Cadence Cosmology — A Quick Clean-Up ★

Three clarifications we need before moving forward

Before we continue the Cadence Cosmology series, we need to pause and clarify three concepts that appeared in the Light Frame Papers but haven’t yet been fully introduced here on Substack.

These three ideas set the stage for everything that comes next.

Let’s get the foundation straight.

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1 · The Cadence Acceleration Scale (a₀)

We showed the average, but not the full story.

Earlier in this series, we mentioned a small acceleration scale that shows up in galaxy rotation curves:

a₀ ≈ 1.2 × 10⁻¹⁰ m/s²

This number appears consistently across real galaxies — including the full SPARC sample of 175 rotation curves. It’s the region where Temporal Stretch (TS) begins to balance Temporal Descent (TD).

But there is another way to estimate a₀ — a geometric prediction we haven’t shown here:

a₀(predicted) = c² / R*

Where R* is the curvature radius of the observable universe.

This predicted value is slightly higher than the observed one.

That mismatch isn’t random and it isn’t an error.

It has a cause.

Which brings us to the second clarification.

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2 · Why the predicted and observed a₀ differ

A faint cosmic rotation (ω)

If the universe had no global rotation, then:

a₀ = c² / R*

would exactly match the data.

But the real a₀ is slightly lower. Cadence Geometry explains this through a tiny torsional term — a faint rotational cadence built into the large-scale structure:

a₀ = c² / R* − ω² R*

Solving for ω gives:

ω ≈ 4.38 × 10⁻¹⁹ s⁻¹

This corresponds to a rotation period of:

T ≈ 4.54 × 10¹¹ years

A half-trillion-year rotation:

• slow enough to evade all cosmological constraints,

• strong enough to explain why the observed a₀ is lower than pure closure predicts.

This is the same rotation period independently found by Szapudi (2025) while analyzing the Hubble tension.

This rotational correction had not yet appeared in earlier Substack posts — so the “a₀ mismatch” was left hanging. Now the missing piece is on the table.

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3 · The δ Exponent in the Cadence Rotation Law

What it is, and why we haven’t derived it here yet

In the cadence rotation law, there is a mass-scaling behavior:

TS_boost ∝ (M / M₀)ᵟ

This δ exponent appears in:

• galaxy rotation curves,

• galaxy clusters,

• strong-lensing systems.

Readers deserve two clear facts.

• Where δ comes from

Astronomers discovered decades ago that galaxy dynamics follow a simple power-law scaling:

δ ≈ 0.25

This is not a free parameter.
It is a real, stable property of rotation-curve data.

• Why we’ve used δ without deriving it (yet)

Because the derivation requires deeper structure:

• the full cadence-field action,

• the cadence scalar,

• the origin-identity rules.

These belong in the later part of this series.

For now:

• δ is observed in real galaxies.

• Cadence Geometry explains why δ must take this value.

• The full derivation will be shown later.

This keeps the early posts readable while giving everyone the correct conceptual grounding.

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Where this leaves us

With these three clarifications:

1. a₀ has both an observed value and a geometric value.

2. The difference between them is a faint cosmic rotation.

3. δ is a real astronomical trend that Cadence Geometry later derives.

The road is clear for the next stages of the series.