V2.757: BSM Phase Space Topology — Dark Matter Census

Experiment

The cosmological constant as a mass-independent particle detector. We map the full 3D exclusion landscape in BSM field space (Delta_n_scalar, Delta_n_fermion, Delta_n_vector), prove that iso-R surfaces are planes, and confront 32 BSM/dark matter models with both Planck and Euclid-era precision.

Key formula: R = |delta_total| / (6 alpha_s N_eff) = Omega_Lambda

where delta and N_eff are linear functions of the field content, making R a ratio of two linear functions whose level sets are planes.

Key Results

1. Analytic result: iso-R surfaces are planes

The constraint R = const in (Dn_s, Dn_f, Dn_v) space is:

-0.0855 Dn_s - 0.1320 Dn_f + 0.4957 Dn_v = -0.055

Proof: R = |delta|/(6 alpha_s N_eff) with both numerator and denominator linear in the field additions. Setting R = const yields a single linear equation. The allowed BSM region at n-sigma is a slab between two parallel planes.

The normal vector (-0.164, -0.254, 0.953) is dominated by the vector component (|a_v| / |a_s| = 5.8). This means the vector direction is the most constrained — any new gauge boson shifts Lambda by 5.7x more per particle than a scalar.

2. BSM particle budget

Maximum additional particles allowed beyond SM + graviton:

Confidence	Max scalars	Max Weyl	Max vectors	(Planck / Euclid)
1 sigma	2 / 1	1 / 0	0 / 0
2 sigma	3 / 1	2 / 0	0 / 0
3 sigma	5 / 1	3 / 1	0 / 0
5 sigma	8 / 2	5 / 1	1 / 0

Zero room for additional vectors. Even at 5 sigma with Planck, only 1 extra vector is allowed. With Euclid, zero at any confidence level.

Best fit: Adding 1 real scalar (axion) reduces tension from +0.42 sigma to +0.23 sigma — the framework slightly PREFERS the existence of one additional light scalar.

3. Allowed BSM phase space volume

Integer lattice points (Dn_s, Dn_f, Dn_v >= 0) inside 2-sigma slab:

Experiment	Allowed theories	Shrinkage
Planck	991	—
Euclid	269	3.7x

Most allowed points have many scalars/fermions compensating a few vectors, occupying a thin slab in the full 3D space.

4. Comprehensive BSM / dark matter model census (32 models)

Compatible with Planck (<1 sigma): 5 models

• SM + graviton: R = 0.6877 (+0.4 sigma) — baseline
• +1 real scalar: R = 0.6830 (-0.2 sigma) — BEST FIT (axion, fuzzy DM)
• +2 real scalars: R = 0.6784 (-0.9 sigma)
• +1 Weyl fermion: R = 0.6805 (-0.6 sigma) — sterile neutrino (Majorana)
• +1 sterile nu: R = 0.6805 (-0.6 sigma) — same as above

Compatible with Euclid (<2 sigma): effectively only +1 scalar and baseline

• +1 real scalar: -0.8 sigma — SURVIVES
• SM + graviton: +1.5 sigma — barely survives
• Everything else: >2 sigma

Excluded at >5 sigma (Planck): 12 models

• MSSM (-38.6 sigma), NMSSM (-39.1 sigma), Split SUSY (-12.5 sigma)
• 4th generation (-11.8 sigma)
• All GUTs: SU(5) (+21.8 sigma), SO(10) (+69.6 sigma)
• All dark gauge sectors: SU(2)_dark (+8.4 sigma), SU(3)_dark (+27.1 sigma)

5. Dark matter compatibility

DM Candidate	Field content	Planck	Euclid	Verdict
Axion / fuzzy DM	+1 scalar	-0.2 sigma	-0.8 sigma	COMPATIBLE
Sterile nu (Majorana)	+1 Weyl	-0.6 sigma	-2.1 sigma	tension (Euclid)
WIMP (Majorana)	+1 Weyl	-0.6 sigma	-2.1 sigma	tension (Euclid)
MDM scalar triplet	+3 scalars	-1.5 sigma	-5.4 sigma	EXCLUDED (Euclid)
Dirac DM	+2 Weyl	-1.5 sigma	-5.6 sigma	EXCLUDED (Euclid)
Dark photon	+1 vector	+4.1 sigma	+15.0 sigma	EXCLUDED
Dark U(1) + Higgs	+1v + 1s	+3.4 sigma	+12.6 sigma	EXCLUDED
All dark SU(N)	vectors + matter	>+3 sigma	>+11 sigma	EXCLUDED

The framework predicts: if dark matter is a fundamental particle, it is most likely a single real scalar (axion-like). This is the ONLY DM candidate that survives Euclid-era constraints at <1 sigma.

6. Per-spin sensitivity asymmetry

Spin	dR per particle	Planck sigma	Euclid sigma	Direction
Scalar	-0.00472	-0.65 sigma	-2.36 sigma	decreases R
Weyl	-0.00725	-0.99 sigma	-3.62 sigma	decreases R
Dirac	-0.01428	-1.96 sigma	-7.14 sigma	decreases R
Vector	+0.02699	+3.70 sigma	+13.49 sigma	increases R

Vector/scalar anisotropy ratio: 5.7x

Vectors carry 31x more trace anomaly per component than scalars (delta_v/n_comp = 0.344 vs delta_s/n_comp = 0.011). This creates the extreme asymmetry.

7. Mass independence (Adler-Bardeen theorem)

The trace anomaly delta = -1/90 per real scalar is a UV-finite topological invariant, protected by the Adler-Bardeen theorem. It is independent of:

• Particle mass (from 10^{-22} eV fuzzy DM to 10^{19} GeV Planck mass)
• Coupling constants
• RG scale

A scalar at ANY mass shifts Lambda by exactly delta R = -0.00472. This mass independence is UNIQUE to this framework — no other approach to dark energy makes mass-independent predictions about particle content.

8. Dual-observable constraint (Omega_Lambda, gamma_BH)

The framework makes TWO zero-parameter predictions from the same (a, c) trace anomaly coefficients:

1. Omega_Lambda probes ‘a’ only (Euler density on FRW background)
2. gamma_BH probes ‘a + c’ (Euler + Weyl on Schwarzschild background)

SM + graviton values:

• Omega_Lambda = 0.6877 (observation: 0.6847 +/- 0.0073)
• gamma_BH = -35.98 (LQG predicts -1.5, a 24x difference)
• Enhancement factor eta = 2.898

Per-spin enhancement:

• Scalar: eta = 4.000 (c dominates a)
• Weyl: eta = 2.636
• Vector: eta = 1.581 (a dominates c)
• Graviton: eta = 11.426 (enormous c)

The per-spin eta values are NON-DEGENERATE, so measuring gamma_BH would break residual degeneracies in the Omega_Lambda constraint alone.

Interpretation

What is genuinely new here

1. Iso-R surfaces are planes. This is an analytic result with a clean proof. The BSM constraint landscape has a simple geometric structure: a slab whose orientation is dominated by the vector component.

2. The dark matter prediction. The framework singles out +1 real scalar (axion-like) as the ONLY dark matter candidate surviving Euclid-era constraints. This is a specific, testable prediction connecting dark energy to dark matter.

3. The best fit improves with one scalar. The SM + graviton prediction is at +0.42 sigma; adding one scalar brings it to +0.23 sigma. The framework slightly prefers the existence of an axion.

4. The phase space shrinkage. Euclid reduces the allowed BSM volume by 3.7x, from 991 to 269 integer lattice points. The framework becomes dramatically more constraining.

What this means for physics

The framework is a particle detector with infinite mass range. Unlike colliders (which need energy above mass threshold) or direct detection (which needs coupling to SM), this framework detects ANY fundamental field regardless of mass or coupling, through its contribution to the trace anomaly.

The key falsification hierarchy:

1. Discovery of ANY new vector boson → framework falsified at >4 sigma
2. Discovery of >2 new scalars → tension
3. Discovery of SUSY → framework destroyed at >12 sigma
4. Euclid measures Omega_Lambda outside [0.682, 0.692] → framework falsified

The key confirmation signatures:

1. Euclid confirms Omega_Lambda = 0.685 +/- 0.002 → consistent
2. Axion discovered with single real scalar DOF → R improves from 0.42 to 0.23 sigma
3. No new vectors found at any energy scale → consistent with zero budget

Honest limitations

1. The graviton contribution (n=10 full metric vs n=2 TT) remains the largest theoretical uncertainty, though V2.328 pins n_grav = 10.6 +/- 1.4.
2. The gamma_BH prediction (-35.98) is currently untestable — it requires measuring subleading corrections to black hole entropy.
3. The “991 allowed theories” count is an overestimate — most lattice points don’t correspond to consistent gauge theories with anomaly cancellation.
4. The trace anomaly of spin-3/2 fields (gravitino) is not included.