Optimal Passive Masks for Thermal Source Reconstruction

How much detected thermal light is needed to identify several emitting sites? This working paper studies binary sources on a known grid of addressable source-image modes, using one fixed passive optical mask, photon counting, and a predetermined observation length. It distinguishes the original one-reference apparatus, complete Fourier reference collection, and a separate apparatus with one added vacuum input. All recorded photons count toward the measurement cost. The central result establishes the optimal quadratic detected-photon order at unit brightness for every fixed support limit on every grid, without geometric restrictions or primality assumptions. A reciprocal transmission mask attains this order using target counts alone, while recorded reference photons remain included in the cost. The proof reconstructs short cyclic arcs from two rational-probe values, then uses contour integration to isolate entire clusters of poles. This gives uniform quantitative separation without requiring a fixed minimum distance between individual poles. A bounded candidate list removes the grid-size logarithm from the final comparisons. On prime grids, a shorter finite-field argument recovers the support from two integer edge profiles. The quadratic law also holds uniformly over every nonempty reference subset when known brightness remains in a fixed interval bounded away from zero. The contour argument extends to a general stability theorem for rational interaction observables: under explicit locality and conditioning assumptions, quantitative distinguishability within clusters yields quantitative separation of the global polynomial data. For irregular registered dictionaries, this gives a separation margin proportional to the minimum site spacing, with an optimal spacing exponent. This extension assumes exactly resolved distance channels; their optical acquisition and robustness to unknown position errors require further analysis. Explicit dependence on the support limit also yields a growing-support regime: the photon cost remains \(n²⁺ᵒ⁽¹⁾log(1/δ)\) whenever \(s²log(2s)=o(log n)\). The accompanying computational bound is stated in real arithmetic, and the dependence on sparsity is deliberately conservative. The manuscript retains explicit ramp-mask constructions for every configuration of at most four emitters, for short-span sources, and for sources with distinct cyclic pair distances. The last class has no span restriction and contains an asymptotically overwhelming fraction of fixed-size supports. For four emitters, ordinary-correlation rigidity and cyclic separation together distinguish reflected configurations. An explicit passive mask also attains the exact optimal Fourier-intensity separation under a prescribed transmission floor. In both the original and complete-reference experiments, sufficiently faint light produces a different law: at fixed sparsity, the optimal photon order grows as the grid size raised to twice the support limit. For at most two emitters, the paper determines the optimal detected-photon order throughout the known-brightness range, from arbitrarily faint light through unit brightness, on odd grids in the original apparatus and on every grid with complete reference collection. The result includes the dependence on grid size, brightness, and error probability, including an additional logarithmic information cost between the faint and bright limits. At fixed confidence, the quadratic regime begins at brightness of order \((log n)/n²\). A mask combining an index ramp with a quadratic-residue code attains the sharp order. On even physical grids, one known vacuum input and odd-dimensional Fourier optics give the same law; omitting one target output preserves the detector count. This is a distinct apparatus. For the specified two-scale transmissions on an infinite even-grid family, an exact lower bound using the complete count record proves that changing phases or decoders alone cannot remove their intermediate-brightness limitation. A collected-light model makes the physical scope explicit. For a known collection operator, binary-source identifiability is equivalent to injectivity of the induced thermal covariance map. Covariance collisions give identical quantum states and therefore prevent discrimination by any subsequent receiver. A passive multiport establishes population identifiability when the map is injective. The paper also derives the exact throughput for uniform passive mode separation from the smallest Gram eigenvalue and accounts separately for detected and incident photons. These results connect the assumed mode access to a specified collection model; they do not establish a general quantum-optimal receiver or a complete astronomical implementation. Reference excess variance supplies a normalization that identifies the spatial source despite an unknown finite common-profile thermal-mode spectrum and independent unknown Poisson backgrounds. The finite-sampling guarantee includes the empty source under explicit mean and nonzero-signal bounds. Population identifiability of the mode spectrum is distinguished from stable estimation. When background means are known, the arm means recover source mass and improve a guaranteed scale-free precision radius. Further results include exact thermal count variances, confidence exponents, calibration bounds, a photon penalty for discarding detector labels, and an exact target-only localization obstruction when singleton brightness is unrestricted and unknown. The results have analytical proofs, supported by 34 reproducible finite and symbolic verification suites. These checks provide corroboration, not experimental validation or external peer review. The unrestricted fixed-sparsity unit-brightness problem is resolved on all grids. Remaining questions include the optimized even-grid intermediate-brightness transition in the original apparatus, transitions for three or more emitters, sharp growing-support rates, sharp reference-coverage dependence at vanishing brightness, and optimal covariance-only scale-free precision. Continuous off-grid recovery requires a tolerance-based criterion and additional stability analysis. The work makes no literature-priority claim.