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The Substrate Ledger Computer Exact Exchange and Symbolic Matter Source-bound operators, Bethe closure, composite channels, and reusable many-body spectra

Sean Brady

Published Sep 21, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.12214.66888

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Quantum information theory (QIT) characterizes the capabilities and fundamental limits of quantum information processing, underpinning quantum communication, computation, and error correction. Formalizing its coding theorems requires connecting finite-block protocols, analytic inequalities, and asymptotic limits within a unified machine-checked framework. Existing developments, however, lack a reusable operational layer that defines codes, error criteria, achievable rates, and capacities independently of their information-theoretic characterizations. In this work, we present LeanQIT, a Lean 4 library for finite-dimensional QIT. It provides composable, kernel-checked interfaces for quantum states and channels, source and channel codes, finite-block performance criteria, hypothesis testing, one-shot quantities, and asymptotic rate constructions. Using this infrastructure, we formalize Schumacher's quantum source-coding theorem, the Holevo--Schumacher--Westmoreland classical-capacity theorem, and the entanglement-assisted classical-capacity theorem together with its strong converse. By separating operational definitions from analytic characterizations and exposing reusable achievability, converse, and asymptotic components, Lean-QIT provides a machine-readable foundation for formal QIT and a compositional knowledge substrate for emerging AI-assisted formalization, automated proof search, and agentic reasoning in quantum information and computation.
This paper provides the first complete, self-contained trace of a single measurement through all twelve stops of the GCD/UMCP validation spine. The worked example uses three identical voltage readings—the simplest possible input—to demonstrate the full protocol: declaring the axiom, freezing the contract, embedding raw data into bounded trace vectors, computing the six kernel invariants (F, ω, S, C, κ, IC), reconciling the integrity ledger via seam budget closure, classifying the regime through four conjunctive frozen gates, and deriving a three-valued CONFORMANT / NONCONFORMANT / NON_EVALUABLE verdict. The twelve stops are organized into the five stages of the fixed discourse spine: Contract (axiom + frozen parameters), Canon (channel choice + embedding + kernel), Closures (ledger debits and credits), Integrity Ledger (seam residual + regime classification), and Stance (verdict derivation + casepack packaging + operational execution via CLI, Python API, and REST endpoint). Two appendices contrast the homogeneous trace with heterogeneous counterexamples. The geometric slaughter appendix demonstrates how a single near-zero channel destroys the composite integrity IC (geometric mean) while leaving fidelity F (arithmetic mean) apparently healthy—revealing the heterogeneity gap Δ = F − IC as a structural diagnostic. The degenerate-limits appendix establishes the precise arrow of derivation from GCD kernel identities to classical results (AM-GM inequality, Shannon entropy, unitarity, exponential map), showing that each classical formula is what remains when degrees of freedom are removed. All numbers are reproducible from the open-source reference implementation at https://github.com/calebpruett927/GENERATIVE-COLLAPSE-DYNAMICS. The paper serves as the canonical entrance to the GCD formalism.