Conserved Computation: Symmetry, Monadic Descent, and the Structural Guarantees of $G_2$ Thermodynamic Logic Working Paper v1.0 — May 2026 | DOI: 10.5281/zenodo.20127517 Summary Why does a $G_2$-symmetric thermodynamic computer have guaranteed correctness properties — lossless phase transitions, exact opcode duality, equal error-correction thresholds? This paper answers that question with a single unifying structure: the adjunction between free computation and locked invariants, made exact by the self-duality of $G_2$. Noether's theorem establishes a bijection between continuous symmetries and conserved quantities via the moment map $\mu: M \to \mathfrak{g}^*$. Category theory identifies this bijection as the counit of an adjunction between the Kleisli category of free trajectories and the Eilenberg-Moore category of invariant algebras. Gauge-fixing is monadic descent: the comparison functor $K: \mathrm{Kl}(\mathbb{T}) \to \mathrm{EM}(\mathbb{T})$ becomes an equivalence precisely when the physics is fully determined by its symmetries alone. We apply this framework to the Adelic Simplicial Architecture (ASA), identifying four consequences of the $G_2$ self-duality ${}^LG_2 \cong G_2$ and the resulting isomorphism $\mathfrak{g}_2 \cong \mathfrak{g}_2^*$: The Noether charge of the $G_2$ symmetry is the Topological Skeleton — the Hopf invariant $Q \in \mathbb{Z}$ classifying field configurations up to homotopy. On Fano-compatible trajectories, $Q$ is exactly conserved by any $G_2$-invariant dynamics. The FTC Commutator Anomaly is the moment map obstruction. The anomaly $C(\gamma, F) = \frac{1}{2}\int_0^1 [F'(\gamma(t)), \gamma'(t)], dt$ (doi:10.5281/zenodo.20025384) vanishes precisely on the zero-level set $\mu^{-1}(0)$ — the Fano lines where the Topological Skeleton is conserved. This is the Marsden-Weinstein reduction condition. The Mirror Square identity and equal CSS thresholds are structurally necessary. Because $\mathfrak{g}_2 \cong \mathfrak{g}_2^*$, creation ($\blacksquare$ Split) and annihilation ($\diamond$ Splat) operators map to the same algebraic space and are exact duals. In a non-self-dual architecture the moment map outputs land in a qualitatively different space from the generators — this is the algebraic origin of X/Z threshold asymmetry in generic error-correcting codes. $G_2$ self-duality eliminates that asymmetry structurally (doi:10.5281/zenodo.20101634, Theorems 2.1 and 3.1). The Maslov-Gibbs Einsum is the monadic descent operator. Driving the inverse temperature $\beta \to \infty$ forces the comparison functor $K_\infty: \mathrm{Kl}(\mathbb{T}) \to \mathrm{EM}(\mathbb{T})$ to become an equivalence of categories. The continuous $G_2$ fluid (BOIL phase) and the discrete Fano crystal (SNAP phase) are then descriptions of the same object in two coordinate systems. Catastrophic forgetting in classical neural networks is a structural Kleisli property: without a mechanism forcing $K$ to become an equivalence, new trajectories can overwrite stored invariants. Keywords $G_2$ self-duality · Noether's theorem · moment map · monadic descent · Kleisli category · Eilenberg-Moore category · Maslov-Gibbs Einsum · thermodynamic computation · Topological Skeleton · Hopf invariant · FTC Commutator Anomaly · catastrophic forgetting · CSS error correction · octonions · Fano plane · Adelic Simplicial Architecture