This entry is a synopsis of the complete 63-page manuscript attached as a PDF. The ZIP contains the LaTeX source and a Rust viewer for six geometric examples.
Parametric programs and covectors
A parameterized model is a map $f:P\times X\to Y$. A reparametrization $h:P\to Q$ preserves it when $f=g\circ(h\times\mathrm{id})$. Sequential composition uses a product of parameters and composes the forward maps along the intermediate interface. Parameter sharing is precomposition by a diagonal, whose reverse derivative sums the cotangents of the shared uses. In the Euclidean model, reverse differentiation sends a smooth map to its forward evaluation and the pullback of an output cotangent. For the parameter-change equation it gives $R_P[f]=Dh^*R_Q[g]$. A metric converts that intrinsic covector into an update direction. Gradient flows are equivariant under transported metrics, and a finite step requires a compatible displacement rule as well.
The smooth stationary stack
Work in the smooth higher topos of sheaves of spaces on smooth test manifolds. A Lie group $G$ acting on a parameter manifold $P$ and preserving its predictor defines a quotient stack $[P/G]$. A point over a test manifold $T$ is a principal $G$-bundle on $T$ with an equivariant weight map. Its evaluation lands in the internal function sheaf $Y^X$. At an ordinary point, the stack retains the stabilizer of a parameter in addition to its orbit.
For a smooth invariant risk $L:P\times D\to\mathbb R$, let $B=[P/G]\times D$ and let $V=[(T^*P\times D)/G]\to B$ be the relative covector bundle. The differential in the parameter direction and the zero section give two arrows $B\rightrightarrows V$. Their homotopy pullback is the stationary stack. It is the quotient of the zero-locus sheaf by $G$. Its families are local stationary weights with gauge-compatible overlap maps. Its construction commutes with pullback of the dataset and satisfies descent. For a change of architecture by an embedding, the smaller stationary equation only checks the restriction of the larger covector to tangent directions of the embedding. A stationary point for the smaller problem can therefore fail the larger stationarity test.
Two explicit global obstructions
For the predictor $f_w(x)=(w^\mathsf{T}x)^2$ with two-dimensional weight, three quadratic evaluations determine the matrix $ww^\mathsf{T}$. The rank-one projector $$ Q_\theta=\tfrac12 \begin{pmatrix}1+\cos\theta&\sin\theta\\ \sin\theta&1-\cos\theta\end{pmatrix} $$ defines a smooth predictor family on a circle. Use those three evaluations as labels in a squared loss. Every task has zero minimum, realized at the two half-angle weights. The optimal family in the effective sign quotient is globally defined. Choosing a global continuous weight would split the degree-two circle cover, contradicting the fundamental group. Equivalently the weight bundle has a nonzero degree-one cocycle.
The second example takes unit weights in three dimensions and the effective sign quotient $\mathbb{RP}^2$. Six quadratic evaluations give exact-fit tasks for every projective point. The ineffective group $\mathbb Z/4$ acts through its quotient $\mathbb Z/2$, adding gauge arrows invisible to prediction. Lifting the effective principal bundle to that presentation is the pullback of $B\mathbb Z/4\to B\mathbb Z/2$. For a central extension with kernel $A$, locally lifted transition functions have a triple-overlap defect in $H^2(T;A)$, and the extension class is represented by a map to $B^2A$. In this example the antipodal cover cannot lift, since a homomorphism $\pi_1(\mathbb{RP}^2)=\mathbb Z/2\to\mathbb Z/4$ cannot reduce to the nontrivial sign monodromy. Thus the effective trained model exists and the specified ineffective presentation has an obstructed coherent lift.
Derived refinement and optimizer histories
For a polynomial risk, the derived critical locus is the derived intersection of the differential graph and the zero section of the cotangent bundle. Its affine model is the Koszul algebra on the partial derivatives. For $L(x,y)=(x^2+y^2)^2/4$ put $q=x^2+y^2$. The residuals are $xq,yq$, and the degree-minus-one cohomology is $\mathbb R[x,y]/(q)$, generated by the relation $y\epsilon_x-x\epsilon_y$. The degree-minus-two cohomology vanishes. The Hessian at the unique real critical point vanishes, so the tangent and obstruction spaces are both two-dimensional. This derived computation and the ordinary smooth quotient concern different types of tests.
A deterministic equivariant optimizer gives a span from the state stack to itself. An $n$-step history is the iterated homotopy pullback of the one-step spans and is equivalent to its initial state and $n$ data inputs for a total update. A single principal bundle presents the entire history. An equivariant update carries it forward and can preserve it even when its final observable predictor is constant. Terminal conditions, such as a stationary final state, are imposed by pullback along the final-state map.
Attention, stochastic agents, and distributed updates
A row-softmax attention matrix is a data-dependent finite Markov kernel on key indices. With independently permuted query and key indices, and the mask conjugated accordingly, the output transforms by the query permutation. The derivative includes variation of both the kernel weights and the value array. A fixed causal mask or fixed position labels restrict the symmetry.
A stochastic agent is a stateful Markov kernel on standard Borel interfaces. Serial composition carries the actual sampled intermediate value, and delayed feedback uses an explicit register. A measurable state map that intertwines transition kernels and initial laws preserves every finite output-history law under a common adaptive protocol using only external history. Equality of output laws can forget internal dynamics.
For federated training, the weighted average of one synchronized gradient step is exactly the centralized step. For quadratic client losses, the difference after two unsynchronized local steps is $$ \eta^2\left(\sum_i\lambda_i H_i g_i-\bar H\bar g\right). $$ This is separate from descent of compatible local datasets and from gauge-invariant aggregation. In the sign-quotient example, averaging two independently chosen representatives of the same predictor can produce the zero predictor.
Predecessors and scope
The main compositional predecessors are Fong, Spivak, and Tuyéras on learners, and Cruttwell, Gavranović, Ghani, Wilson, and Zanasi on gradient-based learning. The smooth and higher constructions use the standard theory of quotient stacks and principal bundles. The derived refinement follows the usual Koszul model of a derived critical intersection, and the stochastic model uses Markov categories. The manuscript places its concrete comparison results and explicit obstruction examples within those literatures.
- Fong, Spivak, and Tuyéras, Backprop as Functor. https://arxiv.org/abs/1711.10455
- Cruttwell et al., Categorical Foundations of Gradient-Based Learning. https://arxiv.org/abs/2103.01931
- Jia et al., Category-Theoretical and Topos-Theoretical Frameworks in Machine Learning. https://arxiv.org/abs/2408.14014v3
- Nikolaus, Schreiber, and Stevenson, Principal Infinity-Bundles. https://arxiv.org/abs/1207.0248
- Vezzosi, Derived Critical Loci. https://arxiv.org/abs/1109.5213
- Fritz, A Synthetic Approach to Markov Kernels. https://arxiv.org/abs/1908.07021
The source and viewer illustrate coordinate-sensitive descent, polynomial XOR, a double cover, federated drift, a degenerate quartic minimum, and antipodal monodromy. The paper gives ordinary mathematical proofs and exact calculations. AI assistance was used in research synthesis, exposition, and software development. This is an anonymous review copy.