A SEQUEL TO "CONFORMAL TRANSFORMATIONS THAT CLICK"
c·t = const, That Clicks
On its own, \(c\cdot t=\text{const}\) says nothing at all — it can be realised in any universe.
Which is exactly why it can be substituted into every equation there is: it is a notation for writing cosmology as short as it will go.
This series measures that shortness, all the way, in the language of information theory.
The backbone of the previous series was "dimensionful is bookkeeping, dimensionless is physics". But the quantities of information theory — bits, operation counts, parameter counts, entropy — are dimensionless from the outset. They carry no units. So a cosmology written in that language can only ever live in the "physics" column. We start again from where bonus episode ② of the previous series asked "is the universe a computer with finite resources?" and answered "the motivation was apt, the implementation missed" — this time not as a verdict, but as compression.
The method
Count, then divide. Memory \(N=\dfrac{\pi}{\ln2}\left(\dfrac{R_H}{\ell_P}\right)^2\), clock \(\ln\dfrac{t_0}{t_P}=140\), operations \(\Omega=\displaystyle\int\frac{2E}{\pi\hbar}dt\), parameter counts — all dimensionless. Divide any two and the expansion law itself comes out.
EPISODES
The universe has computed 0.035 operations per bit
Count the memory, count the operations, divide. Time cancels cleanly and only the equation of state is left standing — the number of operations per bit does not depend on the age of the universe, nor on its size. Radiation gives 1/57, matter 1/42.7, \(c\cdot t=\text{const}\) gives 1/28.5. Reaching "1" would need \(w=-0.977\), and the dark energy we observe already sits on the far side of that.
\(\Omega/N=\dfrac{\ln 2}{3\pi^{2}(1+w)}\) — the 8th characterisation of \(a\propto t\) EPISODE 2 LIVE interactive figureTwo clocks that do not mesh
The universe carries two logarithmic rulers: time, \(\ln(t_0/t_P)=140.24\) steps, and the renormalisation group, \(\ln(T_P/T_0)=73.03\) steps. Divide them and the expansion law appears (\(\bar p=0.513\), just above radiation). \(c\cdot t=\text{const}\) demands \(140.24=73.03\) — a verdict reached with no dynamics at all, using only today's temperature and today's age. Neutrons freeze out at 0.8 MeV after 4.05 years; a free neutron lives 880 seconds.
\(d\ln T/d\ln t=-p\) — the 9th characterisation (the two clocks tick 1:1) EPISODE 3 LIVE interactive figure from a reader's question\(c\cdot t=\)const can be realised in any universe
"A rewriting moves no dimensionless quantity. So how can it fail?" — we push that objection all the way. Push it hard enough and \(c\cdot t\) can be held exactly constant in a radiation universe or a matter universe alike (take the time coordinate \(T=e^{\eta/\eta_0}\); verified numerically to 1.000000). All that differs is what \(T\) is — for radiation, \(T\propto e^{\sqrt t}\), nobody's clock. Exactly one claim survives: that this clock is the age of the universe.
\(T=e^{\eta/\eta_0}\) — the 10th characterisation (three clocks coincide) EPISODE 4 LIVE interactive figureEverything collapses into a single mass
Why use a rewriting that says nothing? Because the equations get shorter. Erase everything a conformal transformation can erase and the expansion of space, the curvature, the temperature and the light all vanish in turn — leaving exactly one thing that varies in time. Cosmic history collapses to "a growing mass overtaking a fixed \(k_BT_0\)" (recombination at \(1+z=1100.9\), neutron freeze-out at 0.800 MeV — exactly the standard numbers). Distances and ages become closed forms, with no integrals.
\(H_0d_L/c=(1+z)\ln(1+z)\) — a prediction with zero dimensionless parameters EPISODE 5 LIVE interactive figureWhat can shortness buy?
Zero description length (Ep.4) against a 0.21 mag mismatch at \(z\simeq1.1\). We convert both into one currency: bits. A parameter costs 1.443 bits under AIC (independent of sample size) and \(\tfrac12\log_2N\) — 5.37 bits for 1701 supernovae — under BIC. The fit loss is \(\Delta\chi^2/(2\ln2)=\) 154 bits, so the ledger closes 149 bits in the red. And yet with fewer than 26 supernovae, \(c\cdot t=\text{const}\) is the better model: shortness pays \(\log N\), misfit costs \(N\). Occam's razor has an expiry date.
One parameter \(=\tfrac12\log_2 N\) bits EPISODE 6 LIVE interactive figureOnly \(10^{-18}\) of the memory is in use
The occupancy turns out to be a ratio of areas — glue together the horizons of every black hole in the universe and you get a sphere 17 light-years in radius. And 99.999999999999986% of today's entropy sits on the gravitational (Weyl) side. The history comes in three steps: at the Planck era the occupancy was \(\approx1\) (full), today the thermal side alone is \(7\times10^{-34}\) (it emptied out), and with black holes \(1.5\times10^{-18}\) (gravity refilled 15.4 orders). A conformal transformation can only move the side that is not in use — how broken the tool is, is the arrow of time.
occupancy \(=\sum A_{\rm BH}/A_H\)Feeding it to gravity
Here \(\tilde G=G/a^2\) is forced, giving \(\dot G/G=-2H_0=-1.45\times10^{-10}\)/yr — 1450 times the lunar-laser-ranging bound. And still nobody can notice (because \(\alpha_G\) does not move). Then the punchline: the two large numbers Dirac used in 1937 to predict \(G\propto1/t\), \(N_1=2.27\times10^{39}\) and \(N_2=4.63\times10^{40}\), turn out to be both invariant when you actually vary \(G\) — you cannot move \(G\) alone, so Dirac's prescription spins in place.
Only black-hole entropy and the strain \(h\) stay put — the tool cannot reach the memory that is in use EPISODE 8 LIVE interactive figureFeeding it to quantum mechanics
Both sides of the Schrödinger equation carry weight \(-5/2\), so not a single character changes — only \(m(t)=m_0t/t_0\). Then free-particle velocity decays as \(1/t\) and a wave packet spreads as \(\Delta x\propto\ln t\). Integrate and the comoving reach of anything with mass saturates at \(v_1t_1\) (1.9 parsecs for a hydrogen atom at recombination); only light runs on forever as \(c\,t_1\ln(t/t_1)\). Tunnelling probability is dimensionless and therefore exactly invariant — the Sun burns at the same rate.
Matter stops walking; light keeps walking EPISODE 9 LIVE interactive figureFeeding it to the atom
Atoms shrink as \(1/t\) here — and in 1918 Einstein killed Weyl's unified theory with exactly this argument. Two reasons the same blade misses: (i) \(\Omega\) is single-valued, so there is no path dependence to begin with; (ii) the adiabatic parameter is \(\hbar H/\Delta E=1.1\times10^{-34}\). Even the softest transition (21 cm) would only break adiabaticity before \(10^{-10}\) s, so throughout the entire era in which atoms exist, the evolution is perfectly adiabatic. The induced line blurring is \(9\times10^{-20}\) of the natural width — 19 orders down.
The Bohr radius shrinks at \(7.2\times10^{-11}\)/yr, yet every possible yardstick has weight \(+1\) EPISODE 10 LIVE interactive figureFeeding it to heat and information
Temperature does not move here (\(\tilde T=aT=\)const), so the cost of erasing one bit is fixed for all of cosmic history (\(1.63\times10^{-4}\) eV). But energy is dimensionful — the price is undefined until you name both a comparison and which bath you dump into. Counting how many bits the universe's whole energy could erase, the answer swings 60 orders with temperature, and at the CMB temperature you can only erase \(10^{-30}\) of what you can write. And "exactly at the Landauer limit" turns out to be reading off an identity, \(E=T_HS\).
writable \(10^{122}\), erasable \(10^{92}\) — the universe is very nearly write-once EPISODE 11 LIVE interactive figureFeeding it to light
Number density, energy density, temperature, photon energy, wavelength — every one of them constant. The exponent of \(a\) in the standard picture and the weight of the quantity are the same number, so they cancel exactly. The photon gas in this picture is completely at rest. Observation still does not budge: the CMB is fixed by just two dimensionless numbers, \(s/n=3.60\,k_B\) and \(\eta=6.1\times10^{-10}\). So redshift flips over entirely — not "the light stretched" but "the receiver grew".
\(\Omega^{D-4}\) — living in four dimensions is why this picture works at all EPISODE 12 LIVE interactive figureFeeding it to the vacuum
Energy density has weight \(-4\), so \(\tilde\rho=a^4\rho\). Applied to the three components, the ordering inverts completely: radiation becomes constant, matter goes as \(\propto t\), and the cosmological constant grows fastest of all at \(\propto t^4\). The very name "cosmological constant" depended on which picture you chose. And yet \(\rho_\Lambda/M_{\rm Pl}^4=1.13\times10^{-123}\) and \(\rho_\Lambda/\rho_m\propto a^3\) are both invariant — neither the cosmological constant problem nor the "why now?" problem moves by a millimetre.
Good puzzles are written in dimensionless form EPISODE 13 LIVE interactive figureFeeding it to fluids and turbulence
Reynolds, Mach, Prandtl, Froude, Weber, Strouhal — every one of them has weight 0. So similarity laws and wind-tunnel tests carry over untouched. The interesting part is the breakdown: the four pieces of \(\mathrm{Re}=\rho vL/\eta\) move as \(a^4,a^0,a^{-1},a^3\) — wildly and separately — yet the exponents sum to \(4-1-3=0\). Kolmogorov's \(-5/3\), the critical exponents and the fractal dimensions are all invariant too: a conformal transformation touches "size" only, and cannot reach "shape".
The Navier–Stokes equations survive this picture completely intact EPISODE 14 LIVE interactive figureFeeding it to phase transitions
The weight table this series has used for thirteen episodes was a classical approximation — in field theory \(\Delta=\Delta_{\rm cl}+\gamma\). For the 3D Ising spin operator the free-field value 0.5 becomes 0.5181489(10), a discrepancy of \(\gamma_\sigma=0.0181489\) (3.6%). In two dimensions it is 12.5%, and in four dimensions it vanishes exactly. Through \(\eta=2\gamma_\sigma\), that discrepancy is measurable in water and in magnets. What broke was not "dimensionless is invariant" but the assumption that weights follow from dimensional analysis.
The ledger entries acquire error bars — weights were something to be measured EPISODE 15 LIVE interactive figureFeeding it to chemistry and biology
Arrhenius factors, equilibrium constants and pH all have dimensionless exponents, so they come through entirely unscathed. Take Kleiber's law apart and the exponent 3/4 is invariant while only the coefficient moves, as \(\times a^{5/4}\) (dimensionful, hence bookkeeping). Heart rate \(\propto M^{-1/4}\) times lifespan \(\propto M^{1/4}\) gives about 1.5 billion beats per lifetime, independent of body mass and therefore invariant. Everything life can measure is dimensionless — so living things cannot tell which picture they are in.
Ep.9's "the atom has nothing to compare against", pushed up to the scale of biology EPISODE 16 LIVE interactive figureOnly one thing ever moves (Part II summary)
Nine fields, one notation, and a count of what moves. The answer had the same shape every time: everything that moved was dimensionful, everything that stayed was dimensionless — without a single exception. The map of weights is complete (from \(+3\) down to \(-4\)), and the range of the tool becomes clear: powerful where size is the protagonist, entirely powerless where shape is. With the caveat from Ep.14 that the weight table itself carries error bars.
Not "safe because dimensionless" but "safe because observable"9600 nodes that never communicated agree to 17 bits
The horizon problem, restated in the language of distributed systems. \(\Delta T/T\sim10^{-5}\) is 16.6 bits of agreement, the causally disconnected regions number 9600, and the product is about 20 kilobytes — a phone would send it instantly. The problem was never the quantity; it was that there was no channel. With \(a\propto t\) the particle horizon diverges, so there is one patch and zero bits to agree on — until you put radiation back in, at which point it breaks at \(z>103\) and \(1.2\times10^4\) nodes return. Inflation solves it not by consensus but by replication.
Coincidence would need \(10^{-48000}\) — the option is gone EPISODE 18 LIVE interactive figureThere are not enough address lines
Holography read as addressing. The universe has \(5.27 imes10^{182}\) spatial cells and can write \(2.96 imes10^{122}\) bits, so only \(10^{-61}\) of the cells can be addressed — and since the ratio goes as \(1/R_H\), the gap widens as the universe grows. Inverted, one bit is responsible for a cube of side 1.96 fm — the size of a proton, the cube-root intermediate scale \((R_H\ell_P^2)^{1/3}\), an unexplained coincidence. Holography is not compression: volume cells were never given addresses at all.
Addresses grow only with area — and the address table will not fit in memory EPISODE 19 LIVE interactive figureIs an identity really not physics?
This series has sorted coincidences as identity, coincidence or physics again and again — here the criterion is stated. Surprise \(=-\log_2( ext{width}/ ext{prior range})\), and the strata separate cleanly: identities at 0 bits, coincidences at a few, real problems at \(10^5\). The factor-22 agreement of \( ho_\Lambda^{1/4}\) and \(m_ u\) is five coin flips; only Koide’s relation, at 15.7 bits, is surprising by orders. And an identity is not a prediction but a consistency check.
Identities are 0 bits — but 0 bits is not the same as meaningless EPISODE 20 LIVE interactive figureActually constraining the light sheets
The previous series closed with the line “what should be constrained is the information on the light sheets — nobody has done that calculation.” Done here. The occupancy \(f=s/(3H/4\ell_P^2)\) saturates exactly at the Planck era and opens to a margin of 33 orders — and that saturation is an identity, 0 bits of surprise. As a constraint it excludes almost nothing. Fix the bit count and you get de Sitter, the address space and you get \(a\propto t\), the occupancy and you get stiff \(a\propto t^{1/3}\); the observed universe is none of them.
Right in form, insufficient in effect — it bites only at the Planck era EPISODE 21 LIVE interactive figureThere are four scales for the arrow of time
Total entropy (+104 orders), memory occupancy (−18), holographic margin (+33), degrees of freedom \(a\) (−1.2). Two up, two down — and all four are dimensionless, so the arrow sits entirely in the physics column. Put in the same units, the denominator grows at 0.872 orders per step and the numerator at 0.745: a difference of only 0.127, which over 140 steps gives exactly Episode 6’s occupancy. The arrow lives in the numerator; the denominator is the stage.
Rewriting the books cannot touch the direction of time EPISODE 22 LIVE interactive figureThe instruction set of the universe-as-computer
The ML rate is proportional to energy, so the operational budget is the energy budget. And ML measures energy above the ground state — the vacuum has nowhere to transition to, removing 68.5% at a stroke; of what remains, 84.1% is dark matter, which interacts only gravitationally. 95.0% of the budget goes to components in which nothing happens. Starlight, the most conspicuous activity there is, accounts for one millionth of the total, and Episode 1’s 0.035 operations per bit falls to one per 580 bits.
The gap Episode 1 flagged as “a spec sheet, not a benchmark”, measured EPISODE 23 LIVE interactive figureThe horizon as error correction
Read the horizon as a code: \(n=2.96 imes10^{122}\) physical bits, \(k=4.47 imes10^{104}\) logical, giving a redundancy of \(6.6 imes10^{17}\) — fourteen orders beyond the quantum surface code. That is Episode 6’s occupancy read inside out, and the heart of it: “empty” and “redundant” are the same number until you name the comparison. Episode 3’s surgery, this time applied to the series’ own figure. Closing with an act of restraint — the logical bit’s length matches nothing, so nothing is said.
AdS/CFT is the precedent, but a cosmological horizon is not established EPISODE 24 LIVE interactive figureHow many bits per second cross the horizon?
Dividing the Bekenstein bound by a crossing time gives \(C=2\pi E/(\hbar\ln2)=6.79 imes10^{104}\) bit/s. Its factor-of-two relation to Episode 1’s \(dN/dt\) is an identity, and the clean form is \(C\cdot t=N\) — the universe has exactly enough bandwidth to move its entire memory once per Hubble time. Which settles Episode 17: the 20 KB could have been sent in \(10^{-96}\) seconds. Bandwidth was never the bottleneck; the wiring was.
Three routes now agree: the universe has power to spare EPISODE 25 LIVE interactive figureAre physical laws a compression algorithm?
\(a\propto t\) is 66 bits, the Einstein equations 512, the Standard Model 33,000; \(\Lambda\)CDM compresses \(10^6\)-fold. But totalled with MDL it breaks: counting \(a\propto t\) as an added constraint or as a replacement flips the same model on the same data from losing by 214 bits to winning by 200. Only \(L( ext{law})\) moved — a language-dependent quantity. What is trustworthy is the parameter count and the residual, and those alone give \(-148\) bits. Compression ratio measures the size of the bet, not the quality.
MDL’s compression ratio and Popper’s falsifiability are one axisThe same numbers, in eight languages (Part III wrap-up)
\(1.5 imes10^{-18}\) three times, \(140\) four times — chased down, they are all one number. Occupancy equals black hole share by a one-line identity from holography. Part III’s 24 headline numbers reduce to 12 independent inputs, and the surprises total 8.4 bits, of which 7.4 is the 1.96 fm already judged a coincidence in Episode 18. Part III did not discover; it restated the same numbers in eight languages — and that is how five identities and exactly one unexplained agreement became visible.
A map of structure, not new physics EPISODE 27 LIVE interactive figureInflation, on the same operating table
Cut “it solves the horizon problem” in two: (A) establishing causal contact, (B) producing a fluctuation spectrum. (A) is cheap — the particle horizon \(ct/(1-p)\) diverges at \(p=1\), so \(a\propto t\) removes the problem with zero e-folds and zero parameters. What survives is (B): the horizon fixes \(N_{\min}=62.1\) and the same \(N\) predicts \(n_s\), against an observed \(N=57.0\pm6.8\) — agreement at 0.75σ. The most famous motivation turns out to be the weakest argument.
Same surgery, different survivor EPISODE 28 LIVE interactive figureVSL — where the surgery went wrong
The \(c\) called “the speed of light” appears in four separate roles (Ellis & Uzan), and VSL fixes \(e\) and \(\hbar\), so its observable content is entirely “\(lpha\) varies” — against \(lpha\) pinned to 32.5 bits in the laboratory and 26.4 at Oklo. Solving the horizon problem demands \(lpha(z)/lpha_0\ge1+z\), ten orders over the nucleosynthesis bound. A phase transition escapes that — and loses every prediction in the observable era at the same stroke.
The failure was not moving c, but continuing to call it c EPISODE 29 LIVE interactive figureMOND — the comparison hidden inside an acceleration
\(a_0\) is dimensionful (weight \(-1\)), so “the acceleration is small” needs a comparison — and \(cH_0\) is sitting right next door, at \(a_0/cH_0=0.18\). The ratio has weight 0, so the coincidence cannot be moved by a conformal transformation; measured in bits it is 5.9, the same stratum as \( ho_\Lambda\) and \(m_ u\). On Episode 5’s scales, MOND wins galaxy rotation curves by 1971 bits — and loses clusters, the Bullet Cluster and the CMB. “Dark matter or MOND?” was never one question.
And the coincidence hides a testable fork: constant \(a_0\), or \(a_0\propto H\)? EPISODE 30 LIVE interactive figureMeasuring varying constants for real
Atomic clocks, the Oklo natural reactor, quasar absorption lines — three different physics, one skeleton: (change in the observable) = K × (Δα/α). Oklo’s precision is a coarse 2% and it matches an atomic clock, because its amplification is \(10^7\) — the 97.3 meV resonance is a difference of MeV-scale quantities. Placed on the logarithmic axis, data cover 29% of cosmic history and 26-bit precision covers 0.1%. And measurement of constants sits entirely in the weight-0 column — which is why it can be the referee.
A cancellation of orders can be a mystery or a tool EPISODE 31 LIVE interactive figurePenrose’s conformal cyclic cosmology
CCC’s central move is exactly Episode 11’s result — with no mass there is no ruler, and with no ruler the conformal factor has no meaning. Measuring its three conditions with this series’ quantities: 31.4% of today’s energy must lose its rest mass; the gluing falls at logarithmic step 348, with today only 40% of the way; occupancy falls from \(1.5 imes10^{-18}\) to \(3.2 imes10^{-22}\). And Episode 16 plus Episode 6 show why CCC has no choice but to bet on information loss.
The theory in Part IV that best withstands the surgery EPISODE 32 LIVE interactive figureWetterich’s cosmon
Episode 4’s picture put \(a(t)\) in by hand; the cosmon has field equations determine \(\chi(t)\). In the ledger it pays 2 parameters = 10.7 bits and buys back up to 408 — the tuning of \( ho_\Lambda/M_{ m Pl}^4\). And it does not die like VSL because one field sets every mass, so the ratios are fixed — \(lpha\)’s 26 bits catch nothing. Judgement moves to \(w=-1.03\pm0.03\).
A notation shortens L(law); a theory pays L(parameters) to reduce L(residual) EPISODE 33 LIVE interactive figureMilne versus R_h=ct
Both have \(a\propto t\), but for \(a=t\) the FLRW curvature is \(R=6(1+k)/t^2\) — exactly zero at \(k=-1\), so Milne is Minkowski in other coordinates. From this follows a three-step test, and every FLRW lands in the “needs a conformal transformation” step. And at \(z=1\) the empty Milne universe fits \(\Lambda\)CDM better than \(R_h=ct\) does — which is why “fits the Hubble diagram” is a weak test.
Tell them apart by looking at k EPISODE 34 LIVE interactive figureConformal gravity (Mannheim)
With the Weyl-squared action, \(S o\Omega^{D-4}S\) — conformally invariant only at \(D=4\), exactly Episode 11’s Maxwell structure. Then the coupling \(lpha_g\) is dimensionless, the symmetry forbids a cosmological constant term (Episode 32’s 408 bits, for free), and the vacuum solution’s linear term crosses over at 44 kpc. The price is ghosts — the conformal factor’s ghost is gauged away and a massive spin-2 ghost arrives instead.
The one theory in Part IV that never made it onto the operating table EPISODE 35 LIVE interactive figureAsymptotic safety and a running G
Pair \(G\) with a scale to form \(g=Gk^2\) and run it: the slope is exactly 2, the classical dimension — and “gravity is \(10^{-38}\) times weaker” turns out to mean only that we look at small scales. At the ultraviolet fixed point \(g\) stops, so \(\eta_N=-2\) exactly: Episode 14’s 3.6% error in the weight table becomes 100% in gravity. The Higgs prediction of 126 GeV against a measured 125.25 is a 4–6 bit surprise.
The same entrance as conformal gravity — physics is in a dimensionless couplingOnto the operating table (Part IV wrap-up)
Nine theories on one table, all given the same surgery: exactly one — VSL — failed to separate (A) notation from (B) an observable claim. The line was not whether the name points at (A) but whether the theory itself can tell them apart. Every prediction sat in a dimensionless quantity, without exception. And something new: six of the coincidences fall between 4 and 7.5 bits of surprise — most likely a selection effect, the band where a result is enough for a paper but not for a consensus.
Good theories have already performed Episode 3’s surgery EPISODE 37 LIVE interactive figureQuantum anomalies — writing into the zero column
Episode 11’s "nothing happens to light" was a classical statement. Quantum theory cannot stay at D=4 (D=4-e with dimensional regularisation, a mu with a cutoff), so Episode 34’s exponent Omega^(D-4) becomes the breaking itself. Alpha is dimensionless only at D=4. The running is 7.1 per cent, which against the laboratory noise floor sits 28.7 bits above the noise — far past the band of coincidences. It does not contradict Episode 30’s "constant to 26 bits" because that is a different question. And the size of the breaking turns out to be a count of the fields.
What broke was not the field but the coupling EPISODE 38 LIVE interactive figureThe conformal factor problem
Rewrite the Einstein action as g = Omega^2 g-hat and the conformal factor’s kinetic term has coefficient (D-1)(D-2) with the sign opposite to an ordinary scalar. It vanishes only at D=1 and D=2; at D=4 it is 6. Wrinkle the factor and the Euclidean action falls as n-squared forever — 8498 bits of path-integral weight at n=50, with no bottom. The fix is to rotate the contour, but there is no derivation from first principles. And the finding: the ghost never disappears, it just moves between the conformal factor and the spin 2.
The place where the tool breaks does not vanish; it moves EPISODE 39 LIVE interactive figureRotating spacetime — a bound in the untouchable column
Kerr carries two labels, M and chi. On the weight table M is -1 and chi is 0 — one is bookkeeping, the other is physics, which makes it the cleanest example of Part II’s "a conformal transformation touches only size". The bound chi<=1, the 0.29289 ceiling on extractable mass, and the fact that entropy at chi=1 is exactly half all live in the dimensionless column and cannot be moved. Measured spins sit just below the bound at 4.3 bits of surprise — back in the band, but explained. And on Episode 33’s three-step test Kerr is Step 3: the first spacetime this tool cannot reach.
Weyl not zero — a spacetime the conformal tool cannot reach EPISODE 40 LIVE interactive figureGravitational entropy — not even halfway along
The Hubble sphere sits exactly at its own Schwarzschild radius (an identity at critical density), and because of it three separate 10^122 numbers coincide: the holographic bound, the entropy of all the mass as one black hole, and Episode 24’s N = C.t. The 5 per cent gap is exactly the ratio of the age of the universe to the Hubble time. The actual entropy is 3.1e104, of which 99.999 per cent is supermassive black holes, leaving 59.3 doublings of headroom — headroom that survives because gravity only attracts, so a uniform gravitational field is at minimum entropy, not maximum.
The three headline numbers were one EPISODE 41 LIVE interactive figureThe Weyl curvature hypothesis
Penrose demanded Weyl = 0 at initial singularities — a condition that switches off exactly half of the 20 Riemann components. Measured in bits, his famous 10^(10^123) is 3.27e122 bits, exactly the same number as Episode 24’s N = C.t and Episode 40’s holographic bound: four headline numbers turned out to be one. And lining beginning and end up on Episode 33’s three-step test, the beginning is Step 2 (the tool reaches) and the final black holes are Step 3 (it does not) — the arrow of time runs from where conformal transformations reach to where they do not.
Four headline numbers were one EPISODE 42 LIVE interactive figureInside a black hole
Conformal transformations do not move light cones, and the event horizon is fixed by causal structure alone — so it has weight 0 and can be neither created nor destroyed. The apparent horizon, defined locally by theta = 0, does move: the one that cannot be moved cannot be located without knowing the whole future, and the one that can be located moves. A Penrose diagram is a conformal transformation and the twin of Episode 1’s notation claim — notation is not worthless, it is simply not a claim. And inside, even the most generous estimate falls 93.9 bits short at solar mass and 126.5 at M87*.
Notation is not worthless; it is simply not a claim EPISODE 43 LIVE interactive figureThe Planck scale
The Planck length has conformal weight +1 — it is just a length, sitting in the bookkeeping column, so "the smallest length is l_P" is not yet a sentence; the sentence is L/l_P >= 1. And this is where the tool really breaks: conformal invariance demands the absence of a scale, and a smallest length is a scale, so the two are incompatible by hypothesis rather than by any subtle failure. The concept survives only as a dimensionless statement — the mass where the Compton wavelength and the Schwarzschild radius have ratio 1. We are fifty doublings away from it.
Excluded by hypothesis — the edge of the tool EPISODE 44 LIVE interactive figureDiscretisation
A lattice spacing is a smallest length, yet conformal field theories are computed on lattices every day. The resolution: conformal invariance belongs to the fixed point the lattice theory flows to, not to the lattice — observables depend only on xi/a, which diverges at criticality, so the spacing drops out. The 3D Ising exponents agree to four digits between continuum and lattice, and universality erases the microscopic content entirely. But the traces die only as (a/xi)^0.83, and the lambda transition of helium still shows a 6-sigma unresolved discrepancy.
The demand was never absence, only irrelevanceThe tool touches exactly half (Part V wrap-up)
Laid side by side, Part V’s "breaking" came in only two kinds: a scale was brought in (the tool reporting correctly), or a structure outside the conformal class appeared (the tool having nothing to say). Neither is a malfunction. And the 20 Riemann components in four dimensions split into Ricci 10, which change, and Weyl 10, which are conformally invariant — exactly half. So the tool cannot reach Kerr not from weakness but because the Weyl curvature is what a conformal transformation preserves. Which means the arrow of time is written entirely in the half the tool cannot touch.
The tool never once malfunctioned EPISODE 46 LIVE interactive figureEvery characterisation of a proportional to t
Collecting every way of saying a proportional to t gives twelve statements, nine of which are identities of one another — mere rewritings by differentiation and integration. The independent inputs are three: the expansion law, the Einstein equations, and the spatial curvature. Tested against observation, q=0 misses by 13 sigma and w=-1/3 by 23 sigma, but H_0 t_0 = 1 misses by only 4.9 per cent, and the Hubble tension straddles it exactly. The difference is integral versus instant — which is why c.t = const looked plausible in the first place.
Where the plausibility came from EPISODE 47 LIVE interactive figureThe map of dimensionless quantities
In 2019 the SI fixed the exact values of seven constants and redefined the units from them; the kilogram prototype was retired and mass is now built from h. That is an official international declaration that dimensionful quantities are bookkeeping — the same line this series drew by hand in Episode 3. But alpha could not be fixed: being dimensionless, it is not determined by any definition of units. The map of dimensionless quantities spans 815 bits, physics has 32 parameters of which exactly one is dimensionful, and 171.7 bits of them are unexplained.
c can be decreed; alpha cannot EPISODE 48 LIVE interactive figureBrevity, fit and beauty
Take "beauty" apart into six components and four reduce to the existing two currencies: symmetry, unification and rigidity to brevity, depth to fit (the CKM matrix’s 9 becoming 4 is worth 26.8 bits). What remained was naturalness — and it turned out to be not a third currency but a choice of prior: rho_Lambda over rho_Planck is 408.4 bits surprising under a linear prior and 8.2 under a log-uniform one. The strong CP problem is different, because theta is an angle. And one component, sensory pleasure, this series simply could not measure.
"Unnatural" was a declaration about the prior EPISODE 49 LIVE interactive figureThe doors left open
Every question opened in 48 episodes and not closed comes to sixteen, falling into four kinds: closed by observation (6), closed by calculation (4), fixed by definition (4), coincidence or not measurable (2). The observational ones are mostly expected to settle in the 2020s and 2030s, so readers will see the answers. But nearly four in ten cannot be closed by data at all — conventions, and things outside the tool. Telling which kind of door it is turned out to be a result in itself. And four judgements never moved across all 48 episodes.
Not settled by observation is not unimportant EPISODE 50 LIVE interactive figureFinale: only one thing moves
Six parts, nine theories, eight places the tool broke, sixteen open doors, four compressions. Applying that compression to the series itself, the six tools become two: separating the dimensionful from the dimensionless, and measuring everything in bits — and one step further, the second was the tool for checking the first. What 50 episodes did was one idea and a way of checking it. The verdict never moved; what moved was the understanding of why that verdict can be reached. Three lines are enough to take away.
Separating what moves from what does notWhat it means for mass to vary
mu = m_p/m_e has weight 0, so no conformal transformation moves it — which is why expansion and shrinking cannot be told apart. But m_e is 100 per cent of Higgs origin while 87 to 93 per cent of m_p is Lambda_QCD, a product of the trace anomaly, so mu is the only ratio straddling the two origins. Digging further turned up a degeneracy: variation of the constants is a three-dimensional problem, and the worst direction of the Fisher matrix is 99.6 per cent quark mass, with a condition number of 1196 (10.2 bits). And that invisible direction is the one that matters most to physics.
Turning “cannot be told apart” into “here is what can” BONUS ② LIVE interactive figureA hierarchy shrinks to its own logarithm
Differentiating dimensional transmutation gives d lnLambda / d lnalpha = H — the gain is exactly the hierarchy. So a hierarchy of B bits shrinks to a description length of log2 B bits, exactly. The cosmological constant problem’s 408 bits become 8.7 through an exponential map, a compression of 47. Out of this comes a second criterion for fine-tuning — is an exponential map available — which reproduces which problems physicists actually treat as real. And Episode 32’s cosmon turns out to sit only 2 bits above that floor.
A 408-bit problem becomes 8.7 bits BONUS ③ LIVE interactive figureWhat a dimensionless quantity is, and whether constants exist
A dimensionless quantity is what you can send by radio — conveyed without shipping anything. But the zero column is not homogeneous: ratios, angles, counts and exponents belong to different groups, and the logarithm turns out to be not a discovery but the isomorphism from the multiplicative group to the additive one. That makes Episode 48’s criterion a theorem: compact means the Haar measure is normalisable and the prior is fixed, non-compact means it is not — which is why the one uncontested fine-tuning problem is the one angle. And re-sorting the constants, about 24 are running functions, 6 are this universe’s state, and what is genuinely invariant is a theorem rather than a constant. One remains, and the axion would remove it.
Perhaps there is not a single constant BONUS ④ LIVE interactive figureIs there hidden structure among the constants?
An exactly-137 fine structure constant is excluded at 1.7 million sigma, yet the physical difference is only 0.026 per cent — the sole effect in the world is a 3-sigma shift in the neutron lifetime. It nags because 137 is an integer, and the surprise of that nearness is 3.8 bits, the bottom of the band. “Changing this is the same as changing that” comes in three kinds: exact degeneracies (theta and the quark phases, the CKM’s five directions) tell you about your notation, observational ones about your instruments, and only hidden relations about physics. And searching for relations has a price — a 4.1-bit prediction beats a 15.7-bit discovery.
“Theory first” meant fixing the measure first BONUS ⑤ LIVE interactive figureThe prior was never ours to choose
The renormalisation group hands out the measure: for a one-dimensional flow the invariant measure is rho proportional to 1/beta, uniquely, and it is RG time itself. Since a verdict that changes with the scale depends on a convention (Episode 3), the prior must be RG-invariant — and that fixes it. The shape of the measure follows from the shape of beta alone: multiplicative is cheap, additive is expensive, which is ’t Hooft’s criterion. Scoring all 20 Standard Model parameters this way, exactly three come out expensive — the hierarchy, strong CP and the cosmological constant, with zero false positives. And it closed two escape routes this series had built for itself.
The shape of beta alone hits all three problemsThe position here is consistent throughout: \(c\cdot t=\text{const}\) is a rewriting, not new physics (proved in Ep.3). That is precisely why it can be substituted into any equation safely. What is under discussion is "shortness", never "correctness". The verdict of the previous series stands unchanged: extrapolated to the early universe at face value, it contradicts nucleosynthesis.
"c·t = const, That Clicks" / a sequel to "Conformal Transformations That Click" (itself a sequel to "Cosmology That Clicks"). Every episode carries both an accessible narrative and the reveal of what it means physically. The quantities of information theory (bit counts, operation counts, parameter counts, entropy) are dimensionless and are therefore unmoved by a conformal transformation — they belong, from the start, to the "physics" column of the previous series' decision procedure. Note that the "operation count" used here is the energetic upper bound from the Margolus–Levitin limit and does not refer to meaningful computation. Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model still under test; Melia and collaborators argue it is favoured by low-redshift data, while extrapolating it into the early universe contradicts big-bang nucleosynthesis (Lewis, Barnes & Kaushik 2016, MNRAS 460, 291). This series treats \(c\cdot t=\text{const}\) purely from the standpoint of notational brevity and does not argue for its correctness. The academic standard remains the \(\Lambda\)CDM model including inflation. ── Each page can be saved as PDF via the browser's Print dialogue. Keep this contents page and the episode files in the same folder (the links are relative). Japanese original: わかる c·t=一定.