The Power Rule as Chaos’s Silent Architect

Synthesis v7.3 — The Inertial Momentum Theorem and the Discrete–Continuous Memory Dichotomy
Collaborators: Mustafa Aksu, ChatGPT, Grok, Gemini, Claude
Date: December 15, 2025

Abstract
We establish Curvature Rank as a universal organizer of bifurcation structure derived from repeated application of the power rule. Across continuous flows, stochastic systems, and discrete maps with fractional memory, memory assumes three distinct physical roles: Drag (time-stretching ghosts), Friction (kinetic modulation with preserved geometry), and Inertial Momentum (deterministic domain-exit clocks). The discrete–continuous dichotomy arises from the presence or absence of timestep damping \(h^\alpha\) in the memory kernel. In finite-memory discrete fractional maps, we prove a sufficient condition for fracture and empirically confirm quantized linear clocks \(\tau_{\text{exit}} = kM + c\) with \(k \in \{1,2\}\). The Power Rule is revealed as chaos’s silent architect — stabilizing in the continuum, destabilizing under discreteness and memory truncation.

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0. Executive Summary

The power rule \( \frac{d}{dx}x^n = n x^{n-1} \) acts as a peeling operator: it removes layers of flatness and exposes the
first non-vanishing interaction order. We formalize this via Curvature Rank and show how it organizes dynamics across
three substrates: continuous flows, stochastic systems, and discrete maps.

The Core Discovery: The Tripartite Role of Memory

Across Targets 1–3, fractional memory produces three distinct physical roles:

• 
  Discrete Maps (Target 1): Inertial Momentum.
  
  In the Caputo-increment map with unit step \(h=1\), the history term enters unsuppressed as a finite-memory convolution.
  
  This produces deterministic “momentum accumulation” and a finite-memory fracture (domain exit), replacing sustained chaos
  
  by metastable windows.
  
• 
  Continuous Flows (Target 2): Drag.
  
  For fractional flows, time discretization introduces a factor \(h^\alpha\) multiplying the history accumulation.
  
  As \(h \to 0\), the same memory kernel produces time-stretching (slowing) rather than destabilizing kicks.
  
  Escape-time scaling follows \(\tau_{\mathrm{esc}} \propto \mu^{-1/(2\alpha_t)}\) in the validated weak-memory regime.
  
• 
  Stochastic Systems (Target 3): Friction.
  
  Memory alters kinetics (escape rates) while preserving the static geometry of barriers; the barrier exponent remains \(p=3/2\).
  

Phase 1 Verdict (Finite-Memory): The map “fracture” is not a numerical artifact. For all tested finite memory depths \(M\),
the first-exit time \(\tau_{\text{exit}}\) scales linearly with \(M\) and exhibits quantized turnover branches
\(\tau_{\text{exit}}(M;x_0) = k(x_0)M + c(x_0)\), with \(k \in \{1,2\}\) observed in the tested regime.
As \(M \to \infty\), \(\tau_{\text{exit}} \to \infty\), so classical behavior is recovered only in the idealized infinite-memory limit.
Finite-memory discrete fractional operators convert history into inertia, while continuous fractional operators convert history into viscosity.

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0.1 Unified Exponent Cheat Sheet

This table summarizes confirmed scaling laws and finalized interpretations.

Quantity	Classical	Fractional Law (Confirmed)	Status / Mechanism
Curvature slowing (flows)	\(\beta(n)=\frac{1-n}{n}\)	\(\beta_{\mathrm{mem}} \approx \beta(n)/\alpha_t\)	Time-stretch (Drag). Valid for \(\alpha_t \ge 0.8\) in tested range.
Deterministic ghost escape (rank 2)	\(\gamma(1)=\frac{1}{2}\)	\(\gamma(\alpha_t)=\frac{1}{2\alpha_t}\)	Confirmed (ABM-PECE scaling reduction) in weak memory.
Stochastic barrier exponent	\(p=\frac{3}{2}\)	\(p(\alpha_t)=\frac{3}{2}\) (invariant)	Friction view wins: memory changes rate, not barrier geometry.
Map fracture clock	Invariant interval for \(r\le 4\)	\(\tau_{\text{exit}}(M;x_0)=k(x_0)M+c(x_0)\), \(k\in\{1,2\}\) observed	Deterministic finite-memory fracture (Inertial Momentum).
Map survival boundary	\(r_\infty \approx 3.57\)	\(r_{\text{surv}}(\alpha,M;N,x_0)\) finite and parameter-dependent	Metastable fracture replaces full Feigenbaum cascade for finite \(M\).

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1. Curvature Rank from the Power Rule

Near a one-dimensional bifurcation \((x^*,\mu_c)\), we have \(f_x(x^*,\mu_c)=0\). Taylor expansion forces the next surviving term:

\[
f(x,\mu)\approx a(\mu-\mu_c)+b(x-x^*)^n+\cdots,\quad n\ge 2.
\]

Definition (Curvature Rank):

\[
\boxed{R(f,x^*) := \min\{k \ge 1 : f^{(k)}(x^*) \neq 0\}.}
\]

1.1 Multiplicity peeling (Leibniz + power rule)

If \(f(x)\approx (x-x_0)^m g(x)\), then \(f^{(m)}(x_0)=m!\,g(x_0)\neq 0\).
The factorial \(m!\) is repeated power-rule peeling: a combinatorial layer count of degeneracy.
Fractional peeling replaces factorials with Gamma ratios; for example, \(D^{\alpha_t}x^m \propto x^{m-\alpha_t}\).

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2. Arnol’d \(A_n\), Codimension, and Selection Rules

For curvature rank \(R=n\), generic scalar singularities fall into the Arnol’d \(A_n\) family with codimension \(n-1\).
Observability requires unfolding parameters chosen transverse to the catastrophe manifold. This explains why stable bifurcation catalogues are finite.

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3. Temporal Geometry: Ghosts and Slowing

Critical points generate slow dynamics (“ghosts”) because the leading restoring term is flat.
For a rank-2 saddle-node, the bottleneck width scales as \(\Delta x \sim |\mu|^{1/2}\).

3.1 Symmetry-protected ghosts

Ghosts can be extended by symmetry cancellations (e.g., pitchfork symmetry \(x \mapsto -x\)) that force \(R=3\),
creating “protected flats” where drift vanishes structurally.

3.2 Memory ghosts: The Time-Stretch Confirmation

We tested whether fractional memory acts as geometric deformation or temporal stretch. Escape time scales as

\[
\tau_{\mathrm{esc}}(\mu)\sim \mu^{-\gamma(\alpha_t)}.
\]

3.3 Empirical window (validated regime)

• 
  Weak memory (\(\alpha_t \ge 0.8\)):
  
  \(\gamma=\frac{1}{2\alpha_t}\) confirmed with sub-percent error in tested ranges, supporting the Time-Stretch (Drag) model.
  
• 
  Strong memory (\(\alpha_t \approx 0.7\)):
  
  deviations observed relative to weak-memory extrapolation, consistent with strong pre-asymptotic corrections or a distinct regime.
  

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4. Universality Thermostats

This section records conceptual extensions and is not part of the validated Target 1–3 numerical evidence.

4.1 Schwarzian thermostat

For smooth unimodal maps, negative Schwarzian derivative restricts stable cycles and supports the classical period-doubling route.

4.2 Holonomy thermostat (Non-Abelian)

In group-valued dynamics, a holonomy deficit \(\Delta_f\) can act as a discrete curvature unit; persistent nonzero \(\Delta_f\)
behaves like a stable “failure to realize” target loop curvature.

4.3 Example: The \(\kappa_3\)-sector

Simulations on a graph with cycle rank \(b_1=3\) revealed a robust \(\kappa_3\)-sector with topological stiffness
\(\kappa_{\mathrm{top}}\approx 3.0\), consistent with a double Neimark–Sacker scenario producing an attracting 2-torus.

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5. Fractional Curvature Rank in Discrete Maps

5.1 Fractional peeling

\[
D^{\alpha_t}x^n=\frac{\Gamma(n+1)}{\Gamma(n-\alpha_t+1)}x^{n-\alpha_t}.
\]

5.2 Explicit Volterra fractional logistic map

We define the discrete fractional map via a Caputo-like Volterra sum:

\[
x_n = x_0 + \sum_{j=0}^{n-1} b_{n-1-j}^{(\alpha_t)}\,[r x_j(1-x_j) – x_j],
\]

with kernel weights

\[
b_j^{(\alpha_t)}=\frac{\Gamma(j+\alpha_t)}{\Gamma(\alpha_t)\Gamma(j+1)} \sim j^{\alpha_t-1}\quad (j\to\infty).
\]

5.3 The fracture mechanism: loss of invariance

In the classical logistic map, \([0,1]\) is an invariant interval for \(r\le 4\).
In the Caputo-increment map, the update depends on a weighted sum of past increments, so invariance of \([0,1]\) is not structurally enforced.
We define domain exit as the event \(x_n\notin[0,1]\). Domain exit marks loss of the classical invariant set and typically
precedes numerical blow-up via the quadratic nonlinearity.

• Ghost windows: for \(\alpha_t=0.98\), periodic windows appear as transient ghosts.
• Survival boundary: beyond a boundary \(r_{\text{surv}}\), the orbit eventually exits \([0,1]\).
• Finite-memory scaling: in the metastable regime, \(\tau_{\text{exit}}\) scales linearly with \(M\).

5.4 Phase 1 hardening: the discrete memory clock

Rigorous first-exit simulations at \(\alpha=0.98\), \(r=3.05\) established a deterministic affine clock law:

\[
\tau_{\text{exit}}(M;x_0) = k(x_0)M + c(x_0),
\]

with two turnover branches \(k\in\{1,2\}\) observed in the tested regime.
For a mid-range initial condition (e.g., \(x_0\approx 0.2\)), the dominant branch is
\(\tau_{\text{exit}} = 2M + 4\), verified over memory depths up to \(M=102400\) with zero replicate variance.

A secondary “one-flush” branch is realized for high initial conditions. For example, at \(x_0=0.88\),
\(\tau_{\text{exit}} = M + 1\) was verified over multiple memory depths, again with exact affine scaling.
A basin scan at \(M=3200\) found the one-flush regime localized in the interval \(x_0\in[0.84,0.95]\),
with the two-flush regime dominating for \(x_0\lesssim 0.83\) on the tested grid.

Interpretation: The natural time unit is one memory turnover. Exit occurs after an integer number of turnovers
once accumulated momentum exceeds the domain margin \(1-x_0\).

Basin interpretation: The one-flush branch (\(k=1\)) corresponds to
initial conditions near the upper domain boundary, where the margin \(1 – x_0\)
is small. A single turnover’s worth of momentum accumulation suffices for exit.
The two-flush branch (\(k=2\)) corresponds to mid-range initial conditions, where
the first turnover’s accumulated momentum is partially “flushed” (replaced by
newer increments) before exit occurs during the second turnover.

Figure (schematic reference): A “clock phase diagram” plots \(\tau_{\text{exit}}/M\) versus \(x_0\) and reveals plateaus near 1 and 2,
corresponding to one-flush (\(k=1\)) and two-flush (\(k=2\)) modes.

Figure 5: The Discrete Memory Clock Phase Diagram. Normalized survival time \(\tau_{\text{exit}}/M\) vs initial condition \(x_0\) for the Caputo-increment logistic map at \(\alpha=0.98, r=3.05, M=3200\) (unclipped, first-exit stopping). Two sharp plateaus appear at \(\tau_{\text{exit}}/M \approx 2\) (dominant two-flush branch) and \(\approx 1\) (one-flush branch localized in \([0.84, 0.95]\)).

5.5 The P-2 bypass phenomenon (mechanistic interpretation)

The Inertial Momentum Theorem provides a natural explanation for the P-2 bypass.
In the classical period-doubling cascade, the P-2 regime has higher orbit amplitude
than P-1, placing trajectories closer to the domain boundary. Under finite memory,
orbits in this high-amplitude regime fall into the one-flush basin
and are ejected before completing enough cycles to be detected as period-2.

This interpretation predicts that P-2 should reappear under conditions that
suppress one-flush exit: either (i) very large \(M\) (pushing exit time beyond
detection horizon), or (ii) modified operators that preserve the invariant interval.

5.6 Critical methodological note: clipping caveat

Clipping (enforcing \(x_n\in[0,1]\)) is not a numerical convenience but a model change.
Clipped simulations can show apparently stable periodic orbits with high clip fractions, meaning observed dynamics are dominated by the clipping rule,
not the fractional map. All Target 1 fracture results reported here use unclipped dynamics with explicit tracking of domain exit.

5.7 Empirical boundary statement

We define the finite-horizon survival boundary as

\[
r_{\text{surv}}(\alpha,M;N,x_0) := \sup\Big\{r:\ x_n\in[0,1]\ \text{for all}\ 0\le n\le N\Big\}.
\]

\[
\boxed{
\begin{aligned}
&\textbf{Target 1 (Discrete Map):}\\
&\text{Finite memory induces a survival boundary } r_{\text{surv}}.\\
&\text{For } r>r_{\text{surv}},\ \text{the Feigenbaum cascade is replaced by}\\
&\text{metastable periodic windows that collapse via domain exit.}
\end{aligned}}
\]

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6. The Inertial Momentum Theorem

This section formalizes why finite-memory discrete fractional maps fracture, and why continuous flows do not.
The key distinction is that discrete maps have unit step \(h=1\) and therefore lack the \(h^\alpha\) damping present in flow discretizations.

6.1 Finite-memory momentum operator

Write the finite-memory Caputo-increment map as a sliding-window convolution:

\[
x_{n+1} = x_0 + \mathcal{J}_n(M),\qquad
\mathcal{J}_n(M) = \sum_{j=0}^{m-1} w_j^{(\alpha)}\,\Delta x_{n-j},
\]

where \(m = \min(n+1, M)\) is the effective memory depth at step \(n\) (the sum includes terms from \(j=0\) to \(j=m-1\), i.e., the \(m\) most recent increments). \(\Delta x_k = f(x_k)-x_k\) is the instantaneous increment and weights satisfy the canonical decay
\[
w_j^{(\alpha)} \sim \frac{j^{\alpha-1}}{\Gamma(\alpha)}.
\]

6.2 Theorem (Finite-Memory Fracture)

Theorem.
Let \(x_n\) evolve under the Caputo-increment map with memory depth \(M\) and
kernel \(w_j^{(\alpha)}\). Suppose there exists \(N_0 \ge 0\) and \(\epsilon > 0\)
such that \(\Delta x_n \ge \epsilon\) for all \(n \in [N_0, N_0 + M – 1]\). Then:
\[
x_{N_0 + M} – x_0 \ge \epsilon \cdot W_M(\alpha), \quad
W_M(\alpha) := \sum_{j=0}^{M-1} w_j^{(\alpha)}.
\]
For \(\alpha_t < 1\), \(W_M(\alpha_t) \sim M^{\alpha_t}/\Gamma(\alpha_t+1)\); for \(\alpha_t = 1\), \(W_M(1) = M\) exactly. In particular, if \(x_0 + \epsilon \cdot W_M(\alpha) > 1\), then domain exit occurs
by step \(N_0 + M\).

Why persistent drift is generic in metastable regimes:
In the metastable window, the orbit shadows an unstable manifold of the classical
map. Memory truncation introduces a systematic bias: the oldest increments
(which would have provided stabilizing negative feedback in the infinite-memory
limit) are discarded, leaving a net positive drift. This asymmetry produces
blocks of persistent positive increments with probability approaching 1 as
the orbit approaches the domain boundary.

Figure 6: The Inertial Momentum Mechanism. (A) Caputo memory weights \(w_j\) for weak (\(\alpha=0.98\)) and strong (\(\alpha=0.80\)) memory. (B) Schematic momentum accumulation \(\mathcal{J}_n\) in the metastable regime (drifts illustrative). High \(x_0\) overflows after one turnover (\(k=1\)); low \(x_0\) requires flushing of early transients (\(k=2\)). Stars mark approximate empirical exit times.

6.3 Quantized clock law and turnovers

Because the operator refreshes on the horizon \(M\), the natural time unit is one memory turnover. Empirically, the exit time obeys

\[
\tau_{\text{exit}}(M;x_0)=k(x_0)M+c(x_0),
\]

with \(k\in\{1,2\}\) observed in the tested regime. Higher-order turnover branches are not excluded in principle but would require
more complex transient structures.

6.4 Momentum distinction: discrete map vs continuous flow

For a fractional flow \(D_t^\alpha x=f(x)\), time discretization introduces a factor \(h^\alpha\):

\[
x_{n+1}-x_n \approx h^\alpha\,\mathcal{J}_n.
\]

As \(h\to 0\), the prefactor suppresses accumulation, converting “inertia” into drag (time-stretching without fracture).
The discrete map (\(h=1\)) lacks this suppression, leaving unbound inertia under finite-memory truncation.

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7. Stochastic Curvature Rank: The Friction Verdict

Prototype: \(dx=(\mu+x^2)\,dt+\sigma\,dW_t\).
We tested two hypotheses for barrier escape scaling:

• Hypothesis A (Friction): \(\Delta V \propto |\mu|^{3/2}\) (classical geometry).
• Hypothesis B (Geometry): \(\Delta V \propto |\mu|^{3/(2\alpha_t)}\) (deformed geometry).

An \(\alpha_t\)-sweep confirmed Hypothesis A: fractional memory alters rates (kinetics) while preserving the barrier exponent \(p=3/2\).

\(\alpha_t\)	\(R^2_A\) (p=3/2)	\(R^2_B\) (p=3/(2\alpha_t))	\(\Delta R^2\)	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

Conclusion: Memory acts as friction: it changes the rate, not the barrier geometry.

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8. Mechanistic Anchor: Fractional Quantum Rank

This section is a theoretical extension bridging fractional findings to quantum mechanics.
The fractional Schrödinger equation replaces dispersion \(p^2\) with \(|p|^{\alpha_s}\).
Near criticality, scaling is governed by competition between kinetic order \(\alpha_s\) and potential order \(n\):

\[
\boxed{R_{\alpha_s,\mathrm{quantum}}=\min(\alpha_s,n).}
\]

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9. Theoretical Closure: The Hybrid Rank Unification Test

This section proposes a unified framework for future tests. The core claim is that “rank” is a unified counter across substrates.
We propose a hybrid scaling law

\[
\beta_{\mathrm{hybrid}} \approx \frac{1-R}{R}\,f(\alpha_t).
\]

An explicit candidate model is a fractional SU(2) coupled rotor system. Measuring escape from a symmetry-protected sector could test whether
rank substitution (\(R\to R_{\text{non-ab}}\)) and memory amplification \(f(\alpha_t)\) jointly explain the exponent.

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10. Final Synthesis

The project concludes with a unified physical interpretation of the Power Rule’s role:

• 
  Continuous flows (Target 2): Memory acts as Drag, rescaling time
  
  and stabilizing ghost dynamics (time-stretch).
  
• 
  Stochastic systems (Target 3): Memory acts as Friction, changing kinetics while preserving
  
  barrier geometry (\(p=3/2\)).
  
• 
  Discrete maps (Target 1): Memory acts as Inertial Momentum. Finite-memory truncation converts
  
  history into a destabilizing gain that produces deterministic domain exit after an integer number of memory turnovers.
  

This confirms the thesis: the Power Rule is a “silent architect,” building stability in the continuum but inducing fracture clocks in discrete finite-memory realizations.

\[
\boxed{
\begin{aligned}
&\text{The Power Rule builds order in the continuum through viscosity,}\\
&\text{but imposes deterministic fracture clocks in discrete finite-memory systems through inertia.}
\end{aligned}
}
\]

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11. Resolutions and Open Questions

11.1 Resolution of key v7.2 questions

Question (v7.2)	Resolution (v7.3)
\(M\to\infty\) limit: does \(r_{\text{surv}}\) converge?	Fracture is inevitable for all finite memory depths. The clock law \(\tau_{\text{exit}} \sim M\) implies that classical chaotic behavior is recovered only in the idealized \(M \to \infty\) limit.
Falsification criteria (Target 1)	Passed in the tested window: \(\tau_{\text{exit}}\) exhibits exact affine scaling with zero replicate variance under first-exit stopping.
P-2 bypass mechanism	Consistent with turnover fracture: one-flush basins can eject high-drift transients early, obstructing sustained observation of certain cascade stages under finite memory.
Strong-memory universality	Evidence of a stability transition for sufficiently strong decay (e.g., \(\alpha\lesssim 0.8\)) in tested configurations; precise \(\alpha_c\) is parameter-dependent and remains a mapping target.

11.2 Open questions (v7.3)

1. 
   Higher turnover branches:
   
   Do \(k\ge 3\) regimes exist for other \((\alpha,r)\) windows or other maps, and what transient structures generate them?
   
2. 
   Rigorous necessity:
   
   The theorem provides a sufficient condition (persistent drift). Under what dynamical conditions is such drift generic in finite-memory fractional maps?
   
3. 
   Operator dependence:
   
   Do alternative discrete fractional operators (e.g., Grünwald–Letnikov variants, implicit schemes, or invariant-preserving constructions) eliminate domain exit while retaining memory?
   
4. 
   Flow scaling precision (rank 3):
   
   For the fractional pitchfork \(D_t^\alpha x=\mu x-x^3\), what is the cleanest operational definition of \(\tau_{\text{relax}}\) that yields stable power-law fits near \(\mu\to 0\)?
   
5. 
   Experimental signatures:
   
   Can finite-memory turnover clocks be observed in physical systems with effective memory cutoffs (viscoelasticity, anomalous transport with aging, or finite-bandwidth feedback systems)?
   
6. 
   Universality of quantized clocks:
   
   Is the discrete structure \(k \in \{1, 2, \ldots\}\) universal across fractional map families, or specific to the logistic nonlinearity and Caputo kernel? Testing on other maps (e.g., fractional Hénon, tent map variants) would clarify.
   

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End of document.