Character-Theoretic Effective Rank: From Schur's Lemma to a Stereoscopic Measurement Apparatus on Continuous Hamiltonian Substrates

We isolate a representation-theoretic mechanism — the Schur-forced effective-rank ceiling — that constrains the covariance spectrum of any detailed-balance Markov generator on a state space carrying a finite-group action. The mechanism is sharpened to a closed-form sampling theorem, extended to a number-theoretic spectral law on coprime-residue lattices, and shown to extend to a continuous Hamiltonian substrate where the same character-theoretic mathematics is realized as a stereoscopic measurement instrument. Main results: 1. Character-Theoretic Effective Rank Bound. The covariance of a G-invariant observable of a G-equivariant detailed-balance Markov generator has at most the sum-of-multiplicities distinct eigenvalues, each appearing with Schur-forced multiplicity equal to the corresponding irrep dimension. 2. Wishart-Nyquist Sampling Theorem. Closed-form lower bound on the effective sample size required for the Schur-forced degeneracies to emerge above the empirical noise floor. 3. Triangle-Wave Overtone Theorem. On coprime-residue lattices for squarefree m, the commutator between the natural tent distance and the multiplicative involution has exact parity-chiral degeneracy at every finite m, fundamental amplitude m times phi (m) divided by pi-squared, and rank-4 block ratios (2i-1) ² over (2i+1) ² on odd harmonics. The boundary block holds asymptotic L-squared fraction 8 over pi-squared; the bulk overtone tower holds the residual. 4. Universal Coupling Invariant. Two independent character-theoretic derivations agree on the dimensionless coupling kappa-squared equals two-thirds, promoting square-root of two-thirds from a constant of one paper to a candidate invariant of the entire operator class. 5. A-Optimality of Texture and the Heisenberg Obstruction. On the rank-4 subspace of a continuous Hamiltonian substrate, the harmonic-mean texture functional attains its unique maximum 2 over square-root of K at the equal-tension barycenter; for two irreps the ceiling is square-root of two. Sharp localization at the barycenter incurs a Fourier-dual cost in the conjugate semantic domain, converting the texture ceiling into an extrinsic-curvature measurement program rather than a quality gate. Empirical verification spans the Fancyland Coliseum energy-gauge accumulator (finite three-player interacting-particle system at fifty thousand matches per cell), the prime gas at base primorial six, coprime-residue lattices up to m equals thirty thousand thirty, and a 768-dimensional semantic embedding manifold of 353 entities. A fifth member of the operator class — an anonymous seven-dimensional detailed-balance Markov substrate at Fancyland LLC — is verified internally with substrate identity, dynamics, and group-action structure reserved as trade secrets. The mathematics underlies a trilogy of US provisional patent applications protecting the corresponding apparatus layers: Arithmetic Qubit (US Provisional 64/031, 440), Holographic Eigen-Solver (US Provisional 64/033, 689), and Tensegrity Interferometer (US Provisional 64/048, 617). Mathematics published openly here; engineering apparatus reserved.