docs/EXPERT_REVIEW.md

Expert review targets

The manuscript is offered for mathematical examination. The original separately assigned AI-agent reviews are available, but no external expert endorsement is asserted.

Contribution to examine

The narrow contribution under consideration is the finite rule

$$Aa=0,qquad(x,A)mapsto(x+a,A+x{a}^T-a{x}^T),qquad{A}^T=-A,Aa=0,qquad\; (x,A)mapsto(x+a,A+xa^T-ax^T),qquad\; A^T=-A,$$

together with its exact all-finite-futures minimum, kernel transport, and sharp rank timing. Existing signed-area and control-theoretic constructions are explicitly acknowledged.

Core proof dependencies

Claim label	Audit focus
lem:decomposition	Positive Euclidean metric; invertibility of the skew restriction to its image.
thm:kernel	Both independence cases; graph-kernel sign; zero-dimensional restrictions.
thm:rho	Projection onto the graph kernel; positivity of the denominator; equality cases.
thm:minimum	All finite futures; endpoint uniqueness; zero controls and zero radius.
thm:rank	Sharp lower bound and construction; residual reset after each rank increase.
thm:exterior	Exact orthogonality of the two exterior terms; high-degree and zero-input cases.
thm:accessibility-general	Exact three-step endpoint map, nonzero Jacobian, and the exceptional cases.
thm:dormant	Inductive preservation of inaccessible summands; distinction from minimal realization.
thm:continuous	Absolutely continuous hypotheses and bounded-variation refinement estimate.
prop:closure	Continuous/Euler approximation and why the extra limit endpoints are not finitely reachable.
thm:spectral-rank-one	Relation to classical positive semidefinite rank-one matrix updates.

Hidden equivalence and priority

Potentially useful comparison areas include abnormal extremals, higher-step control discretizations, nonholonomic numerical integration, endpoint maps, and algebraic reachability. A successful identification should specify which of endpoint, stored area, metric, input labels, legality at every prefix, and finite update are preserved.

The unrestricted free step-two law already supplies area accumulation. Goh-type kernel constraints, the three-dimensional continuous field, and general Euler accessibility/closure phenomena have retrieved ancestors. Those components cannot support broad first-discovery claims. The original audit did not certify exhaustive per-theorem priority.

An exact match in older work, a conjugacy, or a general theorem that immediately covers the main results would materially narrow the contribution. Please distinguish that outcome from a generic encoding into another formalism.

Open targets

• Full reachable endpoint sets and their closures.
• Complete behavioral equivalence and minimal realizations.
• Quantitative approximate-gate statements with explicit admissibility controls.
• The higher odd-dimensional cyclic-state interior conjecture conj:interior.
• An external task that uses the precise mechanism, with a measured or proved advantage.

How to make a review actionable

Reference the original claim label, list preserved or changed hypotheses, and supply the complete argument or the smallest exact counterexample available. For prior art, identify the original theorem, definitions, and source rather than matching only vocabulary. If a proof issue is local, identify downstream statements that depend on it.

Proof audit · Novelty audit · Reproduction