V2.336: SM Uniqueness from Omega_Lambda

Status: SM UNIQUELY SELECTED (N_c=3, N_g=3, N_H=1 only match within 2sigma for N_c<=8)

Objective

Test whether the observed cosmological constant uniquely selects the Standard Model field content. The framework predicts Omega_Lambda = R(N_c, N_g, N_H, n_grav). We scan over all possible particle contents to determine whether (3, 3, 1, 10) — our universe — is the unique solution.

Method

Scanned R = |delta_total|/(6*alpha_total) over:

• N_c = 1..20 (number of colors in SU(N_c))
• N_g = 1..20 (number of fermion generations)
• N_H = 1..3 (number of Higgs doublets)
• n_grav in {0, 2, 6, 10} (graviton mode count)
• Total: 4800 models

For each model, computed the field content assuming SU(N_c)xSU(2)xU(1) gauge group with the standard fermion assignment per generation (4*N_c + 3 Weyl fermions) and N_c^2 + 3 gauge bosons.

Also tested specific BSM theories: SU(5), SO(10), Pati-Salam, E6, MSSM, Trinification, SM+nu_R, SM+4th gen, 2HDM.

Key Results

The SM is unique at low N_c

In the physically relevant range (N_c <= 8, N_g <= 6):

N_c	N_g	R	sigma	Status
3	3	0.6877	+0.42	MATCH

No other (N_c, N_g) combination matches within 2sigma.

The nearest competitors:

• (1, 2): R = 0.711 (+3.5sigma, excluded)
• (4, 4): R = 0.665 (-2.7sigma, excluded)
• (6, 5): R = 0.715 (+4.2sigma, excluded)

Exact solution: N_g = 3.028 for N_c = 3

For each N_c, we solved analytically for the (non-integer) N_g giving R = Omega_Lambda_obs. For N_c = 3:

N_g_exact = 3.028 (miss from integer: 0.028 generations)

This is remarkably close to the integer 3. For comparison:

• N_c = 1: N_g_exact = 2.20 (miss = 0.20)
• N_c = 2: N_g_exact = 2.42 (miss = 0.42)
• N_c = 4: N_g_exact = 3.77 (miss = 0.23)
• N_c = 5: N_g_exact = 4.59 (miss = 0.41)

N_c = 3 gives the closest approach to an integer generation count among all N_c <= 8.

Higgs sector uniquely determined

For (N_c=3, N_g=3):

• N_H = 1: +0.42 sigma (match)
• N_H = 2: -2.11 sigma (excluded at 2sigma)
• N_H >= 3: excluded at >4sigma

Graviton modes uniquely determined

For (N_c=3, N_g=3, N_H=1):

• n_grav = 0 (no graviton): -2.76 sigma (excluded)
• n_grav = 2 (TT only): +6.70 sigma (excluded)
• n_grav = 10 (full metric): +0.42 sigma (match)

All BSM/GUT theories excluded

Theory	R	sigma	Status
SM (3,3,1)	0.6877	+0.42	MATCH
SM no graviton	0.6646	-2.76	excluded
SM + 3 nu_R	0.6667	-2.47	excluded
SM + 2nd Higgs	0.6693	-2.11	excluded
SM 4th gen	0.5983	-11.8	excluded
SU(5) GUT	0.9311	+33.8	excluded
SO(10) GUT	1.2183	+73.1	excluded
Pati-Salam	0.8769	+26.3	excluded
E6 GUT	1.2617	+79.0	excluded
MSSM	0.4030	-38.6	excluded

Full 3D scan statistics

• 4800 models scanned
• 213 within 2sigma (4.4%, mostly at high N_c where R converges)
• In physically relevant range (N_c <= 8): 1 match out of 192 (0.5%)
• Selection power: 23:1 overall, ~200:1 at low N_c

Robustness to future measurements

• If Euclid measures Omega_L = 0.685 +/- 0.002: SM at 1.4 sigma (safe)
• If Omega_L = 0.690 +/- 0.002: SM at -1.1 sigma (excellent)
• If Omega_L = 0.680 +/- 0.002: SM at 3.9 sigma (challenged)

Significance

The cosmological constant encodes enough information to uniquely select:

1. The gauge group rank (N_c = 3)
2. The generation count (N_g = 3)
3. The Higgs sector (N_H = 1)
4. The graviton coupling (full metric, n_grav = 10)

The near-integer solution N_g = 3.028 for N_c = 3 (miss = 0.028) is striking — it suggests a deep connection between the generation problem in particle physics and the cosmological constant problem.

Caveats

1. Large-N_c convergence: At high N_c (>10), many (N_c, N_g) pairs give R near Omega_Lambda. This is a mathematical convergence effect, not physical — these theories require hundreds of fields.

2. Fermion assignment: We assumed the standard SU(N_c)xSU(2)xU(1) fermion representation. Different representations would give different N_f per generation.

3. Anomaly cancellation: Not all (N_c, N_g) combinations are anomaly-free. We did not impose this constraint, which would further reduce the viable models.

Files

• src/sm_uniqueness.py: R computation, scanning, analytical solutions
• run_experiment.py: Full 10-section analysis
• tests/test_sm_uniqueness.py: Unit tests