Ø Frozen Predictions · the document of record

Pre-registration: the Hubble constant from the acceleration floor (KS-45.1)

Freeze date: 2026-08-02 Author: G · Studio G · the420code.org Scope: AP18 (The Floor), AP17, and Rosin Ø Proofs ch. 14 / ch. 19 Switches engaged: KS-45 (2π geometric factor), KS-45.1 (Hubble prediction), KS-45.2 (dipole topology, closed), KS-39, KS-3 Reproduction: verify_cosmology.py in this directory Status: frozen for commit. The hash is the timestamp.

1. The registered value

H₀ = 74.3 ± 1.2 km/s/Mpc

The ± 1.2 is propagated from the empirical acceleration floor a₀, not from the derivation. The derivation contributes no uncertainty because it contains no fitted quantity.

2. The relation, and why it has no free parameter

a₀ = α_apex · c · H₀ / (2π) α_apex = 2 ln(sec ½ + tan ½) = 1.04447620656 H₀ = 2π a₀ / (α_apex · c) α_apex / (2π) = 0.16623355

Every step is stated in AP18 and reproduced here so the coefficient can be attacked rather than accepted:

1. The topology is forced, not chosen. The angular field lives on S², whose Euler characteristic is 2. By Poincaré–Hopf a smooth field on S² must have index-sum 2 — so the one break gives one source, the involution one sink: a dipole, exactly two index-1 zeros. Nothing here is adjustable. (KS-45.2, closed.)
2. The apex acceleration is geometric. The widest closing field line reaches the Hubble radius R_H = c/H₀ and returns at the gentlest curvature 1/R_H, giving c²/R_H = cH₀.
3. The coherent fraction is 1/(2π) — one radian of arc on a closed loop.
4. The apex correction is an integral, not a fit. With the S² dipole tension profile T(θ)/T_apex = 1/sin θ, writing u = θ − π/2 turns 1/sin θ into sec u; averaging over one radian, half to each side:

α_apex = 2 ∫₀^½ sec u du = 2 ln(sec ½ + tan ½) = 1.04447620656

That number is a transcendental constant of the geometry. It cannot be tuned without changing the geometry that produced it.

Zero free parameters. The only measured input on the right-hand side is a₀ itself (or H₀, read the other way).

3. Honest accounting: this is one relation, not two predictions

Fix a₀ from galaxy rotation and the relation predicts H₀. Fix H₀ from the distance ladder and it predicts a₀. It is one claim read in two directions, and any presentation that scores it twice is inflating it. Ø Proofs ch. 19 already states this; it is restated here because it is the first thing a referee will check.

4. Current standing (measurements all predate this freeze)

Read backwards — a₀ ≈ 1.2 × 10⁻¹⁰ m/s² (McGaugh, SPARC) gives H₀ = 74.30:

Read forwards — fixing H₀ and predicting the floor:

The relation takes a side in the Hubble tension. It sits with the local distance ladder and against the CMB inference. That is not a hedge, it is an exposure, and it is the reason this claim is worth freezing.

5. Kill and confirm conditions (binding)

1. KILL — the low branch. If the Hubble tension resolves toward the CMB value, i.e. a converged H₀ below 71.9 (2σ below the registered value), KS-45.1 fires. There is no coefficient available to rescue it: α_apex is an integral of a forced geometry, not a parameter.
2. KILL — the floor. If a₀ is re-determined with systematics under 5 % and the relation a₀/(cH₀) = 0.16623 is violated at 3σ using any independently determined H₀, KS-45 fires.
3. CONFIRM. A converged H₀ in 73.1–75.5 with sub-1 % precision, together with a₀ systematics under 5 %, closes KS-45.1 in the corpus's favour.
4. NO RE-ANCHORING. α_apex is frozen at 1.04447620656. If a future draft changes the coherent window from one radian, or the tension profile from 1/sin θ, that is a new prediction and must be registered as one, referencing this file's hash.

6. What limits the test — stated plainly

The bottleneck is a₀, not H₀. H₀ is now measured to just over 1 %. The empirical acceleration floor carries a random uncertainty near 2 % but a systematic uncertainty commonly quoted at 10–20 %, depending on the galaxy sample, the mass-to-light treatment, and the fitting form. Propagated naively, a₀'s systematic gives H₀ = 74.3 ± 6.2 — far too loose to discriminate anything.

So the ± 1.2 in the registered value is the optimistic branch, and this document says so. The claim becomes sharp only when galaxy-dynamics systematics tighten below about 5 %. That is a rotation-curve problem, not a cosmology problem, and it is where the corpus should be pushing.

Read forwards, however, the test is already live: at the best current H₀ the relation predicts a₀ = 1.187 × 10⁻¹⁰ and Planck's H₀ predicts 1.089 × 10⁻¹⁰. Those differ by 9 %, which is within reach.

7. What this does and does not establish

Does: register a parameter-free number, with its geometry exhibited step by step, against a quantity two independent communities are actively measuring, on the side of the dispute where it can be shown wrong.

Does not: establish that the tension field is the correct account of galactic dynamics (KS-3, KS-40 Bullet Cluster, KS-41 structure formation all remain live and unclosed); establish that a₀ is fundamental rather than emergent; or convert a 0.55σ agreement into evidence of uniqueness. Other relations link a₀ and cH₀ with order-unity coefficients; a match removes rivals, it never proves the derivation.

Copyleft 2026. Don't be a cunt. Be kind.

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