feat(stdlib): AURA BCI builtins — gaussian_kernel_1d·separable_blur·nearest_template·r2 (handoff f125d45c)

[dancinlife]

AURA BCI .hexa-native porting primitives (handoff f125d45c)

Implements the 4 numpy-equivalent kernels the anima AURA-* app set needs to drop its // TODO(f125d45c) stubs and compile/run.

Surface (stdlib/aura/bci_kernels.hexa)

fn	semantics	numpy ref
gaussian_kernel_1d(n, sigma)	row-stochastic n×n Gaussian matrix K[i,j]=exp(-(i-j)²/2σ²), each row ÷ its sum (Σ_j K[i,j]=1)	K=np.exp(-(i-j)**2/(2σ**2)); K/=K.sum(1,keepdims=True)
separable_blur(x, k, g)	separable 2-D blur Y = K·reshape(x,[g,g])·K^T	K @ x.reshape(g,g) @ K.T
nearest_template(y, templates, n_t, d)	argmin_i ‖y−templates[i]‖² (int index, ties→lowest)	int(np.argmin(((T-y)**2).sum(1)))
r2(y, ref, n)	coefficient of determination 1 − SS_res/SS_tot	1 - ((y-r)**2).sum()/((r-r.mean())**2).sum()

No new C builtin required

Pure-hexa port over the existing farr handle API (farr_zeros/get/set/free) + self/runtime/math_pure.exp_pure — mirrors the established stdlib/stats/correlation.hexa and stdlib/consciousness/phi_spatial.hexa precedent (stdlib-as-builtin over farr). add_gaussian_noise (RFC 033) and phi_spatial (RFC 036) already existed as builtins; these four close the remaining gap the handoff named.

Verification — 10/10 falsifiers PASS

hexa run stdlib/aura/bci_kernels_test.hexa → __HEXA_STDLIB_AURA_BCI_KERNELS__ PASS (verbatim stdout committed at .verdicts/aura-bci-kernels/bci_kernels_test_2026_05_30.txt).

• F1 gaussian_kernel_1d(5,1.0) every row sums to 1 (max |Δ| = 2.22e-16)
• F2 gaussian_kernel_1d(3,1.0) center K[1,1] = 0.451863 (numpy 0.45186276187760605, asserted to 1e-9)
• F3 gaussian_kernel_1d(4,1.5) row-stochastic AND edge-asymmetric (K[0,1] > K[1,0] — design, see note below)
• F4 separable_blur(const 7.0) == const 7.0 (partition-of-unity)
• F5 separable_blur(center impulse) center stays max, all ≥ 0
• F6/F7 nearest_template exact → idx 3 · nearest(y≈[1.1,1.1,1.1]) → idx 1
• F8 r2(y==ref) == 1.0 · F9 r2 known == 0.8 (exact) · F10 mean-predictor == 0.0

Note on row-normalized Gaussian asymmetry (honest finding)

The first draft of F3 asserted K symmetry — that FAILED, and the self-test caught it. Row-normalized Gaussian K is intentionally asymmetric at the boundary: edge rows sum to less than center rows pre-normalization, so dividing each row by its own sum makes K[0,1] ≠ K[1,0]. This is the standard weighted-average smoothing operator (every output is a proper convex combination of inputs) and matches numpy K/=K.sum(1,keepdims=True). The implementation was correct; the test invariant was wrong and is now fixed to assert the true row-stochastic + edge-asymmetric property. Module doc-comment updated accordingly.

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