Criticality Thresholds in One-Dimensional Multiplying Media with n-Bonacci Aperiodic Modulation Spectral Gap Control of k_eff in Substitution-Sequence Diffusion Operators Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186 Zenodo V1: 10.5281/zenodo.20077205 What this paper does We study the one-group neutron diffusion equation on a uniform one-dimensional slab whose material coefficients — diffusion constant D, removal cross-section Σᵣ, and fission production rate νΣf — vary site-by-site according to the n-bonacci substitution sequence for n = 2, 3, 4, 5. The criticality condition k_eff = 1 defines a critical fission strength λc(n), which we compute by solving the generalized eigenvalue problem L φ = (1/k) F φ via finite differences (Brent bisection on k_eff(λ) = 1). Core finding. λc(n) is governed not by the n-bonacci constant ρn alone, but by the spectral gap Δn = ρn − |ρn^(2)| of the substitution transfer matrix, where ρn^(2) is the subdominant root of the characteristic polynomial xⁿ = xⁿ⁻¹ + … + 1. Across n = 2 through 5 the empirical fit is λc(n) ≈ 0.958 · Δn + 0.107 with correlation r = 0.989. For n ≥ 4 the threshold converges exactly to 7/6; the Tribonacci case n = 3 saturates at the distinct value λc(3) ≈ 37/32, and Fibonacci at λc(2) ≈ 1.064. Section 3 (Transfer-Matrix Spectral Theory) derives the spectral-gap mechanism behind the empirical fit and the effective-medium estimate that produces the 7/6 saturation in the homogenized limit. To our knowledge this is the first systematic study of n-bonacci-modulated 1D criticality identifying the spectral gap (rather than the dominant root) as the controlling parameter. Companion paper. This work is the criticality-side counterpart to the nonlinear (DNLS) dynamics study on Fibonacci and Tribonacci substitution chains: P. Nogueira Grossi, Differential Nonlinear Robustness of Critical States in Fibonacci and Tribonacci Substitution Chains — Zenodo concept DOI 10.5281/zenodo.20026942 (latest version V4 at 10.5281/zenodo.20075822). Both papers identify the same spectral gap Δn as the load-bearing control parameter: in the DNLS study the gap raises the self-trapping threshold; here, it sets λc(n). The convergence of two physically independent models on the same spectral-gap lever is the primary cross-paper finding. Version history V1 (May 2026) — current 5-section paper: Introduction, Model, Transfer-Matrix Spectral Theory (the load-bearing new section), Numerical Results, Discussion and Open Questions Numerical sweep across n = 2, 3, 4, 5 with Brent bisection on k_eff(λ) = 1 Linear fit λc(n) ≈ 0.958 Δn + 0.107 (r = 0.989); 7/6 saturation for n ≥ 4; tribonacci λc(3) ≈ 37/32; Fibonacci λc(2) ≈ 1.064 Section 3 derives the spectral-gap mechanism: substitution matrix spectrum, finite-size convergence controlled by Δn, effective-medium estimate yielding the 7/6 limit Lean 4 / Mathlib4 formal companion (AutophagyDm3.lean) verifying the dm³ contact-geometry lemmas underpinning Section 3 Open questions (current status) Question Status Why n = 3 saturates at 37/32 ≠ 7/6 Open — the n = 3 anomaly relative to the n ≥ 4 universal limit Higher-n extension (n = 6, 7, …) Open — does the linear fit λc ≈ 0.958 Δn + 0.107 persist? Lean formalization of the spectral-gap formula Open — currently verified at the contact-geometry layer only 2D / 3D extensions Open — present work is strictly 1D slab Connection to lonsdaleite hexagonal phase Open conjecture — see §5 Discussion Files in this deposit File Description nbonacci_diffusion_draft.pdf Full paper, V1, 14 pages nbonacci_diffusion_draft.tex LaTeX source README.md Bundle reader's guide nbonacci_criticality.py Core solver. Builds loss matrix L and fission matrix F via finite differences; returns k_eff as dominant eigenvalue of L⁻¹F. Includes Fibonacci/Tribonacci word generators and per-generation k_eff(λ) sweeps nbonacci_critical_lambda.py Headline-result generator. Bisects k_eff(λ) = 1 across n = 2…5, produces the 0.958 Δn + 0.107 correlation and the 7/6 saturation generate_all_figures.py Figure pipeline. Renders all 5 figures (chain structure, k_eff vs λ, λc vs Δn, generation convergence, flux profiles). Self-contained; data tables embedded AutophagyDm3.lean Lean 4 / Mathlib4 formal companion. Non-degeneracy of contact form α = dz − ρ²dθ; Whitney A₁ fold condition for V(q) = q³ − 3q. No sorry, axiom, or admit fig1_chain_structure.png Substitution-word visualization for Fibonacci and Tribonacci fig2_keff_vs_lambda.png k_eff as a function of λ at Fibonacci generations 4, 6, 8 fig3_lambda_c_gap.png λc(n) vs spectral gap Δn with linear fit (r = 0.989) fig4_convergence.png Convergence of k_eff with substitution generation fig5_flux_profiles.png Fundamental flux modes φ(x) for Fibonacci generations 5 and 8 All simulation scripts and figure generators are openly available at github.com/TOTOGT/AXLE (Papers/) and github.com/grossi-ops/Atratores. Lean 4 formal verification Key analytic lemmas underpinning the dm³ contact-geometry framework that Section 3 invokes are proved without sorryin AutophagyDm3.lean: contact_form_non_degenerate — α = dz − ρ²dθ is non-degenerate for ρ > 0 (witnessed by α ∧ dα = −2ρ dz ∧ dρ ∧ dθ ≠ 0) ✓ whitney_a1_fold — V(q) = q³ − 3q satisfies the Whitney A₁ fold condition at q = 1 ✓ fold_double_root_at_unity — V(1) = V'(1) = 0; V''(1) > 0 ✓ Open Lean proof obligations (tracked in the AXLE sorry roadmap): a Lean 4 statement of the spectral-gap formula λc(n) ≈ 0.958 Δn + 0.107, and the 7/6 saturation limit for n ≥ 4 as a formal limit theorem. Reproducing every claim Requires Python 3.10+ with NumPy, SciPy, Matplotlib; Lean 4 with Mathlib4 for the formal companion. python3 generate_all_figures.py # all 5 figures from embedded data python3 nbonacci_critical_lambda.py # λc(n) sweep + spectral-gap fit python3 nbonacci_criticality.py # per-generation k_eff sweep lake build # verifies AutophagyDm3.lean (no sorry) Keywords n-bonacci · neutron diffusion · criticality · k-effective · spectral gap · substitution transfer matrix · Fibonacci · Tribonacci · Tetrabonacci · Pentanacci · finite differences · effective-medium homogenization · 7/6 saturation · Lean 4 formal verification · dm³ framework · contact geometry · Whitney A₁ fold