This paper sets out the Semantic Interface Model, the classical core under two layers of the Möbius Project: MUSE 2.0, a quantitative reading of "meaning-bearing flow", and TCR 3.0, which keeps from the Theory of Cognitive Relativity the idea that geometry is relative to the observer and defines that geometry, for regular interpreters, as the Fisher geometry of their parameter spaces. A perceptual or communicative channel is modeled as a Markov chain Y → S → X → Z from a task variable Y through a physical state S and a sensor signal X to an internal representation Z. Its semantic energy is E_sem := I(Y; Z), Shannon mutual information in bits or nats. Nothing in this is new physics, and the quantity itself is not new: in the information-bottleneck literature it is the relevant information a representation keeps about a target. Three layers are built on it. The scalar semantic aperture is the information an interpreter's output retains about the task, divided by a reference value; with the interface's own semantic energy as the reference, it is the fraction retained. It lies in [0, 1] under the two normalizations recommended here and can exceed 1 under a third, which is flagged. The tensorial aperture is a task-weighted Fisher information metric on the interpreter's parameter space. It measures how sensitive the interpreter's predictions are to its parameters, not how much semantic energy it uses, and no inequality links it to the scalar form. Cognitive curvature, the term by which TCR 3.0 meets information geometry, is the curvature of the Levi-Civita connection of that metric — not the metric itself, which the 2025 draft conflated with curvature. Worked examples make this concrete. A Gaussian location-scale interpreter, and a linear-Gaussian interpreter reading any representation with finite, nonzero second moment, both have constant curvature −1/2, the classical curvature of the normal family; a two-parameter logistic interpreter is flat when the representation takes two values and, at a computed point, curved when it takes three; and a categorical interpreter over three outcomes has constant positive curvature +1/4 even though it ignores the representation. Curvature in this sense is a property of an interpreter reading a representation, not of the true relation between task and representation, and not of meaning at large. The paper also gives exact semantic energies for two toy interfaces, a growth hypothesis relating semantic energy to physical power and the conditions under which it is testable, a CMOS imaging interface as the device example, and a map of where these definitions sit in the published line. A script checks the worked examples, the toy-channel values and the aperture bounds, the last by a randomized property test. No theorem here is new, and no measurement on a deployed system is reported. The verification script (Appendix A, AGPL-3.0-or-later) needs Python 3 and SymPy and checks the worked examples, the toy-channel values and the aperture bounds; it exits non-zero if any check fails. AI disclosure: the December 2025 working draft was written with GPT-5.1 Thinking (OpenAI) as co-observer. The September 2026 revision (corrections, worked examples, verification script, positioning and prose) was generated by Claude Opus 5.5 (Anthropic) under the author's direction. Its successive drafts were checked in four rounds of adversarial refutation, each round covering mathematics, consistency and scope: eleven passes in all, because the fourth round combined mathematics and consistency in one pass. Each pass was run by a separate Claude Opus 5.5 agent with its own context — the same model that generated the revision. The fourth-round draft was also given an editorial read for internal contradictions by a model of a different family, OpenAI's gpt-5.6-sol run through the Codex CLI; all eleven of its items, eight marked must-fix, were checked against the text and corrected. A fifth round, a single confirmation pass by a separate Claude Opus 5.5 agent, checked the edits made in response to the fourth round and the editorial read. A sixth round, another single confirmation pass by a separate Claude Opus 5.5 agent, checked the corrections the fifth round prompted and a positive-curvature example added afterwards; the corrections it prompted were not themselves re-reviewed. The verification script was re-run on the final text. No human referee has reviewed the text. The author reviewed the deposited text and takes responsibility for it. Working method only; the registered author is the human author alone.
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Toeda Taiko (2026) studied this question.