c·t = CONST, THAT CLICKS EPISODE 21 ／ Setting four scales side by side
Two go up and two go down — and it was no contradiction
There are four scales
for the arrow of time
Total entropy, memory occupancy, holographic margin, degrees of freedom \(a\).
Do they point the same way? We check.
We now have four scales for the direction of time — total entropy (Episode 6), memory occupancy (Episode 6), holographic margin (Episode 20), and the degrees of freedom \(a\) (Episode 2). Oddly, two of them increase and two decrease. They measure the same arrow, and point opposite ways. Today we set the four side by side and locate where the arrow of time actually lives.
01The four scales
| Scale | Change | Orders | Direction | From |
|---|---|---|---|---|
| ① Total entropy \(S/k_B\) | \(\sim1\to3.1\times10^{104}\) | \(+104\) | up | Episode 6 |
| ② Memory occupancy \(S/S_{\max}\) | \(1\to1.5\times10^{-18}\) | \(-18\) | down | Episode 6 |
| ③ Holographic margin | 0 orders \(\to\) 33 orders | \(+33\) | up | Episode 20 |
| ④ Degrees of freedom \(a\) (a-theorem) | \(995.5\to62.0\) | \(-1.2\) | down | Episode 2 |
① and ③ up, ② and ④ down. It looks contradictory. But —
02All four are conformally invariant
| Scale | What it is | Under a conformal transformation |
|---|---|---|
| Total entropy | a bit count | invariant |
| Memory occupancy | bits ÷ bits | invariant |
| Holographic margin | log of a ratio | invariant |
| Degrees of freedom \(a\) | a pure number | invariant |
Conclusion of §02
The arrow of time sits entirely in the "physics" column.
On Episode 16's map of weights, all four are in the weight-0 column.
── Rewriting the books cannot touch the direction of time at all.
This backs up, with four scales, the table from Extra 2 of the previous series: the conformal-factor side has no arrow of time; only the Weyl side does.
03The heart — compare numerator and denominator at the same rate
The directions look opposed because some of these are ratios and some are not. Separate them and put them in the same units — orders of magnitude per logarithmic step.
Denominator (capacity): from \(\sim1\) at the Planck era to \(2.05\times10^{122}\) today
$$\frac{122.31\ \text{orders}}{140.24\ \text{steps}}=0.872\ \text{orders/step}$$Numerator (actual entropy): from \(\sim1\) to \(3.1\times10^{104}\)
$$\frac{104.49\ \text{orders}}{140.24\ \text{steps}}=0.745\ \text{orders/step}$$Difference
$$0.127\ \text{orders/step}$$The thing this episode most wants to say
Numerator and denominator are both growing furiously. The denominator is merely slightly faster — by 0.13 orders per step.
Stack that 0.13 over 140 steps and you get 18 orders of empty space.
matching the measured value of Episode 6
So "two up and two down" is no contradiction — ① (the numerator) goes up, while ② and ③ (ratios) are decided by which side wins. The denominator is set by the expansion, so the direction of the ratio depends on the expansion law.
Figure: how the numerator (actual entropy) and denominator (holographic capacity) grow per logarithmic step. Both grow furiously, and the slopes differ by only 0.13 orders per step. The shaded region is the "empty space", reaching 18 orders after 140 steps.
04So where is the arrow of time?
Episode 6 wrote that "how badly the tool is breaking is the arrow of time". In today's terms — that was the statement measured by the ratio (②). Precisely, the tool breaks because the numerator grows, and it looks diluted because the denominator grows faster. Unmixed, the arrow of time lives only in the numerator.
05The a-theorem is on a different axis
| ①②③ | ④ the a-theorem | |
|---|---|---|
| Direction of what | cosmic time | the renormalisation group (energy) |
| Axis | \(\ln t\) (140 steps) | \(\ln\mu\) (73 steps) |
| Why it decreases | expansion widens the capacity | coarse-graining discards information |
| In common | both are forgetting — a direction you cannot go back along | |
As Episode 2 showed, the two axes are linked by \(d\ln T/d\ln t=-p\). One step of cosmic time is \(p=0.513\) steps of renormalisation group flow. So the decrease of \(a\) can be converted into cosmic time, giving 4.8% of forgetting per step over the active range (steps 74–132).
All three have a direction you cannot reverse, all three are dimensionless, and all three are untouched by a conformal transformation — the arrow of time sits entirely in the column this series has been calling "physics".
① "\(S_{\rm obs}\sim1\) at the Planck era" is an indicative value. How to count the entropy of the early universe is not obvious (particles inside the horizon? horizon entropy?), so read it as an order-of-magnitude argument. The 0.745 orders/step of §03 depends on it.
② \(S_{\rm obs}=3.1\times10^{104}k_B\) is the Egan & Lineweaver (2010) census and depends strongly on the supermassive black hole mass function (same caveat as Episode 6 ①). Most of the numerator's growth is black hole formation, so that uncertainty passes straight into the 0.745.
③ "The numerator never decreases" is the second law for a closed system. Whether the universe is a closed system, and whether the interior of the horizon may be treated as one, are themselves unsettled questions — there is flow across the horizon, so naive application needs care.
④ \(a\) is defined only at conformal fixed points (Episode 2; Extra 7 of the previous series). "\(a\) falls from 995.5 to 62" is a schematic free-field count, not a test of the a-theorem, and "4.8% per step" likewise.
⑤ "The arrow of time is in the numerator" is this series' own account. The debate over the origin of the arrow (the past hypothesis, the Weyl curvature hypothesis, decoherence, cosmological initial conditions) is unsettled and this document endorses none of them — its scope is confirming that the four scales do not contradict each other.
Exercises (solvable with this episode's formulas alone)
- Of the four scales, which increase and which decrease?
Show the answer
Up: total entropy (\(+104\) orders), holographic margin (\(+33\)). Down: memory occupancy (\(-18\)), degrees of freedom \(a\) (\(-1.2\)). No contradiction — some are ratios and some are not. - Find the growth of numerator and denominator in orders per step.
Show the answer
Denominator \(122.31/140.24=0.872\), numerator \(104.49/140.24=0.745\) orders per step. Difference 0.127, which over 140 steps is 17.8 orders, matching the occupancy \(1.5\times10^{-18}\). - Why are all four untouched by a conformal transformation?
Show the answer
Because all four are dimensionless (a bit count, a ratio, the log of a ratio, a pure number) — the weight-0 column of Episode 16's map. Rewriting the books cannot touch the direction of time. - Restate Episode 6's "how badly the tool is breaking is the arrow of time" in today's terms.
Show the answer
That was the statement measured by the ratio (occupancy). Precisely, the tool breaks (the Weyl side grows) because the numerator grows, and it looks diluted because the denominator grows faster. Unmixed, the arrow lives only in the numerator. - (Harder) Is the a-theorem the same arrow as the other three?
Show the answer
The direction is the same (irreversible) but the axis differs — ①②③ live on cosmic time \(\ln t\) (140 steps), \(a\) on the renormalisation-group energy axis \(\ln\mu\) (73 steps). Episode 2's \(d\ln T/d\ln t=-p\) converts between them, giving 4.8% of forgetting per step over the active range. What they share is the shape: coarse-grain and you cannot go back.
Summary — the arrow is in the numerator; the denominator is the stage
Four scales for the direction of time: total entropy (\(+104\) orders), memory occupancy (\(-18\)), holographic margin (\(+33\)), degrees of freedom \(a\) (\(-1.2\)). Two up, two down. The first thing to confirm was that all four are dimensionless — conformally invariant: the arrow of time sits entirely in the "physics" column, untouchable by rewriting the books.
The apparent opposition came from mixing ratios with non-ratios. In the same units — denominator (capacity) 0.872 orders/step, numerator (entropy) 0.745. Both furious; the difference is only 0.127 orders per step. Stacked over 140 steps that is 17.8 orders, exactly Episode 6's occupancy \(1.5\times10^{-18}\).
So the arrow's home is clear — it is in the numerator. The second law constrains \(S_{\rm obs}\), which never decreases. The denominator is a stage set by geometry (expansion), and ratios are just the outcome of the race. Episode 6's "how badly the tool is breaking is the arrow of time" was the version measured by a ratio.
The fourth scale, \(a\), lives on a different axis — the renormalisation-group energy axis, not cosmic time. Episode 2's \(d\ln T/d\ln t=-p\) converts between them, giving 4.8% of forgetting per step over the active range. Different axis, same shape: coarse-grain and you cannot go back.
This document is Episode 21 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. It sets side by side the results of Episodes 2, 6 and 20 and makes no new claim of physics. From the denominator \(S_{\max}=2.05\times10^{122}\) (the horizon's holographic bound), the numerator \(S_{\rm obs}=3.1\times10^{104}k_B\) (Egan & Lineweaver 2010, ApJ 710, 1825) and the logarithmic step count \(\ln(t_0/t_P)=140.24\), the figures 0.872 and 0.745 orders per step, their difference 0.127, and 17.8 orders over 140 steps are computed here (kenshou/calc25.py). "\(S_{\rm obs}\sim1\) at the Planck era" is indicative, and how to count early-universe entropy is not obvious — the 0.745 depends on it. \(S_{\rm obs}\) depends strongly on the supermassive black hole mass function, and that uncertainty passes into the numerator's growth. "The numerator never decreases" is the second law for a closed system, and whether the interior of the horizon may be treated as one is unsettled (there is flow across the horizon). \(a\) is defined only at conformal fixed points; "995.5 → 62" is a schematic free-field count rather than a test of the a-theorem (Extra 7 of the previous series), and "4.8% per step" likewise. The link between the axes, \(d\ln T/d\ln t=-p\), was derived in Episode 2. "The arrow of time is in the numerator" is this series' own account; the debate over the arrow's origin (the past hypothesis, the Weyl curvature hypothesis, decoherence, cosmological initial conditions) is unsettled and none is endorsed here — the scope is confirming that the four scales do not contradict each other. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination. The academic standard remains the \(\Lambda\)CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).