Technical Report v1.16: Generalizing the SSIM Masking Identity Beyond sigma_gt=0: A Necessary-and-Sufficient Contrast-Term Theorem and Its Real Occurrence via 8-Bit Quantization

Technical Report v1.11 proved that SSIM's structure term S is mathematically pinned to exactly 1 whenever sigma_gt=0, regardless of the reconstruction. Technical Report v1.15 extended this to SSIM's contrast term C, proving C=1 exactly whenever sigma_gt=sigma_notta, for any shared value, not only zero. This report completes that result in two ways. First, a previously-unwritten converse proof: substituting C=1 into its defining formula and rearranging yields (sigma_gt-sigma_notta)^2=0, a double root with no other solution, so C=1 is not merely correlated with sigma_gt=sigma_notta -- it is mathematically equivalent to it, making ssim_c_notta=1.0 a safe, exact search key even though sigma_notta is not itself a stored column. Second, a direct database search across all four models in this series finds that nonzero sigma_gt=sigma_notta ties occur at a real, quantified, non-negligible rate (0.019% to 0.342% of blocks), the large majority of which involve genuinely different image content that merely shares the same local variance by coincidence -- not pixel-identical blocks. The root cause is traced empirically to 8-bit pixel quantization: only 2,308,415 distinct sigma_gt values exist across 79,321,389 blocks per model, and 76.1% of blocks share their exact value with 100 or more others, making exact coincidental ties between unrelated blocks statistically unsurprising at this dataset's scale. The contribution is offered as a validated diagnostic method for checking whether this kind of coincidental sub-term tie occurs in a given dataset, not as a claim about how prevalent it is universally or a defect in SSIM, the comp_* feature framework, or 8-bit imaging itself. Every finding is computed against the same four independent, no-join, from-scratch verification databases, read-only, screen-recorded and hash-verified. The complete verification SQL, including a fully-expanded, copy-paste-ready 4-model script, is included to enable independent reproduction. Dataset and Mandatory Citation:- Source: NIH ChestX-ray8 (Hospital-scale chest x-ray database)- Citation: Wang, X., Peng, Y., Lu, L., Lu, Z., Bagheri, M., & Summers, R. M. (2017). "ChestX-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thoracic diseases." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 3462–3471. - Download: https://nihcc.app.box.com/v/ChestXray-NIHCC Contact : [email protected]