Scan notes: rootsasgeometry_passE and relevance to φ-transfer extremality

This note records the main items in the roots_as_geometry_passE package that directly strengthen the current φ-transfer-matrix project.

What the root project contributes that we can reuse verbatim

• Descriptor stability under bounded seed perturbations: the growth/descriptor framework proves that bounded perturbations of the seed induce bounded variation in shell-based observables over finite horizons (finite-speed influence). This supports the claim that digit strings extracted from ratio statistics of shell sizes are physically meaningful observables rather than fragile encodings.

• Operator-specific influence bounds: in the canonical thickening case, the project gives explicit, checkable deviation bounds in terms of horizon and perturbation radius. This provides a concrete robustness story for any “coarse-graining → digit extraction” pipeline.

• *Root-map formalization *: defines a normalization αr(G) and a root polynomial PG(t), and formalizes a root map ℛ that is exact on a restricted orthotope class and computable from D★. Conceptually, this is another instance of: stable descriptor observables → normalized code → reconstructive geometry.

How it plugs into the φ project

The φ paper’s “Universal coarse-graining” section argues:

local constrained growth → measurable scalar ratio statistic λ → Gauss normalization → digit string → transfer cocycle → extremal-growth classification.

The root project supplies the missing physics-native justification step:

• the scalar statistic (shell size / flux) is stable under bounded perturbations (finite-speed influence);
• the resulting digit code is therefore stable (over controlled horizons);
• extremal statements about digit-coded transfer growth become applicable as universal baseline bounds for constrained-growth systems.

Perturbative stability vs. seed stability

These are complementary:

• Seed stability (root project): perturb initial condition; show bounded effects on observables.
• Positive-operator stability \(φ project, added in v0.7\): deform the minimal transfer operator by adding a small nonnegative matrix; show the Perron root increases strictly, so φ remains the minimal exponent for the undeformed minimal operator and remains a strict baseline.

Together they address two typical referee concerns:
robustness to microscopic noise (seed perturbations) and robustness of the extremal baseline (positive deformations).