Complete f=0 cancellation-locus classification

Exact statement

In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the eighteen pair-minimum strata whose first nonzero 3-adic digit can cancel contain exactly one solution, $(4+3)(1+8\cdot3)+3^4=2^8$ , of displayed cost $29=2K+13$ . Together with FPRD-IC-T29, the entire exceptional cancellation locus consists of this control and $(2+3)(1+8\cdot3)+3=2^7$ . Every other solution satisfies $\min(v,y,g)=1+v_3(K-r-u)$ .

StatusProved with complete finite quotients and independent reconstruction

External reviewNo documented external or specialist review of these FPRD results is recorded.

Context

The preceding theorem left eighteen pair-minimum cancellation strata. Closing them completes the exceptional first-digit mechanism in f=0 and gives the same kind of valuation control already known for f=1 and f=2.

Proof or evidence

The modulus 33215 has binary and ternary periods 36 and 12. Its complete quotient makes seven families empty and leaves 23 states; modulo 27 leaves one low/high state. A 2-adic gate forces x=3, and the three factor pairs of 175 give the unique control.

Complementary f=0 cancellation closure →

Verification notes

Independent Python and C++20 programs reconstruct the 40/20/18 canonical counts, all quotient states, the sole terminal factorization, and its thirteen-one excess. No bounded search is used for completeness.

Limitations

This classifies exceptional lowest-digit cancellations, not all f=0 solutions.
Higher minimum layers in f=1,2 and the nonexceptional f=0 strata remain open.
The normalized three-addition grammar does not cover arbitrary integer-complexity formulas.
No external review or novelty determination is recorded.

Open work

Pause layer-by-layer enumeration; reopen this branch only with an effective six-term reduction, a uniform freezing theorem, or another structural bridge.