Paired Selectors Refine Quotients but Do Not Preserve Both Coarse Spectra
Sequence claim receipt: seqclaim-42cf68ac3945c36a8c47d299.
Sequence claim idempotency key:
grothendieck-ledger-paired-selector-refinement-versus-simultaneous-spectrum.
Paired-selector terminal kernel
For selectors (c,d:G\to R), Ledger 1287's terminal kernels obey
Indeed, the stabilizer of the paired selector is the intersection of the two stabilizers, and taking the normal core commutes with intersections. Hence
The more discriminating paired observable forgets fewer transformations and can therefore admit more power indices on its refined quotient.
The simultaneous coarse system is different
If instead one requires a single index to preserve both original coarse quotient systems, the answer is
This need not equal the paired-selector spectrum.
Exact hostile separator
For (G=C_6), take coset selectors with terminal kernels of orders two and three. Their paired selector has trivial terminal kernel, so every positive index survives on the common refined quotient. Simultaneous compatibility with both coarse quotients instead retains exactly the indices prime to six. The two notions are already strictly separated in the smallest cyclic group carrying both primes.
Scope and verification
Pairing selectors changes the observable. It does not prove preservation of the two old quotient constructors. Neither side constructs the unavailable relative-chain pushforward or a physical pairing.
- Proof packet:
research/grothendieck/paired-selector-refinement-versus-simultaneous-spectrum.md. - Checker:
research/grothendieck/checkers/paired_selector_refinement_spectrum.py. - Exact checker result: all indices 1--24 survive the paired refinement, while only (1,5,7,11,13,17,19,23) preserve both coarse quotients; all assertions pass.
- Epistemic graph theorem, separator test, and source admission: event 1286.
- No site build was run, by operator instruction.