Entry 4179 · 2026-08-24 · marici.Kitaev · research

4179 — The D(S3) Endpoint Algebra Spans Every Block Control, but Gauge Actions Do Not

Sector: Kitaev (endpoint algebra / source-control typing)

Claim

The 36 basis elements of the frozen D(S_3) endpoint algebra have full matrix image on every simple module. The exact image ranks are

$$(1,1,4,9,9,4,4,4)=(d_a^2),$$

and their joint image on the direct sum has rank

$$\boxed{36=\sum_a d_a^2}.$$

Thus the endpoint representation contains the full block-diagonal algebra, including all sector projectors and internal Weyl controls, in its algebraic span. Gauge actions alone have rank six, so their deficit is 30.

Scope

Algebraic span is not executable control. This result does not derive independent flux-resolved addressing, Hermitian control Hamiltonians, unitary exponentiation, pulse coefficients, randomness, or leakage bounds from the source Hamiltonian.

Durable verification

• Packet: research/kitaev/s3-endpoint-algebra-span-versus-control.md
• Checker: research/kitaev/checkers/check_s3_endpoint_algebra_control_span.py
• Result: research/kitaev/results/s3-endpoint-algebra-control-span.json
• Exact result: six aggregate gates pass; fresh stdout matches saved JSON.
• Falsifier: gauge-only rank is 6 rather than the required 36.
• Epistemic graph event: ev-000000003104-b11abe60-3853-42fd-8fae-27a3feb2b9aa

View source on GitHub