Technical Report v1.18: Completing the SSIM Sub-Term Picture: A Necessary-and-Sufficient Luminance-Term Theorem and Its Real Occurrence via 8-Bit Quantization

SSIM's structure term S and contrast term C have each been shown, in this series, to lock to exactly 1 under an identifiable condition regardless of image content: S=1 whenever sigma_gt=0 (Technical Report v1.11), and C=1 whenever sigma_gt=sigma_notta, for any shared value (Technical Report v1.15), with a real, quantified, coincidental occurrence rate traced to 8-bit pixel quantization (Technical Report v1.16). This report completes the picture with SSIM's third and final multiplicative term, luminance L. A previously-unwritten proof, mirroring the contrast-term derivation exactly with C1 in place of C2, shows L=(2*mu_gt*mu_notta+C1)/(mu_gt^2+mu_notta^2+C1) equals exactly 1 if and only if mu_gt=mu_notta -- a mathematical equivalence, not a correlation, making ssim_l_notta=1.0 a safe, exact search key. Searching all four models in this series for real mu_gt=mu_notta ties at mu_gt>0 finds them at rates from 0.218% to 6.512% of blocks -- an order of magnitude wider spread than the contrast-term case -- the large majority of which (up to 99.97% for Upscayl High-Fidelity) involve genuinely different content that merely shares the same mean brightness. mu_gt is shown to be even more sharply quantized than sigma_gt: only 68,649 distinct values exist across 80,457,746 blocks, and this GT-only measurement is confirmed, empirically rather than by assumption, to be identical between two independently-queried model databases. The report also surfaces a finding the contrast-term work did not: for two of the four models, concrete blocks exist where L=1 exactly yet the aggregate SSIM score is negative (as low as -0.0854), demonstrating directly that a coincidental tie in one sub-term provides no protection for the aggregate score when the remaining terms are simultaneously degraded. Finally, the report closes a further question: whether all three SSIM sub-terms (L, C, S) can coincidentally align to exactly 1 simultaneously for genuinely different content. Since each term is individually bounded above by exactly 1 and strictly positive, their product can equal exactly 1 only if all three do -- making this population identical to the already-established Pixel Identity Theorem's SSIM=1.0 population, which forces pixel-identical reconstruction. A direct four-model database check confirms this: of 4,734,946 blocks with SSIM=1.0 exactly, all but 1,442 (a SwinIR-only floating-point rounding artifact at machine-epsilon scale, diagnosed and confirmed harmless) also had all three terms bit-exact at 1.0 -- so a coincidental joint alignment for different content is not merely unobserved, it is algebraically impossible. As with the contrast-term report, the contribution is offered as a validated diagnostic method, not a claim about universal prevalence. 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 fully-expanded, copy-paste-ready 4-model scripts, 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]