Every linear transformation A has a set of eigenvalues — the special directions
it stretches or compresses without rotating. The spectral radius ρ(A)
is the magnitude of the largest eigenvalue: the maximum reach of A in a single step.
Definition 11.1 — Spectral Radius
ρ(A) = max { |λ₁|, |λ₂|, …, |λₙ| }
where λᵢ are the eigenvalues of A ∈ ℝⁿˣⁿ
Gelfand's formula: ρ(A) = limn→∞ ‖Aⁿ‖^(1/n)
Power iteration: Aⁿx₀ → ρ(A)ⁿ · v₁ · (v₁ᵀx₀)
where v₁ is the dominant eigenvector
Gelfand's formula reveals a deep structure: you need not compute eigenvalues directly.
Simply iterate A repeatedly and measure how fast the vector norm grows or shrinks.
That growth rate is the spectral radius — which the simulation below demonstrates.
ρ < 1
Contraction
Iterations Aⁿx → 0. System is asymptotically stable. All perturbations decay.
ρ = 1
Critical Boundary
Sustained oscillation or marginal stability. System neither grows nor collapses.
ρ > 1
Expansion
Iterations Aⁿx → ∞. Unstable. Claims or populations exceed their support.
Notice the parallel with Chapter 10's Lyapunov exponent λ:
contraction corresponds to λ < 0, the critical boundary to λ = 0,
and expansion to λ > 0. The spectral radius is the discrete-time cousin
of the Lyapunov exponent — the same mathematics, different clock.
§2 · R₀ — The Spectral Radius of Infection
Epidemiologists rarely say "spectral radius" to the public — they say R₀,
the basic reproduction number. But R₀ is the spectral radius
of the next-generation matrix K of the epidemic model (Diekmann, Heesterbeek & Metz, 1990).
Next-Generation Matrix Construction
K = F · V⁻¹
F = new-infection matrix (who infects whom, at what rate)
V = transition matrix (recovery, death, movement between compartments)
R₀ = ρ(K) = ρ(F · V⁻¹)
R₀ < 1 → epidemic contracts and dies out [Chapter 5 K* not crossed]
R₀ = 1 → endemic threshold K* [commitment boundary]
R₀ > 1 → exponential spread, full U-unfolding
Vaccination and behavioral intervention are operations that reduce ρ(K) below 1 —
forcing the epidemic system into the contraction regime.
Herd immunity is achieved precisely when ρ(K) crosses the K* threshold from above.
The same next-generation logic appears in Chapter 5's immune system:
a B-cell clone expands (ρ > 1) when affinity exceeds K*,
and enters apoptosis (ρ < 1) when affinity remains subcritical.
R₀ and affinity maturation are the same operator at different scales.
The spectral radius is the universal language of proliferation:
B-cell clones, epidemic waves, and population dynamics all live or die
by whether ρ crosses 1. The K operator of Chapter 5 is the threshold ρ = 1.
§3 · The Spectral Radius of an Argument
A Discussion section makes claims that "reach" into adjacent fields —
connecting results to prior literature, broader theory, or clinical implications.
Each claim has a spectral reach: how far does it generalise
beyond the evidence that produced it?
Claim Type
ρ Estimate
Verdict
Example Phrasing
Restatement of result
ρ ≈ 0.4
✓ Safe
"Our data confirm X in this cohort."
Contextualises within one field
ρ ≈ 0.8
✓ Sound
"This extends Smith (2019)'s model of Y."
Cross-field connection
ρ ≈ 1.0
⚠ Critical
"These findings suggest a universal mechanism."
Policy or clinical generality
ρ ≈ 1.4
✗ Overclaim
"This proves that all patients should…"
Paradigm resolution
ρ ≈ 2.1
✗ Divergent
"These results overturn the standard model."
Theorem 11.1 — Discussion Stability Condition
Let {c₁, c₂, …, cₖ} be the claims in a Discussion section,
each with spectral reach ρ(cᵢ). The Discussion is
spectrally stable if and only if:
ρ(cᵢ) ≤ 1 for all i
with at least one claim approaching ρ(cᵢ) → 1⁻ from below —
the critical boundary: ambitious but defensible.
A Discussion with all ρ(cᵢ) ≪ 1 is timid and undersells the work.
A Discussion with any ρ(cᵢ) > 1 is unstable and invites rejection.
The target is the subcritical approach — as in Chapter 9's φ.
The ideal Discussion section operates like the Fibonacci sequence approaching φ:
each claim approaches the boundary of what the evidence supports,
without crossing it. The spectral radius formalises what senior researchers
mean when they say "don't overreach — but don't undersell either."
§4 · AXLE — Algebraic Verification of Structural Claims
The AXLE framework treats a paper's logical structure as a formal system.
Just as Lean 4 can verify a mathematical proof step by step,
AXLE audits an argument for spectral stability:
are claims connected to evidence by warranted, non-expanding inference steps?
AXLE Verification Schema
INPUT: Evidence set E = {e₁, e₂, …, eₙ}
CLAIM: C (a Discussion assertion)
WARRANT: W (bridge principle linking E-type evidence to C-type claim)
AXLE checks:
1. ρ(C | E) ≤ 1 [claim reach does not exceed evidence support]
2. W is cited [warrant is not invisible — show your bridge]
3. C survives E \ {eᵢ} for all i [Lyapunov condition: no single point of failure]
STATUS: VERIFIED ✓ | OVERCLAIM ✗ | UNSUPPORTED ✗
AXLE verification is what peer review should do automatically.
In the absence of a formal proof assistant, Prompt 6.2 below provides a manual
protocol — running AXLE by hand on your own Discussion before submission.
The operator chain completes here: C (compress raw data) → K (threshold selection) →
F (fold results into narrative structure) → U (unfold claims into Discussion) →
ρ(U) ≤ 1 (AXLE verification passes). A paper is spectrally stable when U
unfolds to the boundary of evidence — not beyond.
▶ Power Iteration · Spectral Radius Visualiser
MATRIX TYPE
—
ρ̂ (estimate)
—
ITERATIONS n
0
STATUS
—
Select a matrix type to begin power iteration.
⬡ LLM Prompt Portal · Chapter 11
PROMPT 5.2 · SPECTRAL AUDIT
Discussion Reach Analysis
List each distinct claim in your Discussion section.
For each claim, estimate its spectral reach ρ using this scale:
ρ < 0.5 = restatement of result ·
ρ ≈ 0.8 = field-specific contextualisation ·
ρ ≈ 1.0 = cross-field boundary ·
ρ > 1 = overclaim.
Flag every claim where ρ > 1. For each flagged claim,
suggest a revised version that reduces ρ to ≤ 1
while preserving the core scientific insight.
PROMPT 6.2 · AXLE VERIFICATION
Manual AXLE Protocol
For each major claim C in your Discussion, perform this four-step check:
(a) List the evidence set E = {e₁, …, eₙ} that supports C directly.
(b) State the warrant W explicitly — the bridge principle linking E to C.
(c) Remove each eᵢ one at a time: does C still hold without it?
If removing any single piece collapses C entirely, flag as a single point of failure.
(d) Confirm ρ(C | E) ≤ 1: C does not generalise beyond what E can carry.
Output a verification table: one row per claim, columns C / E / W / Single-POF / ρ estimate / Status (VERIFIED · OVERCLAIM · UNSUPPORTED).
EXTENSION · SPECTRAL ARCHITECTURE
Eigenvalue Design of Your Paper
Treat your paper's argument as a matrix A where entry Aᵢⱼ =
"degree to which claim j depends on evidence piece i."
Describe A qualitatively: is it sparse (few claims per evidence item)
or dense (claims entangled with many evidence sources)?
What is ρ(A) — contraction, critical, or expansion?
How would you redesign the Discussion to achieve ρ(A) → 1⁻,
the subcritical approach that is ambitious but formally defensible?