Cosmological constant

SIMPLIFIED DERIVATION & CONCEPTUAL CONNECTION

Can temporal completion fix the absolute vacuum contribution to curvature?

Vacuum curvature depends on an absolute geometric action coefficient r. Normalized phase responses and relative spectra discard a common offset, so they cannot determine it alone. Temporal averaging can reduce fluctuations while leaving the mean unchanged.

Read the common SMM/DTT foundations and notation →
Jump to the live workbench ↓

The SMM/DTT connection

If vacuum geometry is continuously generated and only completed records are retained, it is natural to ask whether this process changes the gravitating vacuum contribution. The key distinction is between a relative response of completed records and the absolute action that couples to geometry. A successful DTT vacuum mechanism must specify the latter.

Retain the constant term of the geometric action

S_geom/a = ∫d⁴y √|g|(κℓ*²R − r).

After x = ℓ*y, the curvature coefficient is aκ/ℓ*² and the constant coefficient is −ar/ℓ*⁴. Matching both terms to [c³/(16πG)](R − 2Λ) gives:

Λ = r/(2κℓ*²), [Λ] = m⁻².

Λ is a curvature parameter. Its energy-density counterpart is ρ_E = Λc⁴/(8πG), while its mass density is ρ_E/c². These quantities have different units.

Show exactly what normalization removes

Multiply a transfer operator by a common source-independent factor, T_C = exp(−C)T₀. Every eigenvalue acquires that factor, so all eigenvalue ratios and relative gaps are unchanged. Likewise, for N_cell cells:

Z_C[J] = exp(−N_cellC)Z₀[J],
−log Z_C[0] = −log Z₀[0] + N_cellC.

Normalized expectations and positive-order derivatives with respect to sources independent of the factor do not see C. The absolute action does. If volume or measure depends on the metric, the offset may contribute to metric response. That dependence must be specified; it is not recoverable from normalized phase data alone.

A common shift leaves relative gaps unchanged while shifting the absolute vacuum action — Figure Λ1. The obstruction identifies missing information. It does not assert a universal rule for which offsets gravitate. Download SVG

Separate averaging from cancellation

For stationary samples ρₙ with mean ρ̄ and finite variance, the average ρ^(M) = M⁻¹Σρₙ has:

E[ρ^(M)] = ρ̄.
Var[ρ^(M)] = O(1/M) when the covariance is absolutely summable.

More samples can narrow fluctuations without reducing the mean. More generally, any normalized nonnegative weights obey Σpᵢ(ρᵢ + C) = Σpᵢρᵢ + C if the weights are unchanged by the shift. Repeated completion is not, by itself, a cancellation mechanism.

Check the quartic horizon proposal

A dimensionally permitted ansatz is ρ_E = Aac/L⁴. With L = c/H:

ρ_E = AaH⁴/c³.

At a = ℏ, A = 1 and the illustrative H = 2.27 × 10⁻¹⁸ s⁻¹, it gives about 1.04 × 10⁻¹³⁰ J m⁻³. The comparison density defined by ΩΛ = 0.69 is approximately 5.72 × 10⁻¹⁰ J m⁻³. The ratio is about 1.82 × 10⁻¹²¹. This particular horizon law is far too small for that benchmark. Choosing L ≈ 86 μm to match it is an inverse fit, not a predicted length.

Connect gravity and vacuum in a unit-free target

λₑ ≡ Λ[a/(mₑc)]²,
gₑλₑ = GΛa/c³ = r/(32πκ²).

The electron normalization cancels. A parameter-free value for r/κ² from one microscopic model would therefore be a meaningful joint target. The cancellation itself is an algebraic identity, not a new empirical prediction.

Try the workbench

Vary r at fixed κ and ℓ*: Λ changes linearly, including its sign. The “Infer r” control uses the illustrative cosmological benchmark and is labeled calibration. The horizon route is a separate stress test; its density conversion imports measured G. A successful dimensional check cannot rescue the failed numerical scale.

Source basis: C · §§12–14; pp. 19–23. These explanations summarize the stated models; they do not add a new physical selection theorem.

Explore Λ.