Medical image classifiers typically return predictions without a native contract for independently reconstructing their spatial decision computation. Evidence Field is a trainable spatial-program architecture that jointly produces class scores, selected covering programs and signed evidence. Its encoder combines absolute and image-relative response fields with associative spatial moments and learned characteristic sums. A typed classifier scores individual regions and parent-conditioned multiplicative groups; a covering recursion selects scale, partition and composition. We establish exact one-fold coverage, optimality within the registered grammar, a bound relating smooth training to hard inference, signed-score conservation and class-centred offset invariance. Direct contractions and their adjoint specify arithmetic and storage scaling on two- and three-dimensional grids. Native output records bind inputs, parameters, supports, types and contextual dependencies to individual and ensemble predictions. In a separately reported histopathology application, patient-separated training and evaluation across held-out centres provide empirical evidence for the architecture. Trained mechanism and corrected-loss comparisons distinguish predictive effects from implementation correctness; independent readers reconstruct descriptors, routes, scores and probability mixtures. The principal contribution is a coupled representation, decision and evidence architecture whose selected computation can be reproduced independently.
Figures & tables
Fig. 1: EF architecture, read bottom to top. Ck(v) is the best partition/packing continuation score; tracing Fk yields program Πk . The right branch exposes training and independent replay.
Symbols
Meaning
d,N
Grid dimension and number of samples in Ω .
R,L,K
Response fields, learned frequencies and output classes.
H,A,b,J
Grammar depth, partition alternatives, b=2d children, maximum group order.
Mn,Mg,D
Own/group score types and region descriptor width.
v,π(v),mv,ℓv
Region occurrence, parent, midpoint and half-width vector.
TABLE I: Notation used by the architecture and its equations.
Figure 3
Fig. 4: Complete-score operands, partial composition occupancies, signed conservation and class-centred alignment. Group scores retain their member anchors.
Evaluation
N (patients)
EF
CNN
ViT
CAMELYON centre 1
34,904 (8)
0.9356 / 84.46
0.7352 / 58.94
0.8502 / 76.13
CAMELYON centre 2
85,054 (9)
0.9453 / 86.77
0.7656 / 67.60
0.7924 / 67.27
TABLE II: Histopathology companion: raw three-member ensemble AUROC / accuracy (%), fixed argmax. N counts patches, parentheses patients; EF is source-selected paired CE.
TABLE IV: Hash-bound CAMELYON17 implementation audits: two source loss inputs, both held-out centres for replay. Finite-difference error is relative; others absolute. Loss checks have no optimizer steps.
Fig. 5: Worked evidence: central- 322y=0 , seed 11’s 962 class-1 cover and signed margin N(E10−E00) . Units are supports, not cells. Raw ensemble Q1=0.88756339 is incorrect.