CatGT · Catalytic Generative Theory · dm³ Framework

Zeolites — Order Matters

Catalysts do not commute. The operator firing sequence is the selectivity.

HZSM-5 and HMCM-22 are both zeolites with similar Brønsted acidity. Given the same feedstock — ethanol — they produce opposite product distributions: one favours aromatics, the other light olefins. Pore size alone cannot explain this. The decisive factor is the order in which the four dm³ operators fire. ZSM-5: C→K→F→U. MCM-22: C→F→K→U. Same operators; different sequence; different products. This is proposed as a hypothesis, not yet a confirmed mechanism: CatGT models zeolite shape selectivity and Pt–Sn ensemble effects with a DNLS self-trapping bound on the contact manifold \(\mathcal{X}_\text{cat}\) (Theorem 1). Only the algebraic core of that theorem is machine-checked in Lean 4; the physical claims are model-level or open, as tagged below.

C K F U order matters

CatGT · Self-Trapping Selectivity Principle (V5) Zenodo: 10.5281/zenodo.22851704 (V5) Lean 4: 13 theorems · 0 sorries (author’s run, Lean 4.32)

§ 1

The Puzzle of Reversed Selectivity

Zeolites are crystalline aluminosilicate frameworks with nanometre-scale pores. They are the workhorses of the petrochemical industry: fluid catalytic cracking (FCC), methanol-to-olefins (MTO), and ethanol dehydration all run in zeolite beds. The selectivity — which products come out — was thought to be determined primarily by pore size and Brønsted acidity (the density of Al-substituted sites that donate protons).

Zilacleide de Sousa et al. (2014, 2023) broke this picture. HZSM-5 (pore: 5.6 Å 10-ring channels, MFI topology) and HMCM-22 (pore: 7.1 Å supercage + 4.0 Å 10-ring exits, MWW topology) were given the same ethanol feed under matched conditions. Similar Brønsted acidity. Different outcome: HZSM-5 produced mainly aromatics; HMCM-22 produced mainly light olefins (ethylene, propylene). The selectivity was reversed.

"Pore-size arguments alone cannot explain the Sousa contrast. CatGT explains it as a difference in operator firing order: the same four operators fire in different sequences." — earlier CatGT draft, Introduction

The dm³ reading: the four operators C, K, F, U are the same in both zeolites. But the geometric topology of each framework determines the order in which they encounter the reactant. In ZSM-5, the 10-ring channel is the first geometric constraint the molecule meets after adsorption: K fires immediately after C. In MCM-22, the molecule enters the large supercage first and branches (F fires before it hits any size constraint K). Same operators; different sequence. Non-commutativity.

§ 2

The Catalyst Contact Manifold

Definition 1 — Catalyst Contact Manifold (CatGT)

The catalyst contact manifold is \(\mathcal{X}_\text{cat} = (\mathbb{R}^3, \alpha_\text{cat})\) with contact form \[\alpha_\text{cat} = dz - r^2\,d\theta\] in cylindrical coordinates \((r, \theta, z)\), where:

\(r\) = pore aperture radius (Å) — the physical channel width
\(\theta\) = catalytic cycle phase — position in the reaction sequence
\(z\) = reaction coordinate — progress along the channel axis

The Reeb vector field \(R = \partial_z\) satisfies \(\alpha(R) = 1\) and \(\iota_R\,d\alpha = 0\). Its integral curves \((r_0, \theta_0, z_0 + t)\) hold \(r\) and \(\theta\) fixed: they are straight vertical lines, not helices. Because the flow preserves \(\alpha\), it preserves \(\alpha \wedge d\alpha\), so it is volume-preserving and has no attracting set. The contact manifold supplies coordinates only; the confinement claimed below comes from DNLS self-trapping (Theorem 1), a separate mechanism.

The DNLS equation governs energy localisation at the \(N\) catalytic sites along the helix:

DNLS equation on N catalytic sites

i ψ̇_n = -J(ψ_{n+1} + ψ_{n-1}) - λ|ψ_n|² ψ_n

J = inter-site coupling   (tunnelling / diffusivity analogue)
λ = on-site nonlinearity  (binding energy analogue)

IPR = ∑|ψ_n|⁴ / (∑|ψ_n|²)²

IPR → 1/N  :  delocalised (mobile reactant, accessible pathway)
IPR → 1    :  self-trapped (blocked site, selectivity enforced)

Below a self-trapping threshold, excitations are delocalised — the reactant can access the fixed point. Above it, the wavefunction self-traps and the pathway is blocked. The model’s confinement bound is \(r^*(\lambda) = a\sqrt{J/\lambda}\), with \(a\) a lattice length scale (model assumption; its normalization — fixed peak amplitude versus fixed norm — is open). What counts as the threshold \(\lambda_c\), and how it depends on pore size, is also open; see the V5 paper.

§ 3

Theorem 1 — Self-Trapping Selectivity Principle

Theorem 1 — Self-Trapping Selectivity · CatGT_Main.lean (renamed from “Helical Selectivity Principle”)

Let \(a\) be a lattice-spacing length scale and define the self-trapping tube \(\mathcal{T}_\lambda=\{(r,\theta,z): r\le r^*(\lambda)\}\) with \(r^*(\lambda):=a\sqrt{J/\lambda}\).

(i) \(\mathcal{T}_\lambda\) is a coordinate region of \(\mathcal{X}_\text{cat}\), not a dynamical attractor: the flow of \(R\) preserves \(\alpha\wedge d\alpha\) and so has no attracting set, and its orbits are straight lines in \(z\). It is the set of points with \(r\le r^*(\lambda)\). ✓ reeb_alpha_eq_one (only α(R)=1 is machine-checked)

(ii) A DNLS state whose radial coordinate satisfies the self-trapping condition remains confined to \(r(t)\le r^*(\lambda)\); the DNLS self-trapping transition, not the Reeb flow, is the confining mechanism. How this connects to a fixed point \(x^*\) of the pipeline is not established. ⚠ MODEL — not formalized

(iii) Selectivity factor: \[\sigma = 1-\Big(\frac{r^*(\lambda)}{r_\text{pore}}\Big)^2 = 1-\frac{J}{\lambda}\Big(\frac{a}{r_\text{pore}}\Big)^2.\] The algebra is checked; whether \(\sigma\) is the selectivity of a real catalyst is a model claim. ✓ selectivityFactor_eq (algebra only)

Open: the normalization behind \(r^*=a\sqrt{J/\lambda}\) (fixed peak amplitude versus fixed norm) is not settled here; see the V5 paper.

CatGT_Main.lean — core inequality (square-root monotonicity) ✓ CLOSED · author’s run, Lean 4.32

/-- Critical (self-trapping) radius r*(λ) = a·√(J/λ), where `a > 0` is the
    lattice-spacing length scale. J and λ are both energies, so J/λ is
    dimensionless and `a` is what gives r* the dimension of a length
    (Theorem 1's 2026-09-19 correction on the paper's page). NOTE: this is
    the *definition* the theorems below are stated against; that r* is the
    physically correct DNLS self-trapping width is a separate claim, and
    depends on what is held fixed (see the 2026-09-20 note in the header). -/
noncomputable def criticalRadius (a J lam : ℝ) (ha : 0 < a) (hJ : 0 < J)
    (hlam : 0 < lam) : ℝ :=
  a * Real.sqrt (J / lam)

/-- **Self-Trapping Selectivity Principle** — formal core of Theorem 1.
    r² ≤ a²·(J/λ) ⟹ r ≤ r*(λ) = a·√(J/λ).
    NOTE (what this does and does not say): the hypothesis is the squared
    form of the conclusion, so this is the equivalence "r ≤ r* ⟸ r² ≤ r*²"
    for r ≥ 0, i.e. sqrt-monotonicity. It does NOT derive confinement from
    any DNLS dynamics; that is Theorem 1(ii)'s open content. -/
theorem helical_selectivity (a J lam : ℝ) (ha : 0 < a) (hJ : 0 < J)
    (hlam : 0 < lam) (r_state : ℝ) (hr : 0 ≤ r_state)
    (h_confined : r_state ^ 2 ≤ a ^ 2 * (J / lam)) :
    r_state ≤ criticalRadius a J lam ha hJ hlam := by
  unfold criticalRadius
  have h1 : r_state ≤ Real.sqrt (a ^ 2 * (J / lam)) := by
    rw [← Real.sqrt_sq hr]; exact Real.sqrt_le_sqrt h_confined
  have h2 : Real.sqrt (a ^ 2 * (J / lam)) = a * Real.sqrt (J / lam) := by
    rw [Real.sqrt_mul (sq_nonneg a), Real.sqrt_sq ha.le]
  rw [h2] at h1
  exact h1

CatGT_Main.lean — selectivityFactor_eq (algebraic identity only) ✓ CLOSED · author’s run, Lean 4.32

/-- Selectivity factor σ = 1 - (r*/r_pore)² = 1 - (J/λ)(a/r_pore)². -/
noncomputable def selectivityFactor (a J lam r_pore : ℝ)
    (ha : 0 < a) (hJ : 0 < J) (hlam : 0 < lam) (hr : 0 < r_pore) : ℝ :=
  1 - (criticalRadius a J lam ha hJ hlam / r_pore) ^ 2

theorem selectivityFactor_eq (a J lam r_pore : ℝ)
    (ha : 0 < a) (hJ : 0 < J) (hlam : 0 < lam) (hr : 0 < r_pore) :
    selectivityFactor a J lam r_pore ha hJ hlam hr
      = 1 - (J / lam) * (a / r_pore) ^ 2 := by
  unfold selectivityFactor criticalRadius
  have hs : Real.sqrt (J / lam) ^ 2 = J / lam :=
    Real.sq_sqrt (div_nonneg hJ.le hlam.le)
  have hkey : (a * Real.sqrt (J / lam) / r_pore) ^ 2
      = (J / lam) * (a / r_pore) ^ 2 := by
    calc (a * Real.sqrt (J / lam) / r_pore) ^ 2
        = Real.sqrt (J / lam) ^ 2 * (a / r_pore) ^ 2 := by ring
      _ = (J / lam) * (a / r_pore) ^ 2 := by rw [hs]
  rw [hkey]

── CatGT_Main.lean · Sorry Audit · September 2026 ────────────── ✓ ipr_between_zero_and_one IPR ∈ (0,1] (N>0, nonzero norm) ✓ criticalRadius_pos r* > 0 ✓ criticalRadius_antitone r* non-increasing in λ ✓ helical_selectivity r² ≤ a²J/λ ⇒ r ≤ r* (sqrt monotonicity only) ✓ selectivityFactor_eq 1-(r*/r_pore)² = 1-(J/λ)(a/r_pore)² (algebra only) ✓ reeb_alpha_eq_one α(R)=1 ✓ reeb_orbit_advances z-advance over t equals t·α(R) ✓ dnlsNorm_nonneg discrete norm ≥ 0 ✓ catgt_dm3_disk disk-membership facts ✓ ensemble_scaling_forms_diverge (1-1/2)² ≠ 1-(1/2)² ✓ relaxStep_fixed / relaxStep_contracts / relax_iterate_dist Total: 13 closed · 0 admits · 0 sorries Kernel status: author's run, Lean 4.32.0 / Mathlib v4.32.0, repeated on the final file (sha256 19e8df03ad44); not run on the v4.14.0 pinned by the io repository CI.

§ 4

The Non-Commutativity of Catalysts

The same four operators — C (Contact/Adsorption), K (Knob/Pore constraint), F (Fold/Shape filter), U (Unfold/Desorption) — fire in different sequences depending on zeolite topology. This is operator non-commutativity made industrial.

HZSM-5 · MFI topology

C → K → F → U

Molecule adsorbs (C), immediately hits the 10-ring channel constraint (K: 5.6 Å), shape selection occurs (F), product desorbs (U). K fires before F. Aromatics favoured — they fit the channel geometry.

HMCM-22 · MWW topology

C → F → K → U

Molecule adsorbs (C), enters the large supercage and branches (F: 7.1 Å), then hits the 10-ring exit constraint (K: 4.0 Å), product desorbs (U). F fires before K. Light olefins favoured — branching precedes confinement.

MHD Reconnection · plasma

K → F → C → U

Field-line topology constrains trajectories (K), plasmoid instability fires irreversibly above Sǒ≈10⁴ (F), Alfvénic jet compresses outflow (C), plasma restabilises (U). NASA MMS grounded: mean rate 0.14 ± 0.09 V_A.

Three physical systems. Three operator sequences. One contact manifold \(\mathcal{X}_\text{cat}\). The same mathematics describes the zeolite channel, the magnetic current sheet, and — as the BZ chapter shows — the chemical oscillator. What changes between systems is not the operators but the topology that determines their order. The first two firing orders are the operator-order hypothesis of the companion paper (tagged model/open there); the shared vocabulary across the three systems is an analogy, not a proved common theorem.

§ 5

Shared Functional Form — Not a Shared Invariant

The scale \(\sqrt{J/\lambda}\) is the generic crossover between a linear spreading term and a localizing term (healing length, Debye length, Thiele modulus, …), so seeing the same functional form in several domains is weak evidence of shared geometry by itself. The table lists the analogy that has been proposed in each domain and the status of each entry as it is stated in that domain’s own chapter; it is not a claim that one invariant has been derived across them. The V5 catalysis paper does not carry this table.

Domain	\(J/\lambda\) analogue	Observable	Status
Zeolite catalysis (ZSM-5, MCM-22)	\(D/E_b\)	Pore cut-off \(r^*\), selectivity reversal	Model (Theorem 1); Lean checks the algebra only
Metal ensembles (Pt–Sn)	\(t_{ij}/U\)	Ensemble size \(N^*\), propylene selectivity	Corollary 1 — model
DNLS soliton	\(J/\lambda\)	Self-trapping IPR	Derived — direct; Lean ✓
MHD reconnection (NASA MMS)	\(V_A^2/\eta\)	Rate ≈ 0.1 V_A; Sǒ ≈ 10⁴	DustyPlasma.lean 13✓; MMS grounded
Financial markets (CapitalGuard)	\(D_s/\gamma\)	EKF regime radius; live Sharpe 2.43	Implemented v2.1; contactomorphism open
Autophagy / mTOR	\(\mu_{\max} \approx -0.41\)	mTOR limit cycle; Lyapunov W	AutophagyDm3.lean — 0 sorries
Triple-alpha / stellar nucleosynthesis	\(\kappa_\text{nuc}/\Delta T\)	\(T^{40}\) fold at \(T^* \approx 10^8\) K	AutophagyDm3.lean — 0 sorries
Polylaminin / SCI	\(\mu_{\max} \approx -0.65\)	6/8 patients motor recovery; Whitney A₁ fold	Chapter B; ANVISA Phase I Jan 2026
Wavenumber 6 / Saturn hexagon	\(\eta^{-k}\) tribonacci	m=6 azimuthal mode; stable decades	Zenodo 19501888; partial Lean
Enceladus cryovolcanism	\(\kappa_\text{cryo}/\Delta P_\text{sub}\)	Plume periodicity; subsurface operator cycle	In preparation
Moon Base Architecture	\(\kappa_\text{struct}/\Delta P_\text{load}\)	Structural resonance modes	Submitted to NASA
Cymatics / Chladni / turtle shell	\(\omega_n/\gamma_\text{damp}\)	Nodal geometry; scute boundaries	Bienal EXP13; 7 machines; Projeto TAMAR
Faraday rotation / IFE	\(V \cdot B / \gamma_\text{relax}\)	Non-reciprocal phase; Verdet constant	In preparation (GOMC Vol. IV)
Dusty plasma	\(\alpha_\text{dust}/\kappa^*\)	\(d_f \approx 1.6\)–\(1.8\); \(\mu_{\max} = -0.42\)	Vol. III Ch. 3; bridge derivation open
Neural oscillations / HPA axis	\(\mu_{\max} \approx -0.38\) to \(-0.55\)	Circadian limit cycles; cortisol period T*	Cited; derivation in preparation
n-Bonacci criticality thresholds	\(\Delta_n = \rho_n - \|\rho_n^{(2)}\|\)	\(\lambda_c(n) \to 7/6\) for \(n \geq 4\)	Zenodo 20077205; Lean pending

§ 6

Falsifiable Predictions (7, 8, 10)

Predictions 7 and 8 concern the self-trapping model (Theorem 1) and are open or model-level as marked. Prediction 10 comes from the operator-order hypothesis of the companion paper and does not depend on Theorem 1; it is the primary experimental test with existing laboratory equipment.

Prediction 7 — DNLS threshold in zeolite pores · OPEN (formula withdrawn)

The formula previously stated here for \(\lambda_c\) in terms of \(r_\text{pore}\) and \(\sigma_\text{LJ}\) is withdrawn. Its direction was inconsistent with the sentence that accompanied it, and what “threshold” means and how \(N\) enters are unsettled. No replacement formula is asserted here; see Section 4 of the V5 paper.

Prediction 8 — Pt–Sn selectivity scaling · MODEL

The fraction of surviving size-2 Pt-only ensembles on \(\text{Pt}_{1-x}\text{Sn}_x/\text{Al}_2\text{O}_3\) is proposed to scale as \((1-x)^2\) (against the alternative \(1-x^2\); which is correct is open) and is identified in the model with \((r^*/r_\text{pore})^2\). Propylene selectivity is the complement, \(\sigma \approx 1-(1-x)^2\), increasing with Sn loading. Testable by in-situ XAS at \(x = 0, 0.1, 0.2, 0.3, 0.4\).

Removed: a former Prediction 9 (a “Reeb-helix” phase \(\theta(t)=\omega t+\theta_0\)). It depended on Reeb orbits being helical; they hold \(\theta\) fixed, so there is no advancing phase to measure. It was cut, not replaced.

Prediction 10 — Operator order switch in MCM-22 (primary test)

Increasing temperature (350→450 °C) or decreasing feed concentration in HMCM-22 ethanol conversion shifts firing order from C→F→K→U toward C→K→F→U. Signature: reversal of the ethoxy/diethyl-ether vs aromatic-ring DRIFTS sequence at shorter contact times. This is directly testable with existing equipment at any zeolite catalysis lab.

§ 7

The Chemistry Arc — BZ, Turing, Zeolites

CatGT sits in the chemistry arc of the Principia Orthogona alongside the BZ reaction and Turing morphogenesis. The connection is not superficial:

The BZ reaction (ch-belousov-zhabotinsky.html) is a chemical oscillator where the order of autocatalytic and inhibitory steps determines whether the system oscillates or converges. Change the relative timescales and you change the firing order — exactly the operator-order sensitivity that CatGT captures.

The Oregonator (BZ model) and the DNLS equation (CatGT) are both descriptions of coupled nonlinear oscillators on a contact manifold. The BZ reaction runs at the molecular scale in solution; the zeolite runs at the molecular scale on a crystalline surface. Same contact form; different substrate; different Reeb orbit.

The difference is selectivity: the zeolite selects which molecular pathway reaches the fixed point \(x^*\). The BZ reaction selects which temporal mode dominates. The π operator (period T*) governs BZ; the self-trapping selectivity model (Theorem 1) is proposed for the zeolite. They share the operator-chain vocabulary on \(\mathcal{X}_\text{cat}\); that this amounts to one theorem applied twice is not proved here.

Unification Claim (CatGT) — Informal

The BZ reaction, Turing morphogenesis, and zeolite catalysis are three instances of the same contact-geometric structure:

BZ: Oregonator on contact manifold; fixed point = limit cycle; π operator sets period.
Turing: Reaction-diffusion on contact manifold; fixed point = patterned state; K operator selects wavenumber.
Zeolite: DNLS on contact manifold; fixed point = selective product; operator order sets selectivity.

In all three: the contact form \(\alpha = dz - r^2\,d\theta\) is the universal constraint. The operators G = U ∘ F ∘ K ∘ C act on it. The fixed point is \(x^*\). The rate of convergence is μ = −2.

§ 8

Central Open Problems

Conjecture — Global Contactomorphism (Open Problem 8)

There exists a contactomorphism \(\varphi\) from the DNLS phase-space cylinder (equipped with the symplectic structure from the DNLS energy functional at fixed \(\lambda\)) to the contact manifold \(\mathcal{X}_\text{cat}\), preserving \(|\psi| \mapsto r\). Under \(\varphi\): the self-trapping threshold maps to \(r = r^*(\lambda)\); the self-trapped state maps to the boundary of \(\mathcal{T}_\lambda\); the delocalised state maps to the interior.

Status: Genuine open problem in contact geometry. Not a Lean tooling issue. Proof would require symplectic reduction of the DNLS energy functional followed by contactomorphism to \(\ker(\alpha_\text{cat})\). Open — Part II

Open items are stated in prose in the V5 paper (Section 6): the quantitative fit to the Sousa et al. selectivity data, the normalization of \(r^*\), the threshold prediction, the definition of the operators \(C,K,F,U\) and the fixed point \(x^*\), and which Pt–Sn survival form is correct. The earlier “3 ADMITS” block that appeared here quoted declarations that no longer exist.