Changing the metric needs a new test.
The sources contain more than the unit-metric carrier. Correctly normalizing their operators reveals three distinct obstructions to the full default clock, including a geometric family with five bands.
Keep the two metric conventions separate
| Declared evolution | Normalized degree coupling | Scale every metric by s |
|---|---|---|
| Fixed boundary B, with its G-adjoint | √(g(k−1)/gk) | Rates unchanged |
| Mass equation G ċ=A0c | 1/√(g(k−1)gk) | Rates divided by s |
The historical inverse-product formula belongs to the mass equation. C0496 already repairs that distinction. Both use a conserved G pairing; neither chooses a physical norm or energy.
Read the source correction →Which full-generator routes are excluded?
| Source family | Obstruction | Clock scope |
|---|---|---|
| Real S4-equivariant skew generator, any positive invariant metric | At most three positive bands | Even n≥8 |
| Fixed geometry, Gram Rule A or B | Five bands obey u/v+x/y=2; all 120 default assignments fail | Default n=10; band count for n≥12 |
| Degree-weighted K− or K+ | Irrational paired ratios allow at most three clock-grid matches | Even n≥8 |
Copies expand multiplicities and preserve the frequency set. Heterogeneous or coupled copies, a different complex, and a broader declared operator need fresh premises. The geometric result leaves n=8 unresolved.
The three-band default comparison has an exact relaxed residual floor η/κ=1, attained by free centers 3,7,10. These are mathematical witnesses, not adopted couplings. A residual floor does not bound actual endpoint error from below.
Inspect the source-defined geometric candidate
C0173 starts from a stretched tetrahedron with q=r²>0. Rule A uses normalized simplex Gram determinants; Rule B squares them. Both are source-authored candidates. A nonregular geometry can split the spectrum into five bands, so the three-band argument does not apply.
Frequencies use the declared operator's rate unit. The shape controls inspect conditional source formulas; they select no native geometry or clock calibration. The proof covers every q>0, not just these examples.
The exact five-band proof
To match all five positive clock frequencies, their squared targets {1,4,9,16,25} must occupy the five distinct native bands up to one common factor. That factor cancels from the ratio identity. None of the 120 assignments of four distinct targets to U,V,X,Y satisfies U/V+X/Y=2. Degenerate bands fail the frequency count instead.
The minimum rational identity gap 1/144 certifies exact inequality. It is not an operator residual or endpoint-error bound.
The next admission frontier
A heterogeneous or coupled carrier, a different source complex, or an explicitly weaker prepared-window/endpoint contract can leave these families. Each route needs a declared operator and complete encoding, with clock preparation, fresh donors, retained outgoing records, readout and full restart resources still accounted for.
Exact algebra and separate constructor-team checks are available below. Human review and physical calibration remain pending. Earlier finite-window and endpoint-alias results retain their own scope.
New: source subdivision, three compatibility tests and retained complement →