Adaptive Closure Transport — 2I Edge-Space Decomposition & Closure-Derived Lyapunov on V<sub>600</sub>

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Paper B — PDF Adaptive Closure Transport Repository GitHub — selection-layer-papers Paper A (companion) Transport Laws in Closure Dynamics Operator-witness companion The Closure-Response Operator

Role in the programme: The dynamics layer extended into the adaptive regime — where the substrate itself co-evolves with the field it is transporting. This is the regime of non-equilibrium steady states, Nernst-Planck transport, Hebbian learning, and ARIA's couple_tick dynamics. It is the regime in which selection becomes possible: the substrate "remembers" the field it has transported, and subsequent transport is biased by that memory.

What this paper does: Defines a 4-tuple (M, L_M, W, R_hom) with learning rule Ẇ = G(ρ, j, W). The passive regime (G = 0) recovers the transport-law setting; the active regime (G ≠ 0) carries W-dependent feedback. Delivers two unconditional theorems on the cascade-compatible 600-cell substrate, plus a conditional selection hypothesis with six analytic conditions explicitly listed.

Epistemic status: Two unconditional theorems on the worked-example substrate plus a conditional selection hypothesis. ARIA is presented as a candidate active-regime substrate witness; the row-by-row identification of ARIA's update with G(ρ, j, W) is not verified here. Biological / chemical instantiations (microtubule, phyllotaxis, neural plasticity, cell metabolism, DNA methylation) are roadmap, not results — none is derived. Convergence of the full coupled system beyond the worked example is not asserted.

The 4-Tuple and the Learning Rule

An adaptive-closure-transport substrate is a 4-tuple (M, L_M, W, R_hom) with a learning rule Ẇ = G(ρ, j, W) where:

M, L_M

The substrate

The graph and its Laplacian. For the worked example, M is the 600-cell V₆₀₀ with its short-edge graph Laplacian.

The learned conductivity

An edge-space variable. In the passive limit W is fixed; in the active regime W co-evolves with the transported field via the learning rule G.

R_hom

The fast-equilibrium map

The fast equilibrium ρ* = (L_W + φ⁻²I)⁻¹f as a function of W, exhibiting timescale separation against the slow W-dynamics.

The learning rule

A function of the field ρ, the current j, and the substrate state W. Either G ≡ 0 (passive) or G ≠ 0 (active).

Every learning rule is either passive (G = 0, recovering the transport-law setting) or active (G ≠ 0, with W-dependent feedback). A definitional dichotomy classifying flows / learning rules — not operating points.

Two Unconditional Theorems

Theorem 1 — Explicit 2I edge-space decomposition

Identifying V₆₀₀ with the binary icosahedral group 2I via the standard quaternion embedding, the lifted left-action on the 720-edge space decomposes into exactly 6 free 2I-orbits. After complexification, the 720-dimensional edge space decomposes into 9 isotypic components with explicit complex dimensions

{6, 24, 24, 54, 54, 96, 96, 150, 216}

summing to 720. This closes the EdgeDec hypothesis of the cascade-selection statement — the action of 2I on the edges is no longer conditional.

Theorem 2 — Closure-derived Lyapunov, worked example

On the closed positive cone Ω = ℝ^E_M_≥0, the explicit

V_f(W) = ½ ⟨f, (L_W + φ⁻²I)⁻¹f⟩ + ½λ‖W − W₀‖²

is strongly convex (Hessian ⪰ λI), coercive, and induces a globally convergent subgradient flow Ẇ ∈ −∂(V_f + I_Ω)(W) with linear contraction ‖W(t) − W*‖ ≤ e^−λt‖W(0) − W*‖ (Brézis 1973). On the open interior the flow is the unconstrained gradient flow Ẇ = −∇V_f(W); strong convexity gives the Polyak-Łojasiewicz inequality with exponent θ = ½. Hyperbolicity of the fast subsystem is automatic.

Numerical Witnesses — Reproduces in Seconds

verify_2I_edge_action.py — Theorem 1

Builds the V₆₀₀ vertex set with quaternion identity at index 0, verifies closure under quaternion multiplication, spot-checks the group law on 200 random samples, builds the 720 short edges, verifies the lifted action well-defined, and computes the orbit decomposition.

Quantity	Value
V₆₀₀ size	120
is_2I_under_quat	true
group law holds (200 samples)	true
n_edges	720
edges preserved by all g ∈ 2I	true
edge orbit count	6
edge orbit sizes	[120, 120, 120, 120, 120, 120]
free action on edges	true
isotypic dims	[6, 24, 24, 54, 54, 96, 96, 150, 216]
isotypic dims sum	720

verify_lyapunov_selection.py — Theorem 2

Initialises a perturbed W with λ = 0.1 and a localised source f at vertex 0; runs the slow-flow Ẇ = −∇V_f(W) for 2000 explicit-Euler steps at Δt = 0.05.

Quantity	Value
V_f initial	≈ 9.80
V_f final	≈ 0.057
‖∇V_f‖ initial	≈ 1.40
‖∇V_f‖ final	≈ 6.2 × 10⁻⁵
V monotone non-increasing	true
critical point reached (‖∇V‖ < 10⁻³)	true
late-time linear-contraction rate per step	≈ 3.5 × 10⁻²

The exponential late-time contraction rate matches the linear-rate prediction of the Brézis subgradient theorem.

The Conditional Selection Hypothesis

On the reduced slow-manifold W-flow, the selection hypothesis states: under the six analytic conditions listed below, W converges to a critical point of V, generically a local minimum — a candidate selected operating point. The hypothesis is conditional. Only the closure-derived V₆₀₀ worked example of Theorem 2 discharges the six conditions in this paper.

Condition	Status (V₆₀₀ worked example)
Timescale separation	discharged (relaxation rate ≥ φ⁻²)
Lyapunov potential existence	discharged (Theorem 2: V_f strongly convex on Ω)
Hyperbolicity of the fast subsystem	discharged (linear system, smooth in W on Ω°)
Global existence / forward-orbit precompactness	discharged (V_f coercive)
Łojasiewicz-Simon inequality	discharged (Polyak-Łojasiewicz from strong convexity)
Morse genericity	discharged (single non-degenerate critical point in Ω°; constrained-minimum uniqueness on Ω)

For generic non-convex learning rules, a full transversality argument for the cascade substrate is deferred. Convergence of the full coupled system beyond the worked example is not asserted.

Open Items: Closed in This Paper vs Still Open

Closed in this paper

Edge-space decomposition (EdgeDec): Theorem 1 delivers the explicit 6 free 2I-orbits and 9 isotypic components.
Closure-derived Lyapunov on the V₆₀₀ closed positive cone: Theorem 2 gives strong convexity, coercivity, and linear contraction.
Hyperbolicity of the fast subsystem (worked-example case): automatic at every W ∈ Ω° with relaxation rate ≥ φ⁻².

Still open

Lyapunov for biologically-relevant non-gradient learning rules. Existence of a coercive analytic potential V for closure-derived non-gradient learning rules where no global potential need exist.
Hyperbolicity for non-linear fast dynamics. For closure-derived non-linear fast dynamics (residual-driven beyond the linear regime).
Structural stability for non-convex learning rules. A full transversality argument for the cascade substrate.
ARIA repository-level identification. Row-by-row verification of the dictionary mapping ARIA's released kernel trajectory to a learning rule G(ρ, j, W) admitting the closure-derived V_f.
Paper XIX residual → learning rule. Whether the nonlinear closure residual gives rise to a learning rule with a Lyapunov potential.
Edge-space ↔ vertex-space lift. The relation between the explicit 6-orbit / 9-isotypic edge-space structure and the vertex-space 94 + 26 integer / irrational split.

What This Paper Does Not Deliver

Heredity / replication. This paper describes adaptive transport, not life. Adaptive transport is strictly weaker than life: a river shaping its bed is adaptive, a cell is alive. The replication operator R: M → M × M is not introduced here
Measurement / observer content. The framework says nothing about what adaptive-transport systems report about their own state
Subjective experience. Whether an adaptive-transport system has qualia is a philosophical question this paper is silent on. If qualia are physical, they live on an adaptive-closure-transport substrate; the converse is not claimed
A derivation of specific biological learning rules. Microtubules, phyllotaxis, neural plasticity, cell metabolism, DNA methylation are listed as candidate further instantiations — roadmap, not results
Convergence of the full coupled system beyond the worked example case

What This Is, And Is Not

What is delivered. Two unconditional theorems on the cascade-compatible V₆₀₀ substrate. An explicit 2I edge-space decomposition into exactly 6 free orbits and 9 complex isotypic components with dimensions {6, 24, 24, 54, 54, 96, 96, 150, 216} summing to 720 (Theorem 1). A closure-derived strongly-convex Lyapunov V_f(W) on the closed positive cone with linear-contraction subgradient flow at rate ≥ λ (Theorem 2). All six analytic conditions of the selection hypothesis discharged for the worked example.

What is not delivered. A selection theorem for the full coupled system beyond the worked example. A row-by-row identification of ARIA's couple_tick with the 4-tuple framework. A derivation of biological / chemical instantiations — the application catalogue is roadmap, not results. A heredity / replication operator. Subjective-experience claims of any kind.

The selection layer, made concrete on one substrate. Two theorems that close half the conditions, six open items that name the rest.

Reproduction package open-access. Two scripts at github.com/vfd-org/selection-layer-papers: verify_2I_edge_action.py (~10 s) and verify_lyapunov_selection.py (~3 min). numpy + scipy only, no fitted parameters. MIT licence.