Ch. 13 frames a "gap" as A ⋡ B in the partial order of diagrams under the implemented rewrites F. An additive monotone (T-count, spider-count) provably non-increasing under Fcertifies a gap as real rather than conjectured. The numbers below are read directly off the Guppy emit functions — same source of truth the v0.3 predictors use.
n = 3 data qubits
Kernel
cphase
CX
spiders
T-count (UB)
G1 · modal projector
3
9
30
1
G2-real · two-coset combiner
9
22
49
1
G12 · modal-projection scaling
3
9
30
1
G3 · per-window cost
3
9
30
1
n = 4 data qubits
Kernel
cphase
CX
spiders
T-count (UB)
G1 · modal projector
6
18
58
3
G2-real · two-coset combiner
14
35
81
3
G12 · modal-projection scaling
6
18
58
3
G3 · per-window cost
6
18
58
3
n = 5 data qubits
Kernel
cphase
CX
spiders
T-count (UB)
G1 · modal projector
10
26
87
6
G2-real · two-coset combiner
20
47
114
6
G12 · modal-projection scaling
10
26
87
6
G3 · per-window cost
10
26
87
6
n = 6 data qubits
Kernel
cphase
CX
spiders
T-count (UB)
G1 · modal projector
15
39
129
10
G2-real · two-coset combiner
27
64
160
10
G12 · modal-projection scaling
15
39
129
10
G3 · per-window cost
15
39
129
10
n = 7 data qubits
Kernel
cphase
CX
spiders
T-count (UB)
G1 · modal projector
21
51
172
15
G2-real · two-coset combiner
35
80
207
15
G12 · modal-projection scaling
21
51
172
15
G3 · per-window cost
21
51
172
15
What the monotones certify
Spider count is a sum over local emit calls. It is invariant under any rewrite in F that preserves connectivity (spider fusion is the archetype). Strictly increasing across kernels means the diagrams are not inter-rewritable in F.
T-count (upper bound) counts cphase rungs past depth 2 (i.e. past the Clifford fragment {0, π/2, π, 3π/2}). Non-increasing under the Clifford-ZX rewrites (PQP Thm 9.128).
For the current kernel set, G1, G12, and G3-cost are diagrammatically identical (same QFT skeleton); G2-real is strictly larger on every monotone for n ≥ 3. That is a real, certified separation — not a conjecture.
Per-kernel notes
G1 · modal projector: One real-QFT register, no label qubits. Closed form is two delta peaks. kernel page →
G2-real · two-coset combiner: Two label qubits, two cphase ladders, CX(lbl0,lbl1)+H(lbl1)+measure glue. Closed form: four delta peaks. kernel page →
G12 · modal-projection scaling: Same diagrammatic skeleton as G1; G12 is the scaling sweep at N ∈ {64, 128}. kernel page →
G3 · per-window cost: Same shape as G1 plus classical post-selection (a bastard-spider output, no extra quantum gates). kernel page →