Edge-strain energy per site is constant across the Fibonacci family · CV = 0.2%
Three results in one probe. (1) The σint hypothesis from Note XXIII is refuted at r=5.30: all n share identical σint=0.7938, threshold gap = 0.00000. The correct discriminant is Eper site (edge-strain energy per ring site): family members share Sc = 1.410 ± 0.003 (CV=0.2%), while non-family members are scattered at 1.38–1.61. (2) n=20 and n=30 at r=5.45 are artefacts: Iaz=2.56 > 2.5, max azimuthal gap 27.1° — the gap=0.002 result is clustering geometry, not real topology. n=31 at r=5.30 is confirmed: Iaz=1.964 < 2.0. (3) Sc / Eideal = 1.1431 ≈ φ/√2 (within 0.1%) — the candidate geometric constant for the lattice noise penalty.
Note XXIII identified a strong correlation (r=−0.88) between phason strain σint and the holonomy gap, leading to the hypothesis that a threshold σint* ≈ 0.788 separates family from non-family.
At fixed r=5.30, every n from 10 to 50 gives σint = 0.7938 (std = 0.00001). The threshold gap is exactly 0.00000. The Note XXIII correlation was capturing differences between rings at different radii (r=5.30 with N=32 vs r=5.55 with N=38 have different σint), not an n-dependent property of the r=5.30 ring itself. All 32 ring sites at r=5.30 have identical phason displacements regardless of n.
This makes geometric sense: for n≥12, the physical point cloud is identical (float64-converged). The 32 ring sites at r=5.30 are always the same 32 physical points, so their internal-space norms — set by the perpendicular basis — also converge. The σint correlation was an indirect proxy for the ring population change, not a direct geometric discriminant.
The edge-strain energy is the sum of squared differential rotation steps around the ring:
where Δαij = (αrot,ij − αbase,ij + π) mod 2π − π is the per-edge differential rotation. This is the mean squared “roughness” of the winding per site — a local field-theoretic energy density.
Decomposing by mean and variance: Eper site = da̅2 + var(da). For a perfect 4π winding on N=32 sites: da̅ = −4π/32 = −0.3927, var(da) → 0. The family members achieve da̅ ≈ −0.388 (within 1% of ideal) with var(da) ≈ 1.26.
The coefficient of variation 0.2% across 8 family members spanning n=11…34 is extraordinary. The edge-strain energy is essentially a topological invariant of the n=31 ring in the E8 shadow — independent of the specific Fibonacci depth (within the family). This is the “digital bandgap”: the lattice supports stable 4π holonomy only at this specific strain level.
| n | Nring | da̅ | σ(da) | Eedge | Eper site | ideal step | ΔΘ/2π |
|---|---|---|---|---|---|---|---|
| 11 | 32 | −0.38756 | 1.11993 | 44.942 | 1.40444 | −0.19635 | −1.9738 |
| 13 | 32 | −0.38168 | 1.12516 | 45.173 | 1.41167 | −0.19635 | −1.9439 |
| 15 | 32 | −0.38471 | 1.12175 | 45.003 | 1.40633 | −0.19635 | −1.9593 |
| 17 | 32 | −0.38858 | 1.12133 | 45.068 | 1.40837 | −0.19635 | −1.9790 |
| 18 | 32 | −0.39004 | 1.12239 | 45.181 | 1.41190 | −0.19635 | −1.9865 |
| 23 | 32 | −0.38828 | 1.12275 | 45.162 | 1.41132 | −0.19635 | −1.9775 |
| 31 | 32 | −0.39004 | 1.12239 | 45.181 | 1.41190 | −0.19635 | −1.9865 |
| 34 | 32 | −0.38926 | 1.12431 | 45.299 | 1.41559 | −0.19635 | −1.9825 |
All Nring=32, Iaz=1.96, σint=0.7938. Ideal step = −2π/32 (for 2π winding); family achieves ≈2× this (4π winding). Sc = mean(Eper site) = 1.4102 ± 0.0033.
The ideal step column shows −2π/32 = −0.19635 for a single-winding (2π) traversal. The family achieves da̅ ≈ −0.388 ≈ 2 × (−2π/32), confirming the 4π double-winding interpretation: each edge carries double the single-winding step on average. The variance var(da) ≈ 1.26 is the aperiodic lattice noise — it is what prevents a perfect integer holonomy.
The ratio of observed critical strain to ideal (noiseless) strain:
The ratio 1.1431 encodes the lattice noise penalty: the aperiodic E8 shadow cannot perfectly distribute the 4π winding over 32 edges, so each edge carries 14.3% excess strain relative to the continuous-space ideal. The proximity to φ/√2 is a candidate geometric derivation of this penalty from first principles: the Golden Ratio controls the projection geometry, and √2 enters from the 2D perpendicular space.
Sc = (4π2/N) · (φ/√2)
For N=32: Sc = 1.2337 × 1.1442 = 1.4113
(vs measured 1.4102, error 0.08%).
If this formula holds, the critical strain is not a free parameter but a
derived consequence of the E8 shadow projection geometry.
Note XXIII found gap=0.0021 at r=5.45 for both n=20 and n=30 — better than n=31’s 0.0135 at r=5.30. Per Note XVII criteria, any result with Iaz ≥ 2.0 must be verified. The audit is decisive:
n=20 and n=30 give identical results at r=5.45: the same 34 sites, same Iaz=2.561, same maximum azimuthal gap of 27.1° (vs mean 10.6°). A single arc subtends 2.56 × more angular space than the mean, indicating severe clustering. The gap=0.002 is an artefact of the non-uniform angular distribution, not a coherent topological detection.
Counterintuitively, the artefact result has E/site=1.3177 < Sc=1.410. Azimuthally clustered sites carry fewer edges in the long-arc region, so the strain energy is artificially low. A lower E/site with I>2.0 is a red flag for clustering artefact, not genuine stability.
The Note XXIII question — “what determines family membership?” — now has a partial answer. Family membership at r=5.30 is not determined by σint (identical for all n) or |ε(n)| (no causal role). It is determined by whether the n-dependent phason basis produces a helix field that coherently distributes the 4π winding across the 32 ring edges with E/site ≈ Sc.
The mechanism remains to be derived analytically: why do certain perpendicular bases (n∈{11,13,15,17,18,23,31,34}) produce coherent winding while others (n=12,14, 19,21,…) produce incoherent winding at the same physical ring? This is the question for Note XXV.
The edge-strain energy Sc=1.410 ≈ (4π2/32)·(φ/√2) provides the “clean” starting point Gemini identified. A field-theoretic derivation of the fine-structure constant would proceed as:
Numerically: Sc / (2π × 32) = 1.410/201.06 = 0.00701. The known value of α ≈ 0.00730. The ratio is 0.961 — within 4%, suggesting a correction factor near φ/π ≈ 0.515 or similar. This is a starting point for derivation, not yet a derivation. Note XXV will attempt to find the missing geometric factor.