A First-Principles Derivation of Absolute Physical Scale from an Eleven-Dimensional Covariant Master Action

An eleven-dimensional covariant theory may determine dimensionless geometry, spectra and response operators without yet specifying the physical magnitude of every observable. The companion derivations of physical time and the quantum of action provide, on their declared physical components, a positive temporal section and a positive action section selected by the parent dynamics. This paper proves how observable-specific responses of the same parent action acquire physical dimensions and how that construction changes when the finite operational support changes between event epochs. All interacting action terms and constraints are assembled before a single stationary reduction. An observable is specified by a variational port already present in the unreduced parent, an ordered derivative, a physical quotient, a normalization and a readout map. On an admissible finite-capacity stratum at epoch \(k\) and in sector \(s\), a quantity of grade \([O]=M^{p}L^{q}T^{r}\) has the unique realization \[O_{\mathrm{phys}}^{(k,s)}=C_{O}^{(k,s)}\mathfrak{a}_{\star,k}^{p}c_{k}^{\,q-2p}\tau_{\star,k}^{\,r-p+q}.\] The coefficient and the physical sections must be evaluated on the same stationary solution and support, or be related by an explicit structure-preserving transport. Complete direct and factorized expressions are alternative representations rather than multiplicative corrections. A reference-unit atlas represents an already selected quantity but does not by itself generate an object-specific chart component. When the sector and reference-clock actions are block separable and no shared phase-sensitive owner is present, the mixed derivative vanishes and one metrological degree of freedom remains. A declared canonical chart can close the numerical coordinates transparently; that closure is distinct from a numerical prediction based on independently determined physical input. Operational capacity and the sector-dependent maximum regulator level are dimensionless domain data. They can alter the coefficient by changing the admissible support and operator domain, but they do not add a fourth dimensional generator. Fixed-rank continuation is therefore separated from capacity birth, and transported historical values are distinguished from newly stationary responses on an enlarged support. Applications include masses and rest energies, complex poles, widths and lifetimes, gravitational and vacuum observables, the cosmological constant, dimensionless gauge couplings and finite-domain Yang–Mills gap bounds. The theory retains shared-input covariance, distinguishes metrological closure from independent validations and states explicit falsifiers at every interface. Parent dynamics selects physical scale; metrology names its numerical coordinate.