# The Gradient-Free Dual: B-Spline Kernels and the Quantum Geometry of the Invariant

**Abstract.** This note describes a publicly reproducible scaffold for re-expressing a gradient-trained LLM's internal geometry as a *gradient-free* quantum kernel. Quantum kernels are chosen precisely because they train no circuit parameters — sidestepping barren plateaus by construction. Smooth B-spline embeddings, encoded over an orthogonal Base-SAE basis, resist the kernel-method failure mode (exponential concentration). The invariance claim is made falsifiable through centered kernel alignment (CKA) across distinct models. This is a theory note; the reference scaffold and a worked example will follow. The origin-first *synthesis metric* enters as a single, clearly marked hook and is withheld.

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### I. From gradients to a dual

A modern LLM is the product of iterative gradient descent. But its internal geometry — the relational structure of its hidden states — can be *re-read* without any further gradient optimization. The dual is this: the model was *built* by iteration; the kernel *reveals* its geometry through fixed-feature-map state overlaps and a single convex (gradient-free) classical step. We do not retrain anything. We measure what is already there.

### II. Why kernels (no barren plateau)

In quantum kernel estimation, each vector x is mapped to a state |φ(x)⟩ by a *fixed*, non-trainable feature-map circuit. The kernel entry is the fidelity K\_ij = |⟨φ(x\_i)|φ(x\_j)⟩|², estimated via the inversion/overlap test. Because no circuit parameters are optimized, there is no gradient-descent loop and therefore **no barren plateau** in training. The only optimization left is the downstream classical problem (an eigendecomposition, or an SVM dual) — convex, with a unique optimum.

### III. The real adversary: exponential concentration

Kernels do not make the difficulty disappear; they relocate it. Quantum kernels can suffer *exponential concentration* — as qubit count grows, off-diagonal entries collapse toward a fixed value and the Gram matrix degenerates toward the identity, yielding a model whose predictions ignore the input. This is the formal equivalent of barren plateaus, and it is driven by high expressivity, heavy entanglement, global measurements, and noise. The mitigations are: low-expressivity, smooth embeddings; modest qubit counts; projected (local) kernels; and centering the Gram matrix before spectral analysis. Naming this risk openly is the point — a method that hides it is not yet a method.

### IV. Why B-splines

A B-spline kernel of odd order is a valid Mercer kernel: its Fourier transform is sinc^(2n+2) ≥ 0, i.e. smooth and rapidly decaying — a low-pass kernel. Low-frequency, smooth embeddings are exactly the regime that resists concentration. The B-spline is therefore not a stylistic choice but a structural one: it keeps the Gram matrix non-degenerate as the encoding scales.

### V. The orthogonal Base-SAE basis

The encoding directions matter as much as the kernel. Standard sparse autoencoders are deliberately overcomplete and non-orthogonal — superposition is their defining feature. Choosing instead an **orthogonal** basis means each vector is encoded into distinguishable, non-overlapping quantum states. Orthogonality fights concentration *at the level of the basis*, before the circuit is ever touched. This is the origin-first move: do not search for the basis by iteration — know it, and project onto it.

### VI. Making the invariant falsifiable

The candidate invariant is read out classically in two complementary ways. First, the *dominant eigen-subspace* of the centered Gram matrix — reported as a subspace, not as fragile individual eigenvectors. Second, **centered kernel alignment (CKA)** between Gram matrices computed from two different models on the same inputs. CKA is invariant to orthogonal transforms and isotropic scaling; CKA → 1 supports the hypothesis that distinct models induce the *same* kernel geometry; CKA materially below 1 falsifies it. This is the empirical, refutable form of the invariance claim — and it connects directly to the Platonic Representation Hypothesis: as models grow, they appear to measure distance between datapoints in increasingly alike ways.

### VII. What is public, what is not

The full pipeline — extract hidden states → project onto the orthogonal basis → encode via a B-spline feature map → compute the gradient-free Gram matrix on a GPU statevector simulator (cuQuantum / qsimcirq) → spectral readout + CKA — is described here in full, and the reference code with a worked example will follow. The *synthesis metric* — the origin-first transform that makes the extraction synthetic rather than gradient-iterated — is **not** disclosed. It enters the scaffold at a single, well-marked hook. Everything around it is scaffolding; the core remains private.

### Honest limits

This is a scaffold, not a solution. On a classical statevector simulator there is **no quantum speedup** — the value is methodological: a faithful, hardware-portable construction. Exponential concentration must be monitored empirically. The invariance is a *hypothesis under test*, not a result. And the surfaced structure is conditioned on the chosen layer, pooling, and basis — all of which should be swept.

> *A gradient-trained model, re-read without gradients. A curved geometry, kept curved. An invariant, made refutable.*