Math Grounding VI — The Cadence Arc
Weightless Orbits and Why You Fall Into a Planet
The cadence balance between inward and outward
We’ve just seen how the universe keeps two rhythms — an inward one that gathers things into rest, and an outward one that opens the deep field into drift.
But in between those two slopes sits one of the most familiar motions in the sky:
orbit.
Everyone hears the classic line:
“you’re always falling, but you miss the ground.”
Cadence lets us see why that’s true — and why it feels weightless — and also answers the deeper question people always ask next:
“If space isn’t compressing or expanding, then how do I still fall and hit the ground?”
This is the moment to show how the Cadence Star handles both.
Let’s walk through the cleanest version of the math.
1. The Ordinary Equation Everyone Knows
A circular orbit satisfies the basic condition:
LaTeX: {\frac{v^2}{r}=\frac{GM}{r^2}}
Left side: sideways curvature from motion.
Right side: inward curvature from gravity.
Nothing exotic so far.
But cadence lets us read this equality in a new way.
2. The Same Equation in Cadence Language
Every motion tilts cadence between two legs:
TD — Temporal Descent: the inward rhythm of depth, gravity, rest.
TS — Temporal Stretch: the outward rhythm created by motion, sideways curvature.
In an orbit, these two rhythms meet.
Sideways motion creates an outward TS projection
LaTeX: {a_{\mathrm{TS}}=\frac{v^2}{r}}
Gravity creates an inward TD curvature
LaTeX: {a_{\mathrm{TD}}=\frac{GM}{r^2}}
Orbit happens when the two triangles match
LaTeX: {a_{\mathrm{TS}}=a_{\mathrm{TD}}}
At that moment, the inward triangle and the sideways triangle close into a single cadence arc.
Your fall becomes a foundation.
Your path holds steady.
The Cadence Star stays in balance.
3. Whys Orbit Feels Like Nothing
Your body only feels proper acceleration — the part of cadence curvature that isn’t balanced by motion.
Standing on Earth:
TD acts, TS = 0 → you feel weight.Free fall:
TD still acts, but TS from your path matches it → you feel nothing.Orbit:
TD = TS along your path → continuous free fall → weightlessness.
You’re not defying gravity.
You’re matching it.
Your body rides the closed triangle.
4. Why You Fall Into a Planet Even Though Space Isn’t Compressing
This is the part that matters most after The Outward Rhythm.
Gravity is not space pushing or pulling.
It’s cadence tilting inward.
When you stand above a planet, your Light Frame sits on a slope of deeper cadence (TD).
Your proper rhythm increases as you fall inward — not because space shrinks,
but because your cadence is sliding into a deeper layer of the same Light Frame.
This inward slide creates physical acceleration:
LaTeX: {a_{\mathrm{TD}}=\frac{GM}{r^2}}
As long as nothing balances that inward cadence, you fall.
When you orbit, sideways TS cancels TD.
When you stand, no TS cancels anything — so TD acts fully and you feel it as weight.
You don’t fall because space compresses.
You fall because cadence descends.
Your Light Frame is literally being drawn into deeper curvature.
5. The Bridge Back to Drift (The Outward Rhythm)
Here’s the elegant symmetry:
Locally:
sideways TS from motion balances TD → orbit.Cosmically:
outward TS from drift balances the slow global TD → apparent expansion.
Same geometry.
Same Cadence Star.
Just different scale and source of stretch.
This is the missing piece between The Outward Rhythm and The Optical Present:
the same rhythm that carries galaxies outward
also carries you inward —
and the same balance that makes you float
is what shapes the universe at large.