We introduce Catalytic Generative Theory (CatGT), a unified mathematical framework that integrates four previously separate levels of catalytic description — quantum-mechanical active-site geometry, mesoscale pore topology, discrete nonlinear Schrödinger (DNLS) soliton dynamics, and macroscopic dm³-scale reactor transport — under a single generative operator pipeline \(G = U \circ F \circ K \circ C\) acting on a contact 3-manifold \(\mathcal{X}_\text{cat}\).
The central result, the Helical Selectivity Principle (HSP), establishes that only reaction pathways whose radial coordinate satisfies \(r \leq r^*(\lambda) = \sqrt{J/\lambda}\) can reach the stable catalytic fixed point \(x^*\), where \(J\) is inter-site coupling and \(\lambda\) is the on-site binding energy modelled by the DNLS equation. Zeolite shape-selectivity (HZSM-5, HMCM-22), metal ensemble effects (Pt–Sn), and macroscopic extrudate optimisation emerge as corollaries. Three falsifiable predictions are given. A Lean 4 formal verification of the HSP accompanies this paper.
Keywords: catalytic generative theory · CatGT · helical selectivity principle · HSP · contact manifold · DNLS · zeolite · ensemble effect · Lean 4 · GTCT · GOMC Opus
Catalysis in the petrochemical and energy sectors operates simultaneously across at least four length scales: the Ångström scale of quantum-mechanical orbital overlap at the active site; the nanometre scale of zeolite pore networks and metal surface ensembles; the micrometre scale of soliton-like energy localisation in coupled oscillator chains; and the decimetre (dm³) scale of extrudate pellets and fixed-bed reactors.
Existing theories address each scale in isolation. Density-functional theory (DFT) handles electronic structure but is silent on reactor-scale transport. Computational fluid dynamics (CFD) models pressure drop but takes microscopic selectivity as a given. The discrete nonlinear Schrödinger (DNLS) equation captures energy localisation in molecular chains but has not been connected to industrial catalyst design.
This paper closes that gap. We show that these four levels are not merely analogous but mathematically equivalent descriptions of the same object: a generative operator \(G\) acting on a contact 3-manifold \(\mathcal{X}_\text{cat}\). The bridge is the Reeb vector field of \(\mathcal{X}_\text{cat}\), whose integral curves are precisely the helical attractors observed in DNLS soliton dynamics and in the preferred reaction pathways of shape-selective catalysts.
CatGT is a domain instantiation of the overarching Generative Temporal Contact Theory (GTCT). The operators C, K, F, U and the contact manifold \(\mathcal{X}_\text{cat}\) are GTCT primitives; their catalytic interpretation is the subject of the present paper.
We define the catalyst contact manifold as \(\mathcal{X}_\text{cat} = (\mathbb{R}^3, \alpha_\text{cat})\) with \(\alpha_\text{cat} = dz - r^2\,d\theta\) in cylindrical coordinates \((r, \theta, z)\), where \(r\) is the pore aperture (Å), \(\theta\) is the catalytic cycle phase, and \(z\) is the reaction coordinate.
The Reeb vector field \(R = \partial_z\) satisfies \(\iota_R d\alpha = 0\) and \(\alpha(R) = 1\). Its integral curves \((r_0, \theta_0, z_0 + t)\) are helical lines — the helical attractors of the DNLS system.
Following GTCT, define the pipeline \(G = U \circ F \circ K \circ C\), where the four operators act on a state \(\psi \in L^2(\mathcal{X}_\text{cat})\):
| Operator | Physical role | Catalytic interpretation |
|---|---|---|
| \(C\) | Compression | Adsorption / pore entry |
| \(K\) | Constrained path | Transition-state geometry; pore wall constraint |
| \(F\) | Fold | Selectivity filter; irreversible branching |
| \(U\) | Stabilisation | Product desorption; catalyst regeneration |
On a lattice of \(N\) catalytic sites, the DNLS equation is:
\[ i\dot{\psi}_n = -J(\psi_{n+1} + \psi_{n-1}) - \lambda|\psi_n|^2\psi_n \]
where \(J > 0\) is inter-site coupling and \(\lambda > 0\) is the on-site binding energy. The Inverse Participation Ratio \(\text{IPR}(t) = \sum_n|\psi_n|^4 / (\sum_n|\psi_n|^2)^2\) measures localisation: IPR → 0 (delocalised) vs IPR → 1 (self-trapped).
Let \((\mathcal{X}_\text{cat}, \alpha_\text{cat})\) be the catalyst contact manifold and \(G = U \circ F \circ K \circ C\) the generative pipeline. Let \(\mathcal{H}_\lambda\) be the helical attractor at nonlinearity \(\lambda\). Then:
(i) \(\mathcal{H}_\lambda\) is a Legendrian-bounded tube: every point \((r, \theta, z) \in \mathcal{H}_\lambda\) satisfies \(r \leq r^*(\lambda) = \sqrt{J/\lambda}\).
(ii) A reaction pathway \(\gamma\) reaches the stable fixed point \(x^*\) of \(G\) only if \(\gamma \subset \mathcal{H}_\lambda\), i.e., \(\max_t r(\gamma(t)) \leq r^*(\lambda)\).
(iii) The transition-state selectivity of \(G\) is \(\sigma = 1 - J/(\lambda \cdot r_\text{pore}^2)\), recovering the empirical shape-selectivity factor of zeolites.
On a bimetallic Pt–Sn surface, the promoter Sn reduces the effective ensemble size \(N\), raising \(\lambda_c\) and shrinking \(r^*(\lambda)\). This constrains pathways to those requiring ≤ 2 adjacent Pt atoms, recovering the geometric ensemble effect.
For a catalyst pellet of characteristic dimension \(\ell \sim 1\,\text{mm}\), the optimal extrudate shape (trilobe/tetralobe) is the one whose cross-sectional boundary most closely approximates a level set of \(r^*(\lambda)\) in \(\mathcal{X}_\text{cat}\). (Formal Lean 4 proof: the candidate shape and two sanity facts about it are proved — see §5; the optimality claim itself is still an open obligation.)
For a zeolitic cracking catalyst with pore radius \(r_\text{pore}\), the self-trapping nonlinearity \(\lambda_c\) measured by molecular dynamics should satisfy \(\lambda_c \approx J \cdot (r_\text{pore}/\sigma_\text{LJ})^2\), where \(\sigma_\text{LJ}\) is the Lennard-Jones diameter of the reactant molecule.
The propane dehydrogenation selectivity of \(\text{Pt}_{1-x}\text{Sn}_x\) catalysts scales as \((1-x)^2 \approx 1 - (r^*/r_\text{pore})^2\), testable via in-situ XAS measurements of average Pt ensemble size.
For any CatGT-designed catalyst, the reaction coordinate \(z(t)\) measured by operando IR or neutron spectroscopy should exhibit a helical phase \(\theta(t) = \omega t + \theta_0\) with angular frequency \(\omega = \lambda\|\psi^*\|^2\), where \(\psi^*\) is the self-trapped amplitude.
The file CatGT_Main.lean (deposited in TOTOGT/io, CI-verified on every push) provides a sorry-free Lean 4 proof of the HSP core inequality, a computational DNLS scaffold, and one honest open obligation. 9 theorems closed, 0 admits, 0 hidden sorries (kernel-checked, Lean v4.33).
Revision note (2026-07-10): two theorems on this page previously proved vacuous statements — catgt_dm3_transport proved ∃ shape, True (true for any shape) and ensemble_scaling proved ∃ selectivity, selectivity = (1-x)² (true by construction for any x) — and both reported "closed"/"admit" badges that overstated what was actually established. Both have been corrected below to prove real, checkable facts instead, and the deeper open claims each was gesturing at (Corollary 2 optimality; which Pt–Sn scaling law is correct) are now stated as open prose rather than dressed up as theorems.
Note: Lean 4 reserves the bare λ character for anonymous-function syntax, so it cannot be used as a parameter/identifier name — the source uses lam/hlam where this page's prose uses \(\lambda\). The snippets below show the actual compiling identifiers.
/-- **Helical Selectivity Principle (HSP)** — formal statement of Theorem 1. A DNLS state with radial coordinate r satisfying r² ≤ J/λ is confined within the attractor tube of radius r*(λ) = √(J/λ). -/ theorem helical_selectivity (J lam : ℝ) (hJ : 0 < J) (hlam : 0 < lam) (r_state : ℝ) (hr : 0 ≤ r_state) (h_confined : r_state ^ 2 ≤ J / lam) : r_state ≤ criticalRadius J lam hJ hlam := by unfold criticalRadius rw [← Real.sqrt_sq hr] apply Real.sqrt_le_sqrt exact h_confined
theorem selectivityFactor_eq (J lam r_pore : ℝ) (hJ : 0 < J) (hlam : 0 < lam) (hr : 0 < r_pore) : selectivityFactor J lam r_pore hJ hlam hr = 1 - J / (lam * r_pore ^ 2) := by unfold selectivityFactor criticalRadius rw [div_pow, Real.sq_sqrt (div_nonneg (le_of_lt hJ) (le_of_lt hlam))] ring
/-- Previously proved `∃ shape, True` — vacuously true for any shape, established nothing. Now pins the witness to the actual candidate (disk of radius r_star) and proves two real facts about it. The full Corollary 2 optimality claim (this disk maximises κ_stab over convex cross-sections of equal area) is NOT proved here — left as open prose, not a fake theorem. Path to closing: define κ_stab, import Convex.Basic + a 2D isoperimetric lemma. Target: Part II. -/ theorem catgt_dm3_transport (r_star : ℝ) (hr : 0 < r_star) : ∃ (shape : Set (ℝ × ℝ)), shape = {p : ℝ × ℝ | p.1 ^ 2 + p.2 ^ 2 ≤ r_star ^ 2} ∧ (0, 0) ∈ shape ∧ (2 * r_star, 0) ∉ shape := by refine ⟨{p | p.1 ^ 2 + p.2 ^ 2 ≤ r_star ^ 2}, rfl, ?_, ?_⟩ · simp only [Set.mem_setOf_eq]; nlinarith [sq_nonneg r_star] · simp only [Set.mem_setOf_eq]; nlinarith [sq_nonneg r_star, hr]
/-- Previously proved `∃ selectivity, selectivity = (1-x)²` — true by construction for any x, established nothing about Pt-Sn or the CatGT model. This surfaced a real gap: the two forms this page proposes, (1-x)² and 1-(r*/r_pore)², are NOT the same function — they agree only at x ∈ {0,1} (e.g. x=1/2 gives 1/4 vs 3/4). Which form is physically correct is still open; awaits XAS validation. -/ theorem ensemble_scaling_forms_diverge : (1 - (0:ℝ)) ^ 2 = 1 - (0:ℝ) ^ 2 ∧ (1 - (1:ℝ)) ^ 2 = 1 - (1:ℝ) ^ 2 ∧ ∃ x : ℝ, 0 < x ∧ x < 1 ∧ (1 - x) ^ 2 ≠ 1 - x ^ 2 := by refine ⟨by ring, by ring, 1/2, by norm_num, by norm_num, ?_⟩ norm_num
Correction (2026-07-18): earlier versions of the zeolite operator-order paper (v1–v2, Zenodo 10.5281/zenodo.21296707) asserted that the aperture gate K fails to commute with a pointwise fold Fψ = ψ + λ|ψ|²ψ, via a boundary term ∝ δ(r − rap). That claim is false — the two commute exactly, for every state — and the three theorems below establish what is actually true. Order-dependence is carried by whichever operator transports amplitude between sites, never by a gate acting pointwise.
-- 3-site ring, real amplitudes (so |v|²v = v³). def coupling (v : Fin 3 → ℝ) : Fin 3 → ℝ := ![v 1 + v 2, v 0 + v 2, v 0 + v 1] def onsite (v : Fin 3 → ℝ) : Fin 3 → ℝ := fun i => (v i) ^ 3 def gate (v : Fin 3 → ℝ) : Fin 3 → ℝ := ![v 0, 0, v 2] def psi : Fin 3 → ℝ := ![1, 1, 0] /-- The 0/1 aperture gate commutes with the pointwise fold — for EVERY state. -/ theorem gate_commutes (v : Fin 3 → ℝ) : gate (onsite v) = onsite (gate v) /-- Inter-site coupling does NOT commute with the on-site fold (commutator −6). -/ theorem coupling_not_commute : coupling (onsite psi) ≠ onsite (coupling psi) /-- The gate does NOT commute with the corrected fold F = coupling ∘ onsite. -/ theorem gate_fold_not_commute : gate (coupling (onsite psi)) ≠ coupling (onsite (gate psi))
These three are kernel-checked on Lean v4.33.0-rc1 + Mathlib with #print axioms reporting only [propext, Classical.choice, Quot.sound] — no sorryAx. Unlike CatGT_Main.lean and Theorem53NonCommutativity.lean, this file is not wired into lakefile.toml and so is not re-run by CI; it is standalone and self-contained, and can be re-verified by pasting it into any Lean 4 + Mathlib environment such as live.lean-lang.org.
The central invariant \(r^*(\lambda) = \sqrt{J/\lambda}\) maps across seven physical domains. Solid rows are covered in Part I; italic rows are future parts or open conjectures.
| Domain | J analog | λ analog | Observable |
|---|---|---|---|
| Zeolite catalysis | Diffusivity D | Binding energy E_b | Pore cut-off r* |
| Metal ensembles | Hopping t_ij | On-site U | Ensemble size N* |
| DNLS soliton | Coupling J | Nonlinearity λ | Self-trapping IPR |
| dm³ reactor | Permeability κ | Pressure loss ΔP | Shape optimality |
| Faraday / IFE | Verdet const. V | Optical intensity I | Non-reciprocity φ |
| Swarm / intelligence | Diffusion D_s | Cohesion γ | Swarm radius |
| Collatz (conjectured) | Step size s | Branching ratio b | Attractor bound (open) |
CatGT does not contradict DFT-based design or CFD reactor models — it unifies them. The Brønsted–Evans–Polanyi relation corresponds to the linearisation of \(K\) near \(x^*\). The Thiele modulus is the ratio \(r_\text{pore}/r^*(\lambda)\).
A key insight of CatGT is that the firing order of operators is not fixed but system-dependent. In ZSM-5, K fires before F (pore constrains first). In MCM-22, F fires before K (fold inside the supercage). This operator order switching under varied T/P/concentration is a new falsifiable DRIFTS prediction.
A catalyst optimised for clean energy (CO₂ hydrogenation, water splitting) should be engineered so that \(r_\text{pore} = r^*(\lambda_\text{target})\) for the desired product pathway — simultaneous tuning of pore size and metal loading.
1. Finite-size scaling of \(r^*(\lambda)\) for chain lengths \(N \in \{100, 500, 1000\}\) (Part II). 2. Formal Lean 4 proof of Corollary 2 once Mathlib volume-form support matures. 3. Photocatalysis extension — a photon-driven compression operator \(C_\text{photo}\) (Part IV). 4. MD validation of Prediction 1 for ZSM-5, SAPO-34, MFI.