, $V \not\simeq C$ as soon as an invariant differs, and $B \not\simeq V$ when autonomous closure differs. The inference $$U(V) \simeq U(C) \implies B \simeq C$$ is invalid. Equality after $U$ is equality in the quotient semantics $U$ induces. It says nothing about the domain unless $U$ is conservative. Since $U$ is not conservative, the inference fails.

20. The combined invariant

Define $$\mathcal{I}(X) = \bigl( \operatorname{Supp}(X),\ \{\mathrm{Cl}_a(X)\}_{a \in \Sigma} \bigr).$$ This records material support together with the closure profile.

Theorem 20.1 The combined invariant is preserved

If $X \simeq Y$ as fully structured physical systems, then $\mathcal{I}(X) = \mathcal{I}(Y)$.

Proof.

Material support is invariant by Theorem 3.1. Each closure component is invariant by Theorem 7.1.

Corollary 20.2 Two independent obstructions

If $\operatorname{Supp}(X) \not\simeq \operatorname{Supp}(Y)$, or if $\mathrm{Cl}_a(X)$ fails to agree with $\mathrm{Cl}_a(Y)$ for some $a \in \Sigma$, then $X \not\simeq Y$.

There is a substrate obstruction, and there is an organizational-closure obstruction. Neither is a restatement of the other.

21. Substrate and organization are independent

The two invariants should not be collapsed. There can exist systems $X$ and $Y$ with $\operatorname{Supp}(X) \not\simeq \operatorname{Supp}(Y)$ while both are $\Sigma$-closed. A hypothetical self-maintaining artificial organism could instantiate a different substrate and still satisfy the same closure axioms. Conversely, two systems built from similar materials may differ in organizational closure.

Neither sameness of material nor sameness of computation is sufficient for equivalence. The invariant that matters is structured organization.

22. Simulation is weaker than equivalence

Suppose a semantics or simulation functor $S \colon \mathbf{PhysSys} \to \mathbf{Comp}$ sends a biological system $B$ to a computational representation $S(B)$. Even a perfect reproduction of some state-transition structure is only a morphism, or an equivalence, at the semantic level. One cannot infer $B \simeq S(B)$. The two need not even lie in the same category.

If $S$ forgets $\operatorname{Supp}(B)$ or $\mathrm{Cl}_{\Sigma}(B)$, then $S$ fails to be conservative for precisely the structure at issue.

Simulation of dynamics does not imply equivalence of physical organization.

Categorically, this is the difference between preservation of selected observables and equivalence of structured objects.

23. Open-system category theory

The framework is compatible with compositional treatments of open physical and chemical systems. Baez and Pollard model open reaction networks as morphisms, with composition given by joining outputs of one system to inputs of another. Supplementation $X \mid H$ is a higher-categorical analogue of that composition.

The additional structure here is the separation of generic material flows from organizational enabling processes. Only the latter enter the closure invariant.

24. Relation to Rosen

Rosen’s $(M, R)$-systems motivate the same structural separation. A system may stay open to material flow while requiring that its efficient causes be generated inside the organizational network. Later analyses call this closure to efficient causation: the catalysts or processors the system requires are themselves produced within the system.

The inequality $\operatorname{Req}_a(X) \le \operatorname{End}_a(X)$ is an $(\infty)$-topos-theoretic abstraction of that relation. It does not adopt Rosen’s stronger, and controversial, claims about computability. The separation theorem uses only invariance of closure. It does not claim that organisms are noncomputable.

25. Relation to autopoiesis

Maturana’s formulation requires that an autopoietic system recursively produce the components realizing its own network of production, and its boundary. The present formalism encodes this by including modes $a_{\mathrm{boundary}}$ and $a_{\mathrm{production}}$ and requiring both $\mathrm{Cl}_{a_{\mathrm{boundary}}}(X)$ and $\mathrm{Cl}_{a_{\mathrm{production}}}(X)$.

A categorical formalization of autopoiesis is still an active research problem, not a settled theorem. The construction here is an axiomatic formalization, not a claim to be the unique mathematics of life.

26. Main separation theorem

Theorem 26.1 Organism, virus, and computer

Let $B$, $V$, and $C$ in $\mathbf{Sys}_{K,\Sigma}(\mathcal{E})$ represent a cellular organism, a virus, and a conventional computer. Assume:

• Cellular closure: $\mathrm{Cl}_{\Sigma_{\mathrm{org}}}(B)$.
• Viral host dependence: for some $a \in \Sigma_{\mathrm{org}}$, $\neg \mathrm{Cl}_a(V)$, but $\mathrm{Cl}_a(V \mid H)$ for a host $H$.
• Computer industrial dependence: for some $b \in \Sigma_{\mathrm{org}}$, $\neg \mathrm{Cl}_b(C)$, but $\mathrm{Cl}_b(C \mid I)$ for an industrial environment $I$.

Then $B \not\simeq V$ and $B \not\simeq C$. If in addition $\operatorname{Supp}(V) \not\simeq \operatorname{Supp}(C)$, then $V \not\simeq C$ as $K$-typed systems, even when a dependency functor satisfies $U(V) \simeq U(C)$.

Proof.

The first two non-equivalences are Theorem 7.1: $B$ is closed in a mode where $V$ is not, and in a mode where $C$ is not. The third is Theorem 3.1. The possible equivalence $U(V) \simeq U(C)$ does not affect the conclusion, because $U$ is not conservative.

27. Logical form of the conclusion

The invariant takes different values on the three systems: $$\begin{array}{ccccc} B & & V & & C \\ \downarrow \scriptstyle \mathcal{I} & & \downarrow \scriptstyle \mathcal{I} & & \downarrow \scriptstyle \mathcal{I} \\ \mathcal{I}(B) & & \mathcal{I}(V) & & \mathcal{I}(C), \end{array}$$ with $\mathcal{I}(B) \neq \mathcal{I}(V)$ and $\mathcal{I}(B) \neq \mathcal{I}(C)$. A coarse functor may still satisfy $U(V) \simeq U(C)$.

V \not\simeq C

is compatible with a valid analogy $V \sim C$ in a quotient semantics. More importantly, whenever autonomous organizational closure is among the preserved structures,

B \not\simeq V \qquad \text{and} \qquad B \not\simeq C.

28. Final categorical formulation

There is an $(\infty)$-category $\mathbf{Sys}_{K,\Sigma}(\mathcal{E})$ of physically typed, organizationally structured systems, and full subcategories $\mathbf{Aut}_{\Sigma}(\mathcal{E})$, $\mathbf{HostDep}_{\Sigma}(\mathcal{E})$, and $\mathbf{ExtMaint}_{\Sigma}(\mathcal{E})$, cut out respectively by $$\begin{aligned} \mathrm{Cl}_{\Sigma}(X), \\ \neg \mathrm{Cl}_{\Sigma}(X) \land \exists H.\; \mathrm{Cl}_{\Sigma}(X \mid H), \\ \neg \mathrm{Cl}_{\Sigma}(X) \land \exists I.\; \mathrm{Cl}_{\Sigma}(X \mid I). \end{aligned}$$ The second and third predicates have the same logical shape. What differs is the supplement they quantify over — host machinery, or an external maintenance ecology — together with every other invariant the ambient category retains. At a biological scale one may model $B \in \mathbf{Aut}_{\Sigma}$, $V \in \mathbf{HostDep}_{\Sigma}$, and $C \in \mathbf{ExtMaint}_{\Sigma}$. There is then no equivalence $B \simeq V$ or $B \simeq C$ preserving the full $(K, \Sigma)$-structure.

Computational equivalence is strictly coarser than organizational equivalence, which is strictly coarser than equivalence as physically typed systems.

A computer, a virus, and a bacterium become indistinguishable only after one successively forgets the structures that distinguish them. The mathematical question is never simply whether both can be represented computationally.

Which functor is being used, what structure does it forget, and does it reflect equivalences?

Without an affirmative answer to the last question, sameness of representation is not sameness of system. Essay II lifts the image inequality to operadic generation. Essay III requires that generation be thermodynamically feasible and that a viability kernel be inhabited.

References

Jacob Lurie, Higher Topos Theory. Princeton University Press, 2009. Chapters 6–7: $(\infty)$-topoi, effective epimorphisms, truncation, and image factorizations.

Louis Martini and Sebastian Wolf, Internal Higher Topos Theory. The theory of topoi internal to an arbitrary $(\infty)$-topos.

Humberto R. Maturana, “The Organization of the Living: A Theory of the Living Organization.” Autopoietic organization as recursively self-producing organization.

Robert Rosen, Life Itself: A Comprehensive Inquiry Into the Nature, Origin, and Fabrication of Life. Columbia University Press, 1991. Relational biology, machines, and organizational causation.

María Luz Cárdenas, Juan-Carlos Letelier, Claudio Gutierrez, Athel Cornish-Bowden, and Jorge Soto-Andrade, “Closure to efficient causation, computability and artificial life.” Journal of Theoretical Biology 263 (2010), 79–92.

John C. Baez and Blake S. Pollard, “A compositional framework for reaction networks.” Reviews in Mathematical Physics 29 (2017), 1750028.

Tyler B. Jacobson, Melanie M. Callaghan, and Daniel Amador-Noguez, “Hostile takeover: how viruses reprogram prokaryotic metabolism.” Dependence of viral reproduction on host cellular machinery.

Peter D. Nagy and Judit Pogany, “The dependence of viral RNA replication on co-opted host factors.” Nature Reviews Microbiology 10 (2012), 137–149.

Operadic Autopoiesis in an (∞)-Topos

Abstract

Let $\mathcal{E}$ be an $(\infty)$-topos containing an internal symmetric monoidal $(\infty)$-category $\mathbb{P}^{\otimes}$ of open physical processes, and let $\mathbb{O}^{\otimes}$ be the associated internal colored $(\infty)$-operad. Colors are interfaces, resources, and organizational roles. Multimorphisms are composable physical operations.

For a system $X$, write $\mathbb{E}_X^{\otimes} \hookrightarrow \mathbb{O}^{\otimes}$ for the endogenous process operad, $\mathbb{R}_X^{\otimes} \hookrightarrow \mathbb{O}^{\otimes}$ for the required organizational operad, and $\mathbb{G}_X^{\otimes}$ for the least fixed point of an accessible production operator applied to $\mathbb{E}_X$. Operadic autopoiesis is the inclusion $$\mathbb{R}_X^{\otimes} \subseteq \mathbb{G}_X^{\otimes}.$$ Every organizational operation required for the continuation of $X$ is realizable by a recursively self-generated network internal to $X$.

Interaction across a common interface operad $\mathbb{I}^{\otimes}$ is the homotopy pushout $\mathbb{E}_X^{\otimes} \amalg_{\mathbb{I}} \mathbb{E}_H^{\otimes}$. The pushout contains mixed composites, not merely the union of the two operads. A host-dependent replicator then satisfies $\mathbb{R}_V \nsubseteq \mathbb{G}_V$ but $\mathbb{R}_V \subseteq \mathbb{G}_{V \star H}$. A cell satisfies $\mathbb{R}_B \subseteq \mathbb{G}_B$. An ordinary computer, once fabrication, repair, replacement, and physical reproduction are required, satisfies $\mathbb{R}_C \nsubseteq \mathbb{G}_C$.

These properties are invariant under equivalence, local-to-global along effective epimorphisms, and separate autonomous systems from externally dependent ones. A non-conservativity theorem explains why equivalence of dependency patterns, or of computational observations, does not yield equivalence of physical systems.

1. Ambient (∞)-topos

Let $\mathcal{E}$ be an $(\infty)$-topos, interpreted as a generalized space of physical contexts: spacetime regions, experimental conditions, environments, or organizational scales. Systems and processes may vary locally, and they satisfy descent. Internal higher categories in an $(\infty)$-topos may be studied as suitable sheaves of $(\infty)$-categories. The internal theory required below is available in that setting.

2. Internal (∞)-categories

Write $\operatorname{Cat}_{\infty}(\mathcal{E})$ for the $(\infty)$-category of internal $(\infty)$-categories in $\mathcal{E}$. Concretely, an internal $(\infty)$-category may be presented as a complete Segal object $C_{\bullet} \colon \Delta^{\mathrm{op}} \to \mathcal{E}$. Then $C_0$ is the object of objects, $C_1$ the object of morphisms, and $C_n$ the object of coherently composable $n$-fold strings. The Segal equivalences read $$C_n \simeq C_1 \times_{C_0} \cdots \times_{C_0} C_1,$$ and completeness identifies equivalences internally.

3. Internal symmetric monoidal (∞)-categories

Definition 3.1 Internal symmetric monoidal (∞)-category

An internal symmetric monoidal $(\infty)$-category is a commutative algebra $$\mathbb{P}^{\otimes} \in \operatorname{CAlg}\bigl(\operatorname{Cat}_{\infty}(\mathcal{E})\bigr).$$ The multiplication $\otimes \colon \mathbb{P} \times \mathbb{P} \to \mathbb{P}$ is parallel composition. The categorical composition $\circ$ is sequential composition. The symmetric monoidal coherences hold internally in $\mathcal{E}$.

4. Open physical processes

Interpret $\mathbb{P}^{\otimes}$ as the internal symmetric monoidal $(\infty)$-category of open physical processes. Objects are interfaces: chemical species, membranes, material stocks, energy carriers, molecular machines, electrical ports, mechanical ports. A morphism $p \colon A \to B$ is an open process taking the input interface $A$ to the output interface $B$.

Definition 4.1 Open realization

An open process may be represented by a cospan-shaped diagram $L(A) \to X \leftarrow L(B)$, where $X$ is the system performing the transformation and $L(A)$, $L(B)$ are its exposed interfaces. Sequential composition is gluing along a common interface, $X \amalg_{L(B)} Y$. Parallel composition is $X \otimes Y$.

This is the higher-categorical analogue of the structured cospan used for open reaction networks, electrical circuits, and Petri nets. Ordinary pushouts are replaced by homotopy-coherent pushouts.

5. The process (∞)-operad

A symmetric monoidal $(\infty)$-category determines an $(\infty)$-operad. In Lurie’s formalism an $(\infty)$-operad is a map $\mathcal{O}^{\otimes} \to N(\operatorname{Fin}_*)$ with inert lifts and Segal conditions. We use the internal analogue.

Definition 5.1 Process operad

Let $\mathbb{O}^{\otimes}$ be the internal $(\infty)$-operad underlying $\mathbb{P}^{\otimes}$. For colors $x_1, \ldots, x_n, y$ there is an object $$\operatorname{Mul}_{\mathbb{O}}(x_1, \ldots, x_n; y) \in \mathcal{E}$$ of physical $n$-ary operations $(x_1, \ldots, x_n) \to y$. Operadic substitution supplies $$\operatorname{Mul}(y_1, \ldots, y_n; z) \times \prod_i \operatorname{Mul}(x_{i1}, \ldots, x_{ik_i}; y_i) \longrightarrow \operatorname{Mul}(x_{11}, \ldots ; z).$$ A physical causal network is a tree of composable operations, not merely a sequence of maps.

6. Internal suboperads

Definition 6.1 Suboperad

A suboperad $\mathbb{A}^{\otimes} \hookrightarrow \mathbb{O}^{\otimes}$ consists of subobjects $$\operatorname{Mul}_{\mathbb{A}}(\vec{x}; y) \hookrightarrow \operatorname{Mul}_{\mathbb{O}}(\vec{x}; y)$$ for every profile $(\vec{x}; y)$, stable under units, permutations, inert operations, operadic substitution, and restriction in $\mathcal{E}$. Write $\operatorname{SubOp}(\mathbb{O})$ for the poset of suboperads, ordered by inclusion. We write both $\mathbb{A} \le \mathbb{B}$ and $\mathbb{A} \subseteq \mathbb{B}$ for that order.

Lemma 6.2 Intersections of suboperads

An arbitrary intersection of suboperads of $\mathbb{O}^{\otimes}$ is a suboperad.

Proof.

Intersection is computed profilewise, $$\operatorname{Mul}_{\bigcap_i \mathbb{A}_i}(\vec{x}; y) = \bigcap_i \operatorname{Mul}_{\mathbb{A}_i}(\vec{x}; y).$$ Each factor contains the operadic units and is stable under permutation and substitution, so the intersection is as well.

Definition 6.3 Generated suboperad

For a collection $S$ of operations of $\mathbb{O}$, the generated suboperad $\langle S \rangle_{\mathbb{O}}$ is the intersection of all suboperads containing $S$. It is the smallest suboperad containing $S$.

7. Systems as bounded operadic realizers

Definition 7.1 Structured physical system

A structured physical system is a tuple $$X = \bigl( |X|,\ \partial X,\ \mathbb{E}_X^{\otimes},\ \mathbb{R}_X^{\otimes},\ \Phi_X \bigr).$$ Here $|X| \in \mathcal{E}$ is the physical carrier, $\partial X$ its interaction boundary, $\mathbb{E}_X^{\otimes} \hookrightarrow \mathbb{O}^{\otimes}$ its primitive endogenous process operad, $\mathbb{R}_X^{\otimes} \hookrightarrow \mathbb{O}^{\otimes}$ its organizational requirement operad, and $\Phi_X$ its recursive production operator.

8. Endogenous processes

Definition 8.1 Endogenous operation

An operation $f \in \operatorname{Mul}_{\mathbb{O}}(\vec{x}; y)$ is endogenous to $X$ when a physical realizer of $f$ lies inside the causal boundary of $X$. The endogenous suboperad $\mathbb{E}_X^{\otimes}$ contains every operation realizable from structures belonging to $X$, and it is closed under operadic composition: if $f, g_1, \ldots, g_n \in \mathbb{E}_X$, then $f \circ (g_1, \ldots, g_n) \in \mathbb{E}_X$.

9. Organizational requirements

Endogeneity is not enough. A rock has endogenous physical processes. The question is whether the processes required for the organization to persist are generated by that organization.

Definition 9.1 Requirement operad

The requirement operad $\mathbb{R}_X^{\otimes}$ is the suboperad generated by those organizational operations whose continued realization is required for $X$ to remain in the chosen viability class. Generators may include boundary maintenance, repair, catalysis, translation, fabrication, replication, regulation, and energy transduction. The signature is selectable. Different theories of life correspond, in part, to different choices of $\mathbb{R}_X^{\otimes}$.

10. Why one-step generation is not enough

Suppose an organism currently possesses a ribosome. Translation is then an endogenous operation. That does not say whether the system can produce or repair the machinery that realizes translation. Closure has to be recursive, not a snapshot of the operations available at one moment.

11. Production doctrine

Definition 11.1 Production doctrine

A production doctrine for $X$ is a monotone endomap $\Phi_X \colon \operatorname{SubOp}(\mathbb{O}) \to \operatorname{SubOp}(\mathbb{O})$. The value $\Phi_X(\mathbb{A})$ contains those organizational operations whose physical realizers $X$ can construct, maintain, regenerate, or repair using operations from $\mathbb{A}$, together with the primitive endogenous capabilities and the permitted matter–energy inputs. Monotonicity means $\mathbb{A} \le \mathbb{B}$ implies $\Phi_X(\mathbb{A}) \le \Phi_X(\mathbb{B})$.

Assumption 11.2 Accessibility

The operator $\Phi_X$ is accessible, and inflationary once the primitive endogenous operations have been adjoined. For a sufficiently large regular ordinal $\lambda$, $\Phi_X$ preserves the relevant $\lambda$-filtered colimits. Recursive generation may therefore be computed by transfinite iteration.

12. Recursive endogenous closure

Set $\mathbb{G}_X^{(0)} := \mathbb{E}_X$. At a successor ordinal, $$\mathbb{G}_X^{(\alpha+1)} := \Phi_X\bigl(\mathbb{G}_X^{(\alpha)}\bigr).$$ At a limit ordinal $\beta$, $$\mathbb{G}_X^{(\beta)} := \left\langle \bigcup_{\alpha < \beta} \mathbb{G}_X^{(\alpha)} \right\rangle.$$ For sufficiently large $\lambda$ the iteration stabilizes. Write $\mathbb{G}_X := \mathbb{G}_X^{(\lambda)}$, and also $\mathbb{G}_X = \mu \Phi_X$ for this least fixed point.

Proposition 12.1 Fixed point

The operad $\mathbb{G}_X$ is a fixed point of $\Phi_X$: $\Phi_X(\mathbb{G}_X) \simeq \mathbb{G}_X$.

Proof.

By accessibility, $$\Phi_X \Bigl( \operatorname*{colim}_{\alpha < \lambda} \mathbb{G}_X^{(\alpha)} \Bigr) \simeq \operatorname*{colim}_{\alpha < \lambda} \Phi_X\bigl(\mathbb{G}_X^{(\alpha)}\bigr).$$ The right-hand side is the colimit of the successor stages $\mathbb{G}_X^{(\alpha+1)}$. Shifting the ordinal index does not change the colimit, so $\Phi_X(\mathbb{G}_X) \simeq \mathbb{G}_X$.

Proposition 12.2 Least fixed point

The operad $\mathbb{G}_X$ is the least $\Phi_X$-closed suboperad containing $\mathbb{E}_X$.

Proof.

Let $\mathbb{A}$ contain $\mathbb{E}_X$ and satisfy $\Phi_X(\mathbb{A}) \le \mathbb{A}$. Transfinite induction gives $\mathbb{G}_X^{(\alpha)} \le \mathbb{A}$ for every $\alpha$. Successor stages use monotonicity. Limit stages use that $\mathbb{A}$ is itself a suboperad containing every earlier stage, hence contains the suboperad they generate. Therefore $\mathbb{G}_X \le \mathbb{A}$.

13. Operadic organizational closure

Definition 13.1 Operadic autopoiesis

The system $X$ is organizationally closed, written $\operatorname{Aut}(X)$, when $$\mathbb{R}_X^{\otimes} \subseteq \mathbb{G}_X^{\otimes}.$$ Every operation organizationally required by $X$ lies in the recursively generated endogenous operad.

14. Interpretation

The inclusion $\mathbb{R}_X \subseteq \mathbb{G}_X$ does not say that $X$ is materially isolated. Environmental resource morphisms $M_{\mathrm{env}} \to X$ and $E_{\mathrm{env}} \to X$ are allowed. Matter and energy may flow through the system. Closure concerns the generation of organizational constraints and of productive machinery.

Organizational closure is not thermodynamic closure.

This matches the classical autopoietic idea: a living system recursively produces the network and the boundary that constitute it.

15. Equivalence of structured systems

Definition 15.1 Equivalence

An equivalence $X \simeq Y$ of structured systems consists of compatible equivalences $|X| \simeq |Y|$, $\partial X \simeq \partial Y$, $\mathbb{E}_X \simeq \mathbb{E}_Y$, and $\mathbb{R}_X \simeq \mathbb{R}_Y$, conjugating the production operators: $$F_* \circ \Phi_X \simeq \Phi_Y \circ F_*.$$

16. Fixed-point invariance

Theorem 16.1 Equivalences preserve generative closure

If $X \simeq Y$ is an equivalence of structured systems, then $F_*(\mathbb{G}_X) \simeq \mathbb{G}_Y$.

Proof.

At stage zero, $F_*(\mathbb{E}_X) \simeq \mathbb{E}_Y$, so $F_*(\mathbb{G}_X^{(0)}) \simeq \mathbb{G}_Y^{(0)}$. If the claim holds at stage $\alpha$, conjugacy of the production operators gives $$F_* \Phi_X\bigl(\mathbb{G}_X^{(\alpha)}\bigr) \simeq \Phi_Y F_*\bigl(\mathbb{G}_X^{(\alpha)}\bigr) \simeq \Phi_Y\bigl(\mathbb{G}_Y^{(\alpha)}\bigr),$$ which is the successor stage of $Y$. At limit ordinals, $F_*$ preserves the generated colimits in question. By transfinite induction the stages agree, and therefore so do the stable values.

17. Autopoiesis is invariant

Theorem 17.1 Invariance of operadic autopoiesis

If $X \simeq Y$, then $\operatorname{Aut}(X)$ if and only if $\operatorname{Aut}(Y)$.

Proof.

Suppose $\mathbb{R}_X \subseteq \mathbb{G}_X$. Applying $F_*$ and using preservation of requirements together with Theorem 16.1 yields $\mathbb{R}_Y \subseteq \mathbb{G}_Y$. The converse uses $F^{-1}$.

Corollary 17.2 Separation by autopoiesis

If $\operatorname{Aut}(X)$ and $\neg \operatorname{Aut}(Y)$, then $X \not\simeq Y$.

This already separates an autonomously closed cell from an externally maintained machine, inside the present model.

18. Interaction interfaces

Let $X$ and $H$ interact through an interface operad $\mathbb{I}^{\otimes}$, with operad morphisms $i_X \colon \mathbb{I}^{\otimes} \to \mathbb{E}_X^{\otimes}$ and $i_H \colon \mathbb{I}^{\otimes} \to \mathbb{E}_H^{\otimes}$. Interface colors may include membrane receptors, ribosomal binding sites, ATP, nucleotides, polymerases, and transport machinery.

19. Operadic coupling

Definition 19.1 Coupled operad

The raw coupled operad is the homotopy pushout $$\mathbb{E}_X \amalg_{\mathbb{I}} \mathbb{E}_H.$$ Whenever the two endogenous realizations agree on the common interface, there is a canonical morphism $$j \colon \mathbb{E}_X \amalg_{\mathbb{I}} \mathbb{E}_H \longrightarrow \mathbb{O}.$$ Write $\mathbb{E}_{X \star H} := \langle \operatorname{im}(j) \rangle_{\mathbb{O}}$ for the suboperad it generates.

20. Why the pushout is larger than the union

The pushout contains more than $\mathbb{E}_X \cup \mathbb{E}_H$. It contains operations obtained by composing an operation from one system with operations from the other. Schematically, $f_X \circ (g_H, h_X, k_H)$ is legal whenever the colors match. The coupled system has mixed causal pathways that neither system has alone.

21. Mixed trees

Lemma 21.1 Mixed-tree lemma

Every finite operadic tree whose vertices are labeled by operations from $\mathbb{E}_X$ or from $\mathbb{E}_H$, and whose cross-system internal edges have colors in the common interface $\mathbb{I}$, determines an operation of $\mathbb{E}_{X \star H}$.

Proof.

Induct on height. A tree of height zero is a single operation of $X$ or of $H$, hence lies in the image of the pushout. If the claim holds up to height $n$, and $T$ has height $n+1$, each immediate subtree of the root represents, by induction, an operation of the pushout-generated operad. The root operation lies in $\mathbb{E}_X$ or in $\mathbb{E}_H$. Matching edge colors make the operadic substitution defined, and closure under substitution produces the operation corresponding to $T$.

22. Emergent causal capacity

Corollary 22.1 Coupling creates operations

An operation $r$ may satisfy $r \notin \mathbb{E}_X$ and $r \notin \mathbb{E}_H$ while $r \in \mathbb{E}_{X \star H}$.

Proof.

The operation $r$ may be a composite tree with some vertices supplied by $X$ and others by $H$. Lemma 21.1 places that tree in $\mathbb{E}_{X \star H}$.

Coupled capability is not the union of the isolated capabilities.

This is the structure needed for viral use of host-cell machinery.

23. Coupled production

The coupled system has a production doctrine $\Phi_{X \star H}$ whose primitive stage contains $\mathbb{E}_{X \star H}$. Its recursively generated operad is $\mathbb{G}_{X \star H} := \mu \Phi_{X \star H}$.

24. Relative closure

Definition 24.1 Closure relative to a host

The system $X$ is organizationally closed relative to $H$, written $\operatorname{Aut}(X \mid H)$, when $\mathbb{R}_X \subseteq \mathbb{G}_{X \star H}$.

25. Strict host dependence

Definition 25.1 Strict dependence

The system $X$ is strictly dependent on $H$ when $\neg \operatorname{Aut}(X)$ but $\operatorname{Aut}(X \mid H)$. Equivalently, $\mathbb{R}_X \nsubseteq \mathbb{G}_X$ while $\mathbb{R}_X \subseteq \mathbb{G}_{X \star H}$.

26. The virus theorem

Let $V$ be a virus and $H$ a susceptible host cell. Viral replication depends on host translation, and many viruses co-opt host proteins, membranes, metabolism, and other cellular processes for genome replication and assembly. The modeling hypothesis is $$\mathbb{R}_V \nsubseteq \mathbb{G}_V, \qquad \mathbb{R}_V \subseteq \mathbb{G}_{V \star H}.$$

Theorem 26.1 Viral host-dependence

Under these hypotheses, $V$ is a strictly host-dependent organizational system.

Proof.

This is Definition 25.1. The content is in the structure that instantiates the predicates: $\mathbb{G}_{V \star H}$ contains mixed virus–host operations produced by the operadic pushout, which $\mathbb{G}_V$ does not.

27. The cell

Let $B$ be a bacterial cell, and let $\mathbb{R}_B$ be generated by membrane maintenance, repair, translation, replication, metabolism, regulation, and the other modes treated as essential. The biological-autonomy hypothesis is $\mathbb{R}_B \subseteq \mathbb{G}_B$. This is an axiomatization, not a theorem of biology. Inside the theory it says that the cell’s required organizational machinery lies in its recursively regenerative operadic closure.

28. Virus and cell

Theorem 28.1 Virus–cell separation

If $\operatorname{Aut}(B)$ and $\neg \operatorname{Aut}(V)$, then $B \not\simeq V$ as structured physical systems.

Proof.

Operadic autopoiesis is invariant under equivalence by Theorem 17.1. An equivalence $B \simeq V$ would force $\operatorname{Aut}(B)$ if and only if $\operatorname{Aut}(V)$.

29. The computer

Let $C$ be a conventional computer. On a restricted operational signature — instruction execution, memory access, arithmetic — the machine may have substantial endogenous closure. Take instead the physically complete requirement operad $\mathbb{R}_C^{\mathrm{phys}}$ generated by chip fabrication, replacement, repair, structural reproduction, power infrastructure, and assembly. For an ordinary present-day computer, $$\mathbb{R}_C^{\mathrm{phys}} \nsubseteq \mathbb{G}_C,$$ while an industrial infrastructure $I$ supplies the missing operations, $\mathbb{R}_C^{\mathrm{phys}} \subseteq \mathbb{G}_{C \star I}$. Under the physical requirement signature, $\neg \operatorname{Aut}(C)$.

30. Cell and computer

Theorem 30.1 Cell–computer separation

If $\operatorname{Aut}(B)$ and $\neg \operatorname{Aut}(C)$, then $B \not\simeq C$ as operadically structured physical systems.

Proof.

Theorem 17.1.

31. Stronger than a list of parts

A purely material argument says $\operatorname{Supp}(B) \not\simeq \operatorname{Supp}(C)$. The operadic argument says $\mathbb{G}_B \not\simeq \mathbb{G}_C$ as organizational process algebras. The first is a difference of objects. The second is a difference of the higher algebra of transformations among objects.

The distinction is not merely different parts. It is different recursively generated causal algebras.

32. Material typing

The substrate invariant remains available. Let $K \in \mathcal{E}$ be the object of material component types, and let $\tau_X \colon C_X \to K$ be the typing of the components of $X$. Set $\operatorname{Supp}(X) := \operatorname{im}(\tau_X)$.

Proposition 32.1 Support is invariant

An equivalence that preserves material typing induces $\operatorname{Supp}(X) \simeq \operatorname{Supp}(Y)$.

Proof.

As in the typed setting: if $\tau_X \simeq \tau_Y \circ f$ with $f$ an equivalence, the images agree.

Material support is an invariant independent of the closure truth value.

33. The full physical invariant

Define $$\mathcal{J}(X) = \bigl( \operatorname{Supp}(X),\ \mathbb{E}_X,\ \mathbb{R}_X,\ \Phi_X,\ \mathbb{G}_X \bigr).$$

Theorem 33.1 Structured equivalence preserves $\mathcal{J}$

A structured equivalence $X \simeq Y$ induces $\mathcal{J}(X) \simeq \mathcal{J}(Y)$. Any difference among these components obstructs physical equivalence.

Proof.

Support is Proposition 32.1. The operads $\mathbb{E}$ and $\mathbb{R}$, and the operator $\Phi$, are preserved by Definition 15.1. The generated operad $\mathbb{G}$ is preserved by Theorem 16.1.

34. Why a virus and a program can look alike

Let $V$ be a virus and $P$ a self-copying program. There is a coarse dependency pattern $V + H \to V^{\sqcup n}$ and $P + M \to P^{\sqcup n}$. Define a forgetful functor $Q \colon \mathbf{PhysSys} \to \mathbf{Dep}$ that remembers only specification, host, and replication. It is entirely possible that $Q(V) \simeq Q(P)$.

35. Non-conservativity

Theorem 35.1 Forgetting support or the operad destroys conservativity

If $Q$ forgets either material support or the endogenous operad, then $Q$ is not conservative in general. Thus $Q(X) \simeq Q(Y)$ does not imply $X \simeq Y$.

Proof.

Choose $X$ and $Y$ with the same dependency diagram and with $\operatorname{Supp}(X) \not\simeq \operatorname{Supp}(Y)$. Then $Q(X) \simeq Q(Y)$. If $X \simeq Y$ held in the full category, Proposition 32.1 would force the supports to agree.

36. The virus/program analogy

Therefore $Q(V) \simeq Q(P)$ may hold even though $V \not\simeq P$. The analogy lives in a quotient of the structural theory. It is invalid to infer $V \simeq P$ from $Q(V) \simeq Q(P)$, and still more invalid to infer $B \simeq P$ for a bacterium $B$.

37. Localization

The same phenomenon can be phrased as a localization. Let $L \colon \mathcal{E} \to \mathcal{E}_{\mathrm{comp}}$ be an accessible left-exact localization, read as a computational observational modality. Interpret $LX$ as what is observable about $X$ through the chosen computational semantics.

Definition 37.1 Computational equivalence

Systems $X$ and $Y$ are computationally equivalent when $LX \simeq LY$.

Proposition 37.2 Non-conservative localizations identify non-equivalent systems

If $L$ is not conservative, there exist $X \not\simeq Y$ with $LX \simeq LY$.

Proof.

Failure of conservativity is exactly failure to reflect equivalences.

38. Simulation

Suppose a bacterium and a computer simulation satisfy $L(B) \simeq L(C)$. All that follows is equality after computational localization. One cannot conclude $B \simeq C$ unless $L$ reflects equivalences on the relevant class of systems.

Equality after computational localization does not imply equality in the ambient category.

39. Locality

Let $u \colon U \twoheadrightarrow 1$ be an effective epimorphism, and assume the production doctrine is geometric: $$u^* \Phi_X(\mathbb{A}) \simeq \Phi_{u^* X}(u^* \mathbb{A}).$$ Inverse-image functors preserve colimits, so transfinite generation commutes with pullback: $u^* \mathbb{G}_X \simeq \mathbb{G}_{u^* X}$.

40. Descent for closure

Theorem 40.1 Closure descends

If $u^* \mathbb{R}_X \subseteq u^* \mathbb{G}_X$ for an effective epimorphism $u \colon U \twoheadrightarrow 1$, then $\mathbb{R}_X \subseteq \mathbb{G}_X$.

Proof.

Suboperad inclusion is profilewise inclusion of subobjects. For each multimorphism profile, $u^* R(\vec{x}; y) \le u^* G(\vec{x}; y)$. Pullback along an effective epimorphism reflects subobject inclusion, so $R(\vec{x}; y) \le G(\vec{x}; y)$ for every profile.

41. Host coupling is not a change of context

Corollary 41.1 Observation does not create autonomy

If $\neg \operatorname{Aut}(V)$, effective-epimorphic base change alone cannot produce $\operatorname{Aut}(u^* V)$.

Proof.

If $\operatorname{Aut}(u^* V)$ held, Theorem 40.1 would give $\operatorname{Aut}(V)$.

Host infection is not a change of context. It is operadic enlargement by coupling.

The host has to contribute operations. That is the point of the pushout.

42. Three loci

Suppose structured systems form an internal moduli object, or a stack, $\mathbf{Sys}$ over $\mathcal{E}$. The autonomous locus, the merely dependent locus, and the host-dependent locus are the internally definable subobjects $$\begin{aligned} \mathbf{Aut} &= \{ X \mid \mathbb{R}_X \subseteq \mathbb{G}_X \}, \\ \mathbf{Dep} &= \{ X \mid \mathbb{R}_X \nsubseteq \mathbb{G}_X \}, \\ \mathbf{HostDep} &= \{ X \mid \neg \operatorname{Aut}(X) \land \exists H.\; \operatorname{Aut}(X \mid H) \}. \end{aligned}$$ Thus $\mathbf{Aut} \hookrightarrow \mathbf{Sys}$ and $\mathbf{HostDep} \hookrightarrow \mathbf{Sys}$ are internal loci, not external labels pasted onto the mathematics.