{
  "schema": "finite-hermitian-proof-review/1",
  "date": "2026-10-02",
  "lesson_sha256": "a8547e894bab4c4eb862086af6fc078519945e11aa9976ba823bde94cbef551c",
  "verdict": "proved as written; bounded author-instance self-review",
  "whole_course_verified": false,
  "independent_review": false,
  "human_review_claimed": false,
  "provenance": {"model": "OpenAI Codex GPT-6 Astra", "effort": "Ultra", "work": "Proofs, explanations, examples, solutions, self-review and integration"},
  "rights": "CC-BY-SA-4.0",
  "target": "Finite-dimensional positive-definite complex Hermitian spaces, linear-first convention: orthonormal bases and V=U direct-sum U-perp for every subspace U.",
  "supplied_generality": "Gram-Schmidt for every finite or countably infinite independent sequence in any real or complex inner-product space, with equality of each finite prefix span; orthogonal projection and unique decomposition for any finite-dimensional subspace of any such ambient space; finite-dimensional ambient case as a corollary.",
  "consumer": {"course": "RT-FIN", "lesson": "representations-and-complete-reducibility", "revision": "801f186833b9811fa826575ef0717e78a5047ab0", "sha256": "659942bdf47420e6665d59c4e80cf973867730d3b231d7e9c80ab8177bd83712", "loci": ["Proposition 2.1: orthonormal basis for the averaged form", "Following paragraph: invariant orthogonal-complement decomposition", "Section 7: explicit Hermitian prerequisites"]},
  "earlier_proof": {"lesson": "From bases to projections", "revision": "r2", "sha256": "131f01c7611a96c60a0dfdaf08c7b17e0f1a864dd371968ecd858b19f4e49652", "locus": "Extending a basis and choosing a complement, theorem and full proof", "use": "Existence of a finite basis for any subspace of a finite-dimensional space", "body_copied": false, "rights": "CC-BY-SA-2.5"},
  "sources_compared": [
    {"author": "John M. Erdman", "title": "Functional Analysis and Operator Algebras: An Introduction", "version": "2015-10-04", "source_url": "https://web.pdx.edu/~erdman/FAOA/functional_analysis_operator_algebras_web.zip", "archive_sha256": "0c667cfa7420b61dda8f8cb4ed9d619db8abbd1b53d17eafe7d4a2e153342e53", "source": "linalg.tex", "sha256": "a15cabf306adf5457cedce046f98b9474c72b38ab50197b0dc4288e942772096", "loci": "Chapter 1 inner-product convention; Gram-Schmidt proposition, lines 1017-1022; corollary 0001514", "status": "Statements without proofs in those passages", "notation": "Same linear-first convention; M in the source becomes U", "comparison": "The finite decomposition statement is exact; the independent-sequence Gram-Schmidt statement is supplied at full finite/countable generality in Theorem 1", "rights_evidence": "The source master functional_analysis_operator_algebras.tex, line 227, explicitly states CC BY-SA 4.0; the native source-authority record pins the downloaded edition."},
    {"programme": "OA-FND", "title": "Hilbert spaces and compact operators", "sha256": "ee431aa610e598b2ac40bcc48e4d00eb5d666842e5d9e1937be2c3173f6fdbe3", "loci_read": "Conventions; Proposition 1.1; complete completion lemma; Theorems 2.1-2.2; Theorem 4.1 and its proof; finite-dimensional closedness paragraph", "comparison": "General Hilbert projection uses completeness and closedness. It is retained as a broader later result, not imported without bridges into elementary finite-dimensional representation theory. The orthonormal-basis proof uses maximality/Zorn; the separable sentence names Gram-Schmidt without spelling out its recurrence.", "public_section_anchor_verified": "OA-FND-HS-02", "current_local_source_equals_older_directory_pin": false, "body_copied": false},
    {"programme": "DG-CHAR", "title": "Vector bundles and their constructions", "sha256": "c1740f19c0e1d1d589ead8ec8d8e400489c5841a0892323d2500a94ec60d1556", "locus_read": "Section 4, Theorems 4.1-4.2 and proofs", "comparison": "The bundle complement proof gives the Gram-matrix projection in a conjugate-linear-first convention. It does not provide the elementary Gram-Schmidt recurrence needed here. It remains unchanged; no allegation of a theorem error is made.", "body_copied": false}
  ],
  "search_scope": "Bounded existing-programme search: D20 source and admitted companions, current OA-FND and DG-CHAR provider proofs, and the existing B40 basis bridge. Not an exhaustive external literature or novelty search.",
  "overlap": "Standard known results and standard constructive proofs, independently written out here; no novelty claim.",
  "obligations": [
    {"claim": "Gram-Schmidt recurrence", "status": "passed", "reason": "Residual orthogonality is obtained by an explicit finite expansion; a zero residual contradicts prefix independence; positive definiteness legitimizes normalization; both span inclusions are shown."},
    {"claim": "Finite and countable scope", "status": "passed", "reason": "Each construction step is finite. Countable recursion does not invoke convergence of infinite linear combinations. The empty case is explicit."},
    {"claim": "Orthogonal direct sum and projection", "status": "passed", "reason": "Residual lies in U-perp by coefficient calculation; positivity makes the intersection zero; uniqueness proves basis independence; image, kernel, idempotence, self-adjointness and squared-length identity follow explicitly."},
    {"claim": "Exceptional cases", "status": "passed", "reason": "V=0, U=0 and U=V are included. Positive semidefiniteness and infinite-dimensional nonclosed U are explicitly not silently admitted."},
    {"claim": "Exact downstream use", "status": "passed", "reason": "The averaged form is positive definite and invariant. Orthonormal coordinates make the action unitary; invariance preserves U-perp; the unique orthogonal projection commutes with the action."},
    {"claim": "Examples and solutions", "status": "passed", "reason": "Complex projection coefficients, matrix square, adjoint, kernel and decomposition were recomputed; the displayed half-coefficient example also passed exact binary arithmetic checks. This finite check is not the universal proof."}
  ],
  "limitations": ["No independent mathematical certification", "No claim of all RT-FIN prerequisites being closed", "English lesson only", "No replacement of the general Hilbert-space or bundle theorems", "Build and public-byte verification recorded separately"]
}