C0: Contrast Calculus — Reconstruction. Recovery from Multi-AI Conversation Archives

This document reconstructs the mathematics and philosophy of C0 (Contrast Calculus), a relations-first post-object mathematical framework where differences are primary to things. The original development occurred across multiple AI systems (ChatGPT-5 High, Gemini, DeepSeek, Grok, Claude) during mid–late 2025. The ChatGPT sessions containing the deepest mathematical formalization were lost when the LLM Arena context windows closed. This reconstruction draws on surviving material in Claude conversation archives, context transfer packages between AI systems, and partial transcripts. C0 defines six primitives (contrast, void, interface, flip-site, weave, exposure) and three local moves (flip, annihilation, slide) operating under boundary-only semantics: the interior of a weave is never observed; only residual exposure at interfaces is registered. Four fundamental lemmas (slide-invariance, local consistency, observational determinism, no-clone) and four purity invariants (no creation ex nihilo, termination, confluence, interface purity) govern the system. The document preserves the C0/E1 firewall: C0 (core ontology) is clearly separated from E1 interpretations in traditional mathematical language. E1 can read C0 but never writes to C0. Canonical patterns (XOR/Parity, Coincidence, Any-pass), the EBNF minimal syntax, the pre-symbolic practice layer C−1, and the meta-theory of cognition emergence (Δ→∼→‡) are included. Extensions C0-ext-superpose and C0-ext-persist are documented in their partially recovered state. The E1 axiomatic framework (A0–A7 for pointed fusion categories over Z₂×Z₂) and the Python implementation (contrast-calculus v1.0.3 on PyPI) are recorded as frozen interpretive artifacts. This document serves as the foundational reference for the STKWC programme's contrast-native constructions, including Document F (renormalization of topological knowledge) and C0-Crypt (contrast-native cryptography). Additional notes Reconstruction: March 2026. Original development: August–December 2025. Material originally developed with ChatGPT-5 High, Gemini, DeepSeek, Grok, Claude. Lost ChatGPT sessions partially reconstructed from context transfers. Part of the STKWC preprint stack.