Hidden regenerative state in planarians: A geometric model of bioelectric memory using Tangential Action Spaces

Planarian fragments can regenerate with normal gross anatomy after a transient bioelectric perturbation yet display altered outcomes upon re-cutting, implying that regeneration can store a persistent hidden state. Here we formulate an open-path version of Tangential Action Spaces (TAS) for this setting. Regeneration after a given cut is represented as a prescribed coarse anatomical trajectory together with multiple physiological lifts in a higher-dimensional state space. A metric on physiological state space defines a baseline lift, an effective excess-cost functional, and a baseline-relative endpoint displacement that serves as written hidden regenerative state. Re-cutting converts this open-path construction into a challenge readout. Locally, the theory yields a cut-dependent memory co-metric that identifies latent directions that are easy, difficult, or inaccessible to rewrite. We show that this geometry is consistent with published observations of cryptic phenotypes, stable re-challenge ratios, and near-absorbing double-headed outcomes. A reduced rank-one latent-threshold realization fitted to published 8-OH immediate and re-challenge counts identifies a challenge-sensitive cryptic interval below the immediate double-headed threshold and predicts out-of-sample re-challenge penetrances near 15% for nigericin- and monensin-treated immediate single-headed survivors using only their immediate phenotype penetrances. As a mechanistic bridge, a local electrodiffusive in-silico example instantiates a local version of the physiological-state effort metric G. This metric defines the baseline lift and excess rewriting cost, in relative biophysical units, and yields explicit example local write geometry. An illustrative semimechanistic readout based on integrated wound-edge gap-junction contrast and Na/K-ATPase load reproduces the treated-family ordering and similar transfer predictions when the untreated baseline is softly anchored near zero. These quantitative layers are intended as proof-of-concept calibratability and mechanistic-grounding checks rather than full validation of the complete open-path model. The framework therefore turns cryptic regenerative memory into a geometric, costed, and experimentally testable object, yielding predictions about temporal-profile dependence, compensatory cancellation, sign-reversing controls, cut dependence, anisotropic rewriting, and multi-round accumulation of hidden regenerative state.