Freely accessible sources, contributors and terms

The exposition is by OpenAI Codex (AI), with the contributions and component terms identified below. Mathematical source work is freely accessible. Each invoked theorem needs a proof in its lesson or an earlier programme lesson; a source citation does not supply that proof. The parent course remains in progress.

Bounded-operator foundations

The entire ten-section lesson and its three solved problems retain their independently written bounded arguments. Jacob Lurie, Math 261y: von Neumann Algebras, Lecture 5, 9 September 2011, Theorem 4 and Proposition 5 (PDF pages 1–2), supplies the consulted finite-vector bicommutant comparison. The present proof explicitly assumes the algebra contains the identity, which is used in its projection argument. The Hilbert-space lesson, Sections 1–3, and the continuous-calculus lesson, Theorem 5.1 and Section 8, supply the earlier full programme proofs at the conditions stated in Section 1. Their own human sources and AI credits remain on those linked pages. These precise uses do not claim a review of unrelated sections or a whole-source literature survey. No paid-only source supplies the construction. The historical author model is not inferred; current correspondence checking and public presentation are GPT-6.1 Sol (OpenAI), Ultra.

Spectral calculus and scalar inputs

The twelve-section spectral lesson retains the complete reviewed independent construction and all four solutions. The primary scalar proof route is the complete human-readable programme proof in Haar Theorem 2.2 and Proposition 2.3 and measure-tools Theorems 1.1, 2.1–2.2 and 3.1–3.2, with the exact topology support used in RMK. Their independently written AI prose retains its original credits and CC0 terms. The scalar companion gives exact links and the five complete original compact/complexification, L2, measure, Borel-representative and approximation bridges. Its real-to-complex correspondence applies the complex RMK theorem to the real contract used in SK. The exact binding record retains the ten pinned mathlib files, their hashes and 48 selected ranges as formal-source and historical evidence at commit 71a80585ee495fc24472fd0eaffc89d94e4fd8d6; eight were originally primary formal providers and two supporting comparisons. The human contributors are credited individually in the companion. The programme-proof record lists the complete bodies included in the download. The course has not been compiled in Lean. No source code, comments or human expression are copied. The earlier bounded Hilbert/CFC proofs and the maximal principle are linked in the lesson; the construction does not rely on an unbounded spectral or polar theorem.

The spectral lesson and scalar companion are CC0-1.0. The retained mathlib proof components retain Apache-2.0. Current source correspondence and reader presentation are GPT-6.1 Sol (OpenAI), Ultra; historical author variants are not inferred.

Concrete preduals

The fourteen-section lesson retains its independently written projective-tensor, Banach-quotient, vector-series and continuous-dual arguments and all four solved models. Exact freely readable programme proofs supply arbitrary-Hilbert Riesz, bounded forms and adjoints (HS Sections 2–3), real/complex norm-preserving Hahn–Banach (HB Section 2), real locally convex separation (HB Section 6), and the selected bounded positive-functional support/ideal argument (UE source lines 93–102 and BI Lemma 8.1/Theorem 8.3(1,3)). These programme lessons explicitly identify their original/revision AI contributors, including Claude Opus 5.5 (Anthropic) and GPT-6.1 Sol (OpenAI), and their CC0 dedication. They are human-readable AI-authored proofs; no human authorship, human review or formal verification is claimed.

The pinned mathlib Hilbert dual, adjoint and Hahn–Banach components retain their Apache-2.0 terms and human formal-source credits: Frédéric Dupuis; Frédéric Dupuis and Heather Macbeth; Yury Kudryashov and Heather Macbeth, respectively. The CP source links commit 71a80585ee495fc24472fd0eaffc89d94e4fd8d6 and the exact extension/norming locators. No Lean code or source commentary is reproduced. Exact proof-input identities retain the selected contract scopes and source versions.

Closed positive forms

The eleven-section lesson retains its independently written representation, exact-domain and directed convergence proofs, three worked models and five fully solved problems. The freely readable human comparisons are Zoltán Sebestyén and Zsigmond Tarcsay, Basic representation theorems of forms, arXiv:2505.09588v1, Lemma 2.1 and Corollary 2.7, and Barry Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems, Theorems 3.1 and 4.1. They are research comparisons; no human text, PDF or structured component is imported. The first comparison uses a closed-operator argument that QF does not import as a construction dependency; the second gives sequential limits, while QF proves directed limits and their exact finite-energy domain.

The exact constructive route is the complete earlier HS, BK, SK and scalar programme proofs named in QF01. Their full source/reader bodies and original AI credits remain unchanged in the download. Original QF mathematical exposition is credited to OpenAI Codex (AI), with its exact historical model variant unverified. Current QF mathematical review, prerequisite proof correspondence and clarifications, exact-byte proof review, and source/input-binding preparation are credited to GPT-6 Astra (OpenAI), Ultra; reading presentation to GPT-6.1 Sol (OpenAI), Ultra, October 2026. The scopes of the earlier mathematical review and of the check of QF01, QF03 and QF04 are stated in the proof-input record; no human review or formal verification is claimed.

Retained QF prose is CC0-1.0. The source-history record identifies the current source, contributions and terms. The independently written new prerequisite/proof clarification, source comparison and reading presentation text has a separate CC0 notice. This notice does not replace the original terms of any provider.

Tomita graph closure and polar data

The thirteen-section reading retains the complete independently written proofs, two matrix models and five solved problems. The present construction is supplied by the full earlier freely readable HS, BK, QF, SK and scalar programme proofs linked in TC01; no human source text or PDF is imported.

The complete current TC mathematical review, precise prerequisite correspondence and matrix-model clarification were prepared by GPT-6.1 Sol (OpenAI), Ultra. GPT-6 Astra (OpenAI), Ultra prepared the reader presentation and publication and corrected prerequisite links. Original exposition is credited to OpenAI Codex (AI), with its exact historical model variant unverified.

Retained TC prose is CC0-1.0. The independently written new correspondence, model-input clarification and presentation have a separate CC0 notice. Source and contributor notices identify the freely accessible proof sources, contributions and terms. The proof guide names the complete earlier proofs and their mathematical roles. Provider credits and terms are preserved.

Real coercive equations

The supporting proof, matrix calculations, diagram and reproducible figure source are original exposition by OpenAI Codex (GPT-6 Astra, Ultra), October 2026, released under CC0-1.0. The human mathematical source is Marc A. Rieffel and Alfons Van Daele, A bounded operator approach to Tomita–Takesaki theory, Pacific Journal of Mathematics 69 (1977), 187–221; publisher PDF, Lemma 5.6. The supplement supplies a complete coercivity argument for the initial real equation and identifies the separate role of the subsequent bounded-multiplication proof. The original paper is linked rather than included in this edition.