Recovery budget as two nonnegative costs · FPRD-T140

Context

The identity separates completion viability into local scalar loss and a boundary-lift charge, making deterministic-tail conditions explicit.

Definitions

$C^+_{L,q}=2^{L-q}(3^q-2^q)$ is the unique maximal affine constant at suffix length $L$ and weight $q$ .
$\delta_z=C^+_{L,q}-c_z$ , $\mu_z\equiv3^{-p}(r_z-y_u)\pmod{2^L}$ , and $B_u=3^qy_u+C^+_{L,q}-2^Lr_u$ .

Hypotheses and scope

$D=2^K-3^S>0$ .
The suffix must still have the required length and fixed weight.

Proof or evidence

Insert $c_{uz}=3^qc_u+2^jc_z$ and $r_{uz}=r_u+2^j\mu_z$ into $c_{uz}-Dr_{uz}$ , then use $3^pr_u+c_u=2^jy_u$ and $c_z=C^+_{L,q}-\delta_z$ . Nonnegativity gives pruning and the bound $1+\lfloor B_u/D\rfloor$ on surviving lift classes.

Prefix recovery calculus and proof →

Verification notes

The algebra and all stated consequences were rechecked. The checker reran 370,252 split instances and 340 nonnegative-score splits. The page explicitly retains the weight/length feasibility condition for the forced suffix.

Limitations

Finite checks are regression evidence, not the all-parameter proof.
A unique modular lift is not automatically a feasible suffix word.

Open work

Use the exact decomposition only with joint guard–disorder information; marginal counts have already failed to give contraction.

Notes

The proof expands the exact concatenation and guard-lift formulas and separates two nonnegative costs: local completion deficit and boundary lift. The 370,252 checked split instances corroborate the identity but do not prove it. The deterministic suffix must still meet the fixed-weight and length requirements.