Slow Electromechanical Interference Links Division Geometry, Differentiation Bias, and Senescence-like Arrest

This work develops a biophysical theory in which a bioelectric field V ( x, t ) and a cortical stress field σ ( x, t ) are weakly and reciprocally coupled via an overdamped electromechanical coupler. We show that interference between two fast latent modes produces a measurable slow beat f _slow that acts as a tissue-level clock. By sampling the dynamics at “neutral moments”—recurring instants of phase symmetry—we derive a reduced even circle map in which healthy homeostasis corresponds to locking within a specific 2/21 Arnold tongue.

We then introduce a coarse-grained dual-field algebra that collapses the continuum description into three effective blocks (Γ, a, b ) capturing net electromechanical gain and dissipation. In this algebraic picture, ionic and rheological perturbations are represented as smooth deformations of the parametrization space (Γ, a, b ), while the clock variables (Ω, f _slow , K ₂ ) provide experimentally accessible coordinates on those deformations. This construction offers a concrete bridge between molecular-scale regulation and tissue-level mechanics, connecting subcellular control to the emergent seconds–minutes slow clock that constrains division geometry.

Evaluating the membrane-potential profile at neutral moments defines a neutral charge-asymmetry observable Δ Q _n that quantifies left–right voltage imbalance at the division axis and links the slow-phase map to directly measurable bioelectric patterns. The same neutral-map construction admits a rotational interpretation in terms of circle maps and slow precession of the locked orbit: small detunings δ from the ideal 2/21 plateau generate a hierarchy of time scales and predict a scaling law T _dev ~ 1/( f _slow | δ |) relating the fast electromechanical beat to developmental timing.

The theory yields four falsifiable predictions. (P1) Homeostatic epithelia exhibit a narrow shared slow-band peak in voltage and stress with high coherence. (P2) The effective forcing and coupling (Ω, K ₂ ), derived from physical parameters, reside within the 2/21 tongue while avoiding broad low-order resonances. (P3) A weak, frequencyspecific drive at f _slow (phase-targeted entrainment) selectively increases coherence and reduces spindle-angle dispersion in unlocked states, providing a physical basis for bioelectric modulation of regenerative dynamics. (P4) Across conditions with comparable f _slow , the number of neutral compensation cycles required to complete a phenotypic transition scales inversely with the detuning | δ |, linking slow precession of the neutral map to macroscopic developmental time.

Ultimately, this framework treats cancer-like instability and senescence-like arrest not as independent pathologies, but as opposite failures of navigation in a single underlying electromechanical cycle, from persistent unlocking to rigid oversynchronization.

Highlights

• Proposes an overdamped double-oscillator model of tissue electromechanics in which two fast latent modes generate a slow beat ( f _slow ).

• Links the slow beat to an even circle map via neutral moments, identifying a specific 2/21 Arnold tongue that governs stable spindle orientation.

• Introduces a quantitative dual-field algebra in which ionic (pump-like) and rheological (stiffness-like) perturbations act as deformations of coarse-grained blocks (Γ, a, b ), predicting matched shifts in Ω and f _slow .

• Defines a neutral charge-asymmetry observable Δ Q _n at neutral moments, recasting the locking scenarios (P1–P3) as constraints on left–right membrane-potential imbalance that can be computed from existing bioelectric models.

• Relates the slow beat to capture kinetics T _lock , linking fast carrier interference to mitotic (minute-scale) timing through progressive synchronization.

• Derives a scaling law (P4) in which the number of neutral compensation cycles required for a phenotypic transition scales inversely with the phase detuning | δ |, naturally generating a hierarchy of time scales from minutes to days.

• Models “proliferative unlocking” and “senescent overlock” as opposite dynamical failures of the dual-field clock (drift vs. rigidity), providing a unified view of cancer instability and aging.

• Validates a protocol for phase-targeted entrainment, predicting that a weak drive at f _slow selectively recovers coherence and reduces spindle-angle dispersion in unlocked states.