On Computable Images and Films, With an Application to the Entscheidungsproblem

This paper develops a theory of computable images and computable lms analogous to Turing's 1936 theory of computable real numbers, in a layout that mirrors his section by section. The central methodological commitment is the adoption of an RGB colour-native alphabet ΣRGB = 0,. . . , 2553 for the universal machine, in place of the binary alphabet of the original construction. The standard alphabet equivalence theorem ensures the class of computable functions is unchanged; the cost model and the natural physical substratephotonic computationalign with the colour-native alphabet exactly. We dene computable image and lm schemes as functions uniformly computable in their resolution parameter, present a graded sequence of worked examples ending with the Mandelbrot scheme, and construct a universal image-generating machine in 44 states with ve named subroutines and an explicit transition table. We establish the visual analogues of Turing's diagonal and halting results: there exist denable but uncomputable images, the image-halting problem is undecidable, and the Kolmogorov complexity of image schemes is uncomputable. We engage with three post-1936 developmentscomputable analysis (Pour-ElRichards; Weihrauch) for the continuous-limit renement of the discrete model, the BernsteinVazirani equivalence bounding quantum computation to the same class, and Brattka's bound on analog and photonic computation under realistic noiseand provide a Big-O cost stratication across procedural, rendered, and neural-network image classes for classical, quantum, and photonic substrates. We recover Hilbert's Entscheidungsproblem as unsolvable by reduction from the image-halting problem, and hypothesise a visual Entscheidungsproblem for subsequent development. A nal section maps the engineering gap between the abstract universal machine and any physical instantiation under thermodynamic, information-theoretic, decoherence, and precision bounds, concluding that the colour-native alphabet is exceptionally well-aligned to the photonic substrate that most naturally realises universal visual computation. The paper is, in the U-Mentalism programme distinction made precise in the introduction, the foundational contribution in the U. M. 'O' register the mathematical and computability-theoretic substrate that legitimates the wider U. M. 'C' programme of cinematic supercomputation (patent PCT/PT2021/050014; Homem 2021) without exhausting it. Wider here is to be understood in the implementation register only the engineering, hardware, and architectural eort of which the patent is the realised artefact and not in the ontological register, where U. M. 'C' derives from U. M. 'O' as naturally as nature unfolds articiality as natural.