Same coarse code. Different retained histories.
A center-star and a barycentric refinement preserve the old tetrahedron. Matching frequencies permits an unrestricted encoding, but preserving vertex symmetry forbids that complete encoding under the declared metrics.
Compare every retained direction
| Carrier | Total directions | Details | √5 detail pairs | S4-trivial √5 pairs |
|---|---|---|---|---|
| Original tetrahedron | 15 | 0 | 0 | 0 |
| Center-star 1→4 | 29 | 14 | 7 | 1 |
| Barycentric | 149 | 134 | 10 | 0 |
The common coarse code retains seven frequency2 pairs and one harmonic direction. The star adds seven √5 pairs. Barycentric has enough matching frequencies, but none of its √5 modes is invariant under all24 original vertex permutations.
These spectra belong to H=RᵀR+QᵀQ, Q=I−SR and the corresponding weighted generator. They are not raw unit-metric fine spectra.
Which notion of equivalence?
| Comparison | Result | What it retains |
|---|---|---|
| Coarse transfer SsRb | Exact, but lossy | Old15 directions; kills134 barycentric details |
| Full terminal bijection | NO-GO | Dimensions29 and149 differ |
| Star→bary generator embedding | Exists without symmetry | All29 source directions; nonunique spectral encoding |
| S4-equivariant generator embedding | NO-GO under this contract | Required invariant √5 pair is absent from target |
A comparison must preserve both the stated dynamics and the required symmetry representation. Frequency counts alone do not establish history covariance. Changing metrics, rates, symmetry demands or adding carriers requires a new test.
Keep the outgoing record
A reversible transfer can retain all data by adding14 target-detail directions and keeping134 old-detail directions as an outgoing record.
149 input +14 prepared →29 target +134 record =163 directions
(Sb c+zb, ws) →(Ss c+ws, zb)
Both sides have squared norm ||c||²+||zb||²+||ws||². With blank ws=0, the old detail record still exists. This is an information-preserving direct-sum transfer, not an acquired clock, physical gate, complete generator equivalence or free reset.
New coordinates do not create a detail source
If A is H-skew and AS=SA0, the coarse image and its orthogonal detail complement are invariant. A blank detail sector stays blank. Exact inheritance therefore cannot also supply autonomous coarse-to-detail excitation.
A closed skew coupling C can seed detail only with reciprocal coarse feedback and a nonzero inheritance defect. External forcing f instead gives ż=Adz+Qf and a norm-work rate2〈x,f〉H. Either route needs a selected source law and acquisition ledger.
Source C0656 funds its1→4 event with a same-address C4 launch stock3q². That historical ledger is not the new metric detail norm or a conversion into Qf. A moving metric, prepared seed or recorded jump remains another explicitly typed option.
Next: declare the excitation interface
Specify the seed, forcing, coupling or jump; state clock/event conditions, retained records, donor and reset costs; fix the observer. Then test symmetry and generator compatibility before any native or physical claim.
This comparison advances the development lane only. Same-sequence hierarchy coherence remains intact; native metric, observer and detail-source adoption remain open.