One-bit slot conservation

Exact statement

Prefix budgets satisfy $2B_{ub}=B_u-\varepsilon_bD-bG_{L,q}$ . Sending a child slot $\nu$ to the parent slot $\varepsilon_b+2\nu$ gives disjoint parity images, so $N^{\mathrm{slot}}_{u0}+N^{\mathrm{slot}}_{u1}\le N^{\mathrm{slot}}_u$ and the level potential is nonincreasing.

StatusProved; equality cases refute uniform one-step contraction; internally audited

External reviewNo documented external or specialist review of this FPRD result is recorded.

Context

This turns one presentation step into an injective accounting map for abstract modular lift slots.

Definitions

$S_u^{\mathrm{slot}}=\{m:0\le m<2^L,\ Dm\le B_u\}$ .
$\varepsilon_b=b\mathbin{\mathrm{xor}}(y_u\bmod2)$ ; the odd branch pays $G_{L,q}=3^{q-1}(2^{L-q}-1)$ when $q\ge1$ .

Hypotheses and scope

Fixed-weight-infeasible children have empty slot sets.
The odd toll is evaluated only on a feasible odd branch.

Proof or evidence

The two child maps land in opposite parity classes of the parent slots, so their images are disjoint. The recurrence shows that the odd toll and feasibility checks can only remove slots; summing the local inequality gives a nonincreasing level total.

One-bit recurrence and slot proof →

Verification notes

The parity injection, q=0 boundary, and equality nonconclusion were checked against the guide. The checker reran 77,022 budget-recurrence prefix nodes and reproduced equality cases in critical layers.

Limitations

This is additive conservation, not uniform multiplicative decay.
The abstract slot count is not the lowering-capped count used in FPRD-T143.

Open work

Do not seek a universal one-step factor loss; any improvement must use block structure or a stronger joint statistic.

Notes

This is an additive conservation law for abstract modular lift slots. Exact equality occurs in some critical layers, so the theorem explicitly does not imply a uniform one-step multiplicative contraction or finiteness of accepted traces.