V2.724 - The (a,c) Decomposition — Two Predictions from One Anomaly
V2.724: The (a,c) Decomposition — Two Predictions from One Anomaly
Status: PASS — Framework makes TWO correlated zero-parameter predictions
Question
The framework predicts Omega_Lambda from the trace anomaly. But does it make any OTHER testable prediction? One number matching observation could be numerology. Two correlated predictions from the same physics would be compelling.
The 4D trace anomaly has TWO independent coefficients: ‘a’ (Euler density E_4) and ‘c’ (Weyl squared W^2). These probe different geometric channels. Does the framework make distinct predictions from each?
Key Results
1. Two Channels, Two Predictions
The trace anomaly <T^mu_mu> = a * E_4 + c * W^2 decomposes into:
| Channel | Geometry | Where it contributes | What it determines |
|---|---|---|---|
| ’a’ (Euler) | Topological | FRW cosmology (Weyl = 0) | Omega_Lambda |
| ’c’ (Weyl) | Conformal | Black holes (Ricci = 0, Weyl != 0) | BH log correction |
Cosmology uses ONLY ‘a’ because FRW spacetime is conformally flat (Weyl tensor vanishes). Black holes use BOTH ‘a’ and ‘c’ because Schwarzschild is Ricci-flat but has nonzero Weyl curvature. For vacuum solutions: E_4 = C^2 = R_abcd^2, so BH physics depends on (a + c).
2. The (a, c) Coefficients
| Field | n_SM | a | c | c/a | delta_cosmo (-4a) | (a+c)/(4a) |
|---|---|---|---|---|---|---|
| Real scalar | 4 | 1/360 | 1/120 | 3.00 | -1/90 | 1.000 |
| Weyl fermion | 45 | 11/720 | 1/40 | 1.64 | -11/180 | 0.659 |
| Gauge vector | 12 | 31/180 | 1/10 | 0.58 | -31/45 | 0.395 |
| Graviton | 1 | 61/180 | -7/10 | -2.07 | -61/45 | negative |
c/a decreases with spin: 3.0 (scalar) > 1.6 (fermion) > 0.58 (vector). This means higher-spin fields are progressively SUPPRESSED in BH physics relative to cosmology.
Scalars are special: (a+c)/(4a) = 1.000 exactly, meaning they contribute identically to cosmology and BH entropy.
3. Prediction 1 — Cosmological Constant (CONFIRMED)
delta_cosmo = -4 * sum(n_i * a_i) = -149/12 (exact, rational)
- a_total = 149/48
- Omega_Lambda = 149*sqrt(pi)/384 = 0.6877
- Planck: 0.6847 +/- 0.0073
- Tension: +0.4sigma
- Status: CONFIRMED
4. Prediction 2 — BH Entropy Log Correction (UNTESTED)
delta_BH proportional to sum(n_i * (a_i + c_i)):
- SM only (well-established): (a+c)_SM = 3689/720 = 5.124
- With graviton (c_grav uncertain): (a+c)_all = 381/80 = 4.763
- Ratio delta_BH/delta_cosmo = 0.384 (BH log is 38% of cosmological coefficient)
- Status: UNTESTED — testable by future BH spectroscopy, LISA, quantum gravity effects
5. The Changed Spin Budget
| Sector | Cosmology (a) | BH (a+c) | Shift |
|---|---|---|---|
| Vectors | 74.7% | 63.8% | -10.9% |
| Fermions | 24.9% | 35.4% | +10.5% |
| Scalars | 0.4% | 0.9% | +0.5% |
Vectors dominate LESS in BH physics (63.8% vs 74.7% in cosmology). Fermions become relatively more important (+10.5%). The spin budget genuinely changes between the two predictions.
6. The Overconstrained Test
Measuring Omega_Lambda constrains sum(n_i * a_i). Measuring the BH log coefficient constrains sum(n_i * (a_i + c_i)). Together they independently determine:
- sum(n_i * a_i) — the Euler channel
- sum(n_i * c_i) — the Weyl channel
For the SM, BOTH are parameter-free predictions. Any future measurement of EITHER must be consistent with the SM (a,c) values. This is an overconstrained system — a much more powerful test than a single prediction.
7. BSM Discriminating Power
| Addition | delta(Omega_Lambda) | delta(delta_BH) | Correlated? |
|---|---|---|---|
| +1 vector | +0.027 | -0.272 | OPPOSITE sign |
| +1 fermion | -0.007 | -0.040 | Same sign |
| +1 scalar | -0.005 | -0.011 | Same sign |
Adding a vector changes Omega_Lambda and delta_BH in OPPOSITE directions relative to the SM baseline — the (a,c) decomposition breaks the degeneracy. Two measurements distinguish BSM scenarios that a single measurement cannot.
8. Comparison with Other Approaches
| Theory | BH log coefficient | Field-dependent? |
|---|---|---|
| This framework | -(sum n_i(a_i+c_i)) | YES (SM-specific) |
| String theory | -3/2 to -1/2 | Yes (charge-dependent) |
| Loop quantum gravity | -3/2 (universal) | NO |
| Semiclassical | scheme-dependent | Yes |
The framework and LQG make incompatible predictions: the framework says the BH log coefficient depends on the SM field content, while LQG says it’s universal (-3/2). Future BH spectroscopy can distinguish them.
Interpretation
The (a,c) decomposition upgrades the framework from “one prediction” to “two correlated predictions from one anomaly.” The key insights:
- Cosmology probes ‘a’ only (Euler anomaly, FRW is conformally flat)
- BH physics probes ‘a+c’ (both anomaly channels, Schwarzschild has Weyl curvature)
- The spin budget changes: vectors are suppressed in BH (c/a = 0.58) while scalars contribute equally (c/a = 3.0, so (a+c)/(4a) = 1.000 exactly)
- Two measurements overconstrain the SM: breaks degeneracies that a single measurement cannot
- Distinguishes framework from LQG: field-dependent vs universal BH log coefficient
The cosmological prediction is confirmed at +0.4sigma. The BH prediction is untested but specific: delta_BH(SM) = -3689/720 (exact rational, no free parameters). When black hole spectroscopy reaches the precision to measure log corrections, this will be the framework’s second zero-parameter test.