From Darwin to Teleology: A Categorical Final-Cause Calculus for Evolution
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From Darwin to Teleology: A Categorical Final–Cause Calculus for Evolution Andrei T. Patrascu 1 1 FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] We propose a teleological calculus for evolution that generalizes the classical Darwin–Fisher picture by making final causes—what systems keep true about themselves—into mathematical objects with universal properties. In our framework, the state space is a category Cacted upon (laxly) by time T, and viability constraints live in a fibration p : E→ C. An endogenous functor Gt :C →E extracts invariants from the system (e.g. topological features via persistent homology, sheaf gluing compatibilities, symmetry/conservation laws, or behavioral attractors). The present compatible with realizing these constraints at horizon tis the right Kan extension Telt(X0) = RanιΦ (X0), equivalently a (possibly enriched) limit or a largest invariant subcoalgebra. Passing to concrete dynamics xt yields an endogenous bias that selects among feasible futures without introducing exogenous rewards: dxt=f(xt, et) dt−ε∇xLt(xt) dt+√2DdWt, where the coherence deficit Lt is built from Gt (e.g. PH witness distances and sheaf mismatch penalties). Classical selection appears as the scalar collapse L = −fitness (replicator–mutator). Richer choices of Gt produce a ladder of mechanisms: multi-objective Pareto teleology, morphogenetic teleology (global sections; dwell-time ∼eα∆W/D ), behavioral attractors (final coalgebras), niche-construction holonomy (order-of-operations gaps), multi-level coherence (homotopy limits; synergy/variance bounds), and meta-teleology (doctrine updates for new codes). We derive measurable predictions, provide a unifying categorical spine, and outline algorithms to infer Gt from data. This reframes evolution as selection by endogenous invariants, with Darwinian fitness as a special case. Keywords teleology; teleonomy; category theory; Kan extensions; limits and coalgebras; persistent homology; sheaf theory; behavioral attractors; niche construction; multi-level selection; evolutionary theory; variational principles.
2 Notation and Symbols Symbol Meaning C (world) Category of states; objects are system states, morphisms are admissible transformations. TyC (time action) (Lax) monoidal action of time/process on states; (t, X)7→ tX. p:E → C (doctrine) Viability doctrine (fibration of predicates/invariants); b−c : EX→ Cis comprehension (subobject). Gt:C→ E (final cause) Endogenous constraint functor at horizon t extracting invariants (e.g. PH, sheaf gluing, symmetries, attractors). ι:Bt,→C and Φ : Bt→Cι includes Gt -admissible futures; Φembodies evaluation/realization at time t. Telt(X0) = RanιΦ(X0)Teleological object : terminal present compatible with realizing Gt at horizon t(a right Kan extension / limit). Reacht(X0) (reachable set) Concrete reachable set at horizon tin state-space models. Lt (coherence deficit) Derived from Gt (e.g. PH witness distances, sheaf overlap residuals, symmetry/behavioral penalties). dxt=fdt −ε∇Ltdt+√2DdWt Teleological drift (SDE): small endogenous bias toward futures that keep invariants true. holim,colim (multi-level gluing) Homotopy limit/colimit for composing levelwise teleologies (genes, cells, organs, organisms, groups). F:C→C (νF, ω) One-step evolution functor and its final coalgebra (universal behaviors). A? = νS7→ ω−1 ( FS ) ∩ JGtK (behavioral attractor) Largest invariant teleological behavior satisfying Gt ; every system factors uniquely through A?. Θt (invariants) Selected invariants extracted by Gt (e.g. persistent homology bars above threshold). I. INTRODUCTION This paper develops a teleological calculus for evolution: a mathematical framework in which final causes— endogenous constraints that systems keep true about themselves—are first-class objects. We show that Darwinian evolution is recovered as a special (scalar) case, while richer choices of endogenous constraints yield mechanisms that naturally account for morphogenesis, behavioral attractors, niche construction, multi-level coherence, and meta-transitions. We give the categorical spine (right Kan extensions, limits, fibrations, final coalgebras), the corresponding analytic forms (least-violation functionals and stochastic drifts), and measurable predictions. This Section situates the work historically and conceptually, and states the minimal existence and correctness results we rely on in the rest of the paper.
3 A. Evolutionary theory today: Darwin →Modern Synthesis →EES Darwin’s program explains adaptation by variation and natural selection [ 1 ]. The Modern Synthesis formalized this with population genetics (Wright–Fisher, Moran) and quantitative laws such as Fisher’s fundamental theorem [ 216 ] and Wright’s adaptive landscape [ 3 ]. A convenient continuous-time proxy is replicator–mutator dynamics, ˙pi=X j pjMj→i−piX k Mi→k+pi fi(p)−X m pmfm(p)!,(1) for type frequencies p∈ ∆ n−1 , mutation kernel M , and fitnesses fi (possibly frequency-dependent) [5, 155]. The Extended Evolutionary Synthesis (EES) emphasizes additional phenomena: developmental bias and evo-devo,niche construction,multi-level selection, plasticity and genetic assimilation, and major transitions [ 6 , 7 , 9 – 11 , 168 ]. While the EES highlights mechanisms, a unifying mathematics that makes these mechanisms cohere with the classical picture remains desirable. B. Limitations of scalar-fitness and forward-only optimization The scalar-fitness view compresses evolution to a single objective (“increase fitness”). This is often effective but has limitations: (i) Trade-offs and Pareto structure. Many systems operate on a multi-objective front; no single scalar captures all changes without context-dependent weights [12]. (ii) Morphology and global constraints. Organ-level shapes (e.g. lumens, branchings) are global properties arising from local rules; their maintenance and repair show strong teleological signatures (“keep the cavity”) [14, 15, 191]. (iii) Behavioral attractors. Systems re-form routines after perturbations without genome change; these are better modeled as invariant sets of a coalgebraic dynamics [242]. (iv) Niche construction and path dependence. Organisms edit environments; the order of edits and adaptations matters [7, 9]. (v) Multi-level coherence. Genetic, cellular, tissue, organismal, and group-level constraints must cohere; selection at one level can be vetoed by incompatibility with others [11, 168]. These limitations suggest a formulation where what is kept true is not a single scalar but a structured, endogenous set of invariants. C. Final causes vs. efficient causes: Aristotelian roots, categorical forms Aristotle distinguished material,formal,efficient, and final causes. In category-theoretic terms [ 17 – 19 ]: •Efficient cause corresponds to left universal constructions: generation from inputs and push-forward in time (left Kan extensions Lan, colimits, initial algebras; e.g. initial value problems). •Final cause corresponds to right universal constructions: selection by compatibility with constraints (right Kan extensions Ran, limits, final coalgebras; e.g. boundary-value or viability problems). We will use the following canonical form: for a category of states Cwith a (lax) action by a time/process category T, a viability doctrine p : E → C(a fibration of predicates/invariants), and an endogenous constraint functor Gt :C → E at horizon t , the teleological present of an initial state X0 is the right Kan extension Telt(X0) = RanιΦ(X0),(2)
4 where ι : Bt,→ Cincludes Gt -admissible futures and Φ : Bt→ Cembodies them at time t (see §IE for existence). In concrete state spaces, (2) has the analytic least-violation form Telt(X0) = arg min y∈Reacht(X0)Lt(y),(3) with a coherence deficit Lt built from Gt (e.g. persistent-homology (PH) witness distances [ 22 , 244 ], sheaf-gluing residuals [ 23 ], symmetry penalties). This distinction mirrors teleology vs. teleonomy: we do not add an exogenous objective (teleonomy [ 25 , 212 ]), but derive a final cause from invariants extracted functorially from the system itself. All constructions introduced here are non-anticipative: the only “ends” are limits or adjoints computed from present-time diagrams. D. Our claim and program: categorical teleology generalizes Darwin Claim. Evolution can be framed as selection by endogenous invariants. Formally, with Telt defined by (2), we obtain: (a) The classical Darwin–Fisher picture is the scalar collapse Lt=−fitness of (3), recovering (67). (b) Richer Gt (PH, sheaves, coalgebras, base change, holims) yield the L0–L6 ladder of mechanisms within one calculus, with quantitative predictions (hysteresis bounds, persistence-based dwell laws, niche holonomy, synergy inequalities). Operationally, we study baseline dynamics with a small endogenous teleological drift: dxt=f(xt, et)dt−ε∇Lt(xt)dt+√2DdWt,(4) and its stationary law p∗ ( x ) ∝exp{− (Φ( x ) + εLt ( x )) /D} in gradient systems. Equation (4) is not an optimal-control gradient; Lt is constructed from Gt via canonical functors (limits, right adjoints, homology, sheaf sections), ensuring functoriality and stability [17, 18, 23, 244]. E. Minimal existence and correctness results We record the basic results guaranteeing that the categorical objects used above exist and fit their intended roles. Proposition I.1 (Existence of Ran -teleology) . Let ι : Bt→ Cand Φ : Bt→ Cbe functors with Bt small. If Cis complete, then the right Kan extension RanιΦexists, and for each X∈C, RanιΦ(X)∼ =lim(X↓ι)Π −−→ Bt Φ −−→ C,(5) where (X↓ι)is the comma category and Πthe projection. Proof. This is standard [ 17 , Thm. X.3]. Since Bt is small, so is ( X↓ι ). Completeness of Cguarantees the limit (5) exists. The universal property of Ran follows from the universal property of limits via the comma construction. Proposition I.2 (Coalgebraic teleology) . Let F :C → Cadmit a final coalgebra ( νF, ω ), and let JGtK,→ νF be a subobject of behaviors satisfying the endogenous constraints. The operator Φ( S ) := ω−1 ( FS ) ∩JGtK is monotone on the complete lattice of subobjects of νF; thus the largest invariant teleological behavior A?=νΦ(6) exists and is a fixed point of Φ(Knaster–Tarski). Every coalgebra ( X, ξ : X→FX )factors uniquely through A?,→νF. Proof. Subobjects of νF form a complete lattice. Monotonicity: if S⊆S0 , then FS ⊆FS0 (functoriality) and ω−1 ( − )preserves inclusion; intersecting with JGtK is also monotone. Knaster–Tarski yields a greatest fixed point ν Φ. The unique factorization through ( νF, ω )is the defining property of the final coalgebra [241, 242].
5 Proposition I.3 (PH stability for endogenous invariants) . Let f, g : X → R be tame functions defining sublevel filtrations. Then the bottleneck distance dBbetween their persistence diagrams satisfies dBDgm(f), Dgm(g)≤ kf−gk∞.(7) Hence, mapping data to Θ t by persistent homology is 1-Lipschitz, providing a stable (endogenous) source for Gtand Lt. Proof sketch. See [ 244 ] for a full proof. The interleaving distance between filtrations induced by f and g is bounded by kf−gk∞; stability of persistence then yields (7). Proposition I.4 (Teleological drift descends the deficit on average) . Assume Lt is smooth and slowly varying. Under (4), d dtELt(xt)≈ −εEk∇Lt(xt)k2+DE[∆Lt(xt)] ≤0(8) whenever the Laplacian term is dominated by the descent term (e.g. small Dor locally convex Lt). Proof. Itô’s formula for u ( x, t ) = Lt ( x )gives d u =( ∂tu + ∇u· ( f−ε∇Lt ) + D ∆ u )d t + √2D∇u· d Wt. Taking expectations, using E[∇u·dWt]=0and small ∂tu, leads to (8). Remark I.5 (Teleonomy vs. teleology).Teleonomy explains purpose-like behavior via historically selected programs [ 25 , 212 ]. Our calculus explains present selection among futures via endogenous invariants. In static regimes, both agree (selected programs maintain the same invariants). In rapidly repairing systems, niche edits, or multi-level conflicts, the teleological account makes specific, measurable predictions (e.g. order-of-operations gaps) not implied by a single scalar historical fitness. F. Roadmap of the paper Part II formalizes teleological systems, Ran -teleology, least-violation functionals, coalgebraic attractors, and base-change/multi-level gluing, with proofs extending Propositions I.1–I.4. Part III derives the L0–L6 ladder as specializations of Gt (scalar, Pareto, PH–sheaf, coalgebraic, niche/base change, multilevel holims, meta-doctrine updates) and states the associated quantitative signatures. Part IV details mechanisms (constraint-preserving selection, sheaf-gluing selection, persistence-weighted canalization, attractor locking, niche holonomy, cross-level veto, meta-code jumps, constraint-transport speciation, geometric phases, plasticity → assimilation). Part V provides testable predictions and experimental designs (morphogenesis, behavior, niche construction, multi-level, meta-transitions). Part VI discusses limits and outlook, and Appendices contain categorical preliminaries, PH and sheaf details, and algorithmic estimators. Notation. We use Cfor the category of states, Tfor time/process, p : E → Cfor the viability doctrine, Gt :C → E for endogenous constraints, Telt = Ranι Φfor teleology, holim for multi-level gluing, and (νF, ω)for a final coalgebra. II. TELEONOMY VS. TELEOLOGY This section delineates teleonomy—apparent purposefulness explained by historically selected programs— from teleology as developed in this paper—present selection among futures that preserve endogenous invariants. We give precise definitions, a categorical formulation of teleology via universal properties, proofs of endogeneity and reduction to the classical (teleonomic) scalar case, and worked examples (salmon migration, lumen maintenance, niche edits). Throughout, “no mystical foresight” is enforced by adapted, non-anticipative dynamics. A. Teleonomy: program-based, selection-shaped purpose Definition II.1 (Teleonomy [ 27 , 213 , 214 ]) . A system is teleonomic when its goal-like behavior is explained by an internal program (genetic/developmental/neuronal) that was historically selected for producing
6 certain effects. Formally, let g denote (heritable) program parameters and x the phenotype/behavior. A teleonomic explanation attributes observed regularities to the composite gdevelopment/learning −−−−−−−−−−−−−−−→ xmechanism/feedback −−−−−−−−−−−−−−→ effect, together with a population process (e.g. replicator–mutator) that increases the frequency of programs producing fitness-increasing effects. In continuous time, a canonical population law on the simplex ∆n−1is ˙pi=X j pjMj→i−piX k Mi→k+pi Fi(p)−X m pmFm(p)!,(9) with mutation kernel M and (possibly frequency-dependent) performance Fi . In the potential case Fi(p) = ∂piΦ(p)(e.g. symmetric game matrix), (9) is the gradient flow of Φin the Shahshahani metric: Proposition II.2 (Replicator as gradient flow in information geometry [ 40 ]) . Let G ( p ) = diag (1 /pi )be the Shahshahani metric on int(∆n−1)and ΦaC1potential with Pipi∂piΦ(p) = hp, ∇Φi. Then ˙p= gradGΦ(p)⇐⇒ ˙pi=pi ∂piΦ(p)−X m pm∂pmΦ(p)!. Proof. The Riemannian gradient satisfies G ( p ) gradG Φ = Π T∇ Φ, where Π T projects to the tangent space of the simplex {v : Pivi = 0 } . With G−1 ( p ) = diag ( pi )and Π T∇ Φ = ∇ Φ −hp, ∇ Φ i1 , we obtain the stated coordinates. Equation (9) plus Proposition II.2 formalizes the teleonomic stance: apparent purpose follows from programs shaped by past selection (with cybernetic feedback [29, 30]). B. Teleology: endogenous invariants and universal properties We now formulate teleology as a present-time selection among futures that preserve invariants extracted from the system itself. Definition II.3 (Teleological system (categorical form)).A teleological system consists of: (i) a category of states Cwith a (lax) action of a time/process category T; (ii) a viability doctrine p:E → C(a fibration whose fibers EXencode predicates/invariants over X); (iii) an endogenous constraint functor Gt :C → E (at horizon t ) extracting invariants from the system (e.g. topological features, sheaf-gluing compatibilities, symmetry/conservation data, behavioral attractors). Definition II.4 (Teleological object) . Let ι : Bt,→ Cinclude the Gt -admissible futures and Φ : Bt→ C evaluate/embody them at time t. The teleological present of X0∈Cis the right Kan extension Telt(X0) := RanιΦ(X0).(10) Equivalently, if Cis complete, Telt(X0)∼ =lim (X0↓ι)Π −−→ Bt Φ −−→ C.(11) Interpretation. Equation (10) selects, by a universal property, the “most general present” through which all Gt-admissible futures factor. No scalar objective is added. Analytic least-violation form. In a concrete state space x∈ X ⊆ Rn with dynamics ˙x = f ( x, e )and reachable set Reacht(X0), the right-hand side of (11) reduces to the feasible completion Telt(X0) = arg min y∈Reacht(X0)Lt(y),(12) where the coherence deficit Lt is assembled from Gt (e.g. distances of witness maps to representatives of invariants, sheaf overlap residuals, symmetry charges).
7 Teleological drift (non-anticipative). We model small teleological preference as dxt=f(xt, et)dt−ε∇xLt(xt)dt+√2DdWt,(13) which is adapted to the natural filtration (no future information enters). A standard Itô argument yields a mean-descent inequality: d dtELt(xt)≈ −εEk∇Lt(xt)k2+DE[∆Lt(xt)],(14) so E [ Lt ]decreases when noise is not dominant. Thus, in this framework, “final causes” never involve future-peeking: they are realized only as universal constructions on present-time information. C. Endogeneity and reduction to teleonomy Proposition II.5 (Endogeneity and iso-invariance) . Suppose Gt is assembled from right adjoints, limits, and subobject classifiers (and natural transformations) on C. Then for any isomorphism u : X∼ −→X0 in C the square Gt(X)Gt(X0) X X0 Gt(u) u is a pullback in E . In particular, the value of Telt is invariant up to unique isomorphism under re-encoding of the same system. Proof. Right adjoints preserve limits; limits and pullbacks are invariant under isomorphism. Subobjects in a topos (or more generally with a subobject classifier) are stable under pullback. Since Gt is built from these operations and natural transformations, it preserves the iso-invariance property: Gt ( u )is an isomorphism in the fiber and the square is a pullback. The universal property of Ran then identifies Telt up to unique iso. Proposition II.6 (Teleology collapses to teleonomy (scalar case)) . If Gt returns the single scalar predicate “maximize fitness” so that Lt ( x ) = −F ( x ), then (13) becomes a (stochastic) gradient ascent of F . On the simplex, using the Shahshahani geometry, the induced mean-field dynamics is the replicator (9) (with M for mutation). Proof. With Lt = −F , (13) becomes d xt = f d t + ε∇F d t + √2D d Wt . On ∆ n−1 , restrict to the tangent space and use Proposition II.2: the teleological bias equals the Shahshahani gradient of F . Adding mutation Myields (9). No mystical foresight. Equations (10) – (13) contain no backward-in-time influence; Lt is computed from invariants available at (or estimated from) the present/near-past data. Teleology here is compatibility, not retrocausality. D. Illustrative examples (a) Salmon migration (teleonomy vs. teleology). Pacific salmon home to natal streams using multisensory programs (olfactory imprinting, magnetic/solar cues) that were historically selected [ 34 ]. This fits the teleonomic template (program ⇒ effect). The teleological reading adds: the migratory routine itself is a Gt-invariant behavior (a recurrent class). Let Pbe the Markov kernel on route states; for the viable routine A, R(∆t) = X x∈A π(x)P∆t(x, A),kP∆t−Πk ≤ C e−∆t/τmix .(15) After perturbations, R (∆ t )recovers at rate 1 /τmix , expressing behavioral teleology without invoking extra fitness scalars (cf. coalgebraic invariants §II B and [33]).
8 (b) Epithelial lumen maintenance (shape as invariant). Lumen morphogenesis/maintenance requires apico-basal polarity, junctional sealing, and fluid transport [ 36 , 192 , 193 ]. The teleonomic account cites a developmental program; the teleological account extracts the topological hole as an invariant and penalizes its collapse. A persistence-weighted hazard model predicts dwell-time scaling E[τlumen]τ0expα∆W/D,∆W=X b∈B† w(b),(16) linking longer bars (more robust cavities) to exponentially longer maintenance. This follows from a Kramers-type argument applied to the potential εLtin (13). (c) Niche edits (order-of-operations gap). Beavers building dams, or microbes secreting matrix, edit the niche. Let π :C env →Env be a fibration of worlds over environments and f : E→E0 an edit. Teleology ideally commutes with left-exact base change: f∗Ranι0Φ0∼ =Ranιf∗Φ0,(17) but when f∗fails to preserve limits one gets a measurable order effect: ∆NC := d f∗TelE0 t,TelE t(id ×f)∗X0>0.(18) This encodes “adapt → edit 6 =edit → adapt” without assuming new external objectives (cf. niche construction [38, 39]). E. Summary Teleonomy grounds apparent purpose in historically selected programs; its mathematics is the scalar (or few-scalar) case, e.g. (9) and Proposition II.2. Teleology, in contrast, derives present “ends” from endogenous invariants via universal constructions (10) – (11) , realized analytically by (12) and dynamically by (13) . The two views often agree in steady regimes; they diverge in rapid repair, routine re-formation, niche editing with order effects, and cross-level conflicts—where teleology makes quantitative predictions such as (15)–(18). III. CATEGORY THEORY PRELIMINARIES This section gathers the categorical tools used throughout: (i) categories, functors, limits/colimits; (ii) Grothendieck fibrations and Beck–Chevalley change-of-base; (iii) Kan extensions and their pointwise (co)limit formulas; (iv) enrichment over quantales (with the Lawvere metric case); (v) coalgebras and final coalgebras via final sequences; and (vi) sheaves and persistent homology (PH). We keep statements precise and include short proofs or proof sketches. Standard references include Riehl [ 41 ] and Leinster [ 42 ] for general category theory, Borceux [ 43 ] and SGA 1 [ 44 ] for fibrations, Lawvere [ 45 ] and Rosenthal [ 46 ] for enrichment/quantales, Aczel–Mendler [ 47 ] and Adámek–Milius–Velebil [ 48 ] for coalgebras, and Mac Lane–Moerdijk [ 49 ], Kashiwara–Schapira [ 50 ], Edelsbrunner–Harer [ 51 ], Ghrist [ 185 ], and Chazal– De Silva–Oudot [245] for sheaves/topology/PH. A. Categories, functors, limits and colimits Acategory Cconsists of objects Ob (C), morphisms HomC ( X, Y ), identities idX , and associative composition. A functor F :C → Dmaps objects and morphisms preserving identities and composition. A natural transformation η:F⇒Gis a family ηX:F(X)→G(X)natural in X. Limits/colimits. Given a diagram D :J → C, a limit lim D is terminal among cones to D ; a colimit colim D is initial among cocones. Special cases: terminal object 1, products, equalizers, and pullbacks (droids); dually initial object 0, coproducts, coequalizers, pushouts. Existence is assumed when stated. Pullback square (diagram). P X Y Z f g is a pullback if P∼ =X×ZY.
9 B. Grothendieck fibrations and change-of-base A functor p : E → Bis a Grothendieck fibration if for every u : b0→b in Band E∈ E with p ( E ) = b there exists a cartesian lift ¯u : E0→E with p ( ¯u ) = u such that every v : F→E over u◦w factors uniquely through ¯u over w [ 43 , § 8]. Intuitively, objects of E live over base points of B, and each base arrow uhas a good reindexing u∗:Eb→ Eb0. Beck–Chevalley (base change). Given a pullback in the base b0×bc c b0b v0 u0u v and a fibration p : E → B, the Beck–Chevalley condition asserts a canonical isomorphism u0∗ v∗∼ =v0∗ u∗ between the two reindexings when the square is a pullback and the fibration has a chosen cleavage [ 43 , § 8]. Proposition III.1 (Beck–Chevalley for fibrations) . Let p : E → Bbe a Grothendieck fibration with a split cleavage. For each pullback square in Bas above, there is a natural isomorphism of functors Eb→ Eb0×bc , u0∗ ◦v∗∼ =v0∗ ◦u∗, natural in objects and morphisms of the fiber Eb. Proof sketch. Cartesianness of chosen liftings ensures that lifting along u then along v0 equals lifting along v then along u0 : both are the (unique up to unique isomorphism) cartesian lift of the composite arrow into the pullback corner. Naturality follows from the universal property of cartesian morphisms. See [ 43 , § 8.3] or [44, Exposé V]. C. Kan extensions and pointwise formulas Let K :D → Cand F :D → Ebe functors. A right Kan extension RanKF :C → Ewith unit η:F⇒RanKF◦Kis universal among such pairs; dually for LanKF(left Kan extension) [41, § 6]. Proposition III.2 (Pointwise Kan extensions via (co)limits) . If Eis complete and Dis small, then for each c∈Cthe value of the right Kan extension is (RanKF)(c)∼ =lim(c↓K)Π −−→ DF −−→ E,(19) where ( c↓K )is the comma category of objects ( d, α : c→Kd )and arrows h : d→d0 with K ( h ) ◦α = α0 . Dually, if Eis cocomplete, (LanKF)(c)∼ =colim(K↓c)Π −−→ DF −−→ E.(20) Proof. A cone over F indexed by ( c↓K )is exactly the data needed to mediate from c to all Kd , i.e. it is the same as a natural family F ( d ) →X compatible with morphisms in ( c↓K ). The universal property of the limit yields RanKF; see [41, Thm. 6.28] or [42, § 3]. The left case is dual. Comma diagram (for (19)). (c↓K)D E C Π forget α F RanKF D. Enrichment and quantales Aquantale (V ,≤,⊗, I )is a complete lattice with a monoidal structure ⊗ that distributes over arbitrary joins in each variable [ 46 ]. A V-enriched category Aassigns objects Ob (A)and hom-objects A( x, y ) ∈ V with unit I≤A(x, x)and composition A(y, z)⊗A(x, y)≤A(x, z)satisfying associativity/unitality [45].
16 F. Teleological drift and consequences We adopt the (mathematically standard) diffusion convention dxt=f(xt, et)dt−ε∇Lt(xt)dt+√2DdWt,(31) where D > 0is the diffusion scale and Wt a standard Wiener process. (Some literatures absorb the factor into a noise amplitude σ; we keep √2Dfor direct Fokker–Planck correspondence.) Fokker–Planck equation and stationary law. Let p(x, t)be the density of xt. Then ∂tp=−∇·(f−ε∇Lt)p+D∆p. (32) If f=−∇Φ(gradient baseline) and Ltis quasi-static, the Gibbs–Boltzmann stationary solution is p∗(x)∝exp−Φ(x) + εLt(x) D,(33) obtained by setting the probability flux to zero in (32) and integrating. Mean-descent inequality. Itô’s formula for u(x, t) = Lt(x)yields, after taking expectations, d dtELt(xt)≈ −εEk∇Lt(xt)k2+DE∆Lt(xt).(34) Hence, for small Dor locally convex Lt,E[Lt]decreases in time. Large deviations (rare escapes). In the small-noise limit D↓ 0, exit probabilities and transition times concentrate according to large-deviation principles governed by an action functional; teleological bias modifies the quasipotential by εLt(cf. [64]). G. Discrete-time teleological projection (proximal step) For numerical schemes or discrete-time models, a one-step teleological projection can be written as xk+1 = arg min y∈Reachδt(xk) Dψ(ykxk) |{z } geometry +ε δt Lt(y),(35) where Dψ is a Bregman divergence induced by a strictly convex ψ (e.g. squared Euclidean distance, KL on the simplex). This is a standard proximal step [ 61 ]. In the small-step limit and for Euclidean Dψ , (35) recovers a forward Euler discretization of (31) with D = 0; on the simplex, using KL yields a natural-gradient update [60]. H. Compact expression and usage Summarizing, Lt(x) = X b∈Θt(X) wt(b) distbρb(x),Rep(b)+λsheaf Eglue(x) + λsym Esym(x),(36) and the teleological drift is dxt=f(xt, et)dt−ε∇Lt(xt)dt+√2DdWt.(37) In practice: compute Θ t and witnesses ρb from data; assemble Lt via (36) ; evolve by (37) or project by (35) ; and read off predictions (stationary law (33) , mean descent (34) , persistence-weighted dwell times, etc.).
17 VI. VARIATIONAL FORMULATION We now develop a variational view of the teleological calculus. Starting from the endogenous invariant extractor Gt (Sec. V), we define a teleological action functional whose extremals solve a two–boundary problem: given an initial condition and a terminal viability constraint (derived from GT ), the admissible trajectories are those that minimize a least–violation cost. We derive the Euler–Lagrange equations with endpoint constraints, cast the dynamics in Hamiltonian form, and connect to path-integral weights (Onsager–Machlup for diffusions and Feynman-type weights by analytic continuation). Finally, we relate the boundary-value structure to Fermat/Hamilton principles and to the quantum two-state vector (TSVF) formalism—at the level of mathematics, not physics. A. Teleological action functional Let X ⊆ Rn be a (smooth) state manifold, f : X × [0 , T ] →Rn a given baseline drift (possibly f(x, t) = f(x, et)for an exogenous process et), and Lt:X → [0,∞)the coherence deficit induced by Gt (§V). For an absolutely continuous path x: [0, T ]→ X, define the Lagrangian L(x, ˙x, t) := 1 2k˙x−f(x, t)k2+εLt(x),(38) and the teleological action ST[x] := ZT 0L(x(t),˙x(t), t)dt. (39) The boundary data are x(0) = x0, x(T)∈ΣT:= bGTc⊆X,(40) where Σ T is the terminal viability set (comprehension of GT ). Thus, among all trajectories starting at x0 and ending on ΣT, the admissible ones are the minimizers of ST: minimize x(·) ST[x]subject to x(0) = x0, x(T)∈ΣT.(41) This is a classical calculus-of-variations problem with a free terminal point constrained to a submanifold (or more general closed set) [68, 70, 71, 251]. B. Euler–Lagrange equations and endpoint transversality We compute the first variation. Let η∈C1 ([0 , T ]; Rn )be a variation with η (0) = 0 and η ( T ) ∈Tx(T) Σ T (tangent to the terminal constraint). Then δST[x;η] = ZT 0D˙x−f, ˙η−Dxf(x, t)ηEdt+εZT 0∇xLt(x), ηdt =hh˙x−f, ηiiT 0−ZT 0Dd dt˙x−f+ (Dxf)>( ˙x−f)−ε∇xLt(x), ηEdt −ZT 0∂tf(x, t), ηdt, (42) where we used integration by parts, the chain rule, and Dxf for the Jacobian. Stationarity δST = 0 for all such ηyields: Euler–Lagrange (bulk) equation. ¨x−Dxf(x, t) ˙x+ (Dxf(x, t))>( ˙x−f(x, t)) −∂tf(x, t) = ε∇xLt(x).(43) Equivalently, ¨x+(Dxf)>−Dxf˙x−(Dxf)>f−∂tf=ε∇xLt(x).(44)
18 Transversality (terminal boundary condition). The boundary term in (42) is h˙x ( T ) −f ( x ( T ) , T ) , η ( T ) i . For all η(T)∈Tx(T)ΣTthis must vanish, i.e. ˙x(T)−f(x(T), T)⊥Tx(T)ΣT.(45) If ΣT={x:g(x)=0}is a smooth constraint, there exists a multiplier λ∈Rmwith ˙x(T)−f(x(T), T) = Dg(x(T))>λ, g(x(T)) = 0.(46) For inequality constraints h ( x ( T )) ≤ 0, complementarity conditions λ≥ 0, λ>h ( x ( T )) = 0 complete the KKT system (free endpoint variational KKT [68]). Special cases. If f = −∇ Φ( x )(gradient baseline) and Φis C2 , then ( Dxf ) > = Dxf and the ˙x -term cancels in (44), yielding ¨x−D2Φ(x)∇Φ(x) = ε∇xLt(x),(47) a Newton-type equation with teleological force ε∇Lt. C. Hamiltonian form Define the canonical momentum p:= ∂L ∂˙x= ˙x−f(x, t).(48) The Legendre transform H(x, p, t) := p·˙x−L (with ˙x=p+f) gives H(x, p, t) = 1 2kpk2+p·f(x, t)−εLt(x).(49) Hamilton’s equations follow ([70, 71, 251]): ˙x=∂H ∂p =p+f(x, t),˙p=−∂H ∂x =−(Dxf(x, t))>p+ε∇xLt(x),(50) with terminal transversality p ( T ) ⊥Tx(T) Σ T (or p ( T )=( Dg ( x ( T ))) >λ for smooth constraints). The pair ( x, p )is the classical analog of a forward state and a backward costate in optimal control, but note the absence of any exogenous objective: the “potential” εLtis built from Gt. D. Path-integral weights and Onsager–Machlup Consider the teleological SDE dxt=f(xt, t)−ε∇xLt(xt)dt+√2DdWt,(51) with initial condition x (0) = x0 and terminal conditioning x ( T ) ∈ Σ T . The small-noise path probability admits an Onsager–Machlup (OM) representation [72, 75, 76]:[259] Px(·)∝exp−1 4DZT 0 ˙x−f+ε∇Lt 2dt−1 2ZT 0∇·(f−ε∇Lt)dt.(52) Thus the most probable path between constrained endpoints minimizes the OM action. Neglecting the divergence (or absorbing it into a boundary term) and expanding the square, one recovers the quadratic kinetic form plus the teleological potential εLt as in (39) . In imaginary-time quantum formalism, replacing 1/(2D)↔1/~connects (52) to a Feynman weight ei ~R(T−V) dt[74–76]. E. Relation to Fermat/Hamilton principles Fermat-type principle (optics). Let k˙xk = v be fixed (arc-length parametrization) and f≡ 0. If Lt ( x ) = n ( x )(refractive index), then ST [ x ] = RT 0n ( x )d s (optical path length). The Euler–Lagrange equation reduces to Snell’s law and geodesics in the metric n2 d s2 [ 77 ]. Teleology thus generalizes Fermat: with nreplaced by the (endogenous) Lt, rays bend toward regions that keep invariants true.
19 Hamilton’s principle. With f = −∇ Φand Lt ( x ) = V ( x ), (39) becomes RT 01 2k˙xk2−˙x·∇ Φ+ 1 2k∇ Φ k2 + εV d t . Up to a total time derivative, this is RT 01 2k˙xk2−U ( x ) d t with U = −1 2k∇ Φ k2−εV , yielding Newton’s equations. Hence the teleological term acts as a derived potential determined by Gt. F. Two–boundary conditioning and TSVF analogy The variational problem (41) has an initial (pre-selected) condition x (0) = x0 and a terminal (postselected) constraint x ( T ) ∈ Σ T . In stochastic form, the conditioned path measure is proportional to the product of a forward density and a backward harmonic function (Doob h -transform). In quantum mechanics, the analog is the Aharonov–Bergmann–Lebowitz two-state vector hψf||ψii [ 78 , 79 ]. Our setting remains classical/real: the teleological “post-selection” is exactly the constraint Σ T (derived from GT ), and the extremals are the classical trajectories minimizing (39) . The mathematical similarity is the two–boundary character: a forward state x(·)and a backward costate p(·)coupled by (50) and (45). G. Summary The teleological calculus admits a clean variational form: • The teleological action ST [ x ] = RT 01 2k˙x−fk2 + εLt ( x ) d t produces Euler–Lagrange equations (43) and transversality (45). • In Hamiltonian variables p = ˙x−f , (50) is a first-order two–boundary system, with p ( T ) ⊥Tx(T) Σ T . • For diffusions, the Onsager–Machlup weight (52) singles out most probable teleological trajectories between constrained endpoints. • Fermat/Hamilton principles are recovered as special cases, and the two–boundary structure mirrors TSVF at the level of boundary data (without invoking quantum dynamics). VII. COALGEBRAIC TELEOLOGY We develop the coalgebraic form of teleology, in which behaviors are captured as elements of a final coalgebra and the teleological attractor arises as a greatest fixed point of a monotone operator. This setting turns “behavioral routines” into canonical factorizations through a largest invariant subobject A?⊆νF constrained by Gt. A. Coalgebras, final coalgebras, and behavioral semantics Let Cbe a category and F:C→Can endofunctor. Definition VII.1 (Coalgebra; morphism) . An F -coalgebra is a pair ( X, ξ )with ξ : X→FX . A morphism of coalgebras h: (X, ξ)→(Y, γ)is a map h:X→Ywith F h ◦ξ=γ◦h. Definition VII.2 (Final coalgebra) . A coalgebra ( νF, ω )is final if for any ( X, ξ )there exists a unique coalgebra morphism hξ : X→νF such that ω◦hξ = Fhξ◦ξ. We call hξ the behavioral semantics of (X, ξ). In concrete cases, νF collects complete behaviors: e.g. for streams FX = A×X , νF ∼ =Aω ; for deterministic automata FX = 2 ×XA , νF ∼ = 2 A∗ (languages); for labeled transition systems FX = ( PX ) A , νF contains (generally infinitary) behavior trees. Existence of νF holds under standard accessibility hypotheses (e.g. Clocally presentable, Faccessible and preserving limits of inverse ω-chains).
20 B. Endogenous behavioral constraints and the operator Φ Let ( νF, ω )be final. The endogenous constraints extracted by Gt (Sec. IV) determine a subobject of admissible behaviors: JGtK,→νF. We work in a setting where •Sub(νF)(the poset of subobjects of νF) is a complete lattice, •Fpreserves monomorphisms (so direct images of subobjects are well-defined). For a subobject S ,→νF , write it as a mono iS : S→νF . Then FiS : FS →FνF is mono; its inverse image along ω:νF →FνF is again a subobject of νF, denoted ω−1(FS) := {b∈νF |ω(b)∈F(S)},→νF. Define the teleological operator Φ : Sub(νF)→Sub(νF),Φ(S) := ω−1(FS)∩JGtK.(53) Lemma VII.3 (Monotonicity).If S⊆Tthen Φ(S)⊆Φ(T). Proof. S⊆T⇒FS ⊆F T since F preserves monos; hence ω−1 ( FS ) ⊆ω−1 ( FT ). Intersecting with a fixed JGtKpreserves inclusion. C. Greatest fixed point: the teleological attractor By Knaster–Tarski on the complete lattice Sub(νF),Φhas a greatest fixed point. Definition VII.4 (Teleological behavioral attractor).The teleological attractor is A?:= νΦ = \{S⊆νF |S⊆Φ(S)}.(54) Proposition VII.5 (Characterizations of A? ) . (a) A? = Φ( A? ). (b) A?⊆JGtK . (c) ω ( A? ) ⊆F ( A? ) (forward invariance). (d) If S⊆JGtKand ω(S)⊆F(S), then S⊆A?. Proof. (a) Fixed point property by definition. (b) From (53) . (c) A? = ω−1 ( FA? ) ∩JGtK implies inclusion. (d) The hypotheses say S⊆ Φ( S ), hence S is a post-fixed point; the gfp contains all post-fixed points. Descending approximation. Let S0 = JGtK , Sα+1 = Φ( Sα ), and for limit ordinals Sλ = Tα<λ Sα . Then S0⊇S1⊇ ··· stabilizes at A?(at some ordinal ≤ |Sub(νF )|). D. Coinduction and factorization of behavioral routines Theorem VII.6 (Coinduction principle).If R ,→νF is a subobject such that R⊆Φ(R), then R⊆A?. Proof. Immediate from the gfp characterization A?=T{S|S⊆Φ(S)}. Theorem VII.7 (Factorization through A? ) . For any coalgebra ( X, ξ )with unique semantics hξ : X→ νF , if hξ [ X ] ⊆JGtK and hξ [ X ] ⊆ω−1 ( F ( hξ [ X ])) (i.e. the image is forward-invariant), then hξ [ X ] ⊆A? . Equivalently, there exists a unique coalgebra morphism ¯ hξ:X→A?with X νF A? hξ ¯ hξi where i:A?,→νF is the inclusion. Proof. Let S = hξ [ X ] ⊆νF . By hypothesis S⊆JGtK and ω ( S ) ⊆F ( S ), so S⊆ Φ( S ). By Theorem VII.6, S⊆A? . Since hξ factors through its image, we get a unique ¯ hξ with i◦¯ hξ = hξ . Uniqueness follows from mono iand uniqueness of hξ.
21 Interpretation. Abehavioral routine is a coalgebra whose semantics lands in A? ; by Theorem VII.7, its behavior factors through the attractor. Thus routines are precisely those behaviors that are simultaneously admissible (JGtK) and forward-invariant under ω. E. Concrete examples (1) Streams (deterministic routine). Let FX = A×X ; νF ∼ =Aω with ω : σ7→ ( head ( σ ) ,tail ( σ )). Suppose JGtK⊆Aω specifies a regularity (e.g. fixed period p , bounded run-lengths, or a prescribed subshift). For S⊆Aω, Φ(S) = {σ∈JGtK|tail(σ)∈S}. Hence A? is the largest subshift included in JGtK (closed under tail). Behavioral routines are precisely the orbits contained in that subshift. (2) Deterministic automata. Let FX = 2 ×XA (Moore automata). The final coalgebra is 2 A∗ with ω(L)=(1∈L, a 7→ a−1L). If JGtKconstrains languages (e.g. safety property), then Φ(S) = {L∈JGtK| ∀a∈A, a−1L∈S}, so A?is the largest suffix-closed subset of JGtK. (3) Probabilistic routines (discrete-time Markov). Let X be finite; take the (deterministic) coalgebra ( P ( X ) , ξ )with ξ ( A ) = {x∈X : P ( x, A )=1 } to capture almost-sure next-step invariance of sets. If JGtK encodes viability (e.g. acceptable sets), then A? collects the largest almost-surely forward-invariant sets included in JGtK . Return probabilities R (∆ t ) = Px∈A?π ( x ) P∆t ( x, A? )recover to 1 at a rate given by the mixing time on A?(see [88]). F. Logical views: modal descriptions and coinduction Coalgebraic modal logics provide syntaxes whose semantics are invariant under behavioral equivalence. If JGtK is definable by a (coalgebraic) modal theory Tt , then Φcorresponds to closing under “next”-modalities, and A? is the largest model closed under those modalities and included in Tt . Bisimulation-proof principles (coinduction) certify membership in A?(see [82–85]). G. Regularity and computability On finitary functors F (preserving filtered colimits), the descending chain Sn+1 = Φ( Sn )starting at S0 = JGtK stabilizes in at most ω steps on finitary subobjects; decision procedures reduce to checking closure under one-step predecessor and the constraint. For streams/automata examples, this yields standard fixpoint algorithms (e.g. safety games in model checking [ 89 ]). For probabilistic kernels on finite spaces, computing the maximal absorbing class inside JGtK reduces to graph algorithms plus basic linear algebra. H. Summary Coalgebraic teleology treats behaviors in a final-coalgebra universe. The endogenous-constraint subobject JGtK induces a monotone operator Φ( S ) = ω−1 ( FS ) ∩JGtK . Its greatest fixed point A? = ν Φ is the largest forward-invariant set of behaviors consistent with Gt . Behavioral routines are exactly those systems whose unique behavioral semantics factor through A? . This yields both a conceptual picture (routines =coinductive invariants) and practical algorithms (descending fixpoint) with standard correctness guarantees. VIII. NICHE AND MULTI-LEVEL We formalize two structural extensions of the teleological calculus: (i) niche-aware teleology via base change along an environment fibration and the associated Beck–Chevalley isomorphisms and holonomy
22 gaps; and (ii) multi-level teleology by gluing teleologies across levels (genes → cells → organs → organisms → groups) as a homotopy limit ( holim ) of a level-wise diagram. Finally, we encode meta-doctrine updates as the Eilenberg–Moore semantics of a monad M acting on the category of doctrines, capturing major transitions when the very representation of invariants changes. A. Environment fibrations and base change Let Env be a category of environments (objects are environments E , morphisms f : E→E0 are edits of the niche: resource additions, geometric/chemical modifications, social structure changes, etc.). A world fibred over environments is a Grothendieck fibration π:Cenv −→ Env, whose fiber C E := π−1 ( E )is the category of states in environment E . For each edit f : E→E0 , choose a (split) reindexing functor f∗:CE0−→ CE, cartesian with respect to π. The viability doctrine is fibred as well: p:Eenv −→ Cenv with pE:EE→CE. Endogenous constraints are computed fiberwise: GE t:CE→ EE, assembled from canonical right-adjoint/limit constructions internal to the fiber. Teleology in a fixed environment. For a fixed E , write ιE : BE t,→ C E for the inclusion of GE t -admissible futures and Φ E : BE t→ C E the evaluation at horizon t . The environment-relative teleology is the right Kan extension TelE t:= RanιEΦE:CE−→ CE,TelE t(X0)∼ =lim(X0↓ιE)→BE t ΦE −−→ CE.(55) Beck–Chevalley for teleology (base-change compatibility). Consider a pullback square in Env, E0×EF F E0E v0 u0u v and assume πis a split fibration whose reindexings u∗, v∗, u0∗, v0∗ preserve the limits in (55). Then there is a canonical Beck–Chevalley isomorphism of teleologies: u0∗ RanιEΦE∼ =RanιE0×EFu0∗ΦE, v0∗ RanιFΦF∼ =RanιE0×EFv0∗ΦF.(56) Proposition VIII.1 (Beck–Chevalley for Tel ) . Under the hypotheses above, for each X0∈ C E0×EF the canonical mates (56) are isomorphisms. In particular, when F=Eand v= idE, u∗ TelE t∼ =TelE0 t◦u∗.(57) Proof sketch. Use the pointwise formula Ranι Φ( c ) ∼ =lim ( c↓ι ) →BΦ −→ C . Pulling back along a cartesian arrow identifies ( u∗c↓ιE0 )with the reindexed comma category, and preservation of limits by u∗ gives the isomorphism. The naturality squares are exactly Beck–Chevalley squares; see [ 90 – 92 ] for fibered/bicategorical treatments, and [93] for exact squares guaranteeing mates are isomorphisms.
23 Holonomy and order-of-operations gaps. Let γ : E0→E1→ ··· → Ek = E0 be a loop in Env . Transport around γ induces an endofunctor Tγ := f∗ 1◦ ··· ◦ f∗ k :C E0→ C E0 . If all intermediate Beck–Chevalley conditions hold and reindexings preserve the relevant limits, then Tγ◦TelE0 t∼ =TelE0 t◦Tγ⇒no holonomy (commuting transport). When some square fails exactness, define the holonomy gap at X0as ∆hol(γ;X0) := d TγTelE0 t(X0),TelE0 tTγ(X0),(58) measured by any fiberwise metric or divergence (e.g. an enriched cost induced by Lt ). The gap is detectable whenever the Beck–Chevalley mate along some square in γis non-invertible. Proposition VIII.2 (Lower bound by mate defect) . Let θf : f∗Ranι Φ ⇒Ranιf ( f∗ Φ) be the Beck– Chevalley mate for an edit f . If k·k is an operator norm induced by the enriched metric, then along a loop γ, ∆hol(γ;X0)≥c Y f∈γ θf−id , for some c > 0depending on the Lipschitz constants of the reindexings and Tel . Thus any failure of exactness yields a positive gap. Proof idea. Express both sides of (58) as limits over reindexed comma diagrams and compare using the composition of mates; the difference is bounded below by the norm of the deviation from identity. The constant cabsorbs Lipschitz bounds and diameter constraints of the indexing diagrams. B. Multi-level gluing as a homotopy limit Let Lvl be a small category of levels (e.g. a finite poset G→C→O→Org →Grp for genes, cells, organs, organism, group). Suppose we have a diagram of world-fibers C:Lvl −→ Cat, ` 7→ C(`), with restriction functors R`→m :C (`)→ C (m) for `→m in Lvl (e.g. “forget” from organs to cells, or aggregate from cells to organ traits). Each level carries a doctrine p(`) : E(`)→ C (`) and an endogenous constraint functor G(`) t:C(`)→ E(`). The level-wise teleologies are Tel(`) t:= Ranι(`)Φ(`):C(`)→C(`). The level diagram of teleologies. Assemble a diagram Tt:Lvl −→ Cat, ` 7→ C(`),(`→m)7→ R`→m, equipped with a cone of natural transformations {Tel(`) t⇒R`→mTel(`) t} witnessed by compatibility data (e.g. marginalization/aggregation maps). Definition VIII.3 (Multi-level teleology).The multi-level teleology is the homotopy limit Telmulti t:= holim `∈Lvl Tel(`) t,(59) computed in a suitable model for Cat (e.g. as a simplicially enriched category or via a projective model structure on diagrams). Theorem VIII.4 (Existence and universal property of holim ) . If each C (`) admits the limits in (55) , the restriction functors R`→m are right Quillen (or, in a 1-categorical setting, preserve the relevant limits), and Lvl is small, then Telmulti t exists. It represents the terminal cone into the diagram Tt up to weak equivalence. Proof sketch. Use a projective model structure on the functor category [ Lvl,Cat ]or a simplicial framing; levelwise completeness and right Quillen restrictions ensure objectwise fibrancy; then holim is computed by a homotopy end or a Bousfield–Kan formula [94–98].
24 Analytic realization and uniqueness. Let x(`)∈ X(`)⊆Rn`be concrete coordinates and L(`) t(x(`))and C`→mx(`), x(m)(≥0) be, respectively, the level deficit and a coupling penalty quantifying mismatch under restriction R`→m . Consider the multi-level functional Lmulti t(x(`))`:= X `L(`) t(x(`)) + X `→m λ`m C`→mx(`), x(m).(60) Proposition VIII.5 (Existence/uniqueness of the glued minimizer) . If each L(`) t is proper, lower semicontinuous, and µ` -strongly convex, and each C`→m is convex and jointly α`m -smooth, then Lmulti t is strongly convex and has a unique minimizer. This minimizer represents the point of Telmulti t in the concrete model, and it depends Lipschitzly on the data. Proof. Strong convexity is preserved under sums; smooth coupling ensures a unique critical point. The minimizer is the image of the homotopy-limit cone in the concrete (enriched) model; Lipschitz dependence follows from standard perturbation bounds for strongly convex problems. A variance/synergy inequality. Let observables Y(`) at each level be Lipschitz with constant L` in x(`) . If Lmulti t satisfies a Poincaré inequality with constant cP under the steady-state law of the teleological diffusion, then Var"X ` Y(`)#≤cPEk∇P`Y(`)k2≤cPP`L2 `Ek∇L(`) tk2.cP ε d dtELmulti t,(61) linking cross-level output variability to descent of the multi-level deficit (cf. Lemma IV.8 at a single level). C. Meta-doctrine updates via Eilenberg–Moore Let Doct be the category of doctrines (e.g. fibrations p : E → Cequipped with admissible constructions). Ameta-update is an endofunctor M:Doct −→ Doct that adds representational capacity (e.g. new predicates, new levels, new invariants). Assume M carries a monad structure (M, η, µ). Stabilized doctrine and teleology. The category of Eilenberg–Moore algebras DoctM collects stabilized doctrines (p, α :pM→p)compatible with M. For (p∞:E∞→C∞, α)∈DoctM, define G∞ t:C∞→ E∞,Tel∞ t:= Ranι∞Φ∞. Theorem VIII.6 (Universality of stabilized teleology) . Let ( p, α )be an M -algebra and U : DoctM→Doct the forgetful functor. Then Tel∞ t is initial among teleologies computed in doctrines equipped with an M -algebra structure that receive a comparison map from ( p, α ). In particular, any teleology in a pre-update doctrine factors uniquely through Tel∞ tonce the update is absorbed. Proof sketch. Transport the Kan-extension data along the algebra structure α and use the Eilenberg–Moore universal property (algebra morphisms are exactly the M -compatible comparisons). The factorization of teleologies is induced by the universal property of the right Kan extension computed in the EM object [99–101]. A capacity criterion and step-drop. Let κ quantify representational capacity (e.g. dimension of an invariant space, number of levels). Suppose the achievable deficit ¯ L(κ)is differentiable. If ∂¯ L ∂κ <−λ, (62) where λ is the marginal cost of capacity, then an update increasing κ is favored. Under mild regularity, the post-update steady state satisfies EL∞ t≤ELt−(λ+δ)∆κ, (63) for some δ > 0capturing synergies in the EM algebra (improved coupling of invariants), i.e. a step drop in expected deficit at the transition.
25 D. Examples Niche edit vs. adapt first (order effect). Let Ef −→ E0 add a resource; let x7→ TelE t ( x )be computed with Esym fixed. If f∗is not left exact for the relevant limits, then generally f∗ TelE0 t(X0)6∼ =TelE tf∗X0, producing a holonomy gap. Empirically, microbial colonies exhibit “build matrix → adapt” 6 =“adapt → build matrix”: the Beck–Chevalley mate detects the asymmetry. Gene → cell → organ gluing. Let Lvl = {G→C→O} with restrictions RO→C, RC→G . Take CO→C = kRO→C ( x(O) ) −x(C)k2 and CC→G = kRC→G ( x(C) ) −x(G)k2 . Proposition VIII.5 guarantees a unique glued minimizer of Lmulti t; this concretely realizes the holim as a coupled least-squares problem. Meta-update: adding recombination as doctrine. Before the transition, E only expresses clonal inheritance; after applying M , E∞ adds constraints “defined up to recombination equivalence.” The EM teleology Tel∞ t computes present compatibility in the recombination-stable doctrine; by (63) , expected deficit drops if recombination reduces mismatch penalties faster than it costs to maintain. E. Summary Niche-aware teleology is a fibred construction: teleology is computed in fibers and transported along environment edits, with Beck–Chevalley giving conditions for base-change invariance; failures of exactness manifest as holonomy gaps. Multi-level teleology is a homotopy limit of level-wise teleologies, with analytic realizations as strongly convex glued minimizations and variance/synergy bounds. Meta-doctrine updates are formalized by Eilenberg–Moore algebras of a monad M , providing universal stabilized teleologies and a capacity criterion for major transitions. IX. L0: SCALAR FITNESS The lowest rung of the ladder identifies the trivial endogenous-constraint choice in which the only invariant is a single scalar fitness. In the categorical language of Secs. IV–V, this is the collapse Gt(X)≡“maximize a scalar” ⇐⇒ Lt(x) = −F(x), so that teleology prefers states with higher F . In population models on the simplex, this recovers the replicator and replicator–mutator equations of classical evolutionary dynamics. We now give self-contained derivations that place L0 firmly inside the calculus. A. Replicator dynamics as the L0 teleology Let p = ( p1, . . . , pn )be type frequencies on ∆ n−1 = {p∈Rn ≥0|Pipi = 1 } . Let f ( p ) = ( f1 ( p ) , . . . , fn ( p )) be (possibly frequency-dependent) fitnesses and ¯ f ( p ) = Pipifi ( p )their mean. The L0 choice takes Lt(p) = −¯ f(p),(64) so that higher mean fitness corresponds to lower deficit. Geometry on the simplex. Teleology evolves on the constraint manifold Pipi = 1. The natural (Fisher–Shahshahani) Riemannian metric on int(∆n−1)is hu, vip= n X i=1 uivi pi , Tp∆n−1=nu∈Rn:X i ui= 0o.(65) Its associated natural gradient of a smooth scalar Φ( p )is the unique tangent vector grad Φ( p )satisfying hgradΦ(p), uip=∇Φ(p)·ufor all u∈Tp∆n−1. A direct computation gives (gradΦ)i=pi∂piΦ−X k pk∂pkΦ.(66)
32 F. Summary L2 encodes morphology through PH (shape invariants) and sheaves (compatibility). Teleology favors futures that keep cavities and gluing consistent. The theory predicts an Eyring–Kramers dwell-time law in which bar lengths and gluing energies control barrier heights, and cohomological changes produce discrete obstruction jumps. These claims are quantitative, testable, and follow from standard results in PH, sheaf/Hodge theory on covers, and metastability of diffusions. XII. L3: BEHAVIORAL ATTRACTORS At L3 the endogenous constraint Gt selects behavioral invariants: routines that, once formed, are preserved by the one–step evolution. The right categorical language is coalgebra. We specialize the general operator from Sec. VII to concrete dynamics (deterministic, nondeterministic, or Markovian), derive the largest invariant attractor as a greatest fixed point, and connect it to return probabilities and mixing times. This yields quantitative, testable relations between the strength of endogenous routine-keeping and dwell/return statistics. A. Coalgebraic invariants specialized Let F :C → Cbe a one–step dynamics functor (e.g. FX = X for deterministic maps, FX = P ( X ) for transition systems, FX = D ( X )for discrete-time Markov kernels on a finite set X ). An F -coalgebra (X, ξ)models the system: ξ:X→FX encodes the next-step evolution. Assume F admits a final coalgebra ( νF, ω )and that the endogenous constraint Gt singles out a subobject JGtK,→νF of admissible behaviors at horizon t (Sec. VII). The teleological operator on subobjects S ,→νF is Φ(S) := ω−1(FS)∩JGtK,(87) and its greatest fixed point is the teleological behavioral attractor A?:= νΦ = \{S⊆νF |S⊆Φ(S)},(88) the largest subset of admissible behaviors closed under the one–step evolution. Proposition XII.1 (Coinductive characterization) . A behavior b∈νF lies in A? iff there exists a decreasing chain S0⊇S1⊇ ··· with S0 = JGtK and Sn+1 = Φ( Sn )such that b∈Tn≥0Sn . Equivalently, A?is the largest subobject contained in JGtKthat is forward invariant:ω(A?)⊆F(A?). Proof. Immediate from the Knaster–Tarski construction of the greatest fixed point for monotone Φon the complete lattice Sub(νF). Markov specialization. Let ( X, P )be a finite Markov chain (row-stochastic P ). Regard FX = D ( X ), with coalgebra map ξ ( x ) = P ( x, · ). Write B = XN for path space with shift σ ; this carries a final coalgebra for the functor F that sends a path to its head distribution and tail [ 118 , Ch. 1]. The admissible behaviors JGtKmay be given by temporal logic constraints or—in the simplest case—by “stay within A” for some A⊆X. Then Φ(S) = {β∈JGtK:σ(β)∈S} A?={β∈JGtK:σn(β)∈JGtK∀n},(89) i.e. behaviors that remain admissible at all times. Projected to states, A? corresponds to the union of all closed communicating classes contained in A .[ 260 ] Thus, L3 selects the largest recurrent routine consistent with Gt. B. Return probabilities and mixing times Fix a nonempty A⊆X and suppose the chain on X is irreducible and aperiodic with stationary distribution π. Let τA:= inf{t≥0 : Xt∈A},τ+ A:= inf{t≥1 : Xt∈A}.
33 Return probabilities. For an initial law µsupported on A, the t-step return probability to Ais Rµ A(t) := X x∈A µ(x)Pt(x, A).(90) If µ=πA(the stationary distribution conditioned on A), Kac’s lemma yields the mean return time Eπτ+ A|X0∈A=1 π(A)and lim t→∞ RπA A(t) = π(A),(91) i.e. the long-run fraction of visits equals π(A)[118, Thm. 1.7.6]. Mixing times. Let kν−πkTV := 1 2Px|ν(x)−π(x)|. The (total-variation) mixing time is tmix(ε) := inf{t≥0 : max xkPt(x, ·)−πkTV ≤ε}.(92) For reversible Pwith spectral gap γ:= 1 −λ2(P), one has the classical bounds 1 2γlog 1 2ε≤tmix(ε)≤1 γlog 1 ε πmin ,(93) where πmin = minxπ ( x ); see, e.g., [ 119 , Ch. 12] and [ 120 , Ch. 2]. Moreover, the conductance of a set S⊆X, φ(S) := Q(S, Sc) π(S)with Q(S, Sc) = X x∈SX y∈Sc π(x)P(x, y),(94) controls both the spectral gap and exit behavior via Cheeger-type inequalities [ 121 ]: φ2 2≤γ≤ 2 φ , where φ:= minS:π(S)≤1/2φ(S). Exit/dwell from a routine. Let A be a routine region (e.g. the projection of A? to states). Starting from πA, the one-step exit probability is exactly φ(A), hence EπA[τAc]≥1 φ(A).(95) If, additionally, the chain restricted to A is rapidly mixing with spectral gap γA , then for times t 1 /φ ( A ), Rµ A(t)≥1−C e−γAt(µsupported in A),(96) i.e. fast forgetting within the routine and slow exit from it. These inequalities quantify the empirical signature of a behavioral attractor: high short-term return, long dwell, and a separation of timescales governed by γA(internal mixing) and φ(A)(escape). C. Teleological bias and quantitative predictions Teleology at L3 acts by biasing the chain to keep the routine invariant. A simple model is to rescale boundary transitions by a factor e−ε∆Larising from the coherence deficit (Sec. V): e P(x, y) := P(x, y)e−ε∆L(x,y) Z(x), y 6=x, 1−Pz6=xe P(x, z), y =x, Z(x) = X z6=x P(x, z)e−ε∆L(x,z).(97) If ∆L(x, y)>0for exits x∈A,y∈Ac, then conductance decreases: e φ(A)≤e−ε∆φ(A),∆ := inf x∈A, y∈Ac∆L(x, y),(98) and hence E˜πA[eτAc]&e+ε∆1 φ(A).(99) Thus the expected dwell time in a routine increases exponentially with the minimal deficit jump across the boundary. In reversible settings, the internal spectral gap γA of the restricted chain can improve as incoherent micro-transitions are attenuated, further sharpening the two-timescale picture (96).
34 Proposition XII.2 (High-return regime) . Let the chain be reversible and lazy. Suppose (i) the restricted chain on A has gap γA> 0; (ii) the teleological tilt (97) reduces the exit conductance as in (98) . Then there exist constants C, c > 0such that, for t∈[c/γA, c/e φ(A)] and any µsupported on A, Rµ A(t)≥1−C e−γAt. In particular, the window of near-certain return enlarges by a factor &e+ε∆compared to baseline. Sketch. After a time t&c/γA the law inside A is within O ( e−γAt )of πA by (93) . The exit probability per step under e P is e φ ( A )when the law is πA ; a union bound over t steps with e φ ( A ) t≤c gives the claim. Continuous-time diffusion analogue. For the SDE d xt = −∇ Φd t−ε∇Lt d t + √2D d Wt , a routine A corresponds to a potential well. The mean exit time obeys Eyring–Kramers-type laws, scaling as ∼exp{ ∆(Φ + εLt ) /D} ; teleology increases the barrier by ε∆ , yielding the same exponential behavior as (99). D. From coalgebra to statistics: what to measure The coalgebraic attractor A?predicts three empirical signatures: (a) Closure. Estimated one–step maps from data satisfy bω ( A? ) ⊆b F ( A? )(no observed exits from A? within the accuracy of the estimator). (b) High return. For windows t 1 /e φ ( A? ), return probabilities RA? ( t )are near 1, with decay controlled by the internal gap γA?. (c) Dwell scaling. Under manipulations that increase the cross-boundary deficit (e.g. strengthening of an invariant), the mean dwell time in the routine increases exponentially as in (99). E. Summary L3 lifts teleology to the space of behaviors. The greatest fixed point A? = ν Φis the largest admissible and forward-invariant behavior set. In finite-state Markov models, the projection of A? is the union of closed classes contained in the admissible region, and the quantitative fingerprints of a routine—high short-term return, long dwell, two-timescale mixing—are governed by conductance and spectral gaps. Teleological bias shrinks boundary conductance and enlarges dwell windows exponentially in the deficit jump across the boundary. XIII. L4: NICHE CONSTRUCTION At L4 the endogenous constraint acts with the environment: organisms not only adapt to a niche, they also edit it so that their invariants become realizable. Categorically, niche edits are base changes in an environment fibration, and teleology is transported along these edits. When exactness (Beck–Chevalley) fails, one observes an order-of-operations gap (a holonomy) that witnesses genuine path-dependence beyond scalar selection. A. Environment fibration, reindexing, and teleology in fibers Let Env be a category of environments (objects E , morphisms f : E→E0 are edits of the niche). A world fibred over environments is a Grothendieck fibration π:Cenv −→ Env, with fibers C E := π−1 ( E )the categories of states in environment E . A (split) cleavage supplies, for each edit f:E→E0, a reindexing functor f∗:CE0−→ CE,cartesian over f.
35 The doctrine of invariants is likewise fibred, p : Eenv → C env , with fiber pE : EE→ C E . Endogenous constraints are computed fiberwise,GE t:CE→EE, and yield the environment-relative teleology TelE t:= RanιEΦE:CE→CE,TelE t(X0)∼ =lim(X0↓ιE)→BE t ΦE −−−→ CE,(100) where ιE : BE t,→ C E includes GE t -admissible futures and Φ E embodies evaluation at time t (cf. pointwise limit formula). B. Base-change compatibility (Beck–Chevalley) and holonomy Given an edit f:E→E0, there is a canonical 2-cell (mate) θf:f∗ RanιE0ΦE0=⇒RanιEf∗ΦE0,(101) natural in C E . Intuitively, “pull teleology across f ” vs. “teleologize after pulling data across f ”. When f∗ preserves the limits in (100) (i.e. the base-change square is exact), θf is an isomorphism (Beck–Chevalley). Proposition XIII.1 (Beck–Chevalley for L4) . If f∗ preserves the pointwise limits ( X↓ιE0 ) →BE0 t ΦE0 −−−−→ CE0, then θfin (101) is invertible. Equivalently, f∗◦TelE0 t∼ =TelE t◦f∗. Proof. Replace each Ran by its comma-limit presentation and use preservation of limits by f∗ to identify both cones. The universal properties match to give an isomorphism (mate calculus). Order-of-operations gap (holonomy). For a loop of edits γ : E0 f1 −→ E1 f2 −→ ··· fk −→ Ek = E0 , transport induces Tγ:= f∗ 1◦···◦f∗ k:CE0→CE0. Define the holonomy gap at X0∈CE0by ∆hol(γ;X0) := d TγTelE0 t(X0),TelE0 tTγ(X0),(102) measured by any fiberwise metric (e.g. an enriched least-violation cost). If every mate θfi is invertible, then ∆hol(γ;X0)=0. Otherwise one has a quantitative lower bound: Theorem XIII.2 (Quantitative path-dependence) . Let Θ( γ ) := θf1◦···◦θfk be the composite mate around γ . Suppose the fiber carries a metric in which reindexings and limits are L -Lipschitz. Then for some c > 0(depending on Land the diameter of indexing comma categories), ∆hol(γ;X0)≥c Θ(γ)−id . In particular, any non-invertibility of a mate along γ witnesses a strictly positive order-of-operations gap (path-dependence beyond selection). Sketch. Compute both sides of (102) as limits over reindexed comma diagrams. The composite Θ( γ ) compares the two cones; its deviation from identity controls the distance between the induced limits by Lipschitz continuity of reindexing and limits. Positive operator norm of Θ( γ ) −id gives the claimed lower bound. C. Two-arm “adapt vs. edit” experiment Fix Eand an edit f:E→E0. Consider two arms on the same initial state X0∈CE: (a) Adapt→edit: compute Y1:= TelE t(X0), then pull: Z1:= f∗(Y1). (b) Edit→adapt: pull first: ˜ X0 := f∗ ( X0 ), then compute Z2 := TelE0 t ( ˜ X0 ), and finally pull back if desired. The order effect is ∆ord(f;X0) := dZ1, Z2. By Prop. XIII.1, ∆ ord = 0 when θf is invertible; otherwise ∆ ord > 0, with lower bounds as in Thm. XIII.2. Biologically: “adapt → build” 6 =“build → adapt” when edits change the exactness of gluing/aggregation constraints.
36 D. Minimal stochastic model of L4 path-dependence Let X be finite and E parametrize boundary conductances. For each E let PE be a Markov kernel with stationary πE and admissible set AE (the projection of the constraint JGE tK ). The L4 teleology keeps routines by tilting exits from AE as in e PE ( x, y ) ∝PE ( x, y ) e−ε∆LE(x,y) for x∈AE , y∈Ac E . Consider an edit f:E→E0that adds a corridor (new edges). Then, in general, f∗TelE0 t(X0)6=TelE tf∗(X0), because the new corridor changes conductances in a way that is not preserved by pullback (the mate θf fails to be an isomorphism). The mean dwell time difference satisfies E˜πAE0[eτAc E0]−E˜πAE[eτAc E]&1 e φE0(AE0)−1 e φE(AE), so an edit that lowers exit conductance after adaptation can produce a larger dwell than adapting before adding the corridor—an observable order effect. E. What L4 adds beyond L0–L3 L4 shows that even with identical scalar fitness (L0) and identical Pareto surfaces (L1), and even with extant routines (L3), path-dependence can arise purely from the non-exact interaction between niche edits and teleology (non-invertible mates). This gives a structural basis for empirical claims that niche construction generates historical contingencies not reducible to simple selection stories. F. Summary Niche construction is formalized as base change in an environment fibration. Teleology commutes with base change exactly when the Beck–Chevalley mates are isomorphisms; otherwise one observes a holonomy—a strictly positive, quantifiable order-of-operations gap that encodes path-dependence beyond selection. A minimal stochastic model (conductance tilting) shows how dwell and return statistics differ between arms even under identical scalar fitness. XIV. L5: MULTI-LEVEL COHERENCE At L5, endogenous constraints are enforced across levels (genes → cells → organs → organisms → groups). The teleological deficit incorporates cross-level penalties that couple level-specific states into a coherent whole. This section specifies the coupling, proves strong-convexity and convergence properties of the resulting multi-level teleology, and derives variance & synergy inequalities that quantify how cross-level coupling suppresses fluctuations in observables. A. Cross-level penalties and the multi-level deficit Let Lvl = { 1 , . . . , L} index levels. For each level ` , let x(`)∈Rd` be a concrete state, with levelwise deficit L(`) t : Rd`→ [0 ,∞ )(constructed as in Secs. V–XI). Cross-level compatibility is encoded by penalties C`mx(`), x(m)≥0 ((`, m)∈E),(103) for edges ( `, m )of a (typically sparse) level-graph G = ( Lvl, E ). A canonical and analytically convenient choice is the (weighted) quadratic mismatch C`mx(`), x(m):= 1 2 R`→mx(`)−x(m) 2,(104)
37 where R`→m is a linear restriction/aggregation map (e.g. “project organ to cell marginals”). The multi-level deficit is Lmulti tx(1), . . . , x(L):= L X `=1 L(`) tx(`)+X (`,m)∈E λ`m C`mx(`), x(m),(105) with λ`m ≥ 0the coupling weights. The L5 teleological step (continuous-time) is the gradient flow/SDE with respect to (105) (cf. Sec. V): dx(`) t=f(`)(xt, t)dt−ε∇x(`)Lmulti t(xt)dt+√2DdW(`) t, ` = 1, . . . , L. (106) B. Strong convexity, uniqueness, and convergence Assume each L(`) t ( · )is C1 and µ` -strongly convex in x(`) (uniformly in t ), and that C`m ( ·,· )is convex and C1with block-strong convexity in the “difference” direction: there exists β`m ≥0such that ∇C`m(u, v)−∇C`m(u0, v0),(u−u0, v −v0)≥β`m R`→m(u−u0)−(v−v0) 2. Let µmin := min`µ` and suppose G is connected. Consider the block vector x = ( x(1), . . . , x(L) )and the block-quadratic case (104) for concreteness. Proposition XIV.1 (Global strong convexity and unique minimizer) . In the setting above, Lmulti t is strongly convex on Rd1×···×RdLwith parameter µeff ≥µmin +λconn σ2, λconn := min unit z⊥1z>LGz, (107) where LG is the weighted Laplacian of G with weights λ`m , 1 is the all-ones vector on levels, and σ := min(`,m)∈Eσmin ( R`→m ).[ 261 ] In particular, (105) admits a unique minimizer x? t , and the deterministic flow ˙x=−ε∇Lmulti tconverges exponentially to x? twhen tis frozen. Sketch. The block Hessian of (105) is the sum of a block-diagonal matrix diag ( ∇2L(`) t ) µminI and the PSD coupling matrix induced by P(`,m)λ`mR> `→m,−I> R> `→m,−I . The latter is λconn σ2 on the orthogonal complement of the consensus subspace; connectivity and the µmin term remove the nullspace, giving (107) . Exponential convergence of the (frozent ) gradient flow follows from standard strong-convexity arguments [135, Ch. 2]. C. Variance and synergy inequalities Consider the steady state of (106) when f(`) = −∇ Φ (`) and t7→ Lmulti t varies slowly: the stationary law π (d x ) ∝exp− (Φ( x ) + εLmulti t ( x )) /D d x is log-concave with potential whose Hessian is µeffI . By the Bakry–Émery Γ2criterion, πsatisfies a Poincaré inequality with optimal constant cP≤D/µeff : Varπ[f]≤D µeff Eπk∇fk2for all smooth f, (108) see [ 136 , Chs. 3–4], [ 137 , Ch. 3]. Let Y(`) ( x(`) )be an observable at level ` , L` -Lipschitz: |Y(`) ( u ) − Y(`)(v)| ≤ L`ku−vk.For the aggregate observable S(x) = P`Y(`)(x(`))one has VarπhX ` Y(`)i≤D µeff X ` L2 `,(109) because k∇Sk2=P`k∇Y(`)k2≤P`L2 `. Synergy from coupling. By (107) – (109) , increasing any λ`m decreases the variance upper bound through µeff: ∂ ∂λ`m VarπhX ` Y(`)i.−D σ2 (µeff)2Fiedler direction at (`, m)≤0,(110) with strict negativity when the ( `, m )edge participates in the Fiedler eigenvector of LG (algebraic connectivity) [ 139 ]. Thus cross-level coupling generates synergy: tighter coherence (larger λ ) reduces global variability.
38 Perturbative robustness. Small (bounded) perturbations of the potential preserve Poincaré/log-Sobolev inequalities (Holley–Stroock perturbation) [ 138 ]; hence the bounds (108) – (109) are stable under moderate misspecification of Φor L(`) t. D. Summary L5 introduces cross-level penalties C`m to glue levels into a coherent whole. The composite deficit (105) is strongly convex under mild assumptions, ensuring a unique multi-level teleological target and exponential convergence of the gradient flow. In the stochastic regime, the stationary law satisfies a Poincaré inequality with constant D/µeff , yielding the variance & synergy bound (109) : stronger cross-level coupling (larger algebraic connectivity and restriction singular values) suppresses variability of aggregate observables. XV. L6: META–TELEOLOGY At L6 the system is allowed to update its own doctrine: it can enlarge the representational machinery by which endogenous invariants are expressed. Concretely, we model a doctrine update as an endofunctor on the category of doctrines that increases representational capacity (e.g. adding a new code—genetic, regulatory, or symbolic). This meta-level change alters the feasible set of teleological futures and can induce phase transitions in the constraint graph that couples invariants. A. Doctrine updates as capacity-raising functors Let Doct be the category whose objects are doctrines p : E → C(fibred representations of invariants over world states) and whose arrows are doctrine morphisms commuting with the fibrations. A meta-update is an endofunctor Mκ:Doct −→ Doct, parametrized by a capacity index κ∈R≥0 (e.g. number of code symbols, network width, number of regulatory motifs), that adds predicates/constructors admissible in the doctrine. Given p : E → C, write Mκ(p) : Eκ−→ Cκ for the updated doctrine and let Uκ : Eκ→E be the forgetful comparison. The L6 teleology in the updated doctrine is Tel(κ) t:= Ranι(κ)Φ(κ):Cκ→Cκ,(111) with the usual pointwise-limit formula. Denote by L(κ) t the associated coherence deficit (Sec. V) computed in Eκ. Proposition XV.1 (Monotonicity of achievable deficit) . If κ≤κ0 and Mκ refines to Mκ0 via a doctrine morphism that preserves limits and subobjects, then for each initial state X0, inf y∈Reach(κ0) t(X0)L(κ0) t(y)≤inf y∈Reach(κ) t(X0)L(κ) t(y). Proof. The comparison functor Uκ0 ,κ : Eκ0→ Eκ preserves the constructions that define Lt , so any κ-admissible witness remains admissible at κ0and can only reduce (or leave unchanged) the infimum of the deficit. Definition XV.2 (Marginal value of capacity).Let ¯ Lt(κ) := inf y∈Reach(κ) t(X0)L(κ) t(y) be the achievable deficit at capacity κ. Its (right) derivative ∂+¯ Lt/∂κ is the marginal value of capacity.
39 Theorem XV.3 (Adopt-update criterion) . Suppose increasing capacity by ∆ κ incurs implementation cost λ∆κ(per-unit metabolic/maintenance cost), while the teleological calculus minimizes ¯ Lt+cost. If ∂+ ∂κ ¯ Lt(κ0)<−λ, then a doctrine update from κ0 to κ0 + ∆ κ strictly reduces total objective for all sufficiently small ∆ κ > 0. Proof. By definition, ¯ Lt ( κ0 + ∆ κ ) −¯ Lt ( κ0 ) = ∆ κ∂+ ∂κ ¯ Lt ( κ0 ) + o (∆ κ ) . Add the linear cost and use the inequality. B. Genetic assimilation and new codes Acode is a symmetric monoidal functor C: (Σ∗,·, )−→ (Proc,⊗,1), sending strings over an alphabet Σ(DNA bases, regulatory motifs, or symbols in a language) to executable processes (transcription/translation circuits, regulatory programs, or symbolic policies). A doctrine update that adds a code enlarges the alphabet Σand/or the interpretation C , thus increasing κ = log | Σ | and the compositional expressivity. Genetic assimilation (formalization). Let Beh be a category of behavioral policies with deficit Lbeh t , and Gen a category of genotypes/programs with deficit Lgen t. A pair of adjoint functors S:Beh Gen :P, S aP, models assimilation: S compiles behavior into code; P runs code to reproduce behavior. The assimilation error is εassim(b) := dbehb, P ◦S(b), with dbeh a behavioral metric. If S is enriched by a larger alphabet Σ 0 and deeper combinators (larger κ ), then universal-approximation bounds (e.g. for circuits/grammars) give εassim(b)→0as κ→ ∞. Proposition XV.4 (Deficit drop by assimilation) . Assume P is L -Lipschitz and Lbeh t is K -Lipschitz. Then inf g∈Gen Lgen t(g)−inf b∈Beh Lbeh t(b)≤K L inf b∈Beh εassim(b). Consequently, as the code capacity κ grows and εassim ↓ 0, the achievable deficit with code converges to that without code, i.e. learned routines become canalized in code (Waddington assimilation [140]). Proof. For any b , Lgen t ( Sb ) ≤ Lbeh t ( PSb ) ≤ Lbeh t ( b )+ K dbeh ( b, PSb ). Take infima and use dbeh ( b, PSb ) ≤ L εassim(b). Biological/symbolic instances. (i) DNA code. Enlarging Σfrom primitive chemistry to the four-base code and then to extended codon contexts increases κ ; the regulatory genome (cis/trans motifs) further expands expressivity [ 141 , 142 ]. (ii) Symbolic code. Adding a linguistic alphabet and cultural memory (teaching, imitation) realizes a powerful code with cross-generational compilation, accelerating assimilation (Baldwin effect [143]; gene–culture coevolution [144, 145]). (iii) Extended synthesis. Meta-updates are a core mechanism in EES frameworks that emphasize constructive causation [146]. C. Constraint-graph phase transitions Let x be a composite state and consider m constraints ce (invariants to keep true), each acting on a subset Ve⊆V={1, . . . , n}. The L6 deficit takes the generic factor-graph form L(κ) t(x) = m X e=1 λeφ(κ) exVe,(112) with penalties φ(κ) e≥ 0(zero when the constraint is satisfied). The hypergraph H = ( V, {Ve} )is the constraint graph. A code update changes both the local penalties φ(κ) e and the incidence structure (new factors).
40 Random ensembles and sharp thresholds. Let H∼ Hk ( n, m )be a random k -uniform hypergraph ( |Ve| = k ) with density α = m/n ; let SATκ be the event that the minimum of (112) is zero (all constraints satisfiable) at capacity κ. Theorem XV.5 (Sharp threshold (CSP)) . For a broad class of random CSPs (including k -SAT/XORSATtype penalties), there exists a sharp threshold αc(κ)such that lim n→∞ PrSATκ=(1, α < αc(κ), 0, α > αc(κ). Moreover, αc ( κ )is nondecreasing in κ ; increasing code capacity raises the satisfiability threshold. (See [147–149].) Idea. Monotone properties of random discrete structures exhibit sharp thresholds under broad conditions [ 147 ]. Random k -CSP ensembles satisfy these hypotheses, and κ enters through the local constraint class; enlarging κweakly enlarges the satisfying set for each clause, shifting αcright. Topological obstructions and cohomology percolation. For sheaf-based constraints on random covers/complexes, the onset of nontrivial cohomology (obstructions to gluing) also exhibits thresholds: e.g. H1 in random 2-complexes appears sharply at a critical density [ 150 , 151 ]. Crossing this threshold forces ajump in the minimal gluing energy (cf. Sec. XI), constituting a topological phase transition in the constraint graph. Graphical-model perspective. Interpreting (112) as a Gibbs energy on a factor graph links L6 transitions to percolation of long-range dependencies and clustering [ 152 ]. At α≈αc ( κ ), correlation lengths diverge and inference/optimization complexity peaks [153]. D. Summary L6 elevates the calculus to meta level: the system can change how it represents what must be kept true. Doctrine updates that add new codes increase capacity ( κ ), monotonically lowering the achievable deficit and, when the marginal value exceeds costs, being selected for (Theorem XV.3). Formalizing genetic assimilation via adjoint compilation/execution explains canalization: learned routines become encoded as low-deficit code as κ grows (Proposition XV.4). At population/ensemble scales, constraint graphs exhibit phase transitions: sharp satisfiability thresholds and topological obstruction onsets, with αc shifted by capacity (Theorem XV.5). These phenomena—doctrine updates, new codes, and phase transitions—characterize meta-teleology. XVI. PART IV: DERIVED MECHANISMS OF EVOLUTION We derive a family of concrete, testable evolutionary mechanisms that emerge from the teleological calculus developed in Parts I–III. Each mechanism is specified by: (i) a categorical source (which functors/limits/right Kan extensions encode it); (ii) a governing equation (deterministic variational principle, gradient/SDE, or coalgebraic fixpoint); and (iii) predictions (quantitative signatures or inequalities). Throughout, Cis the world category, p : E → Cthe doctrine, Gt :C → E the endogenous constraint functor at horizon t, and Ltthe associated coherence deficit (Sec. V). A. Constraint-Preserving Selection (CPS) Categorical source. CPS arises when the teleology Telt = Ranι Φis computed over the viability kernel diagram: Bt consists of futures whose constraints Gt are maintained, i.e. Gt ( Y ) ∈ EY pulls back to X0 via a witness in the fiber. Pointwise, Telt(X0)∼ =lim(X0↓ι)→Bt Φ −−→ C, so CPS selects those evolutions that minimize the deficit over the reachable set subject to viability [ 154 ].
41 Equation (population form). Let p∈ ∆ n−1 be type frequencies, and gi ( p )the constraint-preserving score gi(p) := fi(p)−εEx∼πiLt(x), where fi is baseline fitness and πi a distribution of phenotypes for type i . The CPS dynamics in the Shahshahani geometry is the replicator ˙pi=pigi(p)−¯g(p),¯g=X j pjgj.(113) Proposition XVI.1 (Lyapunov monotonicity) . If each gi is C1 and ∂gi/∂pj are bounded, then along (113), d dt¯g(p(t)) = X i pigi−¯g2≥0, with equality iff gi= ¯gon the support of p. Proof. Standard replicator calculus (cf. Hofbauer–Sigmund, Akin) with g in place of f shows that ¯g is Lyapunov; the computation is identical to Prop. IX.6 with ¯gin place of Φ. Predictions. (i) Types that achieve low Lt at equal f increase, i.e. CPS explains selection among equals via constraint maintenance. (ii) Under trade-offs, equilibrium satisfies KKT with ∇¯g∈N∆ ; measurable via differential response to perturbations in Gt(c.f. Sec. X). See [155, 156]. B. Sheaf-Gluing Selection (SGS) Categorical source. Let F : Uop →Vect be a morphogenetic sheaf over a cover {Ui} . The doctrine encodes compatibility of local sections; Gtdemands existence of a global section with specified features. The gluing energy Eglue (Sec. XI) yields part of Lt. Equation. Genotype γ determines parameters θ ( γ )for local fields; fitness proxy is penalized by minimal gluing energy: g(γ) := f(γ)−εEmin glueθ(γ),Emin glue(x) = 1 2kPH1r(x)k2.(114) Under mutation Mand selection by g, the quasispecies map updates P(γ). Proposition XVI.2 (Obstruction-induced selective sweeps) . If a mutation γ7→ γ0 changes the cohomology class of r from 0to c6 = 0 (or reduces kck ), then g ( γ0 ) −g ( γ ) = ε ( 1 2kck2 + ··· )(to first order), producing a deterministic bias for γ0proportional to the obstruction jump. Proof. Immediate from (114) and Prop. XI.3: Emin glue jumps by 1 2kck2. Predictions. (i) Alleles that restore global compatibility sweep even if baseline f unchanged. (ii) Spatial assays should see discrete drops in residuals when such alleles fix. See [158–160]. C. Persistence-Weighted Canalization (PWC) Categorical source. Gt carries PH features Θ t ; Lt includes Pb∈Θtwt ( b ) distb . Canalization corresponds to large bars and small residuals. Equation (barrier law). In the teleological SDE d xt = −∇ Φd t−ε∇Lt d t + √2D d Wt , the mean time to morphological failure (loss of target cavities) obeys E[τ]expnε DX b∈Θ∗ wt(b)∆distbo,(115) cf. Theorem XI.1. Predictions. (i) Canalization index Can = Pbwt ( b )predicts exponential dwell times. (ii) Perturbations that increase bar lengths (e.g. extracellular matrix scaffolding) increase robustness. See [158, 161, 162].
48 Major transitions & codes (L6). Doctrine updates Mκ increase representational capacity; when the marginal value of capacity exceeds costs, meta-code “jumps” occur (Sec. XV), consistent with transitions to new inheritance channels and regulatory languages [222, 223]. Theorem XVIII.4 (Teleological unification) . Let T be the 2-category whose objects are doctrines over worlds, 1-cells are doctrine morphisms commuting with Gt , and 2-cells are Beck–Chevalley mates. Then: (a) (Evo-devo) PH+sheaf teleology is a full sub-2-category of T in which right Kan extensions are computed by limits over barcode/sheaf diagrams. (b) (Niche) Niche construction corresponds to base change along 1-cells; holonomy is measured by noninvertible 2-cells. (c) (Multilevel) Multi-level teleology is a homotopy limit in T. (d) (Transitions) Meta-updates are monads Mon T; stabilized doctrines are Eilenberg–Moore algebras. Sketch. Assemble the constructions from Secs. XI–XV: (a) barcode functors and sheaf functors embed into Doct ; (b) base change is functorial, mates are 2-cells; (c) homotopy limits exist under completeness hypotheses; (d) monadicity is standard for doctrine extensions. D. Implications for AI/ML: learning invariants, plasticity →assimilation Learning Gt (invariants) as first-class citizens. Given data ( xi, yi ), define empirical risk with coherence regularization b R(h) := 1 n n X i=1 `h(xi), yi+λb Lth;G,(130) where b Lt penalizes violation of learned invariants G (e.g. group equivariances, conservations, topology). If Greduces the effective hypothesis class complexity from Rn(H)to Rn(HG), standard bounds give R(h)≤b R(h) + 2Rn(HG) + O rlog(1/δ) n!,(131) with Rn the Rademacher complexity [ 224 ]. For group-invariant classes (e.g. G -CNNs), Rn decreases with the orbit size of data augmentations [ 225 , 226 ]. Information-theoretic accounts view Gt as an information bottleneck (IB) that preserves task-relevant invariants while compressing nuisance [227, 228]. Variational inductive biases. Neural ODE/Lagrangian/Hamiltonian nets instantiate the variational structure of Sec. VI by building Lt(or invariants) into the dynamics, improving extrapolation. Plasticity ⇒ assimilation. Complementary learning systems and the Baldwin effect have formal analogs of L6: fast plasticity minimizes Lbeh t , slow consolidation minimizes Lgen t (Sec. XVIJ). In ML, consolidation penalties (e.g. EWC) implement a doctrine update that preserves previously learned invariants while encoding new ones [229–231]. E. Limitations, tractability, and open problems (1) Identifying the “right” Gt from data. Choosing Gt is an inverse problem. A principled approach is structural risk minimization with minimum description length (MDL): select the simplest doctrine that achieves low predictive risk and low violation cost, min (p,Gt)∈M b R(hp,Gt) | {z } fit +λb Lt(hp,Gt) | {z } coherence +βMDL(p, Gt) | {z } complexity , yet model selection is statistically and computationally demanding [ 232 , 233 ]. Causal identifiability of invariants requires interventions across environments (invariance-based causal discovery) [233].
49 (2) Exactness vs. approximation. Computing Ran , holim , and greatest fixpoints is often intractable at scale. Practical pipelines must rely on: •PH at scale: near-linear algorithms and sparsification, with tradeoffs between stability and resolution [234]. •Sheaf solvers: spectral sheaf Laplacians and convex relaxations for gluing, with guarantees only in special cases [235]. •Fixpoints: µ-calculus/model-checking iterations on finite abstractions for routine detection [236]. •Graph solvers: nearly-linear Laplacian solvers for conductance/spectral estimates [237]. A systematic theory of approximate teleology (how approximation errors propagate through Ran / holim ) is missing. (3) Robustness and falsifiability. Many predictions are inequalities (Sec. XVII); noisy biological data demand careful nulls and controls. We emphasized hazard slopes (L2), return windows (L3), order gaps (L4), variance suppression (L5), and change-points (L6). Null results in clean experiments must count against the calculus. (4) No-free-lunch for doctrines. There is no universally superior doctrine across all worlds/environments: any Gt that helps on one distribution hurts on another [ 238 ]. This motivates adaptivity (meta-teleology) and modular doctrines. (5) Ethics and scope. Teleology is a descriptive mathematics of structure-preserving dynamics; it is not a license for goal reification in living systems. In AI, learned invariants must respect privacy, fairness, and robustness constraints. F. Outlook Theory. Three directions look ripe: (i) Approximate Kan extensions with quantitative stability bounds; (ii) Teleological PAC-Bayes bounds coupling invariants and generalization; (iii) a thermodynamics of invariants: work/entropy balances when ε∇Ltbiases dynamics. Biology. Near-term tests include: (i) L2 dwell scaling in lumen/tube systems with targeted ECM/electrical interventions; (ii) L3 routine locking in collectives by minimal leader cues; (iii) L4 order effects in biofilms under matrix gene edits; (iv) L6 change-points in long-term cultures aligned with regulatory-code gains. AI/ML. A program of learning doctrines: libraries of invariants (topological, group-theoretic, conservation, causal) discoverable from data, deployable across tasks via meta-learning, and compressible into codes (weights, grammars). Synthesis. The calculus generalizes Darwin: L0 is the base case; L1–L6 add structure the organism keeps true. Whether or not one prefers the word “teleology,” the mathematics is about invariants and their transport. That is a robust idea—applicable from embryos to swarms to learning machines. Appendix A: Category-Theoretic Proofs A.1 Existence and pointwise form of right Kan extensions Let ι:B,→Cbe a functor with small domain Band let Φ : B→D. Assume Dis complete. Theorem A.1 (Pointwise right Kan extensions exist and are limits).Under the assumptions above, the right Kan extension RanιΦ : C→Dexists and is given pointwise by RanιΦ(c)∼ =lim(c↓ι)proj −−−−→ BΦ −−→ D.(A1) Moreover, for u : c→c0 in C, the induced map Ranι Φ( u )is obtained by functoriality of comma categories and the universal property of the limit. Proof. Standard; completeness of Densures the limit in (A1) exists for each c . The universal natural transformation : Ranι Φ ◦ι⇒ Φis induced by the tip of the comma diagram and satisfies the Kan property by uniqueness of cones (see [239, Ch. X], [240, §3.1]).
50 Proposition A.2 (Stability under reindexing).If H :D → Epreserves limits of shape ( c↓ι )for all c , then HRanιΦ∼ =Ranι(HΦ). Proof. Apply Hto the pointwise formula (A1) and commute Hwith limits by hypothesis. A.2 Beck–Chevalley type results (exact squares) Consider a square in Cat B0B C0C v u0u w together with ι:B,→Cand its pullback ι0:B0,→C0. Let Φ0:B0→Dand Φ=Φ0◦v−1. Proposition A.3 (Beck–Chevalley mate).There exists a canonical natural transformation (the mate) θ:u0∗ RanιΦ=⇒Ranι0u0∗Φ, which is an isomorphism whenever the square is exact (i.e. comma–limit preservation along u0 holds). In particular, if u0preserves the limits in (A1), then θis invertible. Proof. Construct θ by the universal property of Ran and the adjoint mate correspondence (see [ 240 , §5]). Exactness yields a levelwise identification of limit cones, hence θis an iso. A.3 Greatest fixed points for coalgebraic teleology Let ( νF, ω )be a final F -coalgebra in a complete, well-powered category Cin which F preserves monomorphisms. For the monotone endofunction on the subobject lattice Sub(νF) Φ(S) := ω−1(FS)∩JGtK, the greatest fixed point A? = ν Φexists (Knaster–Tarski) and coincides with the largest forward-invariant subobject of JGtK (Prop. VII.5 in the main text). Below we record existence of the final coalgebra under accessible hypotheses. Theorem A.4 (Existence of final coalgebras for accessible functors).If Cis locally presentable and F:C→Cis λ-accessible for some regular cardinal λ, then a final F-coalgebra exists. Proof. By Adámek’s theorem, form the terminal sequence 1 ←F 1 ←F2 1 ← ··· ; accessibility ensures convergence at some stage α and the limit Fα 1carries the final coalgebra structure (see [ 241 , Thm. 5.5]; also [242]). Proposition A.5 (Terminal sequence computation).If F preserves limits of λ -op-chains, the canonical map F(Fα1) →Fα1is an isomorphism for large enough α, and thus (Fα1, ω)is final. Proof. Immediate from preservation and the definition of the limit coalgebra; see [243]. A.4 Homotopy limits for multi-level gluing Let C : Lvl →Cat be a small diagram with levelwise complete fibers and right Quillen restriction functors. Then the homotopy limit holimLvl C exists and can be computed as a homotopy end; we use this in Sec. VIII. A model-categorical construction is given in [ 95 , Ch. 18] but is reproduced here abstractly for completeness. Proposition A.6 (Strictification under injective fibrant replacement).If each C ( ` )admits injective fibrant replacement R` and restriction functors preserve fibrations, then the strict limit of the replaced diagram computes the homotopy limit. Proof. Standard Bousfield–Kan replacement; use injective model structure on the functor category to reduce to objectwise fibrant diagrams. References for Appendix A: [239–243].
51 Appendix B: Persistent Homology — Stability and Continuation Let f, g : X→R be tame filtrations (e.g. sublevel sets of tame functions on a triangulable X ). Denote their persistence diagrams by Dgmk(f), Dgmk(g). Theorem B.1 (Bottleneck stability).For all k≥0, dBDgmk(f), Dgmk(g)≤ kf−gk∞.(B1) Proof. By matching births/deaths across ε -interleavings of filtrations and using the isometry theorem for persistent modules (see [244]). Theorem B.2 (Isometry and interleaving).For q-tame persistence modules M, N, dI(M, N) = dBDgm(M), Dgm(N), where dIis the interleaving distance. Consequently, diagram distances metrize stability of modules. Proof. By the structure theorem for persistence modules and the isometry theorem [245]. Proposition C.1 (Continuation under smooth parameter drift).Let fθ be a C1 family of tame functions and suppose a bar (b(θ), d(θ)) persists without collisions on an interval. Then d dθd(θ)−b(θ)=∂θfθ(xd)−∂θfθ(xb), for critical points xb, xdalong the continuation, up to higher-order terms. Proof. Differentiate the defining equations for critical values along a smooth branch; see [245, §6]. References for Appendix B: [244–246]. Appendix C: A Čech Obstruction Worked Example Consider the circle S1 covered by three overlapping arcs U1, U2, U3 with Uij = Ui∩Uj and cyclic triple overlaps U123 6 = ∅ . Let F be the orientation sheaf with values in R and transition functions tij ∈ {± 1 } on Uij. Suppose t12 =t23 = +1, t31 =−1. Let si∈ F ( Ui )be local sections (real-valued scalars) and define residuals on overlaps rij = si−tij sj∈ F(Uij)∼ =R. Proposition C.1 (No global section & minimal residual).There is no global section s∈ F ( S1 )with s|Ui=sisatisfying rij = 0 for all i<j. The minimal gluing energy Emin glue := min (s1,s2,s3)∈R3 1 2X i<j r2 ij is strictly positive and equals 1 2khk2 for the harmonic representative h of the nontrivial cohomology class in H1(S1;Z2). Proof. The product t12t23t31 = − 1witnesses a nontrivial 1-cocycle in ˇ H1 ( S1 ; {± 1 } )(orientation reversal around the cycle), hence no choice of ( si )can satisfy all rij = 0 simultaneously. In the least-squares metric on C1 the minimal energy is the squared norm of the harmonic component (Hodge decomposition on the nerve), see [247, Ch. 3], [248, §3]. Remark C.1.The explicit minimizer solves the normal equations for the 3 × 3Laplacian associated to the nerve with twisted signs; one finds Emin glue = 1 2 in a normalized setting where each overlap has unit weight. References for Appendix C: [247, 248].
52 Appendix D: Derivations — Noether, Hysteresis, Punctuated Equilibria D.1 Noether’s theorem with teleological defect Consider the teleological action ST[x] = ZT 01 2k˙x−f(x, t)k2+εLt(x)dt on a Riemannian manifold ( X, g ), with f and Lt smooth. Let G act on X by isometries; denote the fundamental vector field by ξXfor ξ∈Lie(G). Theorem D.1 (Noether balance law with defect).If fand Ltare G-invariant, then the momentum map Jξ = h˙x−f, ξX ( x ) ig is conserved along Euler–Lagrange trajectories. If f is G -invariant but Lt is not, then d dtJξ=−εh∇Lt(x), ξX(x)ig.(D1) Proof. Apply the standard Noether argument to the Lagrangian L = 1 2k˙x−fk2 + εLt ; G -invariance implies vanishing Lie derivative. Non-invariance produces the source term (D1) ; see [ 249 , §III], [ 250 , Chs. 9–11], [251, Ch. 8]. D.2 Hysteresis from non-exact base change Let λ∈ Λparameterize environment edits and let Telt ( λ )be the teleology. Consider a closed loop γ⊂Λ. Proposition C.1 (Discrete geometric phase).Let Tγ be the parallel transport of states along γ induced by reindexing functors. If the Beck–Chevalley composites along γ are not all invertible, then the holonomy Tγ6 = id and the order gap ∆ hol ∼ kTγ−idk is strictly positive. In the smooth setting, Tγ = PexpRγA for a connection one-form Awith curvature F= dA+A∧A 6= 0. Proof. Follows from Appendix A.2 and the path-ordered exponential representation of transport; nonexactness produces nontrivial curvature, hence nontrivial holonomy; cf. [184, §2]. D.3 Punctuated equilibria as barrier-jump statistics Let U ( t, x ) = Φ( x, t ) + εL ( x, t )be a slowly varying potential with two metastable wells separated by a barrier ∆ V ( t ). Assume ˙ ∆V is piecewise continuous and that at rare times {tk} the barrier drops by δk>0. Theorem D.2 (Mixed Kramers law with jumps).Let τ be the first passage time between wells in the small-noise limit D↓0. Then P{τ > t} ≈ exp−Zt 0 κ(s)ds, κ(s)K(s) exp−∆V(s) D, with instantaneous Kramers rate κ ( s )and prefactor K ( s ). If at tk the barrier drops by δk , then κ is multiplied by exp(+δk/D), producing punctuated bursts of transitions concentrated near {tk}. Proof. Use adiabatic large deviations/Kramers theory between jumps and match across jump discontinuities of the barrier; details follow [254] combined with Eyring–Kramers asymptotics. Remark D.1.This provides a mechanistic underpinning of Eldredge–Gould punctuated equilibria: long stasis under high barriers, quick transitions when barriers drop, e.g. by doctrine updates or niche edits [252, 253]. References for Appendix D: [184, 249–254].
53 Appendix E: Simulation Algorithms (Pseudocode) We summarize numerical procedures for testing the predictions. Pseudocode uses algorithm/algpseudocode. E.1 Teleological SDE integrator (Euler–Maruyama) We discretize dxt=f(xt, t) dt−ε∇Lt(xt) dt+√2DdWtwith step ∆t. Algorithm 1 Teleological Euler–Maruyama Integrator Require: initial x0, step ∆t, horizon T, drift f(·, t), deficit gradient g(x, t) = ∇Lt(x), noise scale D 1: N← dT/∆te 2: for k= 0 to N−1do 3: tk←k∆t 4: η← N(0, I).standard normal 5: xk+1 ←xk+f(xk, tk)−ε g(xk, tk)∆t+√2D∆t η 6: end for 7: return (xk)N k=0 E.2 Coalgebraic routine finder (finite Markov chain) Given Pon finite Xand admissible set A⊆X: Algorithm 2 Largest Admissible Closed Class Require: Transition matrix P, admissible set A 1: G←directed graph with edges x→yiff P(x, y)>0 2: H←G[A].induced subgraph on A 3: Compute strongly connected components (SCC) of H 4: C← {SCC S⊆A| ∀x∈S:Py∈A\SP(x, y) = 0}.closed SCCs 5: return A?←SS∈CS E.3 PH + gluing pipeline Algorithm 3 Morphogenetic Teleology Features Require: segmented image/mesh x, cover {Ui}, overlap metrics 1: Build filtration Kα(x)(e.g. Vietoris–Rips) 2: Compute barcodes Θand weights w(b)(lengths) (e.g. ripser [255]) 3: For each b∈Θ, compute representative cycle Rep(b)and distance distb 4: Infer local sections sion Ui; residuals rij =si|Uij −sj|Uij 5: Solve least squares min(si)1 2Pi<j krijk2; record Emin glue 6: return Score =Pbw(b) distb+λglue Emin glue E.4 Niche holonomy experiment Algorithm 4 Order-of-Operations Gap Require: initial state X0in environment E, edit f:E→E0, teleology oracles TelE,TelE0, distance d 1: Y1←TelE(X0);Z1←f∗(Y1).adapt→edit 2: ˜ X0←f∗(X0);Z2←TelE0(˜ X0).edit→adapt 3: return ∆ord =d(Z1, Z2)
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