V2.412: Verification of α_s(D=5) — The Formula Fails

Objective

Test the dimensional scaling conjecture α_s(D) = Γ((D-1)/2)/(12·π^{D-3}) at D=5. V2.411 found this formula from D=3,4 data and got 17% deviation at D=5 with C=4. This experiment pushes to C=6 to determine whether the deviation shrinks (confirming the formula) or grows (refuting it).

Key Prediction

α_s(5) = 1/(12π²) = 0.00844343

Results

Phase 1: Convergence Scan

C	α_s (poly)	Deviation	R²
2	0.004609	−45.4%	1.000000
3	0.007019	−16.9%	1.000000
4	0.008784	+4.0%	1.000000
5	0.010089	+19.5%	1.000000
6	0.011082	+31.3%	1.000000

Critical observation: α_s does NOT converge. It crosses the target at C≈4 and keeps increasing. In D=4, α_s always approaches from below and converges monotonically.

Phase 2: Finite Difference Confirmation

The third-order finite difference method agrees with the polynomial fit to <0.01%, confirming the extraction is robust (not a fitting artifact).

Phase 3: C→∞ Extrapolation

Fit α_s(C) = α_∞ + a/C + b/C²:

• α_∞ = 0.01681 (99% above target)
• The extrapolation diverges, not converges

Phase 4: D=4 Comparison

D=4 at the same C values shows monotonic convergence from below:

• C=3: −17.4%, C=4: −10.5%, C=5: −6.6%

This confirms the D=5 behavior is qualitatively different, not a systematic lattice artifact.

Phase 5: Convergence Rate Comparison

C	D=4 dev%	D=5 dev%
2	−30.6	−45.4
3	−18.1	−16.9
4	−10.6	+4.0
5	−6.5	+19.5

D=4 converges monotonically from below. D=5 crosses zero and diverges upward.

Conclusion: The Dimensional Formula Fails at D=5

The conjecture α_s(D) = Γ((D-1)/2)/(12·π^{D-3}) is WRONG.

Evidence:

1. α_s(5) does not converge toward 1/(12π²) — it overshoots and keeps growing
2. The C→∞ extrapolation gives 99% deviation (not approaching zero)
3. The convergence pattern is qualitatively different from D=4 (crossover + divergence vs monotonic approach)
4. Both polynomial and finite-difference methods agree, ruling out fitting artifacts

Implications for the Framework

What this means:

• α_s = 1/(24√π) for D=4 does NOT arise from a simple dimensional formula
• The formula was a 2-point fit (D=3,4) that happened to work — classic overfitting
• The D=4 value 1/(24√π) may still be analytically derivable, but through D=4-specific physics (e.g., conformal properties special to 4D), not a universal dimensional scaling

What this does NOT affect:

• The D=4 result α_s = 1/(24√π) remains confirmed to 0.011% (V2.288)
• The prediction Ω_Λ = 149√π/384 is unaffected (it uses only D=4 physics)
• All cosmological predictions (H_0, species dependence, BSM exclusion) stand
• The framework’s validity in D=4 is not in question

Honest assessment:

The dimensional scaling conjecture was a natural hypothesis worth testing. Its failure teaches us that the D=4 Srednicki coefficient has a D=4-specific analytical origin — perhaps related to conformal flatness of the Einstein static universe, or to the special role of the Weyl tensor in exactly 4 dimensions. The search for an analytical derivation of α_s = 1/(24√π) must focus on 4D-specific mechanisms.

Files

• src/lattice_d5.py — D=5 Srednicki lattice (entropy, polynomial + finite-diff extraction)
• run_experiment.py — 5-phase convergence analysis
• tests/test_lattice.py — 9/9 tests passing