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CLAIM-093: Not every invariance of a stochastic or communicative process implies a Noether conservation law: the symmetry principle can be borrowed, a conservation analogue has to be derived

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Properties8
StatusActive
Roledisclaimed
Warrantdemonstrated
Tagsmathematics, philosophy-of-language
Linepragmatic-transport, formal-apparatus
Workswhat-survives-translation
GroundsTHEORY-211 — In the stochastic-differential-equation model of SGD, with a gradient-noise covariance that depends on the parameters, minibatch noise drives the difference of squared norms along a rescaling symmetry, at a rate proportional to learning rate over batch size, to the point on the symmetry orbit where the gradient-noise traces of the two parameter groups are equal, whenever both groups carry noise; so at stationarity SGD neither keeps the difference fixed, as gradient flow does, nor lets it diffuse, as isotropic noise does, and differs from gradient descent by a finite amount however small the noise
ComplementsCLAIM-107 — Under Markov reconstruction kernels, an observable's expectation is invariant when the observable is harmonic (K f = f), and drifts by at most Σ ε_i when ‖K_i f − f‖∞ ≤ ε_i; on a finite irreducible chain only constants are harmonic

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The claim

A105: “not every invariance of an arbitrary stochastic or communicative process implies a Noether conservation law. Noether’s theorem requires an appropriate variational structure and continuous symmetry. Our translation kernels do not automatically have either. What we can borrow immediately is the symmetry principle. What we would have to derive is an appropriate analogue of conservation.” The derivation it found is Baez and Fong’s theorem for Markov processes, “a genuine conservation-theoretic connection for the continuous-time Markov special case” (CLAIM-107, qualified by CLAIM-094).

Manuscript §4: “Noether’s first theorem relates suitable continuous variational symmetries to conserved currents; its hypotheses do not hold automatically in communicative systems.” §11: “Noether-type results supply conservation criteria for particular stochastic processes, not a universal conservation law of meaning.”

Note of 2026-10-13: an instance in learning dynamics

Two readings give this claim a worked instance outside communication.

  • A derived law, with a condition. In Neural Mechanics (LIT-1012), a differentiable symmetry of a network’s loss gives a conserved quantity under gradient flow only when a further condition holds, ⟨θ, [∂_α ∂_θ ψ]g⟩ = 0 (Theorem 1). The conservation law is derived for the dynamics; it does not come with the invariance alone.
  • A symmetry with no conservation law. Under SGD, a stochastic process whose minibatch gradients share the symmetry exactly, the same quantity is not conserved. For a rescaling symmetry, minibatch noise drives it toward the point where the layers’ gradient noise balances (Ziyin et al., LIT-1013; THEORY-211).

THEORY-211 now grounds this claim. The instance supports the claim’s point that a symmetry principle can be borrowed but its conservation analogue has to be derived, and shows the derivation can come out negative.

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