Non-Archimedean Polydisc Spaces and Applications to Optimisation

Adaptive optimizers carry hidden states that change how visible gradients become parameter motion. We develop optimization geometrodynamics as a variational theory of this hidden geometry. Infimal pushforward eliminates all hidden states realizing the same action and composes across optimizer hierarchies. Under smooth nondegeneracy, it yields hidden susceptibility and the Schur-complement curvature seen after relaxation. For affine pre-reduction perturbations, the induced interaction curvature is the negative-semidefinite operator $-G^*H^{-1}G$, whose mixed entries integrate to finite mechanism contrasts. Our main realization is the determinant-one affine-invariant SPD action map $P\mapsto PA$. We prove a global analytic bundle with closed totally geodesic fibers and a unique analytic nearest-controller section. A strongly convex fiber theorem and an explicit logarithmic action residual give a globally linearly convergent solver from every feasible initializer, together with nonasymptotic value, controller-distance, and residual bounds and observable posterior stopping certificates. A conditional inexact result propagates supplied rigorous residual-error and radius majorants. The dense spectral kernel is confined to an active subspace of dimension $r\le 2m$, yielding an explicit spectral-arithmetic operation bound and a strict dimensional reduction when $r<d$. For nested shape-normalized quadratic actions, canonical multi-secant projections satisfy an exact CAT(0) Pythagorean decrease and recover the determinant-one inverse Hessian shape at the sharp rank threshold $d-1$, provided the scalar gauge $c_H=(\det H)^{1/d}$ is known. These results turn the action bundle into an exact iterative computation with posterior certificates and a finite-identification theory.

Complex constraints often make real-world optimization computationally prohibitive at the scale and speed required for operational decision-making. Here we introduce PolyFormer, a PIML framework that learns compact polytopic representations of the geometry induced by complex constraints. PolyFormer captures constraint-induced geometry and transforms it into efficient polytopic reformulations, reducing the complexity of downstream optimization and enabling the use of off-the-shelf solvers. Neural parameterizations further enable rapid adaptation to varying operating conditions without retraining. Through evaluations across three important problems, i.e., large-scale resource aggregation, network-constrained optimization, and optimization under uncertainty, PolyFormer achieves online solver speedups of up to 6,400-fold and memory reductions of up to 99.87%, while maintaining small feasibility and objective errors. Together, these results establish learned geometric constraint representations as an effective and scalable route to prescriptive optimization under diverse forms of constraint complexity.

Optimizer experiments observe responses to algorithmic configurations without uniquely revealing hidden mechanisms. We develop a causal optimizer interaction calculus that separates pathwise realization, Mobius decomposition, and experimental identification. Under a fixed innovation coupling, every finite-horizon innovation-driven optimizer admits a behaviorally minimal pathwise realization. For any finite effect support and intervention design, an incidence operator gives the complete observational gauge, exact identifiability, sharp quotient stability, held-out predictions, and exact noiseless configuration complexity. Smooth hidden relaxation generates interactions through inverse hidden-state stiffness. Building on this structural law, we prove an observable-readout transfer theorem: arbitrary smooth update or trace readouts inherit an explicit five-term interaction through first and second hidden responses. Unlike the reduced optimal value, a general readout has no universal interaction sign. Its Boolean effects remain exact integrals of continuous interaction curvature and can therefore be identified by factorial interventions. We also derive Gaussian quotient minimax risk, exact confidence sets and tests, misspecification decomposition, certified downstream decisions, and optimal replication. A controlled real-data experiment on a 65-dimensional strongly convex logistic model validates the complete reduced-value chain. Boolean effects and independently integrated curvature agree within 4.21e-11, while nine held-out continuous intensities agree within 8.88e-13. Gaussian campaigns attain the predicted coverage and power, and 4,500 real-minibatch observations reject an order-two interaction model. Neural trace audits provide complementary evidence that the declared response classes remain informative in nonconvex training.