V2.645: SM Uniqueness from the Full BSM Landscape

The Headline

Omega_Lambda does NOT uniquely select the SM from the full landscape.

When all discrete parameters (N_c, N_gen, n_H) are varied simultaneously, 41 theories survive within 2σ in the minimal landscape (880 total), and 358 survive in the extended landscape (6912 total). The SM is among them at +0.42σ, but it is not unique. P(coincidence) ≈ 5%, equivalent to ~2σ.

This is a critical honest result that clarifies exactly what the framework can and cannot predict.

What Omega_Lambda Encodes

Quantity	Value
Theories in minimal landscape	880
After physics filters (anomaly, AF, EWSB)	880
Within 2σ of Omega_Lambda	41
Information content	4.4 bits
P(coincidence)	~5% (~2σ)

The cosmological constant encodes 4.4 bits of information about particle physics — enough to reduce the landscape by a factor of ~20, but not enough to uniquely determine the theory.

Why the SM Is Not Unique (Multi-Parameter Degeneracy)

The formula R = |delta|/(6*alpha) depends on two quantities:

• delta = sum of trace anomaly coefficients (dominated by vectors and fermions)
• alpha = N_eff * alpha_s (total component count)

Higher N_c increases both delta and N_eff. Higher N_gen does the same. The RATIO delta/N_eff can be similar for many (N_c, N_gen) combinations.

Examples of non-SM theories within 1σ of Omega_Lambda:

N_c	N_gen	n_H	grav	R	σ
5	4	5	no	0.6853	+0.09
9	8	1	no	0.6856	+0.13
8	7	4	yes	0.6835	-0.17
3	3	1	yes	0.6877	+0.42

These non-SM theories require large N_c and N_gen to compensate. They are physically disfavored (no evidence for N_c > 3 or N_gen > 3) but are not excluded by Omega_Lambda alone.

The V2.641 Result in Context

V2.641 found N_c = 3 uniquely selected at 8.7σ. That was with N_gen = 3 FIXED (using the Z-width as external input). The key difference:

Constraint set	Survivors (2σ)	SM unique?
Omega_Lambda alone	41	No
Omega_Lambda + N_gen = 3 (V2.641)	~1	Yes (8.7σ)
Omega_Lambda + Z-width + LEP	1	Yes

Omega_Lambda is necessary but not sufficient. It must be combined with independent measurements (Z-width, LEP) to uniquely select the SM.

Parameter Sensitivity

Varying N_c (N_gen=3 fixed)

N_c	R	σ	Exclusion
2	0.6214	-8.7σ	Excluded
3	0.6877	+0.4σ	SM
4	0.7676	+11.4σ	Excluded

With N_gen fixed, N_c is sharply determined. V2.641’s result holds.

Varying N_gen (N_c=3 fixed)

N_gen	R	σ	Exclusion
2	0.8320	+20.2σ	Excluded
3	0.6877	+0.4σ	SM
4	0.5983	-11.8σ	Excluded

With N_c fixed, N_gen is sharply determined. Equally powerful.

Varying n_Higgs (N_c=3, N_gen=3 fixed)

n_H	R	σ	Exclusion
1	0.6877	+0.4σ	SM
2 (2HDM)	0.6693	-2.1σ	Marginal
3	0.6519	-4.5σ	Excluded

2HDM is excluded at 2.1σ — marginal but meaningful. n_H ≥ 3 is excluded at >4σ.

Graviton

Graviton	R	σ
Yes (n=10)	0.6877	+0.4σ
No	0.6646	-2.8σ

Graviton is required at 2.8σ, consistent with V2.326/V2.328.

Extended Landscape (with singlets and vector-like fermions)

Adding 0-5 scalar singlets and 0-3 vector-like fermion pairs:

Quantity	Value
Total theories	6912
Within 2σ	358
P(coincidence)	5.2% (~1.9σ)

Singlet scalars are nearly invisible to Omega_Lambda. Each singlet shifts R by only ~0.005 (delta_scalar = -1/90 is tiny compared to delta_vector = -31/45). The framework cannot exclude or confirm singlet scalars — they are in the noise.

The Non-Circular Cascade (Revised)

The correct cascade requires EXTERNAL measurements:

Step 0: Landscape (N_c, N_gen, n_H, grav)         880 theories
Step 1: Anomaly cancellation                       880  (all pass)
Step 2: Asymptotic freedom                         880  (all pass)
Step 3: EWSB requirement                           880  (all pass)
Step 4: Omega_Lambda (2σ)                           41
Step 5: Z-width (N_gen = 3)                          ~5
Step 6: No extra light particles (LEP)               1   (SM)

Physics filters (steps 1-3) provide NO reduction in this parameterization because all SU(N_c) × SU(2) × U(1) theories are automatically anomaly-free (V2.641) and AF for moderate N_gen.

Honest Assessment

What’s Strong

1. The SM is at +0.42σ — excellent match, no tuning
2. With ANY single parameter fixed (N_c or N_gen), the rest are sharply determined. The framework is predictive along each parameter axis.
3. 2HDM excluded at 2.1σ, 3+ Higgs doublets at >4σ
4. Graviton required at 2.8σ
5. Formula R = 149√π/384 is exact (no adjustable parameters)

What’s Weak

1. 41 theories survive within 2σ — the SM is not unique from Omega_Lambda alone. The multi-parameter degeneracy is real.
2. P(coincidence) ≈ 5% — only ~2σ significance against “random formula matches Omega_Lambda.” Not compelling by physics standards.
3. Physics filters eliminate nothing in this parameterization. Anomaly cancellation is automatic for SU(N_c) × SU(2) × U(1).
4. Singlet scalars are invisible — the framework cannot distinguish SM from SM + singlet. This is a fundamental limitation (δ_scalar is too small).
5. External measurements required — the Z-width and LEP data are needed to break the (N_c, N_gen) degeneracy. The framework alone doesn’t select the SM.

What This Means

V2.641’s claim “Lambda counts colors” is CORRECT but INCOMPLETE. Omega_Lambda determines N_c given N_gen, and vice versa. But it cannot determine both simultaneously without additional input.

The 4.4 bits encoded in Omega_Lambda are real and significant — the framework PREDICTS the observed value with zero free parameters. But the claim should be stated precisely:

“Omega_Lambda + Z-width → SM uniquely” (strong, correct)

NOT: “Omega_Lambda → SM uniquely” (overclaims)

The framework’s value is not that it uniquely selects the SM, but that it EXPLAINS the numerical value of the cosmological constant as a topological invariant of the SM field content. Given the SM, it predicts Omega_Lambda = 149√π/384 ≈ 0.6877, matching observation at 0.4σ.

Files

• src/full_landscape.py — Theory class, landscape scanners, statistics
• tests/test_landscape.py — 8 verification tests (all pass)
• run_experiment.py — Full 8-phase experiment
• results.json — Machine-readable results