Ö = C(E, D) — a structural constant with dual invariance under identity-preserving conditions

This document presents Ö = C(E, D) under identity-preserving conditions as a new structural constant in mathematics, identified by a dual invariance: invariance under permutations of its domain in the sense of Tarski (1986), and invariance under the choice of substrate in which the structure is realised. The combination of these two invariances has not, to the author's knowledge, been previously identified as a single mathematical object.Ö is identified, not derived, in the manner of mathematical constants such as π and e. The identification rests on the necessity of four components — existence E, duality D, composition C, and an identity-preserving condition — together with the dual invariance. A proof of Tarski-invariance is given in Section 2, and a formal characterisation of substrate-invariance is given in the same section.The operational calculus that realises Ö is the pair Phidus = (𝒜, σ), where 𝒜 is a cartesian allegory in the sense of Freyd and Scedrov (1990) and σ is a fully faithful functor from 𝒜 to a binary substrate. The cartesian allegory structure of 𝒜, the fully faithful property of σ, and the Yoneda lemma (Yoneda, 1954) — each treated in full in Section 4 — together fix the operational identity of cells in any Phidus realisation.Sections 6.1 through 6.10 derive operational properties of Phidus from the dual-invariance claim. These properties — omni-agnosticity across domains, multimodal computation in parallel, scale-independence, frequency-independence, append-only causality, dimensionless boundary, structural verifiability without cryptography, intrinsic auditability of AI computation, and trans-disciplinary appearance — are not separable architectural features. Each is derived from the same underlying invariance.Section 7 specifies the first concrete realisation, Phidus 1, in full: its four root requirements R1 through R4, its programming language with twenty-six operations, its hash primitive Keccak used at the system boundary, its TLV frame format with ten core forms, its memory model, its contact-point typology, its information-theoretic confidentiality construction PL3 through Shamir-style splitting, its small-world topology, its deployment tiers, its initial European infrastructure on four providers, its product architecture, and its EP-theorem with five lemmas.The position of this document is that Phidus is a new class of computing machine, motivated by and operationalising the new mathematical theory of Ö. The originality is in the sense of prior publication; establishment in the sense of peer review and acceptance by a research community is a separate matter and remains open.