Datasets:
Hypotheses and unproved extensions
The model is a square-free scalar-twisted algebra with central invertible interchange coefficients, a top-degree detector, and constant-probe block protocols. The results do not classify arbitrary noncommutative inverse problems, general detectors, matrix-valued braiding, or physically constrained light transport.
The third-deficit classification requires m >= 6, independent edge parameters near the exterior point, and exactly the deficit r = m-3. Its proof works in |theta_ij| < 1/10 after q_ij = -exp(theta_ij). The reference scalar-certificate implementation uses the stricter exact real-rational domain |q_ij+1| < 1/20. The five-channel exception is real and exactly verified.
The classification is of the REDUCED analytic germ. It establishes the radical ideal as an intersection of linear primes. It does not show that the original measurement ideal is radical, classify embedded components, or discard nilpotent information in analytic target reconstruction. Pullback along a nonlinear parameter map can change that structure.
Scalar invisibility and full-state rank failure are different. The scalar germ has stars, triangles, and twins; the full-state failure locus is the larger union of double-star subspaces. Equal ranks on two strata do not imply identical geometry or identical lost coordinates.
The rigidity theorem concerns linear changes in logarithmic parameter coordinates. An analytic automorphism preserving the germ has a constrained derivative, but arbitrary higher-order automorphisms are not classified. Relative calibration means reconstructing coordinate geometry from the whole arrangement, not performing an unknown-noisy-parameter experiment for free.
The sharp synchronization count is a count of selected entire twin-component constraints. It is NOT a minimum number of measurements, detector pixels, time steps, or noiseless scalar samples. Learning one such component could require many measurements.
Analytic feedback is now addressed under finite complete-frame and joint-analyticity hypotheses. It cannot enlarge the exact target class beyond that complete frame. Adaptivity may nevertheless lower the number of samples actually used; discontinuous policies, singular reconstruction, nonlinear sensors, statistical priors, noise, and nonreset-only access require separate treatment.
The all-budget core rank equality is generic, not universal at every parameter point, and the stated formula requires r >= 1. At r=0 the detector rank is one. The code for general exact ranks is exponential in m. The O(m^4) scalar certificate count excludes coefficient bit complexity and is not a general ideal-membership algorithm.
The old elliptic singularity is known. The quantum complete intersection, q-factorials, graph decomposition, formal group law, local algebra, and exterior/Pfaffian identities have established antecedents. A targeted search did not establish global priority for the new integration or its theorems.
Written proofs, executable checks, and reproducible files are supplied. Independent peer review, proof-assistant formalization, exhaustive specialist literature review, experimentally realized optical components, and measured GPU advantages are not supplied. Deficits four and higher and the third-deficit scheme structure remain explicitly unproved here.