Question, mathematical claim, and source discipline

The question is whether several constructions called a “clock”, “time”, or “lapse” can be used as the same functional. The answer requires more than a shared word. It requires compatible configuration spaces, prescriptions, normalizations, and dependence on the declared data.

The comparison is motivated by Cheung's Landmark VI, VIII-A, VIII-B, X-A, and Em02-E CheungVI,CheungVIIIA,CheungVIIIB,CheungXA,CheungEm02E. The supplied records separate their classical constructions from their remaining quantum and dynamical obligations. We do not use a documentary separation edge as a mathematical premise. Instead, the definitions needed for the present calculations are written below, and the separation statements are proved directly.

The general relational-coordinate framework is developed in BFHPR,Frob. Hyperbolic coordinate prescriptions and slice-elliptic prescriptions are different choices within that framework. Their equations and boundary data form part of the observable definition.

correction to an overstrong reading. “Not the same functional” does not mean that the numerical values must differ at every configuration or point. Nor does it mean that no future comparison map could relate the constructions. The claim proved here is that they are not interchangeable under the declarations in this paper. Where two expressions can be compared on a common domain, we give a detecting variation or a counterexample to the proposed identification.

correction to an underdetermined reading. The supplied VI excerpts state that a harmonic-coordinate Cauchy prescription is fixed, but do not reproduce all of its defining equations and initial data. Consequently, Section sec:harmonic declares an explicit finite-time harmonic prescription for the mathematical comparison. It does not silently authenticate every detail of the registered VI prescription. A VI instance can use the results only after its actual prescription has been matched to the displayed one. This qualification prevents a separation paper from becoming an unproved identification paper.

domain. Unless explicitly stated otherwise, perturbations below are off shell. They need not solve the coupled Einstein–scalar equations. A counterexample on this kinematic domain is not automatically a counterexample on a smaller solution space.

boundary. The paper does not establish quantum evolution without an external time. It constructs no interacting quantum product, state, field-theoretic instrument, or dynamical continuum limit.