We went looking for 0.68 and found a law instead.
Folio XI's framework says ΩΛ is a locked fraction; the question was which one. The observable-patch construction gives exactly ½ (a pure-cut theorem). Well-separated pairs give 0.83 → 1. The observed 0.68 sits between — so we measured the full f(κ) curve and asked at what separation κ* = d/R two horizon-scale patches hold an account that is exactly 68% locked. The answer was not a number. It was a scaling law.
| R | κ* measured | κ* = 2 + 1.83/R | gap (κ*−2)·R | error |
|---|---|---|---|---|
| 3 | 2.609 | 2.610 | 1.83 | 0.04% |
| 4 | 2.459 | 2.458 | 1.84 | 0.06% |
| 5 | 2.364 | 2.366 | 1.82 | 0.08% |
| 6 | 2.302 | 2.305 | 1.81 | 0.13% |
| 8 | 2.228 | 2.229 | 1.83 | 0.04% |
The discriminator was real: a continuum-geometry reading would have κ* settling at a fixed ratio (≈ 2.3); the constant-gap law instead predicted κ*(8) = 2.229 before the run, and the lattice returned 2.228. The 0.68-locus does not live at a fixed shape — it lives at a fixed absolute distance: about two UV lengths of daylight between the patches, whatever their size.
The transition is ultraviolet. That is the discovery.
State it scheme-independently, because the "1.83" is not: the number of lattice units in the gap depends on how the continuum is discretized, so 1.83 itself is regularization furniture. What survives any scheme is the scaling structure:
In the strict continuum the f(κ) cliff collapses onto κ = 2: touching patches. The structure between ½-locked and fully-locked exists only within a few UV lengths of contact.
Now run the Ledger's reading through it. If ΩΛ is the locked fraction between adjacent causal patches (Folio XI), then the observed 0.68 — strictly between the pure-cut ½ and the separated-pair ≈1 — is only attainable at UV-scale separation. The conjecture this forces is sharper and stranger than the one we started with:
Why this is not crazy on its face: ΩΛ in ΛCDM is famously a "why now / why this value" puzzle precisely because no continuum-IR mechanism naturally produces an O(1)-but-not-1 fraction. A quantity pinned between two universal limits (½ and 1) by UV-scale geometry is exactly the kind of object that could sit at 0.68 without fine-tuning anything in the IR. Why it may yet be wrong: the construction (two balls, EN/MI proxy, one scalar) is a toy of a toy, and "adjacent patches with a Planck gap" is a picture, not a derivation from the postulates.
Three computations that decide it.
Universality: is the gap-law species-independent?
Repeat the cliff scans for free fermions (sine kernel generalizes; the Folio IV thermometer machinery exists) and, harder, for the photon field. If gap(f = 0.68) ∝ ℓUV holds across species with different O(1)s, the law is a property of locking itself; if the structure differs, the scalar toy misled us.
The continuum gap-profile f(g).
The full function — locked fraction vs physical gap at R → ∞ — is the proper object behind the law (our κ* is one contour of it). Computing f(g) and its R-convergence settles whether 0.68 is a stable contour or drifts logarithmically. The FFT machinery makes L = 128 feasible.
From picture to derivation.
"Adjacent causal patches at UV separation" needs to come out of the postulates, not be drawn beside them: why would the horizon's account decompose into nearest-neighbour patch pairs with a cutoff-scale gap? Candidate route: the patch tiling is the coarse-graining the fee (Postulate 4, as amended) imposes at the horizon — the gap as the thickness of the membrane where records cross. Until that argument exists, the cosmological reading stays stamped.
What is measured, what is read into it.
- Measured (and predictively verified): gap(f = 0.68) = 1.83 ± 0.02 lattice units, independent of R across 3…8; the R = 8 value predicted to 0.04% before the run. Scripts and data public: d1_fstar.py, d4_kstar_R35.json, d4_kstar_R8.json, d3_constructions.json.
- Scheme-independent content: the 0.68-locus (and the whole ½ → 1 transition) is a UV-scale structure, not continuum geometry.
- Scheme-dependent: the value 1.83 itself; therefore also its resemblance to ln 2π. Any derivation must produce the scheme covariance, not the number.
- Proxy caveat: f uses EN/MI; EN upper-bounds distillable entanglement, so the locked fraction is if anything underestimated. The interpolation crosses a steep cliff — though the law's 5-point consistency suggests the crossing is well-behaved.
- One scalar, one construction. The cosmological reading additionally assumes the Folio XI closure (Hubble-cutoff, at-par) and the adjacent-patch decomposition — a picture awaiting derivation (§III.3).
- Whatever the gravity reading's fate, the vacuum law stands on its own — a new, sharply tested statement about entanglement structure in free-field vacua, found because a gravity theory needed a number. That is the program working as designed.