Ω_Λ = 2/3 + δ/2π
Bruce Camber, Independent Researcher
In collaboration with Claude (Anthropic)
with critical reviews from Grok and Perplexity, September 2026
With formula-selection analysis by Simon Plouffe (private communication, September 2026)
Abstract
We present a parameter-free geometric conjecture for the dark energy fraction of the universe. Two exact facts about three-dimensional Euclidean geometry — the octahedral volume fraction of the tetrahedral-octahedral honeycomb (f_oct = 2/3) and the normalized angular deficit of five regular tetrahedra around a common edge (δ/2π ≈ 0.020434) — are combined in the conjectured identification:
Ω_Λ = 2/3 + δ/2π = 0.6871
This is within 0.33 sigma of the Planck 2018 measured value of 0.6847 ± 0.0073. No cosmological measurements are used as inputs. No free parameters are fitted to observation. The combination rule — adding a volume fraction from a perfect tiling to a normalized angular deficit from a frustrated packing — and the identification of these geometric quantities with cosmological energy fractions are postulated, not derived from first principles. The physical mechanism connecting Planck-scale geometry to cosmological energy fractions remains the central open question. The conjecture predicts w = −1 exactly as a rigid consequence of the scale-invariance of both geometric inputs. This prediction is falsifiable by DESI, Euclid, and the Rubin Observatory.
I. The Two Geometric Inputs
This conjecture uses exactly two facts about three-dimensional Euclidean geometry. No cosmological measurement enters the derivation.
Input 1: The tetrahedral-octahedral honeycomb
The tetrahedral-octahedral honeycomb — known also as the alternated cubic honeycomb — is a standard space-filling Euclidean construction composed of regular tetrahedra and regular octahedra in the ratio 2:1 by count. It is the natural cellulation of the face-centered cubic lattice and the unique such honeycomb under the constraint that all cells are regular polyhedra of a single edge length. In this tiling, for every octahedron there are exactly two tetrahedra. Since the volume of a regular octahedron is exactly four times the volume of a regular tetrahedron with the same edge length:
V_oct = (√2/3)a³
V_tet = (√2/12)a³
V_oct / V_tet = 4
The volume fractions in the honeycomb are:
f_oct = 4/(4+2) = 2/3 (exact)
f_tet = 2/(4+2) = 1/3 (exact)
This is not an approximation. It is an exact consequence of the geometry of the honeycomb.
Input 2: The tetrahedral angular deficit
The dihedral angle of a regular tetrahedron — the angle between two of its faces — is:
θ = arccos(1/3) = 70.52877…°
Five regular tetrahedra arranged around a common edge span:
5θ = 5 × 70.52877° = 352.64385°
The angular deficit — the gap that prevents perfect closure — is:
δ = 360° − 352.64385° = 7.35615°
This is the Aristotle gap: the angular frustration of five tetrahedra attempting to tile around a common edge in three-dimensional space. Five is the maximum number of regular tetrahedra that can share an edge without overlap. Six would overshoot 360°. Five falls short by 7.356°.
Normalized to a full rotation:
δ/2π = 7.35615° / 360° = 0.020434 (exact)
II. The Conjecture
Ω_Λ = 2/3 + δ/2π
= 0.666667 + 0.020434
= 0.687101
= 68.710%
The dark energy fraction equals the octahedral volume fraction of the perfect three-dimensional tiling, plus the normalized angular frustration of the imperfect one.
III. Verification
Observed value (Planck 2018, TT,TE,EE+lowE+lensing, base ΛCDM):
Ω_Λ = 0.6847 ± 0.0073
Predicted value:
Ω_Λ = 0.6871
Difference:
0.6871 − 0.6847 = 0.0024
In units of observational uncertainty:
0.0024 / 0.0073 = 0.33 sigma
The prediction is within one third of one standard deviation of the observed value.
What was and was not used:
| Quantity | Status |
|---|---|
| Observed Ω_Λ | Not used as input |
| Planck length or time | Not used |
| Age of universe | Not used |
| Comoving radius | Not used |
| Free parameters fitted to data | None |
| Dihedral angle of tetrahedron | Used (pure geometry) |
| Octahedral volume fraction | Used (pure geometry) |
| Number of tetrahedra at edge | Used (pure geometry — forced by 3D space) |
IV. The Physical Interpretation
The tetrahedral-octahedral honeycomb is the only way to tile three-dimensional Euclidean space with regular polyhedra. In this tiling, two kinds of geometry coexist:
The octahedral component — geometrically relaxed. The octahedron closes perfectly around every edge it shares with neighboring octahedra and tetrahedra. It contributes no angular frustration. It is the settled, gap-free component of the tiling.
The tetrahedral component — geometrically frustrated. Five tetrahedra attempt to close around a common edge and fall short by 7.356°. The tetrahedron is the restless, gap-bearing component of the tiling.
The conjecture identifies these two components with the two dominant components of the cosmic energy budget:
Dark energy ↔ octahedral geometry (relaxed, gap-free, driving expansion)
Matter ↔ tetrahedral geometry (frustrated, gap-bearing, resisting expansion)
The 2/3 baseline is the octahedral fraction — the fraction of three-dimensional space that is geometrically at rest. The δ/2π correction is the normalized angular frustration — the small additional fraction contributed by the gap itself, which belongs to the octahedral component because the gap represents space that the tetrahedral arrangement cannot claim.
The identification of the octahedral fraction with dark energy and the tetrahedral fraction with matter is a postulate, not a consequence of the geometry alone. It is motivated by a qualitative parallel: octahedral geometry closes perfectly and contributes no angular frustration, analogous to the non-diluting, geometrically settled character of dark energy; tetrahedral geometry is persistently frustrated and gap-bearing, analogous to the diluting, dynamically active character of matter. This parallel is suggestive but not derivational. Furthermore, the two geometric inputs — f_oct = 2/3 from the perfect honeycomb, and δ/2π from the frustrated five-tetrahedra arrangement — are properties of different geometric configurations. In the tetrahedral-octahedral honeycomb that produces the 2/3, every edge is perfectly closed and no Aristotle gap exists. The gap appears only when octahedra are excluded. A derivation that produces both fractions from a single unified geometric framework — rather than adding them as separate inputs — is the central open problem this conjecture sets before the community. The Dehn invariant, which measures the angular frustration of a polyhedron and determines its dissection properties, may provide such a framework: the regular tetrahedron has non-zero Dehn invariant while the octahedron has zero, and this asymmetry may be the natural origin of both fractions within a single calculation. This remains to be shown.
V. The Connection to the 81018 Model
The 81018 model maps the universe across 202 base-2 doublings from the Planck scale to the observable universe. At each notation, the sphere packing geometry produces tetrahedral and octahedral voids in the ratio 2:1 by count, 1:2 by volume.
The formula Ω_Λ = 2/3 + δ/2π emerges from this structure as a scale-invariant geometric constant — the same at Notation 1 as at Notation 202. It does not depend on which notation we are at. It is a property of three-dimensional space itself, expressed at every scale simultaneously.
In the 81018 framework, all 202 notations are simultaneously present — each notation is not a past epoch but an ongoing structural layer of the present universe. The Planck scale is not the distant past. It is the active foundation of the present moment.
This simultaneity dissolves the coincidence problem. The geometric fractions do not describe the present epoch as opposed to earlier epochs. They describe the complete structure — all notations, all scales, all time — which is always and entirely now.
VI. Predictions
Prediction 1: Dark energy fraction
Ω_Λ = 2/3 + δ/2π = 0.6871
Observed: 0.6847 ± 0.0073. Difference: 0.33 sigma.
Prediction 2: Matter fraction
Ω_m = 1 − Ω_Λ = 1/3 − δ/2π = 0.3129
Observed: 0.3153 ± 0.0073. Difference: 0.33 sigma.
Prediction 3: Dark energy to matter ratio
ρ_Λ/ρ_m = (2/3 + δ/2π)/(1/3 − δ/2π) = 0.6871/0.3129 = 2.196
Observed: 0.6847/0.3153 = 2.172. Difference: within 1 sigma.
Geometric interpretation: the octahedron has four times the volume of the tetrahedron, and the tiling uses two tetrahedra per octahedron, giving a 4:2 = 2:1 baseline ratio corrected upward by δ/2π to 2.196:1.
A note on independence: Prediction 1 is the single independently chosen numerical target. Predictions 2 and 3 follow from Prediction 1 by algebraic rearrangement under the assumption of spatial flatness and two-component closure — they are conditional consequences, not independent predictions. Prediction 4 is an independent physical claim that does not follow from Prediction 1 alone but requires a dynamical assumption about the nature of the geometric sector.
Prediction 4: Equation of state
w = −1 exactly
If the postulated octahedral sector enters Einstein’s equations as a separately conserved constant vacuum-energy density, then w = −1 follows necessarily from the scale-invariance of both geometric inputs — the dihedral angle of a regular tetrahedron does not change with cosmic time, and neither does the octahedral volume fraction of the honeycomb. The derivation of such a stress-energy assignment from the geometric framework is an open problem and is not supplied here. If confirmed, this would make w = −1 a derived consequence rather than a separate postulate.
Testable by DESI, Euclid, and the Rubin Observatory. If w ≠ −1 is confirmed at high significance, the conjecture in its present form is falsified.
VII. What This Is and What It Is Not
What this is:
A parameter-free geometric conjecture that produces four predictions consistent with current observations, derived from two exact facts about three-dimensional Euclidean geometry, with no cosmological measurements used as inputs and no free parameters fitted to data. The combination rule and the identification of geometric fractions with cosmological energy components are postulated. The numerical agreement is real. The mechanism is open.
What this is not:
A derivation. The conjecture identifies two geometric quantities and proposes their sum equals Ω_Λ. It does not explain why the octahedral fraction corresponds to dark energy rather than matter, why the sum rule holds rather than some other combination, or why this geometric identity should hold at the cosmological scale. These are not gaps to be papered over — they are the open problems that make the conjecture worth investigating.
The formula contains no adjustable continuous parameter. Its statistical interpretation is nevertheless limited by the a posteriori selection of the combination rule — the choice to add rather than subtract, square, or otherwise combine the two geometric quantities was made with knowledge of the observational target. A formula-selection audit using the Inverse Symbolic Calculator (Plouffe, private communication, September 2026) confirms that many expressions of comparable algebraic complexity produce the same six decimal digits of 0.687101 — including expressions involving roots of polynomials, hypergeometric functions, Riemann zeta zeros, and combinations of π and γ. The shortest alternative found was −39/10/π² + 17/5/π. None of these alternatives has any geometric motivation connecting to three-dimensional sphere packing. The conjecture’s evidential weight therefore rests primarily on the geometric motivation for the specific formula rather than on the numerical agreement alone. Future high-precision measurements of Ω_Λ from DESI, Euclid, and the Rubin Observatory — if they converge on 0.68710… rather than 0.6847… — would substantially strengthen the case. If they converge elsewhere, the conjecture is falsified. We do not assign a discovery-level probability to the numerical agreement.
VIII. Open Questions
Open Question 1 — The physical mechanism
Why does the octahedral volume fraction of the Planck-scale sphere-packing void structure correspond to the dark energy fraction of the cosmic energy budget? What is the causal or structural connection between the geometry of three-dimensional space at the Planck scale and the energy content of the observable universe at cosmological scales?
A candidate answer to Open Question 1 may lie in the Hexarchon — the four hexagonal cross-sections inherent in every regular octahedron, each encoding one of the four primary irrationals (π, e, φ, √2). If the Hexarchon is the geometric mechanism by which the Planck-scale octahedral structure propagates its properties across all 202 notations simultaneously, then the dark energy fraction is not a coincidence of the present epoch but a structural constant carried by the Hexarchon at every scale. This connection is itself a conjecture — a candidate mechanism awaiting rigorous formalization. If such a derivation exists, it would transform the conjecture’s mechanism from an analogy into a theorem.
Open Question 2 — The absolute energy scale
The conjecture predicts geometric fractions correctly but cannot predict the absolute value of H₀ or ρ_Λ without first solving the cosmological constant problem — the 10^120 discrepancy between the Planck energy density and the observed vacuum energy density. This problem is not specific to our conjecture. It is the deepest unsolved problem in theoretical physics. The conjecture reframes it: rather than asking why ρ_Λ is so small, we ask why the geometric fractions are what they are — and the answer comes from geometry rather than from fine-tuning.
Open Question 3 — The equation of state correction
The conjecture predicts w = −1 exactly as a rigid consequence of the scale-invariance of both geometric inputs. The dihedral angle of a regular tetrahedron does not change with cosmic time. The octahedral volume fraction of the honeycomb does not change with cosmic time. A dark energy density built from true geometric constants cannot evolve — hence w = −1 precisely. Recent DESI DR2 results, combined with CMB data, suggest w may deviate from −1 at approximately 3.1 sigma significance in the quadrant w₀ > −1, w_a < 0. This is not yet settled — the significance depends on the supernova sample and the parametrization used — but it is more than a rumor. If w ≠ −1 is confirmed at high significance by DESI, Euclid, or the Rubin Observatory, the conjecture in its present form is falsified. We regard this as the conjecture’s sharpest and most honest prediction: it is rigid, it is testable in the near term, and it has a clear falsification criterion. We do not introduce a correction of order δ/2π to w in response to current data, as this would constitute a third geometric ingredient added post hoc rather than derived from the framework.
Open Question 4 — Additional predictions
The conjecture must be tested against observations it was not built to match. Candidates include the CMB power spectrum, baryon acoustic oscillations, the growth rate of structure f σ₈, and the redshift evolution of the dark energy fraction. If the geometric framing is correct, these should all be consistent with w = −1 and Ω_Λ = 0.6871.
Open Question 5 — One geometry, not two
The formula Ω_Λ = 2/3 + δ/2π adds a volume fraction from a perfect tiling to a normalized angular deficit from a frustrated packing. These are properties of different geometric configurations — the honeycomb and the five-tetrahedra cluster — and the plus sign connecting them is postulated rather than derived. The most important mathematical challenge this conjecture faces is to find a single geometric framework from which both quantities emerge as parts of one calculation rather than as separate inputs. The Dehn invariant of the regular tetrahedron and regular octahedron offers one candidate: the tetrahedron’s non-zero Dehn invariant and the octahedron’s zero Dehn invariant encode precisely the distinction between frustrated and relaxed geometry that the conjecture invokes qualitatively. Whether this invariant produces the sum 2/3 + δ/2π from a single expression — rather than requiring the two terms to be identified and added separately — is an open mathematical question. If such a derivation exists, it would transform the conjecture from a postulated identification into a theorem. If it can be shown that no such derivation exists, that too would be a result — it would mean the plus sign requires physical rather than purely geometric justification.
IX. The Simplest Statement
Three-dimensional space tiles itself in exactly one way using regular polyhedra — with tetrahedra and octahedra in volume ratio 1:2. The octahedral portion is geometrically relaxed. The tetrahedral portion is geometrically frustrated — five tetrahedra around a common edge fall short of closure by 7.356°. Dark energy is the relaxed fraction plus the normalized angular frustration:
Ω_Λ = 2/3 + δ/2π = 68.71%
No free parameters. No cosmological inputs. Two geometric facts. One formula.
X. Invitation
This conjecture is offered openly for examination, criticism, and development. The numerical agreement is real. The mechanism is being studied. The predictions are falsifiable. The geometry is exact.
The next step belongs to anyone willing to ask: “Why does the shape of space determine the fate of the universe?“
XI. References
[1] Planck Collaboration. Planck 2018 results. VI. Cosmological parameters. Astronomy & Astrophysics 641, A6 (2020). https://doi.org/10.1051/0004-6361/201833910
[2] DESI Collaboration. DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations. arXiv:2404.03002 (2024).
[3] Linder, E.V. Exploring the expansion history of the universe. Physical Review Letters 90, 091301 (2003). https://arxiv.org/abs/astro-ph/0208512
[4] Huterer, D. & Turner, M.S. The DESI results impact the local determination of H₀. arXiv:2606.05358 (2026).
[5] Kessler, D.A., Di Valentino, E., Escamilla, L.A. & Huterer, D. Reconstructing dark energy with fewer assumptions. arXiv:2606.05853 (2026).
[6] Lagarias, J.C. & Zong, C. Mysteries in packing regular tetrahedra. Notices of the American Mathematical Society 59(11), 1540–1549 (2012).
[7] Conway, J.H. & Torquato, S. Packing, tiling, and covering with tetrahedra. Proceedings of the National Academy of Sciences 103(28), 10612–10617 (2006).
[8] Dehn, M. Über den Rauminhalt. Mathematische Annalen 55, 465–478 (1901). [The original paper introducing what is now called the Dehn invariant.]
[9] Camber, B.E. Dark energy from scale invariance: A base-2 Planck-to-cosmos framework. 81018.com/dark-energy-from-scale-invariance/ (2026).
[10] Plouffe, S. Inverse Symbolic Calculator. Private communication, September 2026.