Local phase information and common-field certificates for volume-constrained periodic heat conduction

This technical note studies the worst-outcome reduction of a directional effective-conductivity range after exact local phase labels are revealed in a finite periodic two-phase geometry class with independently supplied volume constraints. The posterior width is characterized by compatible geometry pairs. For a phase-interchange closed half-volume class, one primal-dual field pair shared across all compatible pairs and observation outcomes has optimized worst-pair bound V-R, where V=(a+b)/2 and R=2ab/(a+b). The same equality holds after diagonal pairs are removed whenever a distinct compatible pair remains. In a continuous two-dimensional four-square example, a row observation has true worst posterior width S-R, with S=sqrt(ab), giving a strict certificate gap V-S. The note also gives count-compatible support formulas and finite-field brackets: on a fixed finite class, a certified primal-dual gap at most epsilon for every geometry yields a guaranteed-shrinkage bracket of width at most 4 epsilon. Field construction, integration and optimization costs are additional. The results apply to positive scalar phase conductivities, periodic cell equations, the specified finite geometry classes and exact-label observations. This is the first public technical-note release, version v1.