Operational Utility and Multiscale Invariance of Phase Calculus: A Discrete Foundational Grammar for Dynamics

Phase Calculus is presented in its current stack as a single lifted-object formalism rather than as a loose collection of phase pictures, arithmetic tricks, or downstream branch-specific rules. The native state is the carried lifted object Xi = (b, A, q, theta, kappa, c), q = (u, v), c = evolved by one parent grammar and then read through branch-specific projectors. This demonstration paper does not re-prove the long merged formalisation. Its purpose is narrower: to show, in one readable document, how the same lifted object supports the operator core of Phase Calculus, the selector-closed executable macro-calculus, the completion/shadow branch, the quotient-descent interface to the physics stack, and the four-filter hierarchy including the biological readout. The paper has three central results. First, it restates the single spine of the formalism: primitive roll, lifted state, operator alphabet Q, B, L, selector-closed macro-calculus R, S, T, exact quotient criterion, and arithmetic-completion bridge to the canonical anchor (55, 89) with carried coefficients 1/24 per edge and 1/12 two-sided. Second, it records the computational evidence from the four-filter package: green, blue, red, and yellow are faithful covariant readouts of the same lifted state under the same operator grammar, with exact-zero primary gate errors, machine-precision recovery errors, and sharp negative controls when the state is stripped to product-only or loses its sheet index. Third, it shows how the yellow filter closes two descents at once: the CF01 quantum-geometric-tensor bridge and a native biological scaffold. In this v2 paper the biological count structure is not imported. From the quarter alphabet generated by Q, the three-slot selector block, and the normalized edge residual 24*Eₑdge = 1, the paper derives internally the weight budget (1, 2, 1, 1), the total scaffold size 240, the base-centered counts (56, 72, 56, 56), and the host-indexed bifurcations 240 = (128, 112) -> ( (72, 56), (56, 56) ). The unification claim is therefore precise and limited. One singular lifted object is taken as the native source of the current Phase Calculus hierarchy and of the physics branches that descend from it by quotient projection, while also admitting a biological projection whose scaffold counts are now derived internally and only then compared with the external codon: anticodon / E8 report. The closing section states explicitly what is closed, what remains imported, and what is still not unique.