Kernel Dynamics under Path Entropy Maximization

1 K ernel Dynamics under P ath Entropy Maximization Jnaneshwar Das School of Earth and Space Exploration Arizona State University , T empe, AZ 85287, USA Earth Innov ation Hub jnaneshwar.das@asu.edu Abstract —W e propose a variational framework in which the kernel function k : X × X → R , interpreted as the foundational object encoding what distinctions an agent can repr esent, is treated as a dynamical variable subject to path entropy max- imization (Maximum Caliber , MaxCal). Each kernel defines a repr esentational structure over which an information geometry on probability space may be analyzed; a trajectory through kernel space therefor e corresponds to a trajectory through a family of effectiv e geometries, making the optimization land- scape endogenous to its own tra versal. W e formulate fixed-point conditions f or self-consistent kernels, propose r enormalization group (RG) flow as a structured special case, and suggest neural tangent kernel (NTK) evolution during deep network training as a candidate empirical instantiation. Under explicit information-thermodynamic assumptions, the work requir ed for kernel change is bounded below by δ W ≥ k B T δ I k , where δ I k is the mutual information newly unlocked by the updated kernel. In this view , stable fixed points of MaxCal over kernels correspond to self-reinf orcing distinction structures, with biological niches, scientific paradigms, and craft mastery offered as conjectural interpr etations. W e situate the framework relativ e to assembly theory and the MaxCal literature, separate formal results from structured correspondences and conjectural bridges, and pose six open questions that make the pr ogram empirically and mathematically testable. Index T erms —Maximum Caliber , repr oducing kernel Hilbert space, renormalization group, Fisher -Rao geometry , information thermodynamics, kernel dynamics I . I N T R O D U C T I O N Every inference engine—biological, computational, or learned—operates on a substrate that determines which dif- ferences in the world it can represent. W e call this substrate the kernel : a positiv e-definite function k : X × X → R whose choice induces an inner product on a reproducing ker- nel Hilbert space (RKHS) H k , and—via the representational structure it defines—an effecti ve geometry on the space of probability distributions P and a bound on extractable work via the Sagawa-Ueda generalized second law [1]. Throughout the paper , “kernel” is used in a broadened but still technical sense: not merely as a computational device for nonlinear learning, but as the mathematical object that determines which dif ferences in the world are representable, comparable, or actionable for an agent. This representational reading motiv ates treating kernel change as a change in the agent’ s effecti ve distinction-making capacity , which is the quantity the MaxCal construction is designed to model. In this paper we adopt the k ernel as the primiti ve object from which geometry , dynamics, and information-thermodynamic bounds are deri ved. At fix ed kernel, this static viewpoint recov ers a common exchange rate k B ln 2 per bit under standard assumptions of reversible bookkeeping. The present manuscript focuses on what that static vie wpoint leav es open: a giv en agent operates within a kernel but the question of how kernels themselves change over time was deferred. The present paper addresses that deferral directly . W e ask: if kernels are dynamical variables, what variational principle governs their trajectories? The answer we dev elop is Maximum Caliber (Max- Cal) [2]—path entropy maximization ov er trajectories through kernel space. The resulting frame work has three properties that distinguish it from prior treatments of kernel learning: 1) The optimization landscape is endogenous : each kernel k deforms the geometry of P , so the landscape through which kernels ev olve is itself a function of the trajectory . 2) Fixed points of the dynamics are self-consistent kernels: distinction structures that are self-reinforcing under their own effecti ve geometry . W e interpret RG fixed points as the most structurally direct special case, and biological niches, scientific paradigms, and craft mastery as candi- date higher-le vel analogues of self-consistent kernels. 3) Kernel change has a thermodynamic cost bounded below by k B T per bit of newly unlocked mutual information— a Landauer principle for conceptual change. Relativ e to prior work, the contribution here is not a new kernel-learning algorithm, nor a reformulation of classical information geometry , nor a standard MaxCal model on a fixed state space. Rather, the paper treats the kernel itself as the ev olving object of inference and asks what variational principle governs trajectories through the space of distinction- making structures. The result is a framew ork that is partly formal and partly programmatic: it introduces the kernel-space MaxCal construction, formulates self-consistency and stability conditions, and uses these to organize a set of candidate cor- respondences across physics, learning, biology , and embodied craft. The paper is organized as follows. Section II revie ws the necessary background in MaxCal, RKHS geometry , and the information-thermodynamic assumptions used here. Sec- tion III defines the space of kernels and its topology . Sec- tion IV lifts MaxCal from state space to kernel space. Section V deriv es fixed-point conditions and their stability . Section VI instantiates the framew ork across RG flow , NTK ev olution, adaptive sample-return planning, biological e volu- tion, and craft mastery . Section VII situates the framew ork relativ e to assembly theory . Section VIII separates formal results from conjectural bridges. Section IX poses six open questions. 2 Claim status in this paper . F ormal content includes the definition of kernel space K , the lifting of MaxCal to path measures on kernel trajectories, the self-consistency condition, and the frozen-kernel stability criterion. Structured correspon- dences include the mappings to RG flow and finite-width NTK ev olution. Conjectural bridges include the interpretations in biological e volution, craft mastery , adapti ve field sampling, and scientific paradigm shifts. I I . B AC K G R O U N D A. Maximum Caliber Let Γ denote the space of trajectories γ : [0 , T ] → X over a state space X . MaxCal selects the path distribution p [ γ ] that maximizes the path entropy S [ p ] = − X γ p [ γ ] ln p [ γ ] q [ γ ] (1) subject to dynamical constraints ⟨ f i [ γ ] ⟩ = F i , where q [ γ ] is a reference path measure [2]. The resulting distribution takes the form p [ γ ] ∝ q [ γ ] exp − X i λ i f i [ γ ] ! (2) where λ i are Lagrange multipliers. MaxEnt (Jaynes) is recov- ered when trajectories reduce to single configurations. MaxCal deriv es Green-Kubo relations, Onsager reciprocity , Prigogine’ s minimum entropy production, and master equations as special cases [2]. B. RKHS Geometry and the K ernel Primitive A Mercer kernel k induces an RKHS H k via the feature map ϕ : X → H k , ϕ ( x ) = k ( · , x ) . The kernel is treated as the primitiv e representational object, while Fisher-Rao supplies the canonical information geometry on the associated statistical manifold; kernel change therefore alters the effecti ve geometry of inference by altering the representational substrate on which that manifold is defined. Concretely , the square-root embedding p 7→ √ p ∈ L 2 ( X , ν ) pulls back the L 2 inner product to the Fisher-Rao metric [4] g F ij ( p ) = Z ∂ ln p ∂ θ i ∂ ln p ∂ θ j p dν (3) The Hellinger kernel k H ( p, q ) = R √ p q dν is the unique ker- nel whose induced geometry respects suf ficient statistics [5]. The kernelized Stein discrepancy identifies the score function ∇ x log p ( x ) as the Riesz representativ e of the Stein operator in H k [6]. The central modeling assumption of this paper is that k is prior to all of this structure: the metric, the score function, the mutual information I , and the Sagawa-Ueda work bound W extracted ≤ ∆ F + k B T · I (4) are treated as deriv ed objects once a kernel is fixed. C. Standing Assumptions The following assumptions are used throughout the paper . (A1) X is a Polish (complete, separable, metrizable) space. (A2) ν is a σ -finite reference measure on X . (A3) Every kernel k ∈ K is Mercer (continuous, sym- metric, positive semi-definite) and Hilbert-Schmidt: R R k ( x, x ′ ) 2 dν ( x ) dν ( x ′ ) < ∞ . (A4) The agent–environment system admits a joint distri- bution P agent , en v from which the mutual information I k = I ( A ; E | k ) is computed by restricting the agent’ s representation to the RKHS H k . (A5) Thermodynamic bookkeeping is quasi-static: the Lan- dauer bound δ W ≥ k B T δ I (in nats) is attainable in the rev ersible limit. (A6) All information quantities ( I , D KL ) are measured in nats unless stated otherwise, so that Landauer’ s bound reads k B T per nat (equiv alently k B T ln 2 per bit). I I I . T H E S PAC E O F K E R N E L S Definition 1 (K ernel space) . Let K denote the set of all Mer cer kernels on X : K = { k : X × X → R | k symmetric, positive semi-definite } K is a con vex cone: if k 1 , k 2 ∈ K and α, β ≥ 0 , then αk 1 + β k 2 ∈ K . Products k 1 · k 2 ∈ K as well, making K closed under the operations that arise naturally in kernel composition. A natural metric on K is induced by the Hilbert-Schmidt norm on the associated integral operators T k : L 2 ( ν ) → L 2 ( ν ) : d ( k 1 , k 2 ) = ∥ T k 1 − T k 2 ∥ HS =  Z Z [ k 1 ( x, x ′ ) − k 2 ( x, x ′ )] 2 dν ( x ) dν ( x ′ )  1 / 2 (5) W ith this metric, K is a separable metric space and paths γ : [0 , T ] → K are well-defined as Bochner-inte grable trajectories. A. The Endogenous Landscape The critical structural feature of K is that each point k ∈ K determines a representational substrate relativ e to which an ef fectiv e information geometry on P may be analyzed. A path γ ( t ) ∈ K therefore generates a one-parameter family of Riemannian manifolds ( P , g γ ( t ) ) . The landscape ov er which the kernel ev olves is itself a function of the kernel’ s current value. Writing g k for the metric induced by kernel k , the chain rule giv es d dt g γ ( t ) = D g   γ ( t ) [ ˙ γ ( t )] (6) where D g | k : T k K → Sym 2 ( T ∗ P ) is the Fr ´ echet deriv ativ e of the map k 7→ g k at k , ev aluated in direction ˙ γ ( t ) . This endogeneity—the landscape depends on the current position— distinguishes kernel dynamics from ordinary optimization on a fixed manifold. 3 k 0 k 1 k 2 γ ( t ) g k 0 on P g k 1 on P g k 2 on P geometry changes along γ ( t ) Fig. 1. Illustration of a kernel trajectory γ ( t ) in K . Each kernel k t determines a representational substrate relative to which a metric g k t on probability space P is analyzed, so moving through K simultaneously deforms the geometry on which inference proceeds. I V . M A X C A L O V E R K E R N E L S PAC E W e now lift MaxCal from X to K . Let Π denote the space of paths γ : [0 , T ] → K , and let Q be a reference measure on Π . By (A3), K embeds isometrically into the separable Hilbert space of Hilbert-Schmidt operators on L 2 ( ν ) ; a natural choice for Q is the W iener measure on this Hilbert space, conditioned on the positiv e-semidefinite cone. Definition 2 (Kernel path entropy) . The path entr opy over kernel trajectories is S [ P ] = − Z Π P [ γ ] ln P [ γ ] Q [ γ ] D γ wher e P is a pr obability measure on Π . A. A Minimal T wo-Kernel T oy Model T o make the frame work concrete, consider a finite kernel family K 2 = { k A , k B } and discrete time t = 0 , 1 , . . . , T . Let a trajectory be γ = ( k t =0 , . . . , k t = T ) with k t ∈ K 2 . Choose a Markov reference process Q [ γ ] = π 0 ( k 0 ) T − 1 Y t =0 q ( k t +1 | k t ) . (7) Impose two path constraints: expected cumulativ e switching cost C [ γ ] = T − 1 X t =0 1 [ k t +1  = k t ] , (8) and expected cumulative information gain G [ γ ] = T X t =0 I k t . (9) The MaxCal distribution is then P [ γ ] ∝ Q [ γ ] exp( − λ C C [ γ ] + λ G G [ γ ]) . (10) This yields an effecti ve two-state dynamics with transition odds P ( k t +1 = k B | k t = k A ) P ( k t +1 = k A | k t = k A ) ∝ q ( B | A ) q ( A | A ) exp( − λ C + λ G ∆ I ) , (11) where ∆ I = I k B − I k A . The toy model exhibits an explicit threshold: switching is fav ored when λ G ∆ I > λ C . This is the simplest instantiation of the tradeoff between thermody- namic/transition cost and informational gain. B. Constraints Physically meaningful constraints on kernel trajectories in- clude: 1) Thermodynamic cost constraint. The expected work required to shift the kernel along a path must not exceed av ailable free energy: E P " Z T 0 ˙ W k ( t ) dt # ≤ F where ˙ W k is the instantaneous thermodynamic cost of kernel change (deri ved below in Section IV -D). 2) Fidelity constraint. The kernel must maintain sufficient mutual information about the en vironment’ s rele vant microstates: E P [ I k ( t ) ] ≥ I min ( t ) 3) Consistency constraint. The kernel must remain consis- tent with the agent’ s current generativ e model. Writing p env for the environmental data distribution and q k for the agent’ s model distribution under kernel k (see A4): E P  D KL ( p env ∥ q k ( t ) )  ≤ ϵ C. The MaxCal K ernel Distribution Maximizing S [ P ] subject to these constraints yields P [ γ ] ∝ Q [ γ ] exp  − λ 1 Z T 0 ˙ W k dt + λ 2 Z T 0 I k ( t ) dt − λ 3 Z T 0 D KL ( t ) dt  (12) This is the least-assuming distribution over kernel trajecto- ries consistent with thermodynamic, informational, and con- sistency constraints. The mode of P is the most probable kernel trajectory under the chosen reference measure and constraints, and may be interpreted as the least-assuming path of distinction-change a vailable to the agent model under those assumptions. D. Thermodynamic Cost of K ernel Change The cost of moving from kernel k to k + δ k is the work required to update the agent’ s distinction-making capacity . Creating mutual information between agent and environment requires physical work by Landauer’ s principle [11]; the Sagaw a-Ueda bound (4) quantifies the complementary direc- tion (work extraction from existing correlations). T ogether they imply that acquiring mutual information δ I k that was inaccessible to k but accessible to k + δk costs at minimum (in nats, per assumption A6): δ W k ≥ k B T · δ I k (13) where δ I k = I k + δ k ( A ; E ) − I k ( A ; E ) (14) with A , E the agent and en vironment variables of (A4). Equation (13) may be read as a Landauer -type lower bound for physically realized kernel chang e under assumptions (A4)– (A6): gains in discriminativ e capacity that are physically 4 realized by the agent carry a lower -bounded thermodynamic cost. Combining (13) with the chain rule ˙ I k = ⟨∇ k I k , ˙ k ⟩ HS and Cauchy-Schwarz giv es a speed limit on kernel ev olution:   ˙ k ( t )   HS ≤ ˙ F ( t ) k B T · ∥∇ k I k ∥ HS (15) where ∇ k I k is the Hilbert-Schmidt gradient of I with respect to k , and ˙ F is the rate of free-energy supply . Kernels can only change as fast as free energy permits. Scope of the thermodynamic bound. The bound (13) should be read as an information-thermodynamic lower bound under assumptions (A4)–(A6), not as a claim that arbitrary abstract conceptual change automatically admits a direct calorimetric interpretation independent of embodiment. The intended claim is narrower: when a change in kernel corresponds to a phys- ically realized increase in the agent’ s representational access to en vironmental distinctions, Landauer-style reasoning yields a lower bound on the work required to acquire that addi- tional information. The Sagawa-Ueda relation is inv oked as the complementary result gov erning extraction from existing correlations, not as a substitute for the acquisition bound itself. V . F I X E D P O I N T S A N D T H E I R S T A B I L I T Y Definition 3 (Self-consistent kernel) . A kernel k ∗ ∈ K is self- consistent if it is a fixed point of the MaxCal kernel dynamics: δ S δ γ     γ ≡ k ∗ = 0 That is, k ∗ is the maximum-caliber choice given the geometry that k ∗ itself induces. A. Stability Criterion Stability of k ∗ concerns whether small perturbations k ∗ + ϵ h (with h ∈ T k ∗ K ) grow or decay . Let S ∗ ( k ) denote the optimized path entropy when the kernel is held fixed at k (i.e., S ∗ ( k ) = max P S [ P ] subject to the constraints of Section IV, with the kernel frozen at k ). A self-consistent kernel k ∗ is stable if the Hessian of the self-consistency map satisfies D 2 k S ∗   k ∗ [ h, h ] < 0 for all h ∈ T k ∗ K , h  = 0 . (16) Unstable directions correspond to bifurcations—transitions between distinct distinction regimes. In the two-kernel model of Section IV -A, stability reduces to whether perturbations in the effecti ve gain-cost balance λ G ∆ I − λ C return the system to the same occupancy regime or driv e a transition to the alternati ve kernel; this illustrates in finite dimensions the broader interpretation of unstable directions as transition channels between distinction regimes. Conjecture 1. The set of stable self-consistent kernels forms a discr ete (generically zer o-dimensional) subset of K , separated by unstable fixed points that act as transition states between basins of attraction. The basins of attraction of stable self-consistent kernels are the precise mathematical content of what we informally call niches , paradigms , and mastery domains . V I . S P E C I A L C A S E S A. Renormalization Gr oup Flow as Kernel Dynamics In W ilson’ s renormalization group [7], integrating out de- grees of freedom below a momentum cutoff Λ defines a map R Λ : K → K . The RG flow is the trajectory dk d ln Λ = β ( k ) (17) where β is the beta function. Fixed points of RG flow satisfy β ( k ∗ ) = 0 —scale-in variant kernels at which the distinction between microscale and macroscale collapses. Proposition 1. RG flow can be repr esented as a special case of MaxCal over kernels in which: (i) the constraint is scale in variance of the partition function, and (ii) the refer ence measur e Q is uniform over the r enormalization gr oup orbit. Pr oof status. This statement is currently a structural correspon- dence rather than a full deriv ation. A complete proof requires an explicit map between the RG coarse-graining semigroup and the MaxCal path measure on Π . Conjecture 2. The critical exponents at RG fixed points equal the eigen values of D 2 k S ∗ | k ∗ in the Hilbert-Schmidt metric, so that universality classes correspond to stability basins of self- consistent kernels. B. Neural T angent K ernel Evolution For an infinitely wide neural network, the neural tangent kernel (NTK) Θ( x, x ′ ) gov erns the training dynamics [8]: d ˆ y dt = − Θ · ∇ ˆ y L (18) During training of finite networks, Θ ev olves. The MaxCal kernel framework predicts: 1) The trajectory of Θ( t ) through K maximizes path en- tropy subject to cross-entropy loss reduction at rate ≥ ˙ L min . 2) Fixed points of NTK evolution correspond to networks that have learned a self-consistent representation of their training distribution—feature hierarchies that are self- reinforcing. 3) The thermodynamic cost bound (13) predicts a minimum energy dissipation during NTK ev olution: ˙ W ≥ k B T · ˙ I Θ , which may be empirically testable via GPU power consumption during training. Conjecture 3. The NTK of a tr ained dif fusion model con ver ges to the Hellinger kernel k H of the data distribution—the unique kernel whose geometry respects sufficient statistics [5]. In the NTK case, the appeal of the framew ork is not merely analogical: Θ t is a measurable ev olving kernel, and the framew ork predicts constraints on its trajectory relativ e to information gain and energetic expenditure during training. 5 Empirical pr otocol (falsifiable).: T o test this section’ s claims, one can: (i) estimate Θ t at fixed training interv als for width-scaled model families, (ii) compute a proxy for ˙ I Θ t from held-out representation statistics, and (iii) record wall- power draw to estimate ˙ W ( t ) . The prediction is an inequality trend ˙ W ( t ) ≥ c ˙ I Θ t up to calibration constant c , with tighter agreement at larger width and slo wer learning rate. C. Adaptive Sampling for Dynamic Algal Blooms in a Lake Consider a lake with a time-varying bloom field b t ( x ) (e.g., chlorophyll concentration) ov er spatial location x ∈ Ω . A natural operational realization is a heterogeneous team: an autonomous surface vehicle (ASV) carrying an autonomous underwater vehicle (A UV) with coordinated dock and un- dock. The ASV supplies long-endurance transit, surface- visible fields (e.g., temperature and surface color proxies), and a stable platform for docking, recharge, and data offload; the A UV , when undocked, samples vertical structure (e.g., chlorophyll maximum depth, oxygen, turbidity) that the sur- face cannot see. The key operational dif ficulty is that bloom fronts advect and deform on timescales comparable to the mission duration, while subsurface structure can misalign with surface patches—so the team’ s belief couples two partially ov erlapping observation operators fused at rendezvous. Let σ t ∈ { do ck ed , undo c ked } denote the coordination state. While σ t = dock ed , the A UV is carried and acts as payload; undocking initiates a div e phase with distinct motion costs and information yield. Let k t be the kernel used by the onboard model for spatiotemporal covariance in bloom dynamics (possibly including depth or modality- specific components after vertical profiles are merged at dock), and let u t ( x ) denote posterior uncertainty under k t . Actions decompose into ASV waypoints a ASV t , A UV profile commands when σ t = undo c ked , and feasible dock/undock transitions; subsurface samples accrue only in the undocked phase. Write A t for the joint tuple ( a ASV t , a AUV t , σ t ) . Mission resources impose constraints: X t  c ASV mov e ( a ASV t , a ASV t − 1 ) + c AUV dive ( a AUV t , σ t )  ≤ E max , X t 1 [ collect at t ] ≤ N max . (19) For sample return to base (or shore offload), a feasibility reserve must be maintained; for heterogeneous operation, rendezvous between ASV and surfacing A UV imposes an analogous coupled constraint on relativ e position and time: c return ( a ASV t , base ) ≤ E reserve ( t ) , c meet ( a ASV t , a AUV t , σ t ) ≤ ∆ meet ( t ) . (20) The adaptiv e objectiv e is to maximize expected information gain about future bloom structure per mission cost: max {A t } , { k t } E " X t ∆ I t ( k t , A t ) # − λ E X t c ASV mov e ( a ASV t , a ASV t − 1 ) − λ N X t 1 [ collect at t ] , (21) lake domain Ω base E reserve boundary b t b t +∆ t adaptiv e waypoints return path Fig. 2. Lak e algal-bloom scenario for adaptive sample return. The bloom front advects from b t to b t +∆ t ; adaptive-kernel planning shifts waypoints tow ard moving high-information boundaries while maintaining return-to-base feasibility . The same structure extends to an ASV –A UV team with coordinated dock/undock: surface waypoints place the stack for subsurface profiles, and rendezvous windows replace return-to-base alone as a binding feasibility constraint. where dive costs are subsumed in the energy budget abov e, with dynamics over kernels gov erned by the MaxCal tradeoff in Section IV. Intuiti vely , k t should reweight distinctions tow ard moving bloom boundaries and shear-aligned filaments, and—after each dock—toward consistent cross-depth cov ari- ance, because those structures yield maximal uncertainty re- duction per cycle subject to rendezvous feasibility . This case is operational rather than metaphorical because k t can be interpreted as a task-specific similarity or acquisition function governing what distinctions are worth sensing under budgeted action. This yields a concrete adaptation argument: fixed kernels induce static sampling lattices that under-sample moving bloom fronts, while kernel adaptation concentrates trajectories along information-rich transients subject to return- feasibility constraints. For ASV –A UV teams, undock timing is an additional discrete decision: div es should occur when subsurface uncertainty is high relative to surface-observable structure, and heavy kernel or map updates can run on the ASV while docked. T estable prediction.: Against a fixed-kernel baseline, adaptiv e-kernel planning should produce higher forecast skill at equal energy budget and equal returned sample count, with the largest gains during high-advection intervals; for heterogeneous teams, gains should be most visible on depth- resolved forecast metrics when surface and subsurface fields decorrelate. D. Biological Evolution Biological ev olution instantiates kernel dynamics at the lev el of perceptual and cognitiv e systems. The fitness kernel k f ( genotype , genotype ′ ) encodes which genetic differences translate into phenotypic dif ferences that selection can act on. This kernel co-e volv es with the genome under: dk f dt = − λ δ F fitness δ k + η ( t ) (22) where F fitness is a free-energy-like fitness functional and η is mutational noise. Speciation e vents correspond to bifurcations in kernel space—transitions between basins of attraction of distinct self-consistent fitness kernels. Operationally , k f can 6 be estimated from genotype–phenotype–fitness datasets by fitting local similarity operators that predict fitness response to perturbations. E. Craft Mastery and Embodied Knowledge In [3], the design of a large-scale public artwork was modeled as a product kernel problem: the final artifact is the configuration that simultaneously satisfies the inducti ve kernels of all contributors—mythological, craft, structural, and en vironmental. The MaxCal kernel frame work provides the dynamical substrate for this observation. A concrete illustration is provided by Navagunjar a Reborn , a large-scale Burning Man sculpture whose design and real- ization can be read as a trajectory through interacting kernels rather than ex ecution of a fixed blueprint. In that project, the final artifact emerged from the simultaneous action of mythological, craft, structural, logistical, and environmental kernels: the lead artist’ s symbolic and site-specific priors, Rajesh Moharana’ s dhokra metalwork intuitions, Ekadashi Barik’ s cane-forming expertise, the engineering team’ s load- bearing and fabrication constraints, and the Black Rock Desert’ s wind, transport, and fire-safety requirements. The sculpture’ s realized form was not the maximizer of a single objectiv e, but the configuration that remained viable under all of these distinction-making systems at once. In this sense, the project instantiates the product-kernel view dynamically: iter- ativ e digital modeling, photogrammetric feedback, structural redesign, material substitution, and on-playa improvisation can be interpreted as successive updates to a collective kernel trajectory , progressi vely narrowing the reachable design space until a coherent artifact became the least-assuming surviving path. At the lev el of individual practice, the master artisan’ s embodied kernel appears as a locally stable fixed point in morphological decision space; at the lev el of the collaboration, the completed sculpture is the transient intersection of several such fixed points under hard en vironmental and logistical constraints. A master craftsman’ s k ernel (e.g., Rajesh Moharana’ s dhokra metalwork kernel encoding lost-wax bronze topology) is a stable self-consistent kernel: the distinctions encoded by decades of practice are self-reinforcing under the thermody- namic dynamics of craft execution [3]. Such a kernel is a fixed point because the geometry it induces on morphological space makes it the maximum-caliber choice given the constraints of the tradition. In this domain, a measurable proxy for kernel ev olution is a similarity operator inferred from artifact morphology and process traces across an apprenticeship time series. Apprenticeship is a trajectory through kernel space from an unstable initial kernel tow ard the master’ s fixed point. The thermodynamic cost bound (13) predicts that this trajectory requires sustained metabolic in vestment proportional to the mutual-information gap between novice and master kernels. V I I . R E L AT I O N T O A S S E M B LY T H E O RY Assembly theory [9] measures the assembly index a ( x ) of an object x as the minimum number of recursiv e joining steps required to construct it from basic building blocks. The assembly index is observer -independent and empirically measurable by mass spectrometry . The relationship to MaxCal over kernels is: 1) Ontogenesis vs. mechanics. Assembly theory describes how construction complexity accumulates until a system first acquires the capacity to distinguish microstates—to bear a kernel. MaxCal ov er kernels describes how that kernel evolv es once born. 2) Assembly index as RKHS complexity . W e conjecture that the assembly index a ( x ) is lower -bounded by the RKHS comple xity of the kernel required to represent x : a ( x ) ≥ c · ∥ k x ∥ H + O (1) where k x is the minimal kernel distinguishing x from its chemical precursors. 3) The threshold a ( x ) ≥ 15 . The empirical signature of selection in assembly theory—objects with a ( x ) ≥ 15 are almost certainly products of selection [9]— corresponds, in our framework, to the minimum RKHS complexity required for a self-consistent kernel to exist. Below this threshold, the kernel space is too simple to support self-reinforcing fixed points. A. Conceptual Synthesis The three framew orks compared here share information as a central concept and differ primarily in their primiti ve v ariables: T ABLE I F R A M E W O R K P R I M I T I V E S A N D K E Y O B S E RV A B L E S . Framework Primitive Key observable This work Path over K Kernel trajectory Static kernel limit Fixed k ( x, x ′ ) Mutual information I Assembly theory Construction path Assembly index Maximum caliber Path entropy T rajectory distribution The present paper proposes a synthesis in which kernel dynamics describes ho w kernels traverse the landscape be- tween chemical origin constraints (assembly theory) and stable operating points (static-kernel limit), along least-assuming trajectories selected by MaxCal. V I I I . C L A I M L E V E L S A N D S C O P E For clarity , we separate three claim types used in this paper . 1) Formal definitions/results in this manuscript. Kernel space K , MaxCal lifting to path measures on Π , and the fixed-point condition for self-consistent kernels. 2) Structured correspondences. Mappings from the core framew ork to RG and finite-width NTK e volution. These are mathematically motiv ated identifications, but not complete equiv alence proofs in the present draft. 3) Conjectural bridges. Biological niches, craft mastery , adaptiv e field sampling, and assembly-theoretic thresh- olds are proposed as testable interpretations that require dedicated empirical and model-specific validation. The intended contribution is therefore a unifying variational frame work with explicit testable conjectur es , not a completed final theory of order across all domains. 7 I X . O P E N Q U E S T I O N S 1) Ker nel geodesics. What is the geodesic in ( K , d HS ) between two self-consistent kernels, and does it pass through a saddle point corresponding to a Kuhnian crisis? 2) Quantum kernel dynamics. Does the Fubini-Study metric extension to CP n − 1 support an analogous Max- Cal formulation for quantum kernels, and do quantum phase transitions correspond to fixed-point bifurcations in quantum kernel space? 3) Assembly index and RKHS complexity . Can the conjectured bound a ( x ) ≥ c ∥ k x ∥ H be proved or dis- prov ed for a specific model system (e.g., small organic molecules)? 4) NTK con vergence. Does the NTK of a trained diffusion model con verge to the Hellinger kernel k H of the data distribution in the infinite-width limit? 5) Lake bloom adaptive sampling. Can an online kernel-adaptation policy for dynamic algal blooms im- prov e chlorophyll-forecast skill at fixed energy and fixed returned-sample count relati ve to fixed-kernel sampling—including for heterogeneous ASV –A UV mis- sions with coordinated dock/undock and depth-resolved validation? 6) Paradigm shift thermodynamics. Can the thermody- namic cost bound (13) be used to predict the timescale of scientific paradigm shifts from citation network data— testing the Kuhn corollary as a quantitati ve prediction? A measurable proxy is a time-indexed similarity kernel ov er papers (e.g., embedding or co-citation based) and its trajectory in Hilbert-Schmidt distance. X . C O N C L U S I O N W e have proposed Maximum Caliber over kernel space as a variational principle for the dynamics of distinction-making systems. The frame work organizes renormalization group flow , neural tangent kernel ev olution, biological speciation, and craft mastery as candidate instantiations and analogues of a single kernel-dynamics picture. Fixed points are self-consistent kernels whose stability is set by the curvature of the frozen- kernel path-entropy objectiv e. Kernel change has a thermo- dynamic cost under explicit information-thermodynamic as- sumptions, motiv ating quantitati ve predictions for paradigm- transition timescales and adaptiv e sampling policies. T ogether with assembly theory [9] and Maximum Cal- iber [2], kernel dynamics supports a three-part research pro- gram: assembly theory explains how the first kernel-bearing systems arise; kernel dynamics explains how those kernels ev olve; and the fixed-k ernel limit analyzed here explains what they do while approximately stable, with MaxCal providing the variational principle connecting scales. The present paper is intended not as a completed theory of kernel change across all domains, but as a variational framew ork that makes such change mathematically discussable and empirically targetable. Its value lies in isolating a common question across learning, physics, biology , and craft: what gov erns trajectories through the space of distinction-making structures when representational change itself becomes the dynamical variable? A C K N O W L E D G M E N T S The framew ork builds on intellectual proximity to Sara Imari W alker’ s program on assembly theory and Stev e Press ´ e’ s work on maximum caliber, both at Arizona State Uni versity . The craft-kernel instantiation developed from collaboration with Rajesh Moharana, Ekadashi Barik, and the artisan teams of Odisha documented in [3]. R E F E R E N C E S [1] T . Sagawa and M. Ueda, “Generalized Jarzynski equality under nonequi- librium feedback control, ” Phys. Rev . Lett. , vol. 104, p. 090602, 2010. [2] S. Press ´ e, K. Ghosh, J. Lee, and K. A. Dill, “Principles of maximum entropy and maximum caliber in statistical physics, ” Rev . Mod. Phys. , vol. 85, no. 3, pp. 1115–1141, 2013. [3] J. Das et al. , “Engineering Mythology: A Digital-Physical Framework for Culturally-Inspired Public Art, ” arXiv:2026.xxxxx, 2026. [4] S.-i. Amari, Information Geometry and Its Applications . Springer , 2016. [5] N. N. 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