What Survives, What Is Retracted
This paper contains one exact result that stands, one claim that is retracted, and one exploratory extension that remains open.
The retraction is documented transparently. The exact eigenvalue correspondence below is unaffected.
The Exact Eigenvalue Correspondence
The 600-cell is a regular 4-polytope with 120 vertices and 720 edges. Its vertex graph is 12-regular. The normalised Laplacian of this graph has a complete, exactly computable spectrum.
λ = 15 and ΔC = 15 are independently computed exact quantities whose equality is an observed correspondence. The eigenvalue comes from spectral graph theory; the combinatorial invariant comes from closure-class structure. Neither was derived from the other.
The F4 Fibonacci Extension
Separate from the 600-cell result, this paper explores a hierarchical construction based on F4 root-system geometry with Fibonacci-scaled levels.
At each level of the hierarchy, the effective spectral dimension is computed numerically. The observed trend:
| Level | Vertices | ds (numerical) | Status |
|---|---|---|---|
| Base | 120 | — | Retracted (artifact) |
| F4 Level 2 | 1200 | ~1.72 | Numerical |
| F4 Level 3 | 4960 | ~1.635 | Numerical |
| Conjectured limit | ∞ | φ ≈ 1.618 | Unproven |
The effective spectral dimension descends toward φ from above (ds = 1.635 at V = 4960). This is numerical motivation for a conjectural identification ds → φ. It is not a theorem.
Epistemic Status
This paper operates at three distinct epistemic levels:
| Claim | Status | Basis |
|---|---|---|
| λ = 15 = ΔC | Exact | Independent computation of both quantities |
| 600-cell ds ≈ φ | Retracted | Artifact of coarse time-windowing |
| F4 hierarchy ds → φ | Exploratory | Numerical trend, not proven convergence |
What This Does Not Claim
- That the 600-cell vertex graph has spectral dimension φ — this claim is explicitly retracted
- That the F4 Fibonacci convergence to ds = φ is proven — it is a numerical trend at three data points
- That this paper is a replacement for Paper IV's standalone result — the mass law stands independently of any spectral claim
- That the λ = 15 = ΔC correspondence implies a causal or derivational relationship — it is an observed equality of two exact quantities
An exact spectral fact, a transparent retraction, and an open conjecture.