Spectral response of positive recurrences under memory-filtered balanced coefficient perturbations

A research note. Status: main theorem numerically confirmed (see verify_theorem.py and FINDINGS.md); proof architecture below is a skeleton, not an audited proof; novelty asserted only as "not located," not established.

Alternative accessible title: When Memory Accelerates Recurrence Growth: A Spectral Response Theorem.

1. The model

Two coupled recursions. A driving memory process (AR(1) on the forcing):

$d_{n+1} = a\,d_n + \varepsilon\,q_n, \qquad 0 \le a < 1,$

and a positive companion recurrence whose coefficients it perturbs:

$x_{n+1} = (A + d_n)\,x_n + (B - d_n)\,x_{n-1}, \qquad A, B > 0.$

The perturbation is special. Written on the coefficient pair,

$(A, B) \;\longmapsto\; (A + d_n,\; B - d_n),$

it adds exactly what it subtracts:

$(A + d_n) + (B - d_n) = A + B \quad\text{(conserved for every } n).$

The environment never changes the total coefficient weight; it only transfers influence between the present state $x_n$ and the inherited state $x_{n-1}$. This is the balanced coefficient transfer, and it is what makes the model distinct from ordinary additive off-diagonal disorder.

Transfer-matrix form

The perturbation direction is a rank-one, coefficient-conserving update of a companion matrix. Note $\det M_n = -(B - d_n)$ fluctuates with the forcing — the cocycle is not naturally $\mathrm{SL}(2,\mathbb{R})$, so the Lyapunov exponent carries a determinant/scale contribution that the usual normalized localization cocycles hide. The derivation below works directly with the positive growth of $x_n$ rather than a determinant-one projective cocycle.

Constants

At $\varepsilon = 0$ the deterministic recurrence grows like $\lambda^n$, so the Lyapunov exponent is $\Lambda(0) = \log\lambda$.

2. Main theorem

Assumptions. $q_n = q\circ T^n$ is a bounded, stationary, mean-zero forcing over an ergodic $T$; $\sigma_q$ is its spectral (power) measure; and $\varepsilon$ is small enough that positivity holds, $A + d_n > 0$ and $B - d_n > 0$.

Claim.

with the response kernel

The linear term vanishes because centered forcing makes the first variation mean-zero (Lemma 6); the leading effect is second order.

The factorization — the real content

$M_a$ is exactly the power response of a one-pole (AR(1)) filter; it is everywhere positive and monotone in $|\omega|$. It contains all the $a$-dependence, so the memory parameter cannot move the sign — it only reweights how strongly each frequency is felt.

$P_{\lambda,\beta}$ carries the sign. Its numerator is linear in $\cos\omega$, so it crosses zero at most once on $[0,\pi]$. Hence:

Memory controls strength; recurrence geometry controls sign.

This is sharper than the standard "colored noise changes a Lyapunov exponent": here the what-frequencies-help question and the how-much question separate cleanly into two factors.

3. Exact sub-results

3.1 Frequency transition (fixed recurrence)

For $\lambda > 1$ the response changes sign where the $P$-numerator vanishes:

• If $\cos\omega_c \in (-1,1)$ (Regime I, interior transition): low frequencies $|\omega|<\omega_c$ enhance growth, high frequencies $|\omega|>\omega_c$ suppress it.
• If $\cos\omega_c \notin [-1,1]$ (Regime II): the kernel has one sign across all physical frequencies.
• $\lambda = 1$ (Regime III, critical): see 3.2.

The threshold is independent of the memory parameter $a$ — confirmed to $<10^{-3}$ rad in simulation for $a = 0.2$ and $a = 0.6$ (test T4).

3.2 The exact conservation anchor ($A + B = 1$)

$\lambda = 1 \iff A + B = 1$. Because the transfer is balanced,

so the ratio $r_n = x_n / x_{n-1} \equiv 1$ is an exact invariant for every admissible forcing sequence — not just to second order. The unit-growth state is protected nonperturbatively:

This is why the kernel carries the factor $(\lambda - 1)$: at $\lambda = 1$ the whole second-order response vanishes identically.

3.3 Fibonacci specialization ($A = B = 1$)

Sign-change frequency:

So low/medium frequencies accelerate Fibonacci growth (2nd order); fast oscillations suppress it. For spectrally flat (white) forcing the integrated response changes sign at a memory threshold:

Below $a_c$ the memory is too weak to overcome the harmful high-frequency mass; above it, the low-frequency emphasis of the memory filter makes the net response positive. Both constants are confirmed: the closed-form $\int W\,d\omega$ has its root at $0.20601$, and simulated white-noise $C(a)$ crosses zero between $a=0.1$ ($C<0$) and $a=0.35$ ($C>0$), passing through $\approx 0$ at $a_c$ (test T5).

4. Proof architecture (skeleton)

Work with the ratio $r_n = x_n/x_{n-1}$, which follows the Riccati/Möbius map

• Lemma 1 (bounded stationary memory). $D_\varepsilon(y) = \varepsilon\sum_{j\ge0} a^j q(T^{-1-j}y)$ solves $D_\varepsilon(Ty)=aD_\varepsilon(y)+\varepsilon q(y)$ with $\|D_\varepsilon\|_\infty \le \varepsilon\|q\|_\infty/(1-a)$. Controls positivity.
• Lemma 2 (absorbing interval). For $|d|\le\delta<\min(A,B)$, all positive ratio trajectories enter a common compact interval.
• Lemma 3 (two-step contraction). $F_d$ reverses order, but $F_{d_2}\circ F_{d_1}$ preserves it with derivative $\frac{(B-d_1)(B-d_2)}{((A+d_1)r + B-d_1)^2} < 1$ on the absorbing interval for small $\delta$.
• Lemma 4 (unique invariant graph). The two-step graph transform is a contraction ⇒ unique $R_\varepsilon$ with $R_\varepsilon(Ty)=F_{D_\varepsilon(y)}(R_\varepsilon(y))$; every positive initial ratio synchronizes exponentially.
• Lemma 5 (smoothness). Uniform contraction + analyticity of $F_d$ away from $r=0$ ⇒ $\varepsilon\mapsto R_\varepsilon$ is $C^3$ near $0$.
• Lemma 6 (first variation). Expanding $R_\varepsilon = \lambda + \varepsilon W + \varepsilon^2 V + O(\varepsilon^3)$, $W\circ T = -\beta W + \gamma Z$ with $\gamma = 1 - 1/\lambda$; centered forcing gives $\mathbb{E}[W]=0$, hence $\Lambda'(0)=0$.
• Lemma 7 (second variation). $V\circ T = -\beta V + ZW/\lambda^2 + BW^2/\lambda^3$; solving the linear filters for $Z,W$ and applying the spectral theorem for stationary processes yields the kernel $W_{a,\lambda,\beta}$.

Caveat that must be stated in the theorem. The constants are not uniform as $a\to1^-$: since $\|D_\varepsilon\|_\infty \le \varepsilon\|q\|_\infty/(1-a)$, the admissible perturbation scales like $|\varepsilon| \lesssim 1-a$. The theorem must read "for every fixed $a<1$ there exists $\varepsilon_0(a)>0$," not one $\varepsilon_0$ for all $a\in[0,1)$.

5. Relation to known work (threat assessment)

Established, not claimed here:

• random / weakly-disordered transfer matrices and perturbative Lyapunov exponents;
• correlated diagonal and off-diagonal disorder; correlation-induced mobility edges;
• random Fibonacci recurrences and their Riccati ratio maps;
• regularity/response of Lyapunov exponents under perturbation;
• general linear-response machinery for skew products with contracting fibers.

Not located in targeted searches (⇒ candidate novelty, not proven): the specific balanced-transfer model $(A+d_n, B-d_n)$ with an external AR(1) memory parameter, its closed-form second-order kernel, the clean $M_a\cdot P$ factorization (sign independent of memory), and the Fibonacci constants $\cos\omega_c = -1/2\varphi^2$, $a_c = 1/3\varphi$.

Verdict after literature review (deflated). The real check was done and the general result is established prior art. The form "weakly/randomly perturbed 2×2 recurrence ⇒ Λ(ε)=Λ₀+ε²·(spectral integral)+O(ε³), Λ′(0)=0, kernel integrated against the forcing's spectral density" is owned by weak-noise Lyapunov expansions (Arnold, Papanicolaou & Wihstutz, SIAM J. Appl. Math. 46 (1986) 427–450; and "products of random 2×2 matrices close to the identity," arXiv:1208.6430) and by weak-disorder Lyapunov/localization theory (Thouless-type γ, correlated/colored disorder feeding a power spectrum into the kernel). The AR(1) memory M_a is just a Lorentzian noise spectrum plugged into that known formula. The specific balanced-transfer direction and the golden-ratio constants appear unpublished in this exact form, but they fall out mechanically — the equivalent of evaluating a known general integral for a particular integrand. Honest classification: a new worked example inside a fully established framework, not a new theorem and not a new field. This supersedes the earlier "original special theorem" self-assessment. Full citations and options in FINDINGS-spectral-response.md.

6. What is verified vs. open

Verified numerically (verify_theorem.py, T1–T5 all PASS): the baseline $\log\varphi$; clean $\varepsilon^2$ scaling with $\Lambda'(0)=0$; kernel shape match to $k\cdot W_a(\omega)$ at shape-correlation $1.0000$ for two memory values; $a$-independence of $\omega_c$; the flat-forcing threshold $a_c=1/3\varphi$ (both formula-self-consistent and simulation-confirmed).

Open: audited proof of Lemmas 1–7; the $n$-th-order generalization (transfer across many past states); nonlinear updates; and the practical applications in docs/applications-directions.md.