A Syndrome Algebra for Differentiable Parametric Systems

This is a mathematics paper. It constructs a syndrome algebra for differentiable parametric systems by generalising the syndrome-measurement principle used in quantum error correction (QEC) from a discrete binary setting to a continuous real-valued one. The result is a tool for diagnosing the response of a differentiable system to weight-space perturbations; its concrete application to neural network reliability is the subject of the companion paper "Three Measurable Failure Modes of Large Language Models", this paper is focused on its mathematical formalism.

The starting point is the $[[5,1,3]]$ perfect quantum error-correcting code, whose parity-check matrix maps single-qubit errors to four-bit syndromes. Three replacements take this construction to the continuous setting.

(1) Field replacement: the finite-field arithmetic over $\mathbb{F}_2$ is replaced by row-wise $\ell_2$-normalisation over $\mathbb{R}$.

(2) Basis replacement: the symplectic Pauli error basis is replaced by the right singular vectors of the weight matrix (the SVD basis).

(3) Map replacement: the parity-check matrix is replaced by the averaged Jacobian of the logit map at probe inputs. Under these three substitutions, the syndrome table becomes $S = \mathcal{N}(\bar{J} \cdot V)^\top$, and the discrete $[[5,1,3]]$ code is recovered as the $q=2$ special case.

Seven theorems establish the algebra's main properties:

Theorem 1: the Jacobian is the syndrome table in the linear regime.

Theorem 2: oracle correction is exact on the linear path.

Theorem 3: crossing error (correction in a wrong direction) is positive in expectation.

Theorem 4: the null space of the weight matrix is the confabulation subspace.

Theorem 5: multi-layer measurement achieves identifiability.

Theorem 6 (Jacobian Uncertainty Principle): the product of dictionary variance and observation variance is bounded below by a quantity proportional to the per-direction Jacobian variance.

Theorem 7 (Reliability Principle): for any $[[n,k,d]]_q$ code, normalised robustness, capacity, and expressiveness satisfy a structural trade-off that no architecture can escape.

This paper presents the mathematical foundation for the syndrome algebra framework. Application and experimental validation on neural networks is provided in the companion paper Three Measurable Failure Modes of Large Language Models (see related identifiers). The paper is self-contained: the complete derivation from the $[[5,1,3]]$ quantum error-correcting code to the continuous real-valued syndrome table is provided, with all proofs in the main body; no prior reading of the experimental companion is required.