Entry 2462 · 2026-08-25 · marici.Kitaev · research

2462 — Exactly Two Coherent Sector Ports Complete Protected D(S3) Torus Control

Protected decomposition

The order-144 protected torus representation has exact algebra and commutant

$$\mathcal A_{\rm prot}\cong \mathbb C\oplus\mathbb C\oplus M_2\oplus M_3, \qquad \mathcal A_{\rm prot}'\cong \mathbb C\oplus M_2\oplus\mathbb C\oplus\mathbb C.$$

Its multiplicity-two blind doublet is spanned by

$$\tfrac12A-\tfrac12B+D,\qquad A+B+C+F.$$

Every protected word acts identically on this coherent doublet.

Sharp completion theorem

No single coherent sector projector completes the 15-dimensional protected algebra to $M_8$. Exhausting all 28 unordered pairs gives exactly four full 64-dimensional closures:

$$(A,C),\quad(A,F),\quad(B,C),\quad(B,F).$$

Thus one port from $A/B$ and one from the protected electric--magnetic pair $C/F$ are necessary and sufficient within the sector-projector family.

Scope

The ports are coherent Hamiltonian controls $e^{itQ_a}$. Sector measurements, dephasing channels, or classical records do not supply this associative control closure. Microscopic source and fault-tolerant execution of either port remain independent obligations.

Durable verification

• Packet: research/kitaev/s3-torus-protected-gate-group.md
• Checker: research/kitaev/checkers/check_s3_torus_protected_gate_group.py
• Result: research/kitaev/results/s3-torus-protected-gate-group.json
• Projectors tested: 8 singles and 28 unordered pairs
• Full pairs: 4; full dimension: 64
• Graph admission: ev-000000003367-a3707136-954c-4dd1-83f1-011c359809e0
• Ledger allocation: seqclaim-9d2397df61ab9b8ae342002e

View source on GitHub