Loading…
Honest scope
φ is the measured quantity here. The topological entanglement entropy comes out as γ = log(2+φ); run the Z₂ toric code through the identical code path and you get log 2 instead. Remove φ and the signature changes, so this passes the falsification test.
Fibonacci string-net
…
γ = log(2 + φ) ≈ 1.292
Z₂ toric (surface-equivalent)
…
γ = log 2 ≈ 0.693
Model-swap shift
…
γfib − γZ₂ = log((2+φ)/2)
Full-rank recovery
…
γ vs analytic at full χ
Measured γ: Fibonacci log(2 + φ) vs Z₂ log 2
Fibonacci (d = φ)
Z₂ toric
analytic reference
Spectrum Collapse: γ vs kept bond dimension χ (full-spectrum invariant) topological order = compositional
What this shows: φ in a measured invariant

The topological entanglement entropy γ is the universal constant subtracted off the area law of the entanglement entropy: S(A) = α·|∂A| − γ. It is a measured fingerprint of topological order: γ = log D, where D is the total quantum dimension. For the Fibonacci string-net, D = √(1 + φ²) so γ = log(2 + φ) ≈ 1.292; for the Z₂ toric code, D = 2 so γ = log 2 ≈ 0.693. The bars show the computed γ landing on the dashed analytic reference, and the model-swap shift γfib − γZ₂ = log((2 + φ)/2) is exactly the golden-ratio signature.

The collapse (lower chart) is the deeper point: γ is a full-spectrum quantity. As you truncate the entanglement spectrum to bond dimension χ, γ collapses toward 0: at χ = 1 (a product state) there is no topological order at all, and the true log(2 + φ) is recovered only at full rank. (χ = 0 here denotes the full, untruncated spectrum, the rightmost point.)

Where φ is load-bearing: in the value of the measured invariant itself: log(2 + φ) is golden, log 2 is not. And the collapse makes the honest scope explicit: topological order is a compositional, full-spectrum property, not something any single truncated number captures. Computed live in your browser via WebAssembly.

Honest scope. Both string-nets are simulated. γ is extracted from the exact entanglement spectrum; the analytic references are γ = log(2 + φ) (Fibonacci) and γ = log 2 (Z₂). The collapse curve is a controlled truncation of that spectrum, not measurement noise; it demonstrates that γ is a property of the whole spectrum, recovered only at full rank. Not realized hardware.