Conformal Transformations That Click

In 1918 Weyl let the standard of length be chosen freely at every point, and tried to unify gravity with electromagnetism. Einstein's objection — atomic spectra would stop being sharp — killed it; eleven years later the same idea came back as the gauge of phase. The story of how "gauge" once literally meant a ruler's calibration.g̃ = Ω²g

The universe does not expand; every particle mass grows and atoms shrink to match. That is a real 2013 paper (Wetterich). It works because the only thing a telescope can measure is the ratio "distance between galaxies ÷ radius of an atom".m̃ = a·m

Apply the conformal transformation to c·t = const from the previous series. Two moves turn it into exact Minkowski space — and reading that same metric in cosmic time gives back "the speed of light decreases". Invariance of α turns out to be a theorem, not an extra condition.ds̃² = −c_B(t)²dt² + dx²

Put the conformal transformation into the action and out comes φ²R̃ + 6(∂φ)². The coefficient is 1/6 — exactly the conformal coupling in four dimensions. The scale factor was never a property of spacetime; it was a field living on it.φ²R̃ + 6(∂φ)²

The Friedmann equation is a ball rolling with exactly zero total energy. Radiation is flat, matter is a slope, Λ is φ⁴ — and c·t = const is precisely a mass term. w = −1/3 sits on a triple boundary.U(φ) ∝ −φ^(1−3w)

After the transformation R̃ = 0 and spacetime is geodesically complete — the Big Bang disappears. But curvature is not conformally invariant. Build a dimensionless ratio and you get literally the same expression in both pictures, and it breaks in both. The answer is "only half".N = mc²t/ℏ

Maxwell's action is exactly conformally invariant in four dimensions and nowhere else. Mass snags on the transformation because it brings in a definite length — the Compton wavelength. Both ends of cosmic history are places where mass stops mattering: Penrose's cyclic cosmology.Ω^(D−4)

Quantising requires a scale of resolution — and the symmetry dies the moment you introduce one. The size of the breaking is measured by the β function, so "1/137 becoming 1/128" was that measurement all along. About 99% of your body weight is a by-product.T^μ_μ = (β/2g)F²

Its kinetic term has the wrong sign. Classically harmless, because φ is pure gauge — but a path integral will not accept "it's gauge, so don't look". The action becomes unbounded below: the oldest sore spot in quantum gravity.S_E ∝ −k²

Set "the question itself is misguided" (Flanagan) beside "you are forced to pick one" (Faraoni), and show that the entire dispute is about what counts as an observable. Ten episodes of bookkeeping and physics, folded into a single table.only dimensionless invariants can answer

Episode 8's anomaly has two uses. Demand that it vanish and the dimension of spacetime falls out (D = 26, 10); demand that it be positive and the renormalisation group acquires a direction (the a-theorem). The tool of that proof is episode 4's dilaton, and a turns out to be the entanglement entropy across a sphere — the flow is irreversible because it is forgetting. Joining bonus ②'s information story, sixteen episodes close here.a_UV > a_IR

There were two conformal transformations all along. And a CFT's central object Δ is exactly the Weyl weight this series has been computing — the anomalous dimension is episode 8's anomaly operator by operator, known to seven digits (0.0181489) for 3d Ising. A bootstrap using no dimensionful input yields six critical exponents that water and magnets both obey. And stability turns out to be decided not by the sign of m² but by whether Δ is real.O → Ω^(−Δ) O

G carries units, so it was bookkeeping. The physics is a per-particle α_G=(m/M_Pl)², spanning 10²⁵ from neutrino to top — and it is exactly "how close that particle is to being a black hole". The electron's gravity splits cleanly into the flavour problem times the hierarchy problem. And the phase of a mass is invisible to classical gravity, surfacing only via the gravitational anomaly — in rotating spacetimes alone.R R̃ ∝ Im(Ψ₂²)

Does the mass have an imaginary part? A conformal transformation cannot touch the phase (Ω is real and positive) — but the phase has a second knob, the chiral rotation, which erases it for a single field and makes the mass heavier anyway. What cannot be erased is the phase difference. Then take √m: the charged leptons land dead on 45°, and the electron is 2.27° from massless. For neutrinos, phases cancel masses outright.m = |m| e^(iθγ₅)

In the most primitive language a computer has it becomes one line — the horizon advances one Planck length per Planck time. Cells = ticks = 8.08×10⁶⁰. And "why one?" is answered by the saturation of the strong energy condition: matter contributes nothing to the focusing, and the wiring is that of an empty universe.dR_H/dt = c

Putting the motivation behind c·t=const on trial. The unique expansion law with a constant comoving Hubble radius — the address space does not move. 10¹²² bits of memory, a clock that ticked 140 times, energy per bit exactly on the Landauer limit. But radiation alone cannot be saved (it is conformally invariant) — and the gravitational field itself splits into bookkeeping and physics.it was constraining the wrong thing

Three ways of telling the story do not earn the model a single extra point. Push a ∝ t back to nucleosynthesis and the universe cools too slowly, neutrons are depleted, and helium never forms — and the verdict is identical in all three pictures.n/p = e^(−Q/k_BT_f)