Abstract
A finite amplitude dynamics, a configuration-space propagator, and a mechanical action are different mathematical objects. We give a conditional one-dimensional bridge between them while retaining the assumptions that make each identification possible. Common-additive-shift invariance selects a relative two-mode quadratic energy; a supplied first-order kinetic term and action scale turn its graph Laplacian into reciprocal norm-preserving dynamics. The resulting amplitude action vanishes on solutions and therefore cannot simply be identified with the configuration action. A convergent ordered-jump expansion retains diagonal dwell phases, hopping factors, and time-simplex measure. On a supplied spatial lattice, inverse-square scaling gives second-order fixed-band convergence and strong convergence for each fixed square-integrable state, but not all-mode real-time operator-norm convergence. The raw return-kernel density has two distinct mesh-subsequence limits. A declared Gaussian resolution instead gives an explicit uniform lattice-to-continuum bound, separate finite-volume control, and locally uniform removal of resolution at nonzero time. With independently declared mass and action calibration, the limit has the free mechanical action and exact finite-slice normalization. For smooth positive variable couplings, midpoint edges and uniform site measure select a divergence-form operator. Different measures or orderings retain the same classical principal symbol while changing lower-order terms. The Liouville coordinate and amplitude transformation fixes a local correction, endpoint factors, and the internal measure; these must be carried together. Smooth-orbit lattice and state-level time-slicing estimates give controlled comparisons without asserting a raw variable-coefficient kernel limit. The contribution is the organized conditional bridge and its obstructions, not the invention of the standard component methods or a derivation of geometry, the action-scale value, physical couplings, or empirical probability from primitive complementarity.
Where it sits in the release
The histories paper of the branch. The contribution is the organized conditional bridge and its obstructions — not a derivation of geometry, the action-scale value, physical couplings, or empirical probability.
Download paper (Zenodo) — CC-BY-4.0.