dm³ 103 · Week 09 · Constant: ρ

ρ — The Spectral Radius, Two Different Operators

The Collatz/Syracuse transfer operator, and an exact formula for one of its two natural approximations
dm³ 103 · Week 09 · Constant: ρ
ρ — The Spectral Radius, Two Different Operators
Course: dm³ 103  ·  Constant: ρ  ·  Source: spectral-radius-v2.html

The Syracuse return map \(T\) (the odd-step-only version of Collatz) admits two natural finite approximations, with genuinely different spectral behavior. The probability matrix \(P_M\): row-stochastic scaling of the mod-\(2^M\) transition structure. Since \(n=1\) is a fixed point of \(T\bmod2^M\) for every \(M\), \(P_M\) has eigenvalue \(1/2^{{M-1}}\) exactly, giving \(\rho(P_M)=1/2^{{M-1}}\) — proved exactly, not approximated, and decaying exponentially as \(M\to\infty\) (halving at each modulus doubling).

The other operator: the true Ruelle–Perron–Frobenius transfer operator
\((\mathcal{{L}}f)(x)=\sum_{{T(y)=x}} f(y)/|T'(y)|\), where \(|T'(y)|=3/2^{{v_2(3y+1)}}\) and \(v_2\) is the 2-adic valuation from 102’s CollatzDescent.lean. Its spectral radius equals \(e^\lambda\), the exponential of the Lyapunov exponent: \(\lambda\approx-0.286288\) (computed from mod-243 weighted drift), giving \(\rho(\mathcal{{L}})\approx e^{{-0.286288}}\approx0.7510\) — a genuinely different number, and a genuinely different kind of estimate (numerically computed from a weighted average, not an exact closed form like \(\rho(P_M)\)).

Both operators contract (\(\rho<1\)), but at structurally different rates: \(\rho(P_M)\) decays superexponentially in \(M\) (pure 2-adic erasure of initial precision), while \(\rho(\mathcal{{L}})\approx0.751\) is a fixed rate independent of \(M\), governing genuine long-run statistical behavior of the map. At \(M=7\): \(\rho(P_7)=1/64\approx0.0156\), already 48\(\times\) smaller than \(\rho(\mathcal{{L}})\) — the two operators live on visibly different scales.

This week’s content is grounded directly in the AXLE/Book 3/5 sources cited above — no material in this page depends on the external, unverified source removed from dm³ 102.
-- dm³ 103 · Week 09 · ρ — two distinct spectral radii, both real

-- P_M (probability matrix): ρ(P_M) = 1/2^(M-1)  EXACT
example : (1:ℝ)/2^(7-1) = 1/64 := by norm_num

-- ℒ (Ruelle-Perron-Frobenius transfer operator):
--   ρ(ℒ) = e^λ,  λ ≈ -0.286288 (numerically, mod-243 weighted drift)
--   ρ(ℒ) ≈ 0.7510  — NOT a closed form, genuinely different kind
--   of estimate than ρ(P_M)'s exact formula.
example : (1:ℝ)/64 < 0.02 := by norm_num  -- 48x smaller than ρ(ℒ)≈0.751
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