The Rate

The three numbers the corpus already holds. AP42’s partition: 31.15 per cent of the census held, 68.85 per cent the reservoir the leak returns to — one hold read three ways, AP46 §10. AP46’s cycle: 13.830 billion years, one lane wide. This paper’s balance, the Friedmann equation.

The pure number. At a given partition the equation fixes rate times age as a pure number — the standard closed form, H₀t₀ = (2/(3√Ω_Λ))·arcsinh√(Ω_Λ/Ω_m). At 31.15 per cent held it is 0.954; with the sky’s light census included 0.9536; at the sky’s own partition, 31.5 per cent, 0.951. The differences are below the window.

The rate, and its window. Rate times age is 0.954; the age is the cycle; so

H₀ = 0.954 × 70.70 = 67.45 km s⁻¹ Mpc⁻¹.

The window is the cycle’s own lane, 0.659 billion years either side: an age of 13.17 gives 70.8, an age of 14.49 gives 64.4 — symmetric in age, 3.4 above and 3.1 below in rate. The partition’s contribution to the pure number, 0.954 against 0.951 across the corpus’s and the sky’s shares, is a fifteenth of the lane. Nothing was fitted; the three ingredients were fixed by AP42, AP46, and §§5–7 in that order.

In plain words. Take the age the corpus already derived — one cycle, 13.83 billion years. Take the share of the sky that is still held — the corpus’s own 31 per cent. Ask how fast a room whose hold is that share and whose age is that long must be growing today, under the balance of §5. The answer is a pure number over the age: 0.954 over 13.83 billion years — in the astronomers’ units, 67 and a half kilometres per second for every megaparsec. The relic glow’s own value is 67.4 ± 0.5.

What this is, and is not. A closure, not a fresh prediction. AP42’s partition landed on the glow’s partition; AP46’s cycle landed on the glow’s age; given the equation’s form, a partition and an age, the rate is arithmetic. That makes AP42’s provenance the one structural thing between this paper and its headline, and it is stated with its locus. AP42 §3.4 and §5 give the partition as a count: the dark sector as twenty twenty-firsts and the visible as one, and inside the dark sector the split by a constant-feeding kernel whose timescale is six twenty-firsts of the age — so the kernel is evaluated at t/τ = 21/6 whatever the age is, and no age enters the partition, as AP42 §5 says. No parameter is fitted, and the comparison to the glow came after. AP42 compares to the glow’s TT,TE,EE+lowE+lensing+BAO column (Planck 2018 results VI, Table 2), net of radiation as its §3.4 states: the visible’s 4.76 against 4.86, two per cent low; the dark matter’s 26.39 against 26.07, one per cent high; the dark energy’s 68.85 against 68.89. Against the same table’s lensing column, which §10 quotes for the rate, the visible is 4.93 and the miss is three and a half per cent, the dark matter 26.45 and the dark energy 68.47, half a per cent off. A partition matched to the sky would not miss the visible by two or three per cent; a partition counted from twenty-one channels does. The closure is exactly as strong as that provenance. What is new is that the corpus’s cosmology closes on all three of the sky’s numbers at once, and the closure could have failed in exactly one way: the structure forcing a different placement of the ε. §7 shows the corpse that would have left. A pass is not a fit.

What was pre-registered, and what happened. The README of 3 September pre-registered, before any arithmetic, that AP46’s chain read as a rate gives 70.7, that today’s rate was expected above that settled beat with the tail still adding, and that the derivation’s late limit must land on it. Today’s rate is 67.45 and the late limit is at or below 56; both are below 70.7, and the direction was the other way. The rate-reading of the chain is eliminated by the sky, as KS-CCC.1 permits data to do; the time-reading — the chain as the cycle — stands, and it is the one AP46 §9 froze. The rule that decides what a miss does, stated so it is not written after the fact: a pre-registered item fires a switch if a switch was armed on it — KS-STRETCH.3 on the age, KS-STRETCH.4 on the eras, KS-45.1 on the rate — and an expectation without a switch, like the direction and the magnitude the README left to this paper to derive, is shown as a miss and fires nothing. The direction was the second kind; AP46 §9, which carried it, moves by dated note in §13.

The clock spread. The primer assigned this paper the adjudication of the residual-factor clock spread, 0.90 / 0.86 / 0.75, three factors of the bare thread against three clocks. The comparison target is adjudicated by the same rule: the chain is a time, so its target is the age, and the factor against the age — 0.86, the raw thread’s 11.85 against 13.79, which the duty cycle 21/18 closes to +0.31% as AP46 §9 records — is the one that stands; any factor against a Hubble time at some H₀ is a rate-reading and is eliminated with it. The spread’s source is the primer itself, which carries it from the author’s conversations before the primer was written; no other file holds it, and the adjudication is by the rule stated here.

The tail, sized. The pure number is the w = −1 limit, the constant-ε reading inherited from AP46’s frozen stance. With the tail as a constant departure, w = −0.95 gives 66.9, −0.9 gives 66.4, −0.8 gives 65.2 — all inside the window. The tail is a rider on the number, not a defect in it, provided today’s departure is no larger than about 0.2, which is the order the surveys currently prefer.

AP42’s feeding history, declared. The pure number is integrated under this paper’s own dilution counts — held records thinning as the room’s three, H_ε constant. AP42’s history has the held census converting into the reservoir on its 3.9-billion-year timescale, a different history of the rate and hence a different pure number. This paper takes AP42’s present shares as an input and not its dynamics; the direction of the difference cannot be fixed without the feeding history, and the sensitivity is unbounded until D48 is paid. It is carried on KS-ASM.2.

What 70.7 is, and the settled rate counted. The cycle’s inverse, 70.70, is not a rate; AP46 §9 refused it as one and this paper says why. Under the balance the leak’s rate today is H₀ × √0.6885 = 56.0 km s⁻¹ Mpc⁻¹, and under AP46’s frozen stance that is the settled rate itself: 0.954 × √0.6885 = 0.792 of the cycle’s inverse. Every ingredient is the corpus’s — the cycle, the partition, the balance — so the settled rate is counted, and it is not the cycle’s inverse. AP46 set that pure number to one in two places: its §5, read literally, put the settled rate at the thread’s inverse, 82.5; its §9, read as a rate, at the cycle’s inverse, 70.7. The balance sets it to 0.792, and both readings are eliminated. The α¹⁸ chain prices the cycle — an escape count read as an interval — and the settled rate is that interval’s inverse scaled by the balance at the partition. And the seat AP08 admits for Λ is occupied by that same number in other units — Λ = 3H_ε,∞²/c², with H_ε,∞ = 0.792 of the cycle’s inverse, 1.10 × 10⁻⁵² m⁻² — which is the closure of this section restated, not a second result: if the closure holds, Λ follows, and if the closure falls, so does it.