V2.641 - Lambda Counts Colors — Ω_Λ Determines N_c = 3
V2.641: Lambda Counts Colors — Ω_Λ Determines N_c = 3
The Headline
The cosmological constant measures the number of quark colors.
Among all quantum-consistent gauge theories SU(N_c) × SU(2) × U(1) with 3 generations, only N_c = 3 predicts the observed Ω_Λ. The nearest competitor (N_c = 2) is excluded at 8.7σ; N_c = 4 at 11.4σ.
Why This Is Different from V2.640
V2.640 applied Ω_Λ as a filter (step 2 of 4), creating a circularity concern: using Ω_Λ both as the framework’s prediction and as a selection criterion. This experiment resolves that criticism by applying ALL physics constraints FIRST, leaving Ω_Λ as a pure prediction to be tested.
The Non-Circular Cascade
| Step | Constraint | Type | Survivors |
|---|---|---|---|
| 0 | All SU(N_c)×SU(N_w)×U(1), N_gen=1..10 | — | 840 |
| 1 | Cubic anomaly: d_{abc}(SU(2)) = 0 | QFT theorem | N_w = 2 |
| 2 | Asymptotic freedom: 11N_c > 4N_gen | QCD confinement | 133 |
| 3 | Z-boson width: N_gen = 3 | LEP measurement | 14 |
| 4 | Gauge anomalies: all cancel for any N_c | Algebraic proof | 14 |
| — Everything above is physics. No framework input. — | |||
| 5 | Ω_Λ prediction within 2σ | Framework test | 1 |
Survivor: SU(3) × SU(2) × U(1) with 3 generations = The Standard Model.
The Key Discovery: Anomalies Cancel for ALL N_c
The SM hypercharge assignment generalizes to arbitrary N_c:
| N_c | Y_Q | Y_u | Y_d | Q_u | Q_d |
|---|---|---|---|---|---|
| 2 | 1/4 | 3/4 | -1/4 | 3/4 | -1/4 |
| 3 | 1/6 | 2/3 | -1/3 | 2/3 | -1/3 |
| 4 | 1/8 | 5/8 | -3/8 | 5/8 | -3/8 |
| 5 | 1/10 | 3/5 | -2/5 | 3/5 | -2/5 |
With Y_Q = 1/(2N_c), all four anomaly conditions cancel algebraically:
- SU(N_c)² U(1): Y_Q − (Y_u + Y_d)/2 = 1/(2N_c) − 1/(2N_c) = 0
- SU(2)² U(1): N_c·Y_Q + Y_L = 1/2 − 1/2 = 0
- U(1)³: Cancels via (a+b)³ + (a−b)³ = 2a³ + 6ab² identity
- Gravity × U(1): Cancels by trace condition
This means anomaly cancellation does NOT select N_c. The number of colors is a free parameter from the perspective of quantum consistency.
Ω_Λ as a Colorimeter
| N_c | R = Ω_Λ(predicted) | σ from observation | Status |
|---|---|---|---|
| 2 | 0.621 | -8.7σ | Excluded |
| 3 | 0.688 | +0.4σ | Selected |
| 4 | 0.768 | +11.4σ | Excluded |
| 5 | 0.849 | +22.5σ | Excluded |
| 6 | 0.927 | +33.2σ | Excluded |
N_c = 3 is the unique value within 2σ of Ω_Λ(obs) = 0.6847 ± 0.0073.
The separation from neighbors:
- N_c = 2: ΔR = 0.066, separation = 9.1σ
- N_c = 4: ΔR = 0.080, separation = 10.9σ
Even a 10% measurement of Ω_Λ (σ ≈ 0.07) would distinguish N_c = 3 from its neighbors. Planck’s precision makes the selection overwhelming.
Full Landscape with N_gen Free
Without fixing N_gen, the AF-consistent landscape has 133 theories. Five match Ω_Λ within 2σ:
| N_c | N_gen | N_eff | R | σ |
|---|---|---|---|---|
| 10 | 9 | 994 | 0.685 | +0.0σ |
| 3 | 3 | 128 | 0.688 | +0.4σ |
| 11 | 10 | 1202 | 0.682 | -0.4σ |
| 9 | 8 | 806 | 0.689 | +0.6σ |
| 8 | 7 | 638 | 0.695 | +1.4σ |
However: the Z-boson width (LEP) measures N_gen = 3 (not 7, 8, 9, or 10). With this single measurement, only N_c = 3 survives. The other survivors require 7-10 generations, which are experimentally excluded.
This shows the information decomposition:
- Z-width encodes N_gen = 3 (eliminates 4 of 5 survivors)
- Ω_Λ encodes N_c = 3 (eliminates the remaining 13 of 14 at fixed N_gen)
Physical Interpretation
Why does Ω_Λ depend on N_c? Each quark color contributes:
- 4 × N_gen Weyl fermions (two flavors × two chiralities per generation)
- ~2N_c gauge bosons (from the N_c²-1 gluons)
The ratio R = |δ|/(6α) depends on the balance between fermion and boson contributions to the trace anomaly. N_c = 3 sits at the “Goldilocks” point:
- N_c < 3: too few fermions → R too small
- N_c > 3: too many fermions → R too large
This is not fine-tuning — it’s a discrete choice among integers.
Honest Assessment
What’s Strong
- The argument is completely non-circular: physics constraints first, Ω_Λ as prediction last
- The anomaly cancellation proof is exact (algebraic, not numerical)
- The separation from neighbors is enormous (8.7σ minimum)
- The SM hypercharge generalization to arbitrary N_c is clean and explicit
- Current precision ALREADY suffices (no future measurements needed)
What’s Weak
- The landscape is restricted to SU(N_c) × SU(2) × U(1) with SM-like representations. Exotic matter content (vectorlike fermions, extended Higgs sectors, higher representations) is not scanned.
- With N_gen free, 5 theories survive — uniqueness requires the Z-width input. The framework alone doesn’t predict N_gen.
- The framework takes α_s = 1/(24√π) from lattice computation, which has its own convergence issues (~0.1% uncertainty, V2.288).
- The argument doesn’t explain WHY N_c = 3 is special — it only shows that it’s the one that matches Ω_Λ.
What It Means for the Framework
This is the framework’s strongest argument for non-circularity. The claim is no longer “the SM predicts Ω_Λ” (which could be coincidence), but rather: “among all quantum-consistent theories, only the SM gives the right Ω_Λ, and we can prove this without using Ω_Λ as input.”
The experiment reframes the framework from “a calculation that gets Ω_Λ right” to “a principle that connects the gauge group of particle physics to the cosmological constant.”
Files
src/color_counting.py— Anomaly verification, landscape scantests/test_colors.py— 7 verification tests (all pass)run_experiment.py— Full 7-phase experimentresults.json— Machine-readable results