Temporal continuity and bioelectric pattern recovery: effective signal geometry and conditional microstructural coupling

This paper replaces my August 2025 formulation of bioelectric morphogenesis and quantum coherent microstructures. The earlier paper identified a legitimate cross-scale problem, but it joined a regulated temporal vector, tissue voltage and a coherent microtubular field before the biological and mathematical bridges between them had been earned. Version 2.0 keeps the continuity first foundation, removes the unsupported unification claim and rebuilds the work around measurable bioelectric pattern recovery.

I begin with a conserved temporal current and keep it separate from the biological equations. Tissue voltage is governed by membrane charge balance, ionic response and intercellular current transport. A target voltage pattern is defined as a stationary solution of that system rather than as an assumed morphological setpoint.

The first central result is a pattern recovery theorem. Linearisation about a target state produces a conductivity and setpoint operator. When membrane capacitance is positive, tissue conductivity is positive definite and the spatial operator has a positive spectral gap on the physical voltage space, the recovery energy decreases monotonically. The result applies in both a cell network description and its justified continuum limit. Recovery is therefore controlled by the complete tissue operator and its boundary conditions, not by one local damping coefficient.

The ordinary bioelectric branch is diffusive and does not define a Lorentzian signal geometry. A finite propagation branch appears only when the intercellular current has a measured relaxation time. Using a Maxwell Cattaneo current law gives a corrected telegraph equation containing the full ionic contribution to the damping. Its positive quadratic energy yields a second stability theorem.

The second central result is an effective signal metric theorem. On a two dimensional tissue sheet, the normalised principal action defines a Lorentzian densitised inverse tensor whenever the current relaxation time, capacitance and conductivity are positive. In three dimensional sheet spacetime, the determinant of that tensor fixes the volume factor and therefore the complete effective metric seen by the voltage perturbation. This is a biological signal metric. It is not the Einstein metric. In the ordinary diffusion limit the determinant vanishes, the principal tensor becomes degenerate and no Lorentzian metric is claimed.

The paper also gives a controlled reconstruction result. A direct current relaxation measurement fixes the current relaxation time. Two resolved voltage mode pairs at distinct spatial wave numbers, together with an independent capacitance measurement, then recover the tissue conductivity and local setpoint stiffness. The measured modal sum supplies an independent consistency test. Without the direct current measurement the modal inverse is generally two valued, and without capacitance only parameter ratios are recovered. These obstructions are stated explicitly. Observation covariance is propagated through the inverse, and a singular value recoverability ratio distinguishes formal identifiability from a practically useful reconstruction.

The latest Chronoflux order of Admissibility, Recoverability and Persistence is applied operationally. Admissibility fixes positivity, resolution and branch conditions. Recoverability requires an injective observation map modulo genuine reference freedoms. Persistence requires the same admissible and well-conditioned branch to survive while the measured pattern remains within tolerance.

Temporal continuity may modulate capacitance, conductivity, current relaxation or ionic source terms through declared response coefficients. Those coefficients are not fixed by covariance and may all be zero. Any nonzero claim must survive temperature, ion concentration, mechanics, tissue remodelling and other biological nuisance models.

Microtubular electrical behaviour is retained only as a conditional classical microstructure sector. A driven nonlinear electrical mode can support oscillation, resonance or memristive response without quantum coherence. Cross-scale coupling is therefore tested through the observability rank of the joint tissue and microstructure system. The earlier helicity-like invariant and coherence time formula are not retained.

Quantum promotion requires a separately identified Hamiltonian mode, failure of a controlled classical stochastic model, direct interference or equivalent coherence evidence, and a measured environmental decoherence law. Electrical oscillation alone does not cross that gate.