feat(Analysis/CStarAlgebra/CStarMatrix): operator-monotone rpow/log/sqrt specializations

[allenhaozhu]

Adds three discoverability lemmas for operator-monotonicity on CStarMatrix n n A, specializing the existing C*-algebra-level results from CFC.monotone_rpow / CFC.log_monotoneOn / CFC.monotone_sqrt:

CStarMatrix.monotone_rpow {p : ℝ} (hp : p ∈ Set.Icc 0 1) :
    Monotone (fun M : CStarMatrix n n A => M ^ p)

CStarMatrix.log_monotoneOn :
    MonotoneOn (CFC.log : CStarMatrix n n A → CStarMatrix n n A)
      {M | IsStrictlyPositive M}

CStarMatrix.monotone_sqrt :
    Monotone (CFC.sqrt : CStarMatrix n n A → CStarMatrix n n A)

In the course of implementing these specializations, three shortcut instances are also registered to bypass a typeclass-diamond issue. CStarMatrix n n A derives CStarAlgebra from the underlying CStarAlgebra A, but the typeclass search for ContinuousFunctionalCalculus ℝ (CStarMatrix n n A) IsSelfAdjoint does not find the path through the Matrix diamond automatically. Explicit shortcut instances enable ergonomic use of CFC.rpow, CFC.log, etc.:

instance instContinuousFunctionalCalculusRealIsSelfAdjoint
instance instNonUnitalContinuousFunctionalCalculusRealIsSelfAdjoint
instance instNonnegSpectrumClassReal

Reviewer note: I am NOT attached to this PR — if reviewers prefer that downstream users call CFC.monotone_rpow directly and accept the typeclass-search cost, please close. The PR is offered as a convenience on the assumption that matrix-shaped users may search the matrix namespace first and benefit from the diamond shortcut. The shortcut instances may also belong in a separate PR if reviewers prefer.

Extracted from LTFP-Lean (mechanical formalization of Bach's Learning Theory from First Principles), where matrix-side ergonomics matter for downstream proofs.

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AI-assistance disclosure

Per the Mathlib AI-use policy:

• Tool. Claude Code (Anthropic) with the Claude Opus 4.7 model.
• Use. I specified the three target specialization lemmas. The assistant placed them in the right file, discovered the typeclass-diamond issue while implementing (adding 3 shortcut instances to resolve it), drafted the proofs, and verified the build under lake build Mathlib.Analysis.CStarAlgebra.CStarMatrix.OperatorMonotone. I reviewed the final code.
• Vouching. I have read the declarations and can defend the proofs without further AI assistance. I welcome reviewer feedback on:
  • whether the three specialization lemmas justify their own file or should be merged into CStarMatrix.lean
  • whether the typeclass-diamond shortcut instances belong in this PR or a separate one
  • whether the convenience-layer is justified at all vs. having downstream users call CFC.* directly

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[github-actions]

PR summary a3bf4892db

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Analysis.CStarAlgebra.CStarMatrix.OperatorMonotone (new file)	2794

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Declarations diff

+ instContinuousFunctionalCalculusRealIsSelfAdjoint
+ instNonUnitalContinuousFunctionalCalculusRealIsSelfAdjoint
+ instNonnegSpectrumClassReal
+ log_monotoneOn
+ monotone_rpow
+ monotone_sqrt

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

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No changes to strong technical debt.

No changes to weak technical debt.