Treat symbolic dimensions as a domain with obligations

Let X have shape [B,4] and W shape [4,3], with integer B between one and sixty-four. The product has shape [B,3] because the two contraction dimensions equal four. If a bias has shape [3], row broadcasting can preserve that output shape under the chosen operator semantics. Giving two unrelated input axes the same symbol asserts a relationship; it is stronger than seeing both equal eight in one example. Independent symbols should remain independent unless the contract or a checked condition links them.

Shape inference should also track the domain in which expressions are meaningful. Dividing a dimension by two requires more than writing a fraction in a type annotation when the target dimension must be an integer. Zero-length dimensions, symbolic rank, and value-dependent output sizes have different support across systems and operations. This lesson deliberately uses positive batch sizes and fixed rank. If a production API accepts empty batches, test and represent them explicitly rather than quietly importing this narrower domain into its interface.

A contiguous X of shape [B,4] can be logically reshaped to [B,2,2] because both contain four B elements. By contrast, grouping the rows into blocks of four to obtain [B/4,4,4] requires B divisible by four. If B is six, silently truncating B/4 discards a contract obligation; it does not solve the reshape. A general implementation could use a different representation or handle a remainder, but that is a different program from the unmasked full-block path.

PyTorch 2.9 export and dynamic-shape documentation describe recorded shape conditions associated with a captured program. JAX's shape-polymorphism documentation also explains that its symbolic reasoning is intentionally limited. Failure to prove an inequality can therefore mean analysis is inconclusive, even when a human can establish it mathematically. Conversely, a sample input satisfying a condition does not prove every declared input satisfies it. Keep the proof, runtime check, rejected input, and unsupported analysis outcome as distinct cases.

Construct shape tests around each obligation: the minimum and maximum batch, a valid multiple of the tile size, and a nearby invalid remainder. Check shared-symbol relationships by deliberately varying only one input dimension. The expected response outside a specialized domain might be rejection, a guarded alternate artifact, or a remainder-capable path. It must be specified rather than inferred from a successful in-domain run. An unchecked assumption is particularly dangerous when it controls memory indexing instead of merely changing a diagnostic message.

The modular lab isolates the remainder arithmetic used by one divisibility condition. Reset, set both modulus m=4 and additive step a=3, predict residues for candidate batches B=3n, then Step four times. Each half-second Step advances one addition, giving residues three, two, one, zero. Reset and set both m=4,a=4 to compare batches divisible by four at every positive step. This bounded integer orbit does not propagate tensor shapes, prove range constraints, or validate a real exported artifact.