Criticality Thresholds in One-Dimensional Multiplying Media with n-Bonacci Aperiodic Modulation

Criticality Thresholds in One-Dimensional Multiplying Media with n-Bonacci Aperiodic Modulation Spectral Gap Control of kₑff 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 kₑff = 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 kₑff (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 kₑff (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 nbonaccidiffusiondraft. pdf Full paper, V1, 14 pages nbonaccidiffusiondraft. tex LaTeX source README. md Bundle reader's guide nbonaccicriticality. py Core solver. Builds loss matrix L and fission matrix F via finite differences; returns kₑff as dominant eigenvalue of L⁻¹F. Includes Fibonacci/Tribonacci word generators and per-generation kₑff (λ) sweeps nbonaccicriticalₗambda. py Headline-result generator. Bisects kₑff (λ) = 1 across n = 2…5, produces the 0. 958 Δn + 0. 107 correlation and the 7/6 saturation generateₐllfigures. py Figure pipeline. Renders all 5 figures (chain structure, kₑff 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 fig1chainₛtructure. png Substitution-word visualization for Fibonacci and Tribonacci fig2ₖeffᵥsₗambda. png kₑff as a function of λ at Fibonacci generations 4, 6, 8 fig3ₗambdacgap. png λc (n) vs spectral gap Δn with linear fit (r = 0. 989) fig4convergence. png Convergence of kₑff with substitution generation fig5fluxₚrofiles. 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: contactformₙondegenerate — α = dz − ρ²dθ is non-degenerate for ρ > 0 (witnessed by α ∧ dα = −2ρ dz ∧ dρ ∧ dθ ≠ 0) ✓ whitneyₐ1fold — V (q) = q³ − 3q satisfies the Whitney A₁ fold condition at q = 1 ✓ folddoubleᵣootₐtᵤnity — 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 generateₐllfigures. py # all 5 figures from embedded data python3 nbonaccicriticalₗambda. py # λc (n) sweep + spectral-gap fit python3 nbonaccicriticality. py # per-generation kₑff 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