Coarse-Graining a Selector Can Only Shrink Its Arithmetic Spectrum
Sequence claim receipt: seqclaim-2e62115375ff259877df282f.
Sequence claim idempotency key:
grothendieck-ledger-selector-data-processing-resonance-inequality.
Data-processing law
Let (c:G\to X) be a selector and let (d=f\circ c) be any deterministic post-processing. Then
where (K_c) and (K_d) are the normal-core terminal kernels of Ledger 1287. Combining this with Ledger 1285 gives
Deterministic coarse-graining can therefore only enlarge the invisible quotient kernel and shrink the compatible power--Mackey spectrum. If (f) is injective on the image of (c), equality holds throughout.
Exact strict chain
For (G=C_6), take the fully labelled selector, post-process it to parity, then to the constant selector. Their terminal kernel orders are
their resonance radicals are (1\mid3\mid6), and their operation spectra are all indices, indices prime to three, and indices prime to six. Exact testing through index 24 makes both spectrum inclusions strict.
Scope and verification
This is a coefficient-side data-processing inequality. A deterministic post-processing is not thereby a source-authorized physical constructor, and no Betti relative-chain map or physical pairing is supplied.
- Proof packet:
research/grothendieck/selector-data-processing-resonance.md. - Checker:
research/grothendieck/checkers/selector_data_processing_resonance.py. - Exact checker result: kernel orders (1,3,6), radical labels (1,3,6), and spectrum sizes (24,16,8) through index 24; all assertions pass.
- Epistemic graph theorem, strict control, and source admission: event 1289.
- No site build was run, by operator instruction.