The Power Rule as Chaos’s Silent Architect

Synthesis v6.5 — Curvature Rank, Ghosts, Universality, and Unified Rank Tests
Collaborators: Mustafa Aksu, ChatGPT, Grok, Gemini, Claude
Date: December 2025

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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:

• Local bifurcations: rank selects normal forms; transversality explains why stable bifurcation catalogues (e.g., Arnol’d \(A_n\)) are finite.
• Ghosts: rank controls slowing and “hesitation time”; symmetry can extend ghosts by canceling low-order terms.
• Universality: rank is necessary but not sufficient for Feigenbaum scaling; smooth distortion constraints (e.g., negative Schwarzian for common unimodal classes) act as a thermostat selecting routes.
• Non-abelian generalization: rank becomes commutator depth; holonomy acts as a thermostat parallel to Schwarzian, enforcing selection rules via sector barriers.
• Fractional generalization: rank becomes memory-weighted (Volterra kernels); classic cascades can fracture; ghosts become noise-fragile.
• Quantum/topological generalization: rank becomes band-touching/dispersion order; a flagship mechanistic anchor is the Fractional Schrödinger Equation (Section 12), where \(p^2\) is replaced by \(|p|^{\alpha_s}\).
• Unification goal: a single “hybrid test” should measure ghost escape scaling under simultaneous non-abelian (commutator) and fractional (memory) constraints (Section 13).

Empirical anchors (current runs, v6.0 notes):

• Universality fracture (explicit Volterra fractional logistic, temporal order \( \alpha_t \)): at \( \alpha_t=0.95 \), a single structure change near \( r\approx 2.8 \) is observed and no period-doubling cascade develops (reported \( \delta_{\alpha_t} \) undefined/NaN for cascade ratios).
• Deterministic ghosts: at \( \alpha_t=0.95 \), fitted \( \gamma \approx 0.502 \) versus weak-memory ansatz \( 1/(2\alpha_t)\approx 0.526 \) (few-percent agreement); at \( \alpha_t=0.7 \), fitted \( \gamma \approx 0.43 \) while \(1/(2\alpha_t)\approx 0.71\) (strong deviation).
• Stochastic scaling attempt: failed (\(R^2<0\)) because the run sat in the diffusion-dominated regime \( \sigma^2 \gg \Delta V \). This is a boundary result: fractional ghosts are noise-fragile unless parameters satisfy a barrier-controlled condition (Section 10).

Non-abelian anchor (\(\kappa_3\)-sector): on a complete-graph example with cycle rank \(b_1=3\), simulations reveal a robust \(\kappa_3\)-sector with \( \kappa_{\mathrm{top}}\approx 3 \) and persistent holonomy deficit \( \Delta \approx 0.5 \), consistent with a double Neimark–Sacker scenario producing an attracting \(2\)-torus (two near-zero Lyapunov exponents) embedded in a 4D center manifold (Section 4.5).

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0.1 Unified Exponent Cheat Sheet (working heuristics)

This table summarizes organizing exponents and deformations. These are models, not universal laws; v6.0 data already shows breakdown in strong memory.

Quantity	Classical	Weak-memory / heuristic deformation	Status / caveat
Curvature slowing	\(\beta(n)=\frac{1-n}{n}\)	\(\beta_{\mathrm{mem}}(\alpha_t,n)=\beta(n)\,f(\alpha_t)\)	\(f(\alpha_t)\) must be fitted/validated; candidates in Section 3.2
Deterministic ghost escape (rank-2 saddle-node)	\(\gamma(1)=\frac12\)	\(\gamma(\alpha_t)\approx \frac{1}{2\alpha_t}\) (time-stretch ansatz)	Works near \(\alpha_t\to 1\); fails for strong memory (\(\alpha_t\approx 0.7\))
Barrier height (rank-2 saddle-node, classical)	\(\Delta V \sim |\mu|^{3/2}\)	\(\Delta V_{\alpha_t}\) model-dependent	Time-fractional systems need care; use empirical crossover tests
Diffusion vs barrier boundary	\(\Delta V/\sigma^2 \gg 1\)	\(|\mu|_{\mathrm{crit}}(\alpha_t)\) inferred by scans	Weak-memory scaling laws may exist; strong memory requires measurement
Sector barrier vs memory	(model-dependent)	\(E_{\mathrm{barrier}}(\alpha_t)\propto \Gamma(\alpha_t)^{-1}\) (hypothesis)	Interpreted as destabilization of sector protection at strong memory (Section 9.7)
Quantum effective rank (FSE)	\(E(p)\propto p^2\)	\(E(p)\propto |p|^{\alpha_s}\) with \(1<\alpha_s\le 2\)	Near criticality, scaling is governed by competition between kinetic order \(\alpha_s\) and band/potential order \(n\) (Section 12.1)

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0.2 Data Integrity, Uncertainty, and Reproducibility (recommended)

• Uncertainty reporting: whenever fitting an exponent (e.g., \(\gamma\)), report mean and uncertainty (standard error or bootstrap CI). Example protocol: fit on log-spaced \(\mu\) values; bootstrap resample residuals (e.g., 1000 resamples) to estimate \(\delta\gamma\). Do not claim precision without error bars.
• Convergence controls: fractional memory experiments require explicit convergence tests in memory cutoff \(M\) and transient length \(N_{\mathrm{trans}}\) (Section 9.5).
• Stochastic regimes: always verify barrier-controlled vs diffusion-dominated regime by comparing \(\sigma^2\) to an estimated barrier scale (Section 10).

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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 to appear:

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

Definition (Curvature Rank):

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

1.1 Multiplicity peeling (Leibniz + power rule; and the fractional analogue)

If \( f(x)\approx (x-x_0)^m g(x) \) with \( g(x_0)\neq 0 \), then \( f^{(k)}(x_0)=0 \) for \(k<m\) and \( f^{(m)}(x_0)=m!\,g(x_0)\neq 0 \). The factorial \(m!\) is repeated power-rule peeling: a combinatorial layer count of degeneracy.

Fractional peeling replaces factorials with Gamma ratios (Section 9.1):

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

1.2 Resonance hierarchy (how rank is “made” by symmetry; and the non-abelian parallel)

Curvature rank often reflects a cancellation depth: symmetries suppress lower-order terms until the first non-cancelled interaction appears at order \(k\). A pitchfork provides a concrete anchor: a \( \mathbb{Z}_2 \) symmetry \(x\mapsto -x\) forbids even terms, canceling the quadratic term and forcing the first non-vanishing nonlinearity to be cubic. Thus symmetry makes \(R=3\) in the local normal form.

This scalar “cancellation depth” is the analogue of commutator cancellation depth in non-abelian systems (Section 8): derivatives peel monomial layers; Lie brackets peel symmetry layers.

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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

\[
\boxed{\mathrm{codim}(A_n)=n-1.}
\]

Observability requires unfolding parameters chosen transverse to the catastrophe manifold (non-degenerate unfolding Jacobian). This is the technical hinge behind finite, stable bifurcation catalogues.

2.1 Selection rules via energy gaps (and why noise matters)

Attempting “wrong-rank” routes can incur a stability/energy penalty—an energy gap that suppresses certain pathways and enforces others. In holonomy-sector systems this becomes explicit via a barrier (Section 4.4). Noise can “pay” the gap: when stochastic forcing exceeds the barrier scale, dynamics jump sectors rather than follow the smooth route (Section 10).

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3. Temporal Geometry: Ghosts and Slowing

Critical points generate slow dynamics (“ghosts”) because the leading restoring term is flat. For generic normal forms:

Bifurcation	Leading term	\( \Delta x \sim \)	\( \lambda-1 \sim \)	\( \frac{d\lambda}{d\mu}\sim \mu^\beta \)	Ghost character
Saddle-node	\(x^2\)	\(|\mu|^{1/2}\)	\(|\mu|^{1/2}\)	\(\mu^{-1/2}\)	Tight bottleneck
Pitchfork	\(x^3\)	\(|\mu|\)	\(|\mu|\)	\(\mu^{0}\)	Flat / neutral
Higher-order	\(x^n\)	\(|\mu|^{1/n}\)	\(|\mu|^{(n-1)/n}\)	\(\mu^{(1-n)/n}\)	Extended / sluggish

Curvature Slowing Exponent:

\[
\boxed{\beta(n)=\frac{1-n}{n}.}
\]

3.1 Symmetry-protected ghosts

Ghosts can be extended not only by curvature rank, but by symmetry cancellations that create near-neutral “protected flats” where leading drift terms vanish by structure rather than fine tuning.

3.2 Memory ghosts: unify \(f(\alpha_t)\) as a model-selection problem

Fractional memory can amplify or reshape ghost behavior. A compact descriptor is

\[
\boxed{\beta_{\mathrm{mem}}(\alpha_t,n)=\beta(n)\,f(\alpha_t).}
\]

Two competing models for \(f(\alpha_t)\) are explicitly tracked (do not mix them silently):

• Model A (time-stretch, weak-memory): \( f_A(\alpha_t)=1/\alpha_t \). This predicts very strong amplification as \(\alpha_t\to 0\).
• Model B (\(\Gamma\)-anchored, non-singular): \( f_B(\alpha_t)=1/\Gamma(2-\alpha_t) \). This stays mild even as \(\alpha_t\to 0\).

Key point: these models disagree strongly (e.g., at \(\alpha_t=0.5\), \(f_A=2\) while \(f_B\approx 1/\Gamma(1.5)\approx 1.13\)). Therefore \(f(\alpha_t)\) must be treated as an empirical selection (or a piecewise regime model), not as a single assumed law.

Recommended Figure (Figure 1): plot fitted \(\gamma(\alpha_t)\) (with error bars) versus \(\alpha_t\), overlaying (i) \(\gamma_A(\alpha_t)=1/(2\alpha_t)\) and (ii) \(\gamma_B(\alpha_t)=1/(2\,\Gamma(2-\alpha_t))\). This one plot clarifies the entire memory-amplification story.

3.3 Empirical window (current runs)

• \(\alpha_t=0.95:\) fitted \(\gamma\approx 0.502\), close to Model A prediction \(0.526\).
• \(\alpha_t=0.7:\) fitted \(\gamma\approx 0.43\), contradicting Model A prediction \(0.71\). This indicates strong memory changes the dynamics beyond “time-stretch.”

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4. Universality Thermostats: Schwarzian, Substrate, Holonomy

4.1 Schwarzian thermostat (smooth unimodal universality)

For many smooth unimodal maps, negative Schwarzian and sufficient smoothness/distortion control restrict stable cycles and support the classic period-doubling route when the critical point is nonflat (typically rank 2). Rank 2 is necessary but not sufficient; convergence to a Feigenbaum fixed point requires additional expansion/regularity hypotheses.

4.2 Substrate-independence of universality

Universality depends primarily on rank, unimodal geometry, and smooth distortion structure—not on substrate (interval vs lattice vs graph reductions). When two systems share the same effective interaction rank and distortion control, large-scale scaling laws tend to align even if microscopic implementation differs.

4.3 Holonomy thermostat (non-abelian generalization)

In gauge- or group-valued dynamics, loop observables constrain instability routes. For a loop \(f\) with prescribed flux \(F_f\), define a gauge-invariant holonomy deficit

\[
\boxed{\Delta_f := \frac{\mathrm{Tr}(H_f)}{2}-\cos\frac{F_f}{2}.}
\]

A persistent nonzero \(\Delta_f\) behaves like a discrete curvature unit: a stable “failure to realize” target loop curvature under available dynamics.

4.4 Topological stiffness, sector barriers, and selection rules

Define topological stiffness

\[
\kappa_{\mathrm{top}} := \frac{C^*}{C_{\mathrm{iso}}}.
\]

Define a sector barrier separating “screened” vs “geometric” holonomy sectors:

\[
\boxed{
E_{\mathrm{barrier}}
=
\min_{\text{Target holonomy}} C^*
–
\min_{\text{Screened sector}} C^*.
}
\]

Interpretation: the screened sector is a false vacuum stabilized by commutator screening; the barrier is the cost of enforcing full geometric law (a concrete selection rule on allowed routes).

4.5 Example: the \(\kappa_3\)-sector (cycle rank \(b_1=3\)) with NS anchor

Observed sector invariants:

• Topological stiffness: \( \kappa_{\mathrm{top}}\approx 3.0 \), naturally aligned with cycle rank \(b_1=3\).
• Holonomy deficit: \( \Delta \approx 0.5 \) on frustrated loops, consistent with flux locking near \(F \approx -2.10\,\mathrm{rad}\approx -2\pi/3\) since \( \cos(F/2)=0.5 \).
• Barrier: persistence under perturbations implies \(E_{\mathrm{barrier}}>0\) (sector protection).

Neimark–Sacker mechanism: dynamics are consistent with a double Neimark–Sacker event: two complex-conjugate pairs cross the unit circle at nearby couplings \(K_1^*\approx 0.24048\) and \(K_2^*\approx 0.24052\), with \( \Delta K \approx 4.7\times 10^{-5} \). This yields a 4D center manifold supporting an attracting invariant \(2\)-torus (two near-zero Lyapunov exponents; remaining exponents strongly negative).

Tuning vs symmetry (open): the double crossing occurs in a narrow parameter window. Whether it requires fine tuning or is enforced by graph symmetry (e.g., paired rotational modes forced by degeneracy) remains to be tested.

Relational-geometry interpretation (compressed): \(\Delta\) and \(\kappa_{\mathrm{top}}\) generate proto-geometry on the configuration space:

• Curvature length: a face scale \(r_f\sim 1/|\Delta|\) (in model units).
• Discrete angles: between loops via \( \phi_{f_1,f_2}=\arccos(\mathrm{Tr}(H_{f_1}H_{f_2}^\dagger)/2) \).
• Proto-time: relaxation toward sector invariants defines an arrow via monotone decrease of loop mismatch.

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5. Tent Maps: Topological Chaos Without Smooth Scaling Geometry

Piecewise-linear tent maps can be topologically chaotic yet fail to support smooth renormalization due to slope discontinuity at the peak. Even in the fully chaotic case, a topological conjugacy to the logistic map does not yield Feigenbaum scaling geometry because:

• the conjugacy need not be smooth (e.g., derivatives can vanish at boundaries),
• the cascade conjugacy fails to be differentiable along critical orbits,
• renormalization iterates a doubling operator on function spaces whose norms are sensitive to derivatives; slope discontinuities obstruct this smooth RG construction.

Thus: topological conjugacy can preserve orbit structure while scaling geometry fails.

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6. Neimark–Sacker (Discrete Hopf) as Emergent Rotational Rank

A supercritical Neimark–Sacker (NS) bifurcation is the discrete-time Hopf analog: a complex-conjugate pair crosses the unit circle, producing quasi-periodic motion on an invariant circle. On the \(2\)D center manifold, the linearized dynamics is a rotation and induces an almost-complex operator \(J\) with \(J^2=-I\) in the center directions.

A double NS yields a \(4\)D center manifold that can support an invariant \(2\)-torus \(T^2\) (two incommensurate angles), matching the Lyapunov signature used to anchor the \(\kappa_3\)-sector example in Section 4.5.

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7. Beyond 1D (compressed)

Rank-2 folding mechanisms propagate into higher-dimensional chaos, but Jacobian mixing introduces additional exponents and can change quantitative universality details while preserving cascade topology. Practical rule: curvature rank controls which cascade occurs; dimensional embedding controls how scaling is metrically realized.

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8. Non-Abelian Curvature Rank (Lie-bracket peeling)

In group-valued dynamics, effective nonlinearity arises from Lie brackets. The Baker–Campbell–Hausdorff expansion is the non-abelian analogue of a Taylor series:

\[
\log(e^A e^B)=A+B+\tfrac12[A,B]+\tfrac{1}{12}[A,[A,B]]-\cdots
\]

Definition (Non-Abelian Curvature Rank):

\[
\boxed{
R_{\mathrm{non\text{-}ab}}
=
\min\{k\ge 2:\ \text{all nested commutators of order }<k\ \text{cancel in effective dynamics, and order }k\ \text{enters}\}.
}
\]

8.1 Lie algebras as structural origin (one sentence version)

Derivatives peel monomial layers in Taylor expansions; Lie brackets peel symmetry layers in non-abelian expansions—so commutator depth is curvature rank in disguise.

8.2 Concrete example (coupled rotors / spins; checkable)

Consider two spins with Hamiltonian

\[
H = J_1 S_1^z + J_2 S_2^z + g\, S_1^x S_2^x.
\]

Since \( [S^z,S^x]=iS^y \), the first bracket \( [S_1^z,S_1^x] \) is order 2, while \( [S_1^z,[S_1^z,S_1^x]]\sim [S_1^z,S_1^y]\sim S_1^x \) is order 3. If parameters/symmetry force order-2 effects to cancel in the effective dynamics (e.g., by averaging or coupling structure), then the first dynamically relevant nonlinearity is order 3, giving \(R_{\mathrm{non\text{-}ab}}=3\). This produces non-abelian “flatness” and extended transients analogous to scalar rank-3 ghosts.

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9. Fractional Curvature Rank — Protocols and Interpretation

9.1 Fractional peeling (Gamma scaling)

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

9.2 Explicit Volterra fractional logistic map (notation cleaned)

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

\[
\boxed{
x_n
=
x_0
+
h^{\alpha_t}\sum_{j=0}^{n-1} b_{n-1-j}^{(\alpha_t)}\, f(x_j),
\qquad
f(x)=r x(1-x).
}
\]

Kernel definition:

\[
\boxed{
b_k^{(\alpha_t)}
=
\frac{1}{\Gamma(\alpha_t)}\frac{\Gamma(k+\alpha_t)}{\Gamma(k+1)}
\sim k^{\alpha_t-1}\ \text{as }k\to\infty.
}
\]

What is \(h\)? \(h>0\) is an effective time-scale parameter for the Volterra discretization. In “pure map” experiments one may set \(h=1\) (or absorb \(h^{\alpha_t}\) into \(r\)). If a different normalization is preferred (e.g., \(h^{\alpha_t}/\Gamma(\alpha_t+1)\)), it can be converted by rescaling \(r\) and/or \(h\). The key structural feature is the power-law memory kernel \(b_k^{(\alpha_t)}\).

Memory cutoff: truncate to last \(M\) terms for \(O(M)\) updates. For \(\alpha_t\) close to 1 the kernel decays extremely slowly, so \(M\) must be tested for convergence (Section 9.5).

9.3 Measuring \(\delta_{\alpha_t}\): robust onset detection

1. Discard \(N_{\mathrm{trans}}\ge 10^4\); sample \(N_{\mathrm{samp}}\approx 2000\).
2. Period candidate \(P=2^k\) must satisfy recurrence error \( \max|x_{t+P}-x_t|<\varepsilon \) with \( \varepsilon \in [10^{-5},10^{-4}] \) and persist for at least \(20P\) steps.
3. Optional: cluster into \(P\) clusters and require clear separation (e.g., silhouette \(\gtrsim 0.7\)).
4. Declare chaos only if: no stable period detected up to \(P_{\max}\), MLE \(>0.01\) is sustained, and autocorrelation decays (e.g., \(C(\tau)<0.1\) by \(\tau\approx 100\)).

\[
\boxed{
\delta_{\alpha_t}(k)=\frac{r_k(\alpha_t)-r_{k-1}(\alpha_t)}{r_{k+1}(\alpha_t)-r_k(\alpha_t)}.
}
\]

9.4 MLE for memory maps (critical correction)

With cutoff \(M\), treat the system as a delay-state vector \(X_n=(x_n,x_{n-1},\dots,x_{n-M})\). Compute maximal Lyapunov exponent (MLE) via Benettin renormalization in \((M+1)\)-dimensions using the linearization of the truncated Volterra operator (Toeplitz-like Jacobian induced by kernel weights). Linearizing only the scalar \(x_n\) is incorrect.

9.5 Why the cascade fractures near \(\alpha_t=0.95\): working mechanisms + tests

Current observation: at \(\alpha_t=0.95\) a visible Feigenbaum cascade is absent in the explicit-memory logistic family (reported “stub” behavior).

Working mechanisms (testable):

• Near-nondecaying memory: since \(b_k^{(\alpha_t)}\sim k^{\alpha_t-1}\), at \(\alpha_t=0.95\) decay is \(k^{-0.05}\), essentially flat. The cumulative weight \(W_n=\sum_{k=0}^{n-1}b_k^{(\alpha_t)}\) grows like \(n^{\alpha_t}\), so the map becomes strongly history-dependent on long runs, smearing sharp period resolution.
• RG fixed point suppression: long memory can damp local expansion needed for renormalization contraction toward a Feigenbaum fixed point.
• Cutoff artifact (most important to falsify): if \(M\) is too small, the “true” memory is not represented; for \(k^{-0.05}\) kernels, convergence may require \(M\gg 10^3\).

Minimum falsification test: hold \(\alpha_t=0.95\) fixed and run a systematic \(M\)-sweep (e.g., \(M=200,500,2000,10^4\)) plus a transient sweep (e.g., \(N_{\mathrm{trans}}=10^4,10^5\)). If the cascade appears as \(M\to\infty\), the fracture was numerical. If it does not, the fracture is dynamical.

9.6 Empirical boundary statement (current status)

\[
\boxed{\alpha_{c,\mathrm{eff}} > 0.95\ \ \text{for a visible Feigenbaum-like cascade in the current explicit-memory logistic family and current resolution}.}
\]

This is an effective threshold: it depends on model definition, detection criteria, cutoff \(M\), and finite-time observation windows.

9.7 Fractional memory and non-abelian sector barriers (hypothesis; interpretation tightened)

\[
\boxed{
E_{\mathrm{barrier}}(\alpha_t)\propto \Gamma(\alpha_t)^{-1},
\qquad \alpha_t\in(0,1].
}
\]

Interpretation (destabilization, not “stiffening”): as memory strengthens (\(\alpha_t\to 0\)), \( \Gamma(\alpha_t)\to \infty \) and \( \Gamma(\alpha_t)^{-1}\to 0 \), predicting that topological sector barriers vanish. This aligns with the empirical theme that strong memory increases susceptibility to noise and intermittent hopping between regimes.

Concrete test: fractionalize the relaxation dynamics in the \(\kappa_3\)-sector system and measure the minimal perturbation required to escape the screened sector as a function of \(\alpha_t\). The hypothesis predicts a monotone destabilization with decreasing \(\alpha_t\).

9.8 Theoretical interpretation: RG fixed-point dissolution (conjecture)

Feigenbaum universality in classical unimodal maps is organized by a renormalization operator \( \mathcal{R} \) acting on a smooth function space. (The scaling constant in classical RG is often denoted \(\alpha_F\); it is not the fractional order \(\alpha_t\).)

For fractional-memory maps, composition is replaced by a memory-composition operator \( \mathcal{F}_{\alpha_t} \) induced by the Volterra kernel. A schematic generalized renormalization can be written as

\[
\mathcal{R}_{\alpha_t}[f](x)=\frac{1}{\alpha_F}\,\mathcal{F}_{\alpha_t}\!\left[\mathcal{F}_{\alpha_t}[f]\right](\alpha_F x).
\]

Conjecture: the “fracture” corresponds to dissolution of a hyperbolic fixed point \(f_{\alpha_t}^*\) of \(\mathcal{R}_{\alpha_t}\). As \(\alpha_t\downarrow \alpha_c\), the spectral gap of the linearized operator closes (leading eigenvalue approaches 0 or loses hyperbolicity), and universality dissolves.

Testable prediction: numerically approximate the leading eigenvalue of the linearized RG operator around an empirically estimated fixed point. A collapse of this eigenvalue toward marginality should correlate with the onset of cascade smearing/fracture.

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10. Stochastic Curvature Rank: barrier-crossing vs diffusion regime

Prototype (rank-2 saddle-node with noise):

\[
dx = (\mu + x^2)\,dt + \sigma\,dW_t.
\]

In barrier-controlled regimes, mean escape times typically follow Arrhenius form

\[
\tau \sim \exp\!\left(\frac{\Delta V}{\sigma^2}\right),
\qquad
\Delta V \sim |\mu|^{3/2}\ \text{(classical rank-2 saddle-node)}.
\]

If \( \sigma^2 \gg \Delta V \), escapes are diffusion-dominated and barrier scaling cannot be extracted:

\[
\boxed{\Delta V/\sigma^2 \ll 1 \ \Rightarrow\ \text{diffusion regime}.}
\]

v6.0 boundary result: the stochastic scaling attempt failed because \(\sigma^2\) exceeded the barrier scale by orders of magnitude. This failure is informative: it marks the diffusion regime and quantifies the need for smaller \(\sigma\) or larger \(|\mu|\) to test barrier-based laws.

10.1 Weak-memory crossover scaling (conditional)

Any formula of the form \( |\mu|_{\mathrm{crit}}(\alpha_t)\sim \sigma^{c(\alpha_t)} \) depends on a model for how memory changes \(\Delta V\). Because strong-memory ghost data contradicts simple time-stretching, treat crossover laws as weak-memory only unless validated.

Recommended practice: determine \( |\mu|_{\mathrm{crit}}(\alpha_t) \) empirically by scanning \(\mu\) at fixed \(\sigma,\alpha_t\) and identifying where escape statistics switch from diffusion-like to Arrhenius-like.

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11. Quantum / Topological Rank (bridging)

Band touchings act like “quantum bifurcations.” In a two-band model with Bloch Hamiltonian \(H(k)=\mathbf{h}(k)\cdot\boldsymbol{\sigma}\), the gap behaves as \(E_{\mathrm{gap}}(k)=2|\mathbf{h}(k)|\). Near a touching \(k_c\), if \(E_{\mathrm{gap}}(k)\sim |k-k_c|^n\), then \(n\) is a quantum rank (touching order) controlling scaling.

Topological invariants change only when the gap closes, just as dynamical topology changes only at bifurcations. Berry phase and Chern number are holonomy/curvature integrals in parameter space: classical holonomy deficits \(\Delta_f\) (Section 4.3) mirror Wilson-loop constraints in state space, while Berry holonomy constrains phases in Hilbert space.

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12. Mechanistic Anchor: Fractional Quantum Rank (FSE)

The Fractional Schrödinger Equation (FSE) is a flagship model where the quantum kinetic term becomes non-local. In path-integral language, it arises by extending Brownian paths to Lévy-flight paths, replacing the standard dispersion \(p^2\) with \(|p|^{\alpha_s}\) (space-fractional order \(1<\alpha_s\le 2\)).

\[
i\hbar \frac{\partial \psi(\mathbf{r},t)}{\partial t}
=
D_{\alpha_s}\,(-\hbar^2\Delta)^{\alpha_s/2}\,\psi(\mathbf{r},t)
+
V(\mathbf{r},t)\,\psi(\mathbf{r},t).
\]

12.1 Corrected rank interpretation: competition of scaling dimensions

The kinetic operator imposes fractional dispersion \(E(p)\propto D_{\alpha_s}|p|^{\alpha_s}\). If another structure (band touching or potential) introduces an order \(n\), the effective critical scaling is determined by competition between kinetic scaling order \(\alpha_s\) and the competing order \(n\):

\[
\boxed{
R_{\alpha_s,\mathrm{quantum}}
=
\min(\alpha_s,\ n)
\quad\text{(scaling-dimension rule)}.
}
\]

Interpretation: fractional dispersion can cap the effective quantum rank at \(\alpha_s\) if the kinetic term is the softest scaling. This replaces the earlier “\(n-\alpha_s\)” heuristic, which is generally not correct for space-fractional operators.

12.2 The fractional curvature regulator \(D_{\alpha_s}\)

\(D_{\alpha_s}\) sets the stiffness of the fractional kinetic term and controls how non-locality reshapes propagation and critical response. A practical program is to measure tunneling/transport rates versus \(\alpha_s\) in a toy potential and compare to predictions from the scaling rule above.

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13. Theoretical Closure: The Hybrid Rank Unification Test

The core claim is that “rank” is a unified counter across substrates:

• Scalar rank \(R(f,x^*)\) from derivatives (Sections 1–3),
• Fractional rank from memory order \(\alpha_t\) and kernels (Section 9),
• Non-abelian rank from commutator depth \(R_{\mathrm{non\text{-}ab}}\) (Section 8),
• Quantum rank from band-touching/dispersion order (Sections 11–12).

13.1 Generic measurement target

\[
\tau_{\mathrm{escape}}(\alpha_t, R_{\mathrm{non\text{-}ab}})
\ \propto\
\mu^{-\gamma_{\mathrm{hybrid}}(\alpha_t, R_{\mathrm{non\text{-}ab}})}.
\]

13.2 Hybrid scaling hypothesis (rank substitution + kernel regulator)

\[
\boxed{
\beta_{\mathrm{hybrid}}
\approx
\frac{1-R_{\mathrm{non\text{-}ab}}}{R_{\mathrm{non\text{-}ab}}}\cdot f(\alpha_t),
}
\]
with \(f(\alpha_t)\) determined by the same model-selection logic as Section 3.2 (time-stretch vs \(\Gamma\)-anchored vs fitted).

13.3 Explicit proposed model (computable): fractional SU(2) coupled rotors

Define a minimal SU(2) two-spin system with Hamiltonian

\[
H = J_1 S_1^z + J_2 S_2^z + K\, S_1^x S_2^x.
\]

Fractionalize time evolution using a Caputo time derivative:

\[
D_t^{\alpha_t}|\psi(t)\rangle = -\frac{i}{\hbar} H |\psi(t)\rangle,
\qquad 0<\alpha_t\le 1.
\]

Observables (one concrete choice):

• Non-abelian interaction rank: determine the first commutator depth that contributes to dynamics under the chosen symmetry/initial conditions (often \(R_{\mathrm{non\text{-}ab}}=2\) unless cancellations enforce higher depth).
• Holonomy / “twist” proxy: use a gauge-invariant correlator deviation (example):
  
  \[
  
  \Delta_{\mathrm{ent}}(t)=\langle S_1^z S_2^z\rangle – \langle S_1^z\rangle\langle S_2^z\rangle.
  
  \]
  
• Escape time: prepare an initial metastable state \( |\psi_0\rangle \) near a separatrix (or near a symmetry-protected plateau) and define \(\tau\) as the time until \( \Delta_{\mathrm{ent}}(t) \) drops by a factor \(1/e\) (or crosses a chosen threshold).

Prediction template: fit \( \tau(\alpha_t,K) \) or \( \tau(\alpha_t,\mu) \) to a power law over a controlled window and test whether rank substitution and memory amplification jointly explain the exponent.

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14. Applications and Practical Observables

• Time-series ghosts: fit escape times \(\tau(\mu)\) to extract \(\gamma\) with error bars; compare competing \(f(\alpha_t)\) models.
• Fractional universality: for maps with memory, always run \(M\)-convergence and transient-length convergence before asserting “fracture.”
• Noise fragility: map the diffusion/barrier boundary by scanning \((\mu,\sigma)\) at fixed \(\alpha_t\); report regime diagrams.
• Quantum rank (FSE): measure transport/tunneling rates versus \(\alpha_s\) and test the scaling competition \( \min(\alpha_s,n) \).
• Minimum figure set: (i) fractional “stub” vs classical cascade bifurcation diagram, (ii) \(\gamma(\alpha_t)\) with error bars vs models, (iii) diffusion vs barrier regime map, (iv) \(\kappa_3\) sector invariants vs parameter \(K\).

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Glossary (AI-friendly)

• Curvature Rank: \( R(f,x^*)=\min\{k\ge 1:\ f^{(k)}(x^*)\neq 0\} \).
• Curvature Slowing Exponent: \( \beta(n)=(1-n)/n \).
• Memory ghost model: \( \beta_{\mathrm{mem}}(\alpha_t,n)=\beta(n)\,f(\alpha_t) \) with competing \(f\)-models (Section 3.2).
• Neimark–Sacker (NS): discrete Hopf bifurcation; creates rotational center manifold and quasi-periodic motion.
• Non-abelian rank: \(R_{\mathrm{non\text{-}ab}}\), first commutator depth entering effective dynamics.
• Holonomy deficit: \( \Delta_f=\mathrm{Tr}(H_f)/2-\cos(F_f/2) \).
• Topological stiffness: \( \kappa_{\mathrm{top}}=C^*/C_{\mathrm{iso}} \).
• Sector barrier: \( E_{\mathrm{barrier}}=\min_{\mathrm{target}}C^*-\min_{\mathrm{screened}}C^* \).
• Diffusion-dominated regime: \( \sigma^2 \gg \Delta V \), where barrier scaling is unobservable.
• Volterra kernel: \( b_k^{(\alpha_t)}=\Gamma(k+\alpha_t)/(\Gamma(\alpha_t)\Gamma(k+1)) \).
• MLE: maximal Lyapunov exponent computed in \((M+1)\)-dimensional delay-state for cutoff \(M\).
• FSE: Fractional Schrödinger Equation with kinetic order \(\alpha_s\in(1,2]\).
• Quantum effective rank (FSE): \(R_{\alpha_s,\mathrm{quantum}}=\min(\alpha_s,n)\) (Section 12.1).

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Appendix A (Optional): Division-Algebra Speculation (clearly non-theorem)

Single NS produces a 2D center manifold with a canonical almost-complex structure \(J\) satisfying \(J^2=-I\), suggestive of the \(\mathbb{C}\) rung. A double NS produces a 4D center manifold with two rotational generators and an attracting \(T^2\), suggestive (but not proven) of quaternionic-like structure. A rigorous connection would require showing a closed algebra of compatible complex structures (or an \(\mathbb{H}\)-module structure) on the relevant invariant subspace. This remains an open, exploratory direction.