LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Rational or dyadic stored real blocks have a finite exact discriminant-sign evaluator

Statement

Suppose the stored entries of $B = (\begin{array}{cc} a & b \\ c & d \end{array})$ have supplied exact rational encodings $a = p_{a} / q_{a}$ , $b = p_{b} / q_{b}$ , $c = p_{c} / q_{c}$ , $d = p_{d} / q_{d}$ , with integer numerators and positive integer denominators. Put $Q = q_{a} q_{b} q_{c} q_{d}$ and $A = p_{a} (Q / q_{a})$ , $B_{0} = p_{b} (Q / q_{b})$ , $C_{0} = p_{c} (Q / q_{c})$ , $D = p_{d} (Q / q_{d})$ . Compute the integer $Z = (A - D)^{2} + 4 B_{0} C_{0} .$ Its integer sign is exactly Exact sign predicate for a real 2x2 characteristic discriminant and is found after finitely many exact integer operations. Finite binary floating values with their signed-integer/power-of-two encodings are a special case. This evaluator does not assume arbitrary stored real numbers have rational encodings. If an exact encoding or another terminating exact sign evaluator is unavailable, retain schur-type-undecided.

Facts & Assumptions

Given: The exact finite integer encodings stated above, with positive denominators. Integer arithmetic is exact and is separate from the floating-point operationwise model; no bit-complexity estimate is claimed.

[F1]

Exact sign predicate for a real 2x2 characteristic discriminant defines the sign of $(a + d)^{2} - 4 (a d - b c) = (a - d)^{2} + 4 b c$ .

Proof

1.1givenconstruct

$Q$ is a positive integer and each $Q / q_{j}$ is an integer product of the other three positive denominators. The displayed construction therefore uses a fixed finite number of integer additions, subtractions, products and sign comparisons, each on finite integers. Powers of two encoding binary exponents give such finite numerators and positive denominators as well.

2.1F1step 1.1algebra∎

Direct substitution gives $(a - d)^{2} + 4 b c = Z / Q^{2}$ . Since $Q^{2} > 0$ , the discriminant sign in [F1] equals the integer sign of $Z$ , including the exact zero case. Thus the finite recipe supplies the promised sign without a floating sign oracle; the unspecified-real branch remains undecided.

Depends on

Exact sign predicate for a real 2x2 characteristic discriminant

Used by

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Sources

James Demmel, Math 221 Lecture 16: Hessenberg reduction and implicit Q (standard reference, not scraped)
David Bindel, CS 6210 Matrix Computations, Hessenberg QR iteration (standard reference, not scraped)