# 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:

| Measurement | Value | Distance from 74.3 ± 1.2 |
|---|---|---|
| H0 Distance Network, April 2026 | 73.50 ± 0.81 | **0.55 σ** |
| SH0ES Cepheid ladder | ≈ 73.0 ± 1.0 | 0.83 σ |
| TRGB / CCHP | ≈ 69.8 ± 0.8 | 3.12 σ |
| Planck CMB (ΛCDM) | 67.4 ± 0.5 | **5.31 σ** |

**Read forwards** — fixing H₀ and predicting the floor:

| H₀ input | predicted a₀ | vs 1.2 × 10⁻¹⁰ |
|---|---|---|
| 73.50 (H0DN 2026) | 1.1871 × 10⁻¹⁰ | −1.08 % |
| 67.4 (Planck) | 1.0886 × 10⁻¹⁰ | −9.29 % |

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.

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