The Book of Butterflies - Part 1.The Spine Preview

RSM presents a framework that separates geometry-layer invariants from kinetics-layer reparameterizations in pattern-forming carriers. The invariant sector is defined by a mirror-even divider skeleton extracted as a ridge of a symmetrized occupancy density, and by a discrete Z2 holonomy computed as a Wilson-style loop product of an odd channel constrained to flip under the declared mirror involution. The kinetics sector is represented by event-density (dilation) fields that may reweight windowed estimators without changing the divider topology. A novel extension introduces a leg variable derived from the convex–concave structure of the dilation potential and uses it to gate the effective sign of curvature-pocket contributions to Z2 holonomy. This yields a falsifiable mechanism by which chirality and density accumulation can invert while the mirror-even skeleton and holonomy class remain stable, resolving “seam loss” appearances as estimator-level window breakage in the presence of re-timing. The second implication of this paper is that “curvature sign” is not, by itself, a reliable surrogate for chirality source strength when the system traverses a convex–concave divider in the dilation potential. Formally, the claim is that the relevant charge is the leg-gated sign q(p)=ℓ(ξp)sign⁡K(p), not sign⁡K(p) alone. If this holds, then many cross-domain confusions vanish: two experiments can report “opposite” chirality attribution to pocket types and both be correct, because they were observing opposite legs of the same divider grammar. This would be a correction with reach across any field that uses curvature-based heuristics to infer transport direction, handedness, or stability class.Related Work - The current manuscript, which focuses on the invariant‑spine framework (mirror‑even skeleton, discrete Z2 holonomy, and leg‑gated pocket charge), is a self‑contained excerpt of that monograph and has been assigned the DOI 10.5281/zenodo.18367132. The present contribution constitutes a compact, peer‑reviewed preview of a substantially larger body of work. The full monograph pages +200p in length has been deposited on this platform under the DOI 10.5281/zenodo.18284900 and will become publicly accessible once the review process is complete - very soon - so stay tuned! Readers are invited to consult this paper as a preview / teaser (DOI 10.5281/zenodo.18367132) for a concise overview, and to refer to the full monograph for exhaustive derivations, extensive numerical case studies, and the complete falsifier suite.Copyright (C): Miikka Kirjonen - All Rights Reserved - CC-BY-NC-ND