Derive symmetric + PSD for Gram forms trans(F)*F and F*trans(F)

[petlenz]

Summary

trans(F)*F (right Cauchy-Green) and F*trans(F) (left Cauchy-Green / Finger) are symmetric and positive-semidefinite for any F — no assumption on F required. The CAS currently derives neither annotation.

This is the highest-value case: C = FᵀF is the workhorse strain measure in the finite-strain / spectral pipeline, and downstream sym()-projector and eigen simplifications key off the symmetric annotation.

Math

• Symmetric: (FᵀF)ᵀ = Fᵀ(Fᵀ)ᵀ = FᵀF. Likewise (FFᵀ)ᵀ = FFᵀ.
• PSD: xᵀ(FᵀF)x = (Fx)ᵀ(Fx) = ‖Fx‖² ≥ 0. (Positive-definite iff F is invertible — not generally known, so only PSD is derivable unconditionally.)

Current behavior (probed)

trans(F)*F   -> is_symmetric=0  is_positive_semidefinite=0   (expected: sym, PSD)
F*trans(F)   -> is_symmetric=0  is_positive_semidefinite=0   (expected: sym, PSD)

No false positives — the annotation is simply dropped (sound but incomplete).

How to fix

Binary hook in the tensor·tensor mul operator, mirroring the existing orthogonal trans(Q)·Q → I fold. In include/numsim_cas/tensor/tensor_operators.h, tag_invoke(mul_fn, L&&, R&&) (the tensor×tensor overload), right after the orthogonal branch (~lines 164-169) and before returning the generic mul_base result:

// Build the product first (via mul_base visitor), then annotate.
// Gram form: trans(X)·X and X·trans(X) are symmetric PSD for any X.
if (lhs.get().rank() == 2 && rhs.get().rank() == 2 &&
    (is_trans_of(lhs, rhs) || is_trans_of(rhs, lhs))) {
  // ... obtain `result` from the mul_base visitor ...
  if (result.is_valid()) {
    result.data()->set_space({Symmetric{}, AnyTraceTag{}});
    result.data()->tensor_algebra_assumptions().insert(positive_semidefinite{});
  }
  return result;
}

• is_trans_of(a, b) already exists in this file (matches permute_indices_wrapper with {2,1} over b).
• Post-construction set_space is hash-safe today because the n_ary hash excludes the space annotation — see the identical note on the existing skew branch (lines 118-124).
• Do not insert positive_definite — F invertibility is not tracked.

How to test

tests/TensorSpacePropagationTest.h (and/or the #111 matrix in tests/TensorAnnotationMatrixTest.h):

auto F = mk("F", 3);              // general, no assumption
EXPECT_TRUE(is_symmetric(trans(F) * F));
EXPECT_TRUE(is_positive_semidefinite(trans(F) * F));
EXPECT_TRUE(is_symmetric(F * trans(F)));
EXPECT_TRUE(is_positive_semidefinite(F * trans(F)));
EXPECT_FALSE(is_skew(trans(F) * F));           // safety: not skew
// Regression guard already present: A·B of two DISTINCT symmetric tensors
// must stay non-symmetric (do not over-generalize the pattern).

Also verify sym(trans(F)*F) now short-circuits to trans(F)*F.

Part of the tensor space-annotation completeness effort (umbrella below). Related: #228 (algebra-property annotations), #111 (annotation matrix).

[petlenz]

Tracked under the annotation-completeness umbrella #396 (full gap table + tiered rollout plan there).