We identify the mathematical structure selected by a single invariance principle, Axiom A0, together with four constitutive constraints on any embedded physical observer: a finite-energy, finite-memory record-keeping system that updates a causally ordered internal state solely through local interactions at finite speed along an internally generated proper-time worldline. A0 asserts that physical reality consists only of structures invariant under physically inert transformations. From A0 and the observer constraints, the construction first obtains an operational quotient: physical states are equivalence classes of histories under all admissible observer-protocols. The finite-dimensional quantum reconstruction then proceeds under explicitly named regularity conditions R0-R4: finite randomized closure, countable protocol base, sectorwise continuous accessibility, operational isotropy, and operational local completeness. Under these conditions, the irreducible nonclassical sector is represented by complex Hilbert space together with U(1) phase, tensor-product composition, the Born rule, unitary dynamics, record-coherence complementarity, and local U(1) gauge form. The operational-scale analysis identifies the finite resources accessible to embedded observers. In particular, finite control energy imposes a Margolus-Levitin ceiling on distinguishability-changing operations, n_max(T,E_op)=2E_op T/(pi hbar), giving a concrete bound on protocol refinement. Model-dependent coherence, timing-noise, and thermodynamic estimates are separated from this theorem-level control bound; the proposed laboratory test is the scaling of the saturation pulse count against calibrated drive energy. Extending the same observer constraints to spacetime structure gives an operational causal order from finite ordered records and finite-speed locality. With standard continuum and causal-reconstruction imports, this leads to Lorentzian metric structure, the local-flatness content of the equivalence principle, and, through explicitly identified thermodynamic and geometric imports, Einstein’s field equations. In algebraic quantum field theory, assuming the cyclic separating vacuum as a structural input, the observer-accessible algebra is organized by modular theory and crossed-product constructions. In Rindler, de Sitter, and related horizon settings, this yields the type-II algebraic structure associated with finite generalized entropy and the Bekenstein bound. General curved-background extensions remain conjectural. The same type-II structure has been obtained independently in modern gravitational-algebra constructions. A final extension constrains local first-order relativistic matter form. Under the stated locality, covariance, and record-faithfulness conditions, the Clifford relation and Dirac form appear as the surviving first-quantized matter structure, while the full particle spectrum remains undetermined. The framework is not a theory of everything. It does not derive particle multiplicities, coupling constants, masses, gauge-group specifics, cosmological parameters, or initial conditions. Rather, it constrains the admissible mathematical form of physical law under the assumption that real observers are finite, embedded physical systems. All imported results are explicitly identified, and the epistemic status of each major step is recorded in the dependency graph.