Deterministic Ghost Scaling

A methodological report on power-law memory in flows, stochastic barriers, and discrete maps

Version: 1.0 (December 15, 2025)

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Executive summary

This document consolidates a three-target study of fractional (power-law) memory and its impact on dynamical systems. A central methodological lesson is that the same kernel can play qualitatively different roles depending on model class:

• Target 2 (continuous-time flow near saddle-node): memory acts like drag (time-stretching), yielding an exact scaling exponent \(\gamma(\alpha_t)=1/(2\alpha_t)\).
• Target 3 (stochastic barrier crossing): heavy-tailed kinetics make MFPT unreliable; survival analysis with censoring (Kaplan–Meier) is the correct observable.
• Target 1 (discrete fractional logistic map): memory can cause domain exit and finite-time divergence. Critically, clipping can manufacture bounded periodic behavior and must be treated as a modeling choice, not a harmless numerical fix.

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Summary of quantitative findings

Target	System	Key result	Evidence type	Confidence
2	Caputo saddle-node normal form	\(\gamma(\alpha_t)=1/(2\alpha_t)\) (weak-memory class)	Exact scaling + ABM–PECE verification	High
3	Stochastic fractional escape	Survival-based rate extraction supports friction/kinetics (classical barrier exponent \(p=3/2\)) over geometry deformation across an \(\alpha_t\)-sweep	Kaplan–Meier + monotonic \(\Delta R^2\) trend (Appendix A)	Medium–High
1	Discrete Caputo-increment logistic map	Finite-horizon survival boundary and metastable “ghost” windows at \(\alpha_t=0.98\). Example: at \(M=12800\) and horizon \(N=100000\), survival holds for \(r\le 2.94\) and fails for \(r\ge 2.96\) (finite-horizon definition). Observed metastable plateau lifetimes near \(\tau_{\text{exit}}\approx 2M\) in several runs (empirical).	r-scans + survival scans + M-sweeps + divergence/overflow diagnostics	Medium (M- and horizon-dependent)

Confidence criteria:

• High: supported by an exact analytical result and numerical verification with convergence tests.
• Medium–High: systematic numerical evidence (e.g., monotonic trends across parameter sweeps) but lacking closed-form derivation.
• Medium: reproducible numerical observation but with significant parameter dependence (e.g., \(M\), horizon) that affects interpretation.

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Common notation and modeling choices

• \(\alpha_t\in(0,1]\): fractional order (memory strength).
• \(M\): memory cutoff (history length). This can be (i) a numerical approximation to the infinite-memory kernel and/or (ii) a physical finite-memory horizon.
• Clipping: forcing \(x_n\) into a range (e.g., \([0,1]\) or \([-A,A]\)). Clipping defines a different dynamical system and can qualitatively change conclusions.
• Out-of-bounds (oob) fraction: fraction of iterates leaving \([0,1]\). This is tracked even when the map is run unclipped.

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Target 2: Deterministic Ghost Scaling \(\gamma(\alpha_t)\)

2.1 Model and topological escape definition

We study the fractional saddle-node normal form with Caputo derivative:

\[
D_t^{\alpha_t}x(t) = \mu + x(t)^2,\qquad x(0)=0,\qquad 0<\alpha_t\le 1,\qquad \mu\to 0^+. \]

To avoid metric-threshold artifacts, escape is defined relative to the natural bottleneck scale \(\sqrt{\mu}\).
For fixed \(\kappa\gg 1\), define the topological escape time

\[
\tau_{\mathrm{esc}}(\mu;\alpha_t) = \inf\left\{t>0:\ x(t)\ge \kappa\sqrt{\mu}\right\}.
\]

2.2 Exact scaling symmetry and weak-memory universality class

Use the scaling transformation

\[
x(t) = \sqrt{\mu}\,y(s),\qquad s=\mu^{\frac{1}{2\alpha_t}}\,t.
\]

This yields a \(\mu\)-free rescaled equation

\[
D_s^{\alpha_t}y(s)=1+y(s)^2,\qquad y(0)=0.
\]

Therefore

\[
\tau_{\mathrm{esc}}(\mu;\alpha_t)=\mu^{-\frac{1}{2\alpha_t}}\,s_{\mathrm{esc}}(\alpha_t,\kappa),
\]

and the weak-memory universality class is

\[
\boxed{\gamma(\alpha_t)=\frac{1}{2\alpha_t}.}
\]

2.3 Prefactor estimate (constant-flux / ramp approximation)

Near the bottleneck, approximate \(D_s^{\alpha_t}y(s)\approx 1\), giving

\[
y(s)\approx \frac{s^{\alpha_t}}{\Gamma(\alpha_t+1)}.
\]

Thus \(y(s_{\mathrm{esc}})\approx \kappa\) implies

\[
s_{\mathrm{esc}}(\alpha_t,\kappa)\approx\big(\kappa\,\Gamma(\alpha_t+1)\big)^{1/\alpha_t},
\]

and

\[
\tau_{\mathrm{esc}}(\mu;\alpha_t)\sim \big(\kappa\,\Gamma(\alpha_t+1)\big)^{1/\alpha_t}\cdot \mu^{-1/(2\alpha_t)}.
\]

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Target 3: Stochastic Escape and Fractional Survival Statistics

3.1 Model and why MFPT fails

We consider the heuristic fractional Langevin normal form:

\[
D_t^{\alpha_t}x(t)=\mu + x(t)^2 + \sigma\,\eta(t),\qquad \mu<0, \]

where \(\eta(t)\) is idealized white noise. For \(\alpha_t<1\), first-passage distributions can be heavy-tailed, so the mean first-passage time (MFPT) can be extremely large or effectively undefined within finite horizons. Consequently, MFPT-based regressions can be biased when censoring is significant.

3.2 Canonical observable: survival function

The robust observable is the survival function \(S(t)=\Pr(\tau>t)\).

A common parametric form for fractional kinetics is Mittag–Leffler survival:
\[
S(t)\approx E_{\alpha_t}\!\left(-\lambda t^{\alpha_t}\right),
\qquad
E_{\alpha}(z)=\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+1)}.
\]

3.3 Kaplan–Meier estimator and censoring

Given escape times \(\tau_i\) and censoring flags \(\delta_i\in\{0,1\}\), the Kaplan–Meier estimator is

\[
\widehat{S}(t)=\prod_{t_j\le t}\left(1-\frac{d_j}{n_j}\right),
\]

and we report censoring fraction \(f_c=\#\{\delta_i=0\}/N\).

3.4 Extracting a rate scale \(\lambda\) from survival

A practical extraction fits mid-range survival via

\[
-\ln\widehat{S}(t)\approx \frac{\lambda}{\Gamma(\alpha_t+1)}\,t^{\alpha_t}.
\]

3.5 Barrier scaling hypotheses: friction vs geometry

• Hypothesis A (friction/kinetics): barrier geometry remains classical, \(\log\lambda\sim -C|\mu|^{3/2}/\sigma^2\).
• Hypothesis B (geometry deformation): exponent changes, \(\log\lambda\sim -C|\mu|^{3/(2\alpha_t)}/\sigma^2\).

The reported \(\alpha_t\)-sweep results (Appendix A) show \(\Delta R^2=R_A^2-R_B^2>0\) for all \(\alpha_t<1\) and increasing as \(\alpha_t\) decreases, supporting Hypothesis A over Hypothesis B in the tested regime.

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Target 1: Fractional Logistic Universality (Discrete Map Operator)

4.1 Canonical discrete fractional map (memory on increments)

Let \(g(x)=r\,x(1-x)\) and define increments \(\Delta x_k=g(x_k)-x_k\). Define the discrete Caputo-increment map by

\[
x_n
= x_0
+ \frac{1}{\Gamma(\alpha_t)}
\sum_{k=0}^{n-1}
\frac{\Gamma(n-1-k+\alpha_t)}{\Gamma(n-k)}
\,[g(x_k)-x_k],
\qquad 0<\alpha_t\le 1. \]

This representation matches the standard “\(x_n\) from past increments” form and avoids confusion between \(x_n\) and \(x_{n+1}\) indexing. Equivalent formulations exist, but consistency matters because off-by-one errors can silently change the dynamics.

4.2 Off-by-one kernel sanity test (decisive)

For any \(\alpha_t\in(0,1]\), the first iterate must satisfy \(x_1=g(x_0)\). This holds because at \(n=1\) the sum contains only the \(k=0\) term with weight \(w_0^{(\alpha_t)}=1\):

\[
x_1 = x_0 + 1\cdot[g(x_0)-x_0]=g(x_0).
\]

This identity is independent of \(\alpha_t\) and is a decisive diagnostic for detecting indexing mistakes in the kernel implementation.

4.3 The critical modeling trap: clipping vs unclipped dynamics

For \(\alpha_t<1\), the map is not guaranteed to preserve the classical invariant interval \([0,1]\). Therefore:

• Unclipped map answers: does the fractional operator intrinsically preserve boundedness?
• Clipped map answers: what happens if we enforce bounds (a different dynamical system)?

In practice, unclipped runs exhibited domain exit and sometimes overflow/NaN. Clipping could create “stable” periodic behavior but often with very large clip fractions, indicating the observed behavior is dominated by the clipping rule rather than the raw fractional map.

4.4 Diagnostics that proved necessary in practice

1. Long-horizon simulation: metastable behavior can look asymptotic at short horizons.
2. Phase-error period metric: more robust than brittle recurrence checks for smeared attractors.
3. Out-of-bounds and survival tracking: track oob fraction vs \([0,1]\), maximum \(|x_n|\), and survival time (finite steps until non-finite).
4. M-sweeps: slowly decaying kernels near \(\alpha_t\approx 1\) make finite-\(M\) effects dominant.

4.5 Finite-horizon survival boundary and metastable “ghost” windows at \(\alpha_t=0.98\)

A key empirical observation is that stability in \(r\) depends on the horizon \(T\) (or number of iterates \(N\)) and on \(M\). We therefore define a finite-horizon survival boundary:

\[
r_{\mathrm{surv}}(N;\alpha_t,M,x_0) = \sup\{r:\ \text{orbit remains finite through }N\text{ iterates}\}.
\]

Example (unclipped, \(\alpha_t=0.98\), \(M=12800\), \(x_0=0.2\), horizon \(N=100000\), scan step \(dr=0.02\)):

• Stable survival through \(N=100000\) for \(r\in[2.80,2.94]\).
• First observed divergence at \(r=2.96\) (finite survival time \(\approx 3.8\times 10^4\) steps).
• Metastable plateau behavior for \(r\gtrsim 2.98\) with typical survival times near \(\tau_{\text{exit}}\approx 2.56\times 10^4\approx 2M\) (empirical).

Thus, “periodic windows” detected in short runs can represent metastable ghosts: they can mimic period-\(2^k\) structure for many steps and then collapse into divergence at longer horizons.

4.6 The “Period-2 bypass” phenomenon (observational, not yet a theorem)

In some coarse scans and short-to-moderate horizons, the classical sequence
\[
P_1\to P_2\to P_4\to P_8\to\cdots
\]
was replaced by an observed progression closer to
\[
P_1\to P_4\to P_8\to \text{divergence},
\]
with no clear \(P_2\) window detected under those settings.

This may indicate a genuine restructuring of accessible periodic windows under memory, but it may also be a resolution/horizon artifact: \(P_2\) could exist but be too narrow, too transient, or too sensitive to \(M\), \(x_0\), and horizon to be seen in current scans. This is an open question.

4.7 Why “momentum” is a useful metaphor (grounding)

Why “momentum”? In the discrete Caputo-increment map, the current state \(x_n\) depends on a weighted sum of all past increments \(\Delta x_k=g(x_k)-x_k\). When increments have persistent sign over long epochs (e.g., during approach to a fixed point or during divergence), the accumulated history acts like inertia: the orbit can “overshoot” where a memoryless map would settle, potentially exiting \([0,1]\). This is qualitatively different from the continuous-time drag picture, where memory primarily slows dynamics rather than amplifying deviations.

4.8 What would falsify or revise the Target 1 conclusions

• Survival convergence: if \(\tau_{\text{exit}}(r;\alpha_t,M,x_0)\to\infty\) as \(M\to\infty\) for \(r\) in classical bounded windows, then finite-\(M\) metastability may be a truncation artifact rather than a property of the infinite-memory map.
• P-2 reappearance: if a clear \(P_2\) window appears under finer \(r\)-resolution, larger \(M\), longer horizons, or different \(x_0\) ensembles, then the “P-2 bypass” interpretation must be revised.
• Operator dependence: if an alternative discrete fractional operator preserves \([0,1]\) (without clipping) and restores a bounded cascade, then the divergence route is not universal across fractional-map constructions.

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Synthesis: drag, friction, and momentum

• Flows (Target 2): memory acts as drag/time-stretching, enforcing \(\gamma(\alpha_t)=1/(2\alpha_t)\).
• Noisy barriers (Target 3): memory alters kinetics; survival analysis is essential; data favor a friction/kinetics interpretation (Appendix A).
• Discrete maps (Target 1): memory can destabilize boundedness and produce metastable ghost windows; clipping can qualitatively change conclusions.

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Lessons learned (methodological)

1. Clipping is a modeling choice, not a numerical fix. For discrete fractional maps, enforcing bounds can completely change qualitative dynamics. Always run unclipped first and report oob and survival metrics.
2. MFPT fails under heavy tails. For fractional kinetics, survival analysis with explicit censoring handling (Kaplan–Meier) is essential.
3. Topological escape definitions avoid threshold artifacts. Scale escape criteria with the natural problem scale (e.g., \(\sqrt{\mu}\) for saddle-node ghosts).
4. M-convergence is non-negotiable near \(\alpha_t\approx 1\). Slowly decaying kernels make finite-\(M\) effects dominant; always sweep \(M\).
5. Divergence is data, not failure. Domain exit and blow-up times are meaningful observables that reveal the map’s true dynamics.

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Reproducibility checklist

• Report \((\alpha_t,r,x_0)\), \(M\), horizon \(N\) (or \(T\)), and any transient/keep split.
• Report whether clipping is used; if yes, the bounds and clip fraction.
• For maps: report oob fraction vs \([0,1]\), max \(|x_n|\), and \(\tau_{\text{exit}}\) (finite steps until non-finite).
• For stochastic runs: report censoring fraction \(f_c\), Kaplan–Meier curve \(\widehat{S}(t)\), and fitting window.
• Always include \(\alpha_t=1\) baseline checks for each pipeline.
• Perform M-sweeps and horizon-sweeps whenever metastability is suspected.

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Open questions and future directions

1. Strong-memory regime: does a distinct universality class emerge for small \(\alpha_t\), or are deviations pre-asymptotic?
2. P-2 bypass mechanism: true qualitative restructuring or detection artifact?
3. \(M\to\infty\) limit: does \(r_{\mathrm{surv}}(N;\alpha_t,M,x_0)\) converge as \(M\to\infty\) for fixed horizon \(N\), and does survival persist at arbitrarily large horizons?
4. Operator dependence: which discrete fractional map definitions preserve boundedness without clipping?
5. Physically consistent noise: do fluctuation–dissipation-consistent GLE formulations replicate the Target 3 verdict?

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Related work snapshot (starting points)

• Wu & Baleanu (2014), “Discrete fractional logistic map and its chaos”, Nonlinear Dynamics 75:283–287. DOI: 10.1007/s11071-013-1065-7.
• Edelman (2013), “Fractional Maps as Maps with Power-Law Memory”, arXiv:1306.6361.
• Edelman (2014), “Fractional Maps and Fractional Attractors. Part II”, arXiv:1404.4906.
• Metzler & Klafter (2000), fractional/anomalous diffusion review entry point.
• Barkai & Silbey (2000), “Fractional Kramers Equation”, J. Phys. Chem. B 104.

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Appendix A: Target 3 \(\alpha_t\)-sweep results

Representative \(\alpha_t\)-sweep results (as reported; include your experimental details such as \(\sigma\), \(T_{\max}\), and fit window in your final archive):

\(\alpha_t\)	\(R^2_A\) ( \(p=3/2\) )	\(R^2_B\) ( \(p=3/(2\alpha_t)\) )	\(\Delta R^2 = R^2_A – R^2_B\)	censor_frac
1.00	0.930	0.930	0.000	0.33
0.95	0.893	0.886	+0.007	0.30
0.90	0.869	0.854	+0.015	0.24
0.80	0.779	0.742	+0.037	0.08
0.70	0.851	0.802	+0.049	0.002
0.60	0.821	0.750	+0.070	0.00

Interpretation: \(\Delta R^2>0\) for all \(\alpha_t<1\) and increases as \(\alpha_t\) decreases, supporting Hypothesis A (friction/kinetics) over Hypothesis B (geometry deformation) in the tested regime.

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