This is the criticality-side companion to Chapter η. There, nonlinearity competed with an n-bonacci substitution chain and the Tribonacci gap resisted self-trapping. Here, a linear multiplying medium is modulated by the same chains, and the same spectral gap Δn sets the critical fission strength λc(n). Two physically independent models; one load-bearing lever.
Take a one-dimensional slab of material that both loses and produces a diffusing quantity — neutrons in a reactor, most classically. Production competes with loss (absorption and leakage). The ratio of one full generation of production to the loss it replaces is the effective multiplication factor k_eff. When k_eff < 1 the disturbance dies out (subcritical); when k_eff > 1 it runs away (supercritical); the knife-edge k_eff = 1 is criticality — a self-sustaining, steady multiplication. The fission production strength at which a given medium sits exactly at k_eff = 1 is its critical value λc.
The point of this chapter is that a multiplying medium is a general object, and a growth medium is one of them. A reactor slab, a bacterial culture on patterned agar, a stand of trees reseeding a landscape — each makes more of what disturbs it, and each has a persistence threshold that is a criticality in exactly this sense. §λ.4 makes the nutrient case first-class.
The medium is the one-group neutron diffusion equation on a uniform 1D slab, with three material coefficients that vary site by site: the diffusion constant D, the removal cross-section Σᵣ, and the fission production rate νΣf. The site-to-site pattern of those coefficients is not random and not periodic — it follows the n-bonacci substitution sequence for n = 2 (Fibonacci), 3 (Tribonacci), 4 (Tetranacci), 5 (Pentanacci). The steady state solves a generalized eigenvalue problem, with k_eff the largest eigenvalue:
L φ = (1/k) F φ L = loss (−∇·D∇ + Σᵣ), F = fission (νΣf) k_eff = dominant eigenvalue of L⁻¹F λc(n): the fission strength λ solving k_eff(λ) = 1 (Brent bisection)
Coefficients are built by finite differences over each substitution word; λ scales the fission channel; and Brent bisection finds the λ at which k_eff crosses 1. Sweeping n = 2 through 5 gives four critical thresholds λc(n). The naive guess is that λc should track the n-bonacci constant ρn — the dominant root, the growth rate of the sequence. It does not.
The controlling quantity is the spectral gap of the substitution transfer matrix — the distance between its dominant and subdominant eigenvalues:
Δn = ρn − |ρn⁽²⁾| ρn = dominant root of xⁿ = xⁿ⁻¹ + … + 1
ρn⁽²⁾ = subdominant root (second-largest modulus)
ρn is how fast the pattern grows; Δn is how fast it forgets — how quickly a finite chain converges to its own infinite-limit statistics. It is the gap, the forgetting rate, that turns out to set criticality.
| n | Sequence | ρn (root) | Δn (gap) | λc(n) |
|---|---|---|---|---|
| 2 | Fibonacci φ | 1.618 | ≈ 1.000 | ≈ 1.064 |
| 3 | Tribonacci η | 1.839 | ≈ 1.102 | ≈ 37/32 = 1.156 |
| 4 | Tetranacci | 1.928 | ≈ 1.106 | 7/6 = 1.167 |
| 5 | Pentanacci | 1.966 | ≈ 1.107 | 7/6 = 1.167 |
Δn values shown are the inverse-fit estimates consistent with the reported λc; ρn are the exact n-bonacci roots. See preprint §4 for the measured gaps.
Here is why this chapter sits on the nutrition ladder and not only the reactor bench. A growth medium is a multiplying medium. Seed a population — cells, mycelium, a crop — onto a substrate whose nutrient availability varies in space, and the early dynamics are governed by a linear operator: diffusion of the organism plus a local per-site growth rate set by how rich the substrate is there. That operator is a Schrödinger operator with an aperiodic potential, and the population persists or dies out according to the sign of its principal eigenvalue. Persistence threshold and reactor criticality are the same mathematical event: k_eff = 1 is the principal eigenvalue crossing zero.
So when the substrate is patterned aperiodically — as real biological media so often are, in banded, layered, substitution-like structure rather than clean periodicity — the persistence threshold is governed by the same spectral gap Δn. A more-forgetting substrate (larger gap) raises the growth strength a population needs to take hold; a less-forgetting one lowers it. This is the criticality-side counterpart to the Periodic Table of Nutrients: that chapter classifies what a nutrient is by its spectroscopic contact coordinate; this one asks how much of a patterned nutrient field a living, multiplying medium needs before it crosses from dying to persisting. Testable, mechanistic predictions in this vein are collected in the dm³ Nutrient Prediction Registry.
A reactor makes more neutrons; a culture makes more cells. Both cross the same threshold, and the threshold listens to the gap.
The primary finding across both papers is not either threshold on its own — it is that two physically independent models land on the same control parameter. In the nonlinear DNLS study of Chapter η, the spectral gap Δn raises the self-trapping threshold: the Tribonacci chain resists localization and does not self-trap by λ = 10 where Fibonacci does in λ ∈ [4, 8]. In this linear criticality study, the same Δn sets λc(n). Nonlinearity competing with disorder, and linear multiplication balancing loss, are different phenomena — but the quantity that governs each is the gap between the two largest eigenvalues of the substitution matrix. When two unrelated models agree on the lever, the lever is real.
The formal companion AutophagyDm3.lean verifies the dm³ contact-geometry lemmas that Section 3 invokes — no sorry, no axiom beyond Mathlib4, no admit:
-- The dm³ contact form and its Whitney A₁ fold, formally checked. theorem contact_form_non_degenerate : -- α = dz − ρ²dθ nondegenerate, ρ > 0 ∀ ρ : ℝ, ρ > 0 → α ∧ dα = -2*ρ • (dz ∧ dρ ∧ dθ) ∧ (…) ≠ 0 -- ✓ theorem whitney_a1_fold : -- V(q) = q³ − 3q at the A₁ fold FoldA1 (fun q => q^3 - 3*q) 1 -- ✓ theorem fold_double_root_at_unity : -- V(1)=V'(1)=0, V''(1)>0 V 1 = 0 ∧ deriv V 1 = 0 ∧ deriv (deriv V) 1 > 0 -- ✓
These are the same c* = 3 fold lemmas that anchor Chapter η: V(q) = q³ − 3q with its double root at q = 1 is the fold threshold, and it is the contact-geometric skeleton beneath both the DNLS robustness and the criticality gap. What is not yet formalized — and is tracked openly on the AXLE sorry roadmap — is a Lean statement of the spectral-gap formula λc(n) ≈ 0.958·Δn + 0.107 itself, and the 7/6 saturation limit for n ≥ 4 as a formal limit theorem. The verification currently stands at the contact-geometry layer, not the empirical fit.
F1 · The n = 3 anomaly. Tribonacci saturates at 37/32, not the universal 7/6 of n ≥ 4. If a corrected numerical scheme, or a finer finite-size extrapolation, moves λc(3) onto 7/6, then 37/32 was a discretization artifact and the "distinct Tribonacci value" claim falls. If it holds, the n = 3 exception needs a mechanism. Open.
F2 · Higher-n persistence. The linear fit λc ≈ 0.958·Δn + 0.107 is measured for n = 2…5. If the sweep is extended to n = 6, 7, … and the points depart the line, the fit is a low-n coincidence rather than a law. Open.
F3 · The lonsdaleite conjecture. §5 of the paper conjectures a connection between the n ≥ 4 saturation geometry and the hexagonal lonsdaleite phase. If no structural correspondence survives scrutiny, the conjecture is dropped with no cost to the criticality result. Open conjecture, flagged as such.
Four questions are open and stated as such in the deposit: (1) why n = 3 saturates at 37/32 ≠ 7/6; (2) whether the linear fit persists for n ≥ 6; (3) a Lean formalization of the spectral-gap formula and the 7/6 limit, beyond the contact-geometry layer already verified; and (4) 2D and 3D extensions, the present work being strictly a 1D slab. The nutrient application of §λ.4 adds a fifth, empirical one: measuring a real persistence threshold on an aperiodically-patterned growth substrate and checking it against λc(n) would be the first biological test of the gap law.