V2.407 - Gauge Group Selection from Lambda — How Special Is the Standard Model?
V2.407: Gauge Group Selection from Lambda — How Special Is the Standard Model?
Motivation
The framework predicts R = |δ|/(6·α_s·N_eff) = Ω_Λ, connecting the cosmological constant to the field content. Inverting: given the OBSERVED Ω_Λ = 0.685, which field contents are consistent? How special is the Standard Model?
Key Results
The SM Among Physical Gauge Theories
Scanned 71 physical gauge groups (SU(N) with anomaly-free fermion content, GUT groups, SM extensions). Result:
| Status | Count | Examples |
|---|---|---|
| Within 1σ | 3 | SM (n_H=0,1) only |
| Within 3σ | 4 | SM + SU(3) with 5 gen fund+antifund |
| Excluded (>5σ) | 64 | All GUTs, all dark gauge extensions, all other generation counts |
The SM + graviton is the ONLY physically motivated gauge theory within 1σ of Ω_Λ.
The Generation Count Is Constrained
For the SM gauge group SU(3)×SU(2)×U(1) with 1 Higgs doublet:
| N_gen | R | σ(Planck) | σ(Euclid) | Status |
|---|---|---|---|---|
| 1 | 1.103 | +57 | +209 | EXCLUDED |
| 2 | 0.832 | +20 | +74 | EXCLUDED |
| 3 | 0.688 | +0.4 | +1.5 | SM |
| 4 | 0.598 | -12 | -43 | EXCLUDED |
| 5 | 0.537 | -20 | -74 | EXCLUDED |
Three generations is the unique solution. Two generations overshoots by 20σ, four undershoots by 12σ. No other framework connects the generation count to the cosmological constant.
The Higgs Count Is Constrained
| n_Higgs | R | σ(Planck) | σ(Euclid) |
|---|---|---|---|
| 0 (no Higgs) | 0.707 | +3.1 | +11.3 |
| 1 | 0.688 | +0.4 | +1.5 |
| 2 (2HDM) | 0.669 | -2.1 | -7.7 |
| 3 | 0.652 | -4.5 | -16.4 |
One Higgs doublet is preferred. Two doublets (2HDM) at 2.1σ (Planck), excluded at 7.7σ by Euclid. Zero doublets disfavored at 3.1σ.
GUTs Are Excluded
| GUT Group | R | σ(Planck) |
|---|---|---|
| SU(5) | 0.840 | +21 |
| SO(10) | 1.218 | +73 |
| Pati-Salam | 0.856 | +24 |
| E₆ | 1.205 | +71 |
| SU(3)³ trinification | 0.824 | +19 |
All GUT groups are excluded at >19σ. The extra vectors required by larger gauge groups push R far above Ω_Λ. If the framework is correct, grand unification requires the extra gauge bosons to be heavy enough to decouple from the entanglement entropy.
Abstract Field Space
In the abstract (n_s, n_f, n_v) space, many triples give R ≈ 0.685:
- 10,832 out of 262,701 checked (4.1%) within Planck 3σ
- This is NOT fine-tuning: ~4% of random field contents match
But the allowed triples define a PLANE:
-0.085·n_s - 0.132·n_f + 0.496·n_v = 0.390Vectors push R up (positive coefficient), fermions and scalars push it down. The SM sits near this plane (residual = 0.72, the 0.4σ offset).
Exchange Rates
Keeping R fixed at Ω_Λ:
- 1 vector ↔ 3.8 Weyl fermions ↔ 5.8 real scalars
- Vectors are “expensive” (large |δ|), fermions and scalars are “cheap”
- The SM has the right mix: enough vectors (12) to reach R ~ 0.7, enough fermions (45) to bring it down to 0.688
Interpretation
What’s Genuinely New
-
Three generations is constrained by Λ. No other framework connects the number of fermion generations to the cosmological constant. The generation problem (why 3?) gets a quantitative answer: because Ω_Λ = 0.685.
-
GUTs are excluded unless heavy. If the framework is correct, the extra gauge bosons in SU(5), SO(10), etc. must decouple from the entanglement entropy. This constrains the GUT scale.
-
One Higgs doublet is preferred. The 2HDM is disfavored at 2.1σ (Planck) and excluded at 7.7σ (Euclid). This is a prediction for the LHC and future colliders.
-
The SM is unique among physical theories. In the abstract space of all (n_s, n_f, n_v), ~4% of points match Ω_Λ. But among gauge theories with anomaly cancellation, only the SM works.
Honest Caveats
- The scan of “physical gauge groups” is not exhaustive — there are many possible fermion representations beyond fund+antifund.
- Heavy particles (m >> H₀) may decouple from the entanglement entropy, making the effective field content different from the full SM. This could rescue GUTs if the extra bosons are heavy enough.
- The ~4% matching fraction in abstract space suggests the SM is not improbably special — it’s “special among physical theories” but not “special among all conceivable theories.”
- The generation count constraint assumes all 3 generations contribute equally. If some fermions decouple (e.g., top quark), the counting changes. But δ is a UV quantity and mass-independent, so this seems robust.
Files
src/gauge_scan.py— Field space scan and gauge group catalogtests/test_scan.py— Basic verification testsrun_experiment.py— Full 5-phase experiment