V2.406: Correlated Lambda × gamma_BH Fingerprint

Key Result

The SM trace anomaly simultaneously determines three observables from one input:

Observable	Value	Status
Ω_Λ	0.6877	Planck: +0.4σ
γ_cosmo (de Sitter)	−149/12 = −12.417	Not measurable
γ_Schw (Schwarzschild)	−63/10 = −6.300	Not measurable (yet)

Both gamma values are exact rationals. No other quantum gravity approach makes correlated predictions for all three.

The Two-Gamma Prediction

The trace anomaly has two independent coefficients (a, c):

• Conformally flat spacetimes (de Sitter, FRW): Weyl tensor W = 0, so γ_cosmo = −4a_total
• Schwarzschild: Weyl tensor W ≠ 0, both contribute: γ_Schw = −4a + 2c

For SM + graviton:

• a_total = 149/48
• c_total = 367/120
• γ_cosmo = −149/12 (exact)
• γ_Schw = −63/10 (exact)

The Weyl tensor reduces |γ| by 49.3% for Schwarzschild relative to de Sitter.

Quantum Gravity Landscape Comparison

Approach	γ_Schw	γ_cosmo	Λ prediction	Matter-dependent?
This framework	−6.30	−12.42	0.6877 (zero-param)	Yes
LQG	−1.50	−1.50	not predicted	No
String theory	~−2 ± 1	N/A	landscape	Yes (model-dep)
Asymptotic Safety	~0	N/A	running	No

Key distinguishers:

• vs LQG: |γ_Schw| differs by factor 4.2×; LQG’s γ is universal, ours depends on matter content
• vs strings: our γ is exact, theirs is model-dependent and only computed for BPS BHs
• vs asymptotic safety: our γ is large (−6.3), theirs is ~0

The c/a ≈ 1 Puzzle

An unexpected finding: c/a = 0.985 for SM + graviton, just 1.5% from unity.

If c/a were exactly 1, then γ_Schw = γ_cosmo / 2 exactly. The actual ratio is 0.507, deviating from 1/2 by only 0.7%.

Scanning over generation count: c/a crosses unity between N_gen = 3 and N_gen = 4. The SM sits just below the crossing. This near-unity is unexplained — it’s not required by any known symmetry.

BSM Correlated Shifts

Adding a particle shifts both Λ and γ by calculable, correlated amounts:

Particle	ΔΛ/Λ_obs	Δγ_cosmo	Δγ_Schw	slope dγ/dΛ
+1 scalar	−0.007	−0.011	+0.006	1.6
+1 Weyl	−0.011	−0.061	−0.011	5.8
+1 vector	+0.039	−0.689	−0.489	−17.5

The slope dγ/dΛ differs by particle type. Vectors have steep negative slope; scalars have gentle positive slope. If both Λ and γ are measured and disagree with SM, the type of missing particle is identifiable.

Observational Prospects

• Now: Ω_Λ testable via Planck/Euclid. Framework at +0.4σ.
• γ_BH for astrophysical BHs: Undetectable (log correction is 10^{−74} of leading term)
• Planck-mass PBHs: Remnant mass differs between framework (0.92 M_Pl) and LQG (0.72 M_Pl) — detectable if PBH evaporation endpoints are ever observed
• Theoretical discriminator: γ_Schw = −6.3 vs LQG’s −1.5 is a factor 4.2 difference — distinguishable in the literature right now

What This Means

Strengths:

1. The framework predicts three correlated observables from one input. No competitor does this.
2. γ_Schw = −63/10 and γ_cosmo = −149/12 are exact rationals — clean, precise predictions.
3. The correlated structure enables particle identification (not just detection) from cosmological observables.

Weaknesses:

1. γ_BH is not measurable with foreseeable technology for astrophysical BHs.
2. The Solodukhin formula −4a + 2c may miss zero-mode contributions (Sen 2012).
3. The entire structure assumes Λ_bare = 0 and Jacobson thermodynamics.
4. DESI tension (w ≈ −0.75 at 4.5σ) threatens the framework globally.

Bottom line: This experiment establishes that the framework’s predictions form a correlated fingerprint in the (Λ, γ_cosmo, γ_Schw) space. While γ_BH is not currently measurable, the correlation itself is unique and the theoretical distinction from LQG (factor 4.2 in |γ|) is sharp. The c/a ≈ 1 coincidence is new and warrants investigation.