An operation's input-output function does not, by itself, determine its computational role within a represented process. That role also depends on the processes producing its inputs and receiving its outputs, a philosophical stance termed deep learning externalism (DLE). We develop a mathematical account of represented computation that distinguishes functional equivalence, retained predecessor distinctions, restricted receiver capability, and context-dependent effective architectures. The resulting equivalences depend on which computational relations a scientific description must preserve. We prove that two injective producer contexts can preserve every predecessor distinction while changing whether fixed local and attention receiver families are equivalent. We investigate these relations in a dimension-labelled PreNorm Transformer and their evolution under training. Residual bypasses restrict complete-state information identifications to two merges per block, while finer branches can change receiver accessibility. Exact gradient-descent models yield covariance-dependent geometric attenuation under learned producer routing and algebraic (t−1/2) attenuation under Q/K-softmax training, with optimal bounded-receiver recovery errors derived in both cases. Coupled producer-attention dynamics establish how learning jointly reorganizes the presented features and receiver allocation. Finally, present functional or receiver-profile equivalence need not determine future behavior under an optimizer. In an input-conditioned attention model, two accessibility measurements form an exact reduced predictive state, whereas one alone does not.
Figures & tables
Figure 1: The left column passes from a represented cut to its realized composite and the equivalence relation induced on predecessor states. The right column starts from a complete training state, its transition kernel, and a selected observable, and derives the information required to predict that observable. The two constructions retain different information about the same represented computation.
Figure 2: Two injective producers present the same predecessor information differently. With fixed local and attention receiver families, the encoded presentation gives identical distinction profiles. The separated presentation gives the attention family access to v , which is unavailable to the token-1 local receiver. Thus receiver equivalence changes without loss of predecessor information.
Figure 3: A represented interface has two external relations. The producer prefix and receiver continuation are not determined by the isolated carrier state.
Figure 4: Two residual merges in one PreNorm Transformer block. At a fine cut inside either branch, the carried residual remains separately present. Whole-state identification can occur at the two merges, while normalization, attention routing, hidden FFN computation, and their receiving gradients have finer, receiver-relative semantics. Backpropagation transports covectors through the reversed represented graph.
Figure 5: The derived gradient-flow field for the coupled producer orientation c=coss and attention allocation p . The dashed curve is the moving allocation optimum p∗(c)=(1+c)/2 , not a set of joint equilibria. Trajectories approach aligned producer features and saturated source allocation at different rates. Parameters used for illustration: γs=γα=τ=1 .
Figure 6: Illustrative exact discrete learning trajectory of the producer-dependent Q/K model, showing p+ and p− , which equal the optimal unit-Lipschitz recovery errors for the second source on the two input regimes. The same learned attention parameters produce different receiver accessibility because the upstream representations differ.