Formal correction paper and complete artifacts
The attached PDF and source archive contain the complete paper, *Corrections to Operadic Autopoiesis: Conservativity, Inhabitation, Budgets, and Viability*, version 1.0.0, with definitions, propositions, theorems, proofs, references, a source-location correction map, and executable counterexamples. This journal body is a condensed reading guide; result numbers refer to the complete attached paper.
The target is the three-essay edition 1.0 attached to Organisms, Viruses, and Computers: Closure, Operadic Autopoiesis, and Thermodynamic Viability. The target's current label is Editorially reviewed. No conclusion below depends on that label or on an attribution concerning authorship or automated preparation.
Principal corrections
1. Conservativity is not recovery of objects
A conservative functor reflects equivalences of existing morphisms. It need not satisfy $F(X)\simeq F(Y)\Rightarrow X\simeq Y$. Theorem 3.2 proves independence of these two properties: the functor from a discrete two-object category to the terminal category is conservative but identifies the two objects. Conversely, a one-object category with a noninvertible idempotent maps to the terminal category and trivially reflects object-equivalence classes without being conservative. The conservative functor from groups to sets also identifies the underlying sets of the nonisomorphic groups $C_4$ and $C_2\times C_2$.
Thus Essay II, Theorem 47.1, is false. Full faithfulness, or conservativity together with a suitable lifting condition on equivalences, is a sufficient replacement. The distinct claim about genuine reflective localizations in II, Proposition 37.2, can be recovered by using a nonequivalence localization unit; the full paper gives that argument.
2. Nonzero kernels need not supply viable states
Proposition 4.1 distinguishes $A\not\simeq0$, $\operatorname{im}(A\to1)=1$, and a specified section $1\to A$. In the infinity-topos of sheaves of spaces on the real line, the subterminal object for the proper nonempty open interval $(0,1)$ is noninitial but has neither full support nor a global section. Identity dynamics on that object give a nonzero greatest fixed point with the same defect. This corrects III, Definition 20.1. A physical existence statement should specify its context and require the appropriate inhabitation or state-and-policy witness.
3. Feasibility is resource-graded
Proposition 6.2 shows that a fixed cost sublevel is generally not a suboperad. In the free unary operad on $f$ with cost $c(f^n)=n$, budget one permits $f$ and forbids $f\circ f$. Actual feasible realizers instead compose into the sum of the input budgets. Proposition 6.3 separates an infimum from an attained cost: realizers costing $1+1/n$ have infimum one but none meets budget one. The numerical cost inequality can remain valid. III §§9-11 require resource grading, attainment or slack assumptions, and compatible physical interfaces.
4. Seeded closure and separate witnesses are insufficient
For $\Phi(A)=E\vee A$, Proposition 5.1 gives $\mu\Phi=E$. This monotone inflationary operator satisfies the order-theoretic generation assumptions yet adds no regeneration capability. The distinction between possessing initial machinery and replacing its realizers therefore requires an explicit adequacy condition on production and requirements. Moreover, individual witnesses $\forall r\,\exists\pi_r$ do not provide one schedule $\exists\pi\,\forall r$. The finite resource-aware model below supplies a common policy witness.
5. Generated pushout images lose interface information
Under the monomorphism and existence hypotheses stated in Theorem 7.1, fixed suboperads $E,H\subseteq O$ and a span over $O$ satisfy $$ \left\langle\operatorname{im}(E\amalg_I H\to O)\right\rangle=E\vee H. $$ The left side contains both operands and therefore their join; conversely the pushout maps through the join. For fixed ambient embeddings, this generated image no longer records the interface parameter. The full pushout and its realization map, or explicitly interface-restricted compositions, must be retained when interface permission matters.
6. Viability needs a realization semantics
The monotone predecessor construction has a greatest fixed point, but internally passing from a serial successor relation to an infinite path needs choice or a specified selector. A continuous-time interpretation must additionally rule out finite-time accumulation of events. Stochastic survival requires a chosen probability semantics: a possible surviving successor at each step is compatible with probability zero of infinite survival, as the chain with one-step survival probability $0.99$ demonstrates.
A corrected finite theorem
Theorem 9.1 uses finite configurations containing resource stocks, machine status and regeneration-deadline counters, together with an explicit transition table and unit-time steps. Starting with the viable set $V$, set $$ K_0=V,\qquad K_{n+1}=V\cap\{s:\exists u,\ T(s,u)\in K_n\}. $$ The descending sequence stabilizes after at most $|S|$ strict decreases. From a specified state in the stable kernel, finite ordered search selects a control back into that kernel. Recursion produces a single infinite trajectory satisfying the encoded resource constraints and every repair deadline. Conversely, states on any infinite viable trajectory form a postfixed set and belong to the greatest fixed point. State-and-control isomorphisms preserving time, resource accounting and obligation labels transport these witnesses.
The executable mutual-repair example has 18 states. Its viable kernel contains 16 states with one unit of replenishment per tick and is empty without replenishment. The verification output includes an explicit policy and eventual cycle. This is a finite mathematical model, not an experimental organism or a universal classifier of life.
Energetic obstruction and retained results
Theorem 10.1 repairs the power obstruction by requiring a finite initial store $B_0$ and the cumulative feasibility inequality $C_\pi(T)\le B_0+E_{\mathrm{in}}(T)$. If available asymptotic power is strictly below the infimum of asymptotic viable-policy requirements, division by $T$ gives a contradiction. Strictly positive maintenance power is an additional physical hypothesis, not a consequence of nonnegative entropy production. Average-power invariance must preserve physical time and the policy class.
The image-invariance and effective-epimorphic descent arguments survive, as does transport of least and greatest fixed points by conjugating order isomorphisms. Biological separation remains conditional on justified invariant values, a consistent organizational signature and explicit physical boundaries.
References
- Emily Riehl, *Category Theory in Context*, Dover, 2016; updated author-hosted edition, Lemma 5.6.1: https://emilyriehl.github.io/files/context.pdf
- Jacob Lurie, *Higher Topos Theory*, Princeton, 2009; updated 2017 text, §6.2.3: https://www.math.ias.edu/~lurie/papers/HTT.pdf
- Alfred Tarski, A lattice-theoretical fixpoint theorem and its applications, *Pacific Journal of Mathematics* 5 (1955), 285-309: https://msp.org/pjm/1955/5-2/pjm-v5-n2-p11-p.pdf
- F. William Lawvere, Metric spaces, generalized logic, and closed categories, 1973; TAC reprint, 2002: https://tac.mta.ca/tac/reprints/articles/1/tr1.pdf
- John C. Baez and Kenny Courser, Structured cospans, *TAC* 35 (2020), 1771-1822: https://arxiv.org/abs/1911.04630
- Jordan M. Horowitz, Kevin Zhou, and Jeremy L. England, Minimum energetic cost to maintain a target nonequilibrium state, *Physical Review E* 95 (2017), 042102: https://arxiv.org/abs/1703.04722
- Riccardo Rao and Massimiliano Esposito, Nonequilibrium thermodynamics of chemical reaction networks, *Physical Review X* 6 (2016), 041064: https://arxiv.org/abs/1602.07257
Evidence and responsible preparation
The artifacts are corrections-operadic-autopoiesis-v1.0.0.pdf and corrections-operadic-autopoiesis-source-v1.0.0.zip. The source archive includes the full Markdown and LaTeX, offline MathML HTML, correction map, bibliographic metadata, Python standard-library checker, exact results, reproducible build instructions, and checksum manifest. All finite checks passed. The general proofs are mathematical arguments, not proof-assistant-certified results. No biological experiment or independent referee approval is asserted.
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