Mathematical Grounding V — The Three Ratios
How the Universe Keeps Its Time
The Three Ratios: LFR, LFC, LGR
Why three? Because you need:
a local read on how stretch vs. depth splits (LFR)
a relational read that links two observers across distance (LFC)
a global (universe) read that sets the cadence against gravity and scale (LGR)
Together they keep the beat even as the universe drifts.
LFR: Light-Frame Ratio (local balance)
What it measures: the local tilt between stretch and depth — how the nearby frame is leaning right now.
Definition (local):
LaTeX: \mathrm{LFR}(x) = \frac{dR_{\mathrm{TS}}/d\theta}{dR_{\mathrm{TD}}/d\theta}
Balance:
LaTeX: \mathrm{LFR} = 1
→ TS and TD trade curvature evenly (cadence-flat rest).
Range (drift form):
LaTeX: \mathrm{LFR}(D) = 1 + \varepsilon(D),\qquad 0<\varepsilon\ll 1
ε(D) is the small surplus that accumulates with distance. Nearby, it’s tiny; deep range makes it visible (your “edge of the universe” effect).
What to remember: LFR is here-and-now. It tells one observer how their local frame splits TS vs. TD.
LFC: Light-Frame Coupling (two-point rhythm)
What it measures: how two observers’ cadences line up across distance — the shared optical present between A and B.
Definition (two-point):
LaTeX: \mathrm{LFC}(A!\leftrightarrow!B) = \frac{d\theta_B}{d\theta_A}
Gradient → apparent “fly-by” acceleration (optical movie):
LaTeX: a_{\mathrm{cad}}(r) = -\frac{d\,\mathrm{LFC}(r)}{dr}
When LFC → 1 at the perpendicular fly-by midpoint, your movies sync (the hypotenuse rhythm). Away from that point, the movie stretches (red) or compresses (blue) depending on sign.
What to remember: LFC is between-us. It defines the link that makes reunion consistent even when the movies look lopsided.
LGR: Light-Frame Gravitation Ratio (field calibration)
What it measures: how cadence bends with gravitational potential — the global setting that keeps LFC honest over scale.
Definition (calibration form):
LaTeX: \mathrm{LGR}(\Phi) = \frac{d\theta(\Phi)}{d\theta_\infty}
Here dθ(Φ) is the local cadence beat at potential Φ; dθ∞ is the reference far from mass. Near wells, LGR > 1 (TD wins); far out, LGR → 1 (TS re-levels).
What to remember: LGR is field-wide. It sets how gravity tilts the page so LFC doesn’t drift apart over cosmic distance.
Beneath all three ratios lies the invariant cadence slope:
LaTeX: C_0 = \frac{1}{c}
the universal beat that TS, TD, and the Light Frame preserve.
How they interlock
LFR (local) tells you the tilt of your own frame.
LFC (relational) tells you how two frames talk through light.
LGR (global) sets the calibration so that talk stays coherent across the mass and scale of the universe.
Local tilt (LFR) → linked by LFC → two-point rhythm → set by LGR → global coherence
In words: LGR steadies LFC; LFC reveals LFR.
Quick Reads
LFR = 1 → cadence-flat (TS and TD trade evenly)
LFC ≈ 1 → movies align at midpoint
LFR(D) = 1 + ε(D), ε ≪ 1 → non-linear optical movie
LGR(Φ_deep) > LGR(Φ_shallow) → Δτ persists unless matched by a counter-descent
Closing
Hold the picture like this: LFR tells you how your floor is tilted, LFC tells you how your floor talks to mine, and LGR tells us how the whole building leans.
The universe doesn’t lose time; it redistributes it — locally, between us, and across the field.
Together these three ratios keep the rhythm coherent across the mass and cadence of the universe — the ruler by which we see time itself held in balance.