R12 Compiler-Prior No-Go

Status: exact collapse theorem. Recurrence can provide a useful compiler prior or optimization bias, but it has no intrinsic sample, parameter-bit, or compute generalization advantage over a fair uniform acyclic realization.

1. Theorem

Let theta contain p learned bits and let a recurrent evaluator process a length-L input by

s_i = U_theta(s_(i-1), x_i),
y = O_theta(s_L),

where the state uses b bits at fixed precision, one transition costs work c and depth d, and the learner infers theta from dataset D and prior pi.

There is a uniform acyclic compiler that, for every L, emits

O_theta o U_theta[x_L] o ... o U_theta[x_1].

For every dataset, learner random seed, and input, the compiled evaluator has the identical output. The constructor receives no scale-specific target advice and shares the same learned parameter source nodes across all copied cells.

2. Resource ledger

Materializing every activation in parallel can use Theta(Lb) memory; sequentially scheduling the acyclic graph recovers recurrent peak memory. The real difference is reusable program description or instantiated graph area, not learned information or function class.

3. Consequences

• A single acyclic forward invocation is not a meaningful comparator if it is forbidden to contain the compiled transition chain.
• A streaming one-pass comparator already includes recurrence.
• Fixed parallel depth can yield classical circuit-depth separations, but that is a depth claim and may reach open TC0 versus NC1 questions for strong transformer-like comparators.
• Any claimed sample or extrapolation advantage must come from the training protocol, structural prior, optimization dynamics, or a separately counted resource, not from the presence of a loop alone.

4. Small finite collapse witness

One state bit, one learned bit, two events, and horizon two already demonstrate the identity. Let theta=0 assign a=NOT, b=RESET0 and theta=1 assign a=RESET0, b=NOT, starting from zero. One observed transition identifies theta; ab returns theta and ba returns 1-theta. A recurrent evaluator uses two calls to one cell. The compiler uses two copied cells sharing the same learned bit. There is no finite horizon at which this equivalence fails.

5. Decision

Reject recurrence itself as the outstanding nonlinear-learnability separation. Preserve tied recurrence as a favorable matched control and count its compact reusable program description honestly. No CPU experiment can falsify this identity theorem; experiments may only test whether a frozen training protocol makes the shared algorithm easier to learn at matched resources.