Test exact high-dimensional rotations and reflections

[isomorphisms]

Scope

Tests-first extension of #45 for dimension-generic orthogonal transformations, with concrete R^128 execution rather than only R^3 type instantiations.

Tests first

Commit 1b28c6fb07803d233ef976caf483cce345ae6fe6 adds exact fixtures requiring:

• firstAxisReflection : OrthogonalTransform 128 Reversing;
• firstPlaneQuarterTurn : SpecialOrthogonal 128;
• two reflections landing in SO(128);
• exact coordinate images of a sparse R^128 sample;
• squared-norm preservation;
• dot-product preservation;
• reflection involution (H^2 = I);
• four quarter turns giving identity (G^4 = I).

Minimal implementation

Commit 54d74753e8c97eed8bd8054528a88672d428aea6 adds only the exact coordinate realizations already implied by the named generators:

• zeroVector for structural high-dimensional fixtures;
• first-axis reflection (x, rest) -> (-x, rest);
• first-plane quarter turn (x,y,rest) -> (-y,x,rest).

It does not invent a generic evaluator for every OrthogonalTransform; quaternion coordinate action and arbitrary-angle Givens/Householder transforms remain separate work.

Independent verification

HIGH_DIMENSIONAL_VERIFICATION.md records redundant exact checks against a 128x128 SymPy matrix construction and an independently constructed NumPy/SciPy path. Both agree on determinant, orthogonality, images, norms, dot products, involution, and fourth power. The note also records the literature boundary (Golub--Van Loan; DeTurck et al.).

Macaulay2/homology packages are deliberately not treated as ceremonial oracles for these elementary O(128) identities; they are more appropriate when we begin checking algebraic varieties, quotients, homology/cohomology, or theorem-level topology.

Verification state

Exact source head 214ceaffdb38389bef65b8fa63f73f24d66a609e passed Edric type-system smoke run 33278922443. The real Idriç edric008 compile/execute receipt is verified at the current head. The SymPy and NumPy/SciPy checks remain recorded Markdown evidence rather than rerunnable CI.

Relation to the named-space alternative

This is the cumulative tip of #44 → #45 → #47 and therefore retains #45's rank-indexed Euclidean vector/direct-dot model. Independent #46 retains the competing named-space covector/vector model with no implicit metric. They also collide at edric008; neither supersedes the other.

[isomorphisms]

Strengthened the concrete R^128 oracle after review: the sample now has a nonzero coordinate 128, so the test cannot pass merely by getting the first 2–3 coordinates right while mishandling the distant tail. Exact expectations are now norm^2 = 250 and x·y = 190, with the final coordinate preserved under both the first-axis reflection and first-plane Givens quarter turn. SymPy exact matrices and a separate NumPy/SciPy construction both agree. Current head: 214ceaf; the real Edric type-system smoke is running on this head.

[isomorphisms]

Classification: SUPERSEDED by #51.

#51 retained the exact strengthened R128 oracle from this stack: the distant-coordinate reflection and quarter-turn fixtures, integrated into the reconciled named-space/Euclidean model without reviving the edric008 fixture collision.

No unique implementation from this PR remains pending here.