Lossless Retention and the Physical Meaning of a Comparison

Abstract

What must persist when a physical system retains a comparison? Exact recovery of its inputs, preservation of numerical order, and conservation of energy impose different conditions. We exhibit a reversible count update with a lossless scalar reading whose order reverses, and give an energy-conserving implementation with an explicit excitation reserve. An elementary additive boundedness theorem identifies the stronger continuation condition that forces a common scale factor. An additive two-step join then selects the golden ratio. For a separate seven-clock quantum construction, complete energy continues to report an old comparison, while captured record energy has rank two on its source space. An explicit full-field bound excludes its identification with one squared linear amplitude. These results isolate the physical identification needed to connect retained information, an additive amount, and successive recognition events. The formation Hamiltonian and preparation are supplied; their selection as the physical law remains open. The captured-energy result holds for dimensionless time 0 < u <= 10^-14 and proves exact nonidentity on that interval. Its rank obstruction concerns a common multiplicative gain without an additive baseline; an affine baseline-plus-amplitude representation is not excluded. The source archive includes complete LaTeX, exact verification scripts, results and a hash manifest. Fresh verification passed 30 scalar/2-by-2 checks and 91 finite-clock checks. The full-field bounds are analytical; no photon-number cutoff or trajectory simulation is used.