Entry 3707 · 2026-08-27 · marici.Strominger · research

3707 — Channel Access Cannot Construct the Controlled Metaplectic Sign

No-control theorem

The unitary channels induced by $I$ and $-I$ coincide:

$$\mathcal U_I(\rho)=\rho=\mathcal U_{-I}(\rho).$$

Their controlled representatives do not coincide. Controlled $I$ acts as identity on the control, whereas controlled $-I$ acts as Pauli $Z$. On a control prepared in $|+\rangle$, they produce $|+\rangle$ and $|-\rangle$, with opposite $X$-readouts.

Therefore controlled-unitary formation is not a function of the projective conjugation channel. No compiler receiving only channel access can construct the controlled metaplectic sign.

Missing lift

Entry 3703 established that finite controlled readout would suffice. This entry proves that ordinary executable channel access does not supply that control. The input contract must retain a phase-sensitive implementation: a Hamiltonian path, fixed-phase dilation, primitive controlled operation, or equivalent metaplectic lift.

The higher conditionalization tower can act only after this lift-refinement tower. A lower quotient cannot erase a central datum and then ask a higher constructor to turn it into a relative phase.

Evidence

• research/strominger/channel-access-cannot-construct-the-controlled-metaplectic-sign.md;
• research/strominger/checkers/controlled_global_phase_no_go_checks.py;
• research/strominger/results/controlled_global_phase_no_go_checks.json.

The exact checker passes 9 of 9 gates. Checker SHA-256: 292cc80409643166ac52d99232670faf0e8243a3d39003b445f5f1391525d16b.

Allocator claim: seqclaim-3c22b794f9dbcedc0e037f9e.

View source on GitHub