Abstract
When one says “that galaxy is a billion light-years away,” it is a measured quantity. This paper asks how it was measured——the answer is by stacking a ladder of seven rungs. No new mathematical theorem and no new law is claimed.
First, how far does direct measurement reach? At Gaia’s parallax precision of $20 \mu$as, $10\%$ precision reaches only to $\mathbf{5000}$ pc (Section 2).
Second, this is the core of the paper. The observable universe is $1.4\times10^{10}$ pc, so we are $\mathbf{6.4}$ orders short——all the rest is ladder (Section 2).
Third, there are seven rungs. Parallax $\to$ spectroscopic parallax $\to$ main-sequence fitting $\to$ Cepheids $\to$ TRGB $\to$ type Ia supernovae $\to$ Hubble’s law (Section 3).
Fourth, the errors stack. If each rung adds only $5\%$, the top rung reaches $\mathbf{12.41\%}$ (Section 3).
Fifth, at the top rung two routes disagree. $73.04\pm1.04$ against $67.36\pm0.54$, that is $\mathbf{4.8471\sigma}$——once in $798{,}192$ by chance (Section 4).
Sixth, the separator is ladder or ruler. One stacks upward from the local; the other assumes an acoustic horizon (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——trigonometric parallax, the rungs of the distance ladder, the period--luminosity relation, and the two measured values of $H_0$ are all standard. We do not build cosmology——all we use is one division and one root of a sum of squares. We assert no cause for the disagreement——whether it is new physics, a systematic error, or neither is not treated at all. We only count how many $\sigma$ the difference is. We do not say which is right——this paper endorses neither value. We do not enter calibration detail——the metallicity dependence of Cepheids, the calibration of the TRGB, and supernova standardisation are not treated. The parallax reach depends on premises——the $\mathbf{5000}$ pc follows from two conventions: $\sigma_\pi=20 \mu$as and “$10\%$ relative error”. Change $\sigma_\pi$ and it moves in proportion. The error stacking is illustrative——the “$5\%$ per rung” of Section 3 is an assumption chosen to show the structure, not the actual error of each rung. If systematics correlate across rungs, a sum of squares is not enough. We do not call $4.85\sigma$ a discovery——it is merely a division of two published values, resting on the premise that the systematics have been correctly estimated. Relation to earlier papers: Paper 250 showed that measuring one question with two instruments can reverse the sign——that concerned the coding of the corpus’s own text; this concerns a physical constant, the same figure with a different object. Paper 255 separated two things called “accuracy,” only one removed by calibration——the ladder here has each rung calibrated against the one below, which is the side where systematic error does not vanish. Paper 169 showed that Sciama’s famous coincidence is an identity, using the Hubble constant——this paper treats the measurement of that constant itself. Paper 190 measured “rare” on a logarithmic scale——the “once in $798{,}192$” here is likewise read logarithmically. What is added is computing the parallax reach at $10\%$ as $5000$ pc and measuring the gap to the observable universe as $6.4$ orders, tabulating the seven rungs, showing that $5\%$ per rung stacks to $12.41\%$, and computing the $H_0$ disagreement as $4.8471\sigma$, once in $798{,}192$.
1. Notation and the Question
Setting: There is in effect only one way to measure a distance to a star directly——trigonometric parallax. Measure the apparent shift $\pi$ produced by the Earth’s orbit and the distance is $d[\mathrm{pc}]=1/\pi[\mathrm{arcsec}]$.
Question: How far does that reach? And how is everything beyond it measured?
Observation 1.1 Parallax alone is geometry. It measures a triangle and assumes nothing about the star——every other rung uses an assumption about the nature of the object.
2. The Core: We Are $6.4$ Orders Short
Setting: Take Gaia’s parallax precision as $\sigma_\pi=20 \mu$as. The relative error is $\sigma_\pi/\pi=\sigma_\pi d$.
Check 2.1:
| Distance [pc] | Parallax [mas] | Relative error |
|---|---|---|
| $10$ | $100.0000$ | $0.02\%$ |
| $100$ | $10.0000$ | $0.20\%$ |
| $1000$ | $1.0000$ | $2.00\%$ |
| $\mathbf{5000}$ | $0.2000$ | $\mathbf{10.00\%}$ |
| $10000$ | $0.1000$ | $20.00\%$ |
| $100000$ | $0.0100$ | $200.00\%$ |
Observation 2.2 A relative error of $10\%$ reaches to $\mathbf{5000}$ pc; $1\%$ reaches to $500$ pc. At $10^{5}$ pc the error is $200\%$——larger than the distance itself.
Check 2.3: We compare with the observable universe.
| Quantity | Value |
|---|---|
| Parallax reach at $10\%$ precision | $5.0\times10^{3}$ pc |
| Radius of the observable universe | $1.4\times10^{10}$ pc |
| Ratio | $\mathbf{2.8\times10^{6}}$ |
| In orders | $\mathbf{6.4}$ |
Observation 2.4 This is the core of the paper. Direct measurement covers a mere $3.6\times10^{-7}$ of the universe——the remaining $99.99996\%$ is all measured by ladder.
Observation 2.5 A point to write honestly——the figure $5000$ pc follows from two conventions, $\sigma_\pi=20 \mu$as and “$10\%$”. Halve $\sigma_\pi$ and the distance doubles——what does not change is the structure, that the orders do not suffice.
3. Seven Rungs, and the Errors Stack
Check 3.1:
| Rung | What is assumed | Reach |
|---|---|---|
| $1$. Trigonometric parallax | geometry only (direct) | $\sim10^{3}$ pc |
| $2$. Spectroscopic parallax | absolute magnitude from spectral type | $\sim10^{4}$ pc |
| $3$. Main-sequence fitting | cluster HR diagrams overlay | $\sim10^{5}$ pc |
| $4$. Cepheids | the period--luminosity relation | $\sim10^{7}$ pc |
| $5$. TRGB | the red-giant branch tip is constant | $\sim10^{7}$ pc |
| $6$. Type Ia supernovae | peak luminosity is nearly constant | $\sim10^{9}$ pc |
| $7$. Hubble’s law | redshift is proportional to distance | beyond |
Observation 3.2 Only rung $1$ assumes nothing. Every rung from the second is calibrated against the one below——move the first and all seven move.
Check 3.3: Taking each rung to add $5\%$ independently (the first $2\%$), we watch the accumulation.
| Rung | Accumulated relative error |
|---|---|
| $1$. Trigonometric parallax | $2.00\%$ |
| $2$. Spectroscopic parallax | $5.39\%$ |
| $3$. Main-sequence fitting | $7.35\%$ |
| $4$. Cepheids | $8.89\%$ |
| $5$. TRGB | $10.20\%$ |
| $6$. Type Ia supernovae | $11.36\%$ |
| $7$. Hubble’s law | $\mathbf{12.41\%}$ |
Observation 3.4 If independent, the sum of squares gives $\mathbf{12.41\%}$. Six additions of $5\%$ do not reach $30\%$ because variances add——the same single line stands here too: summing independent contributions grows the standard deviation only as a square root.
Observation 3.5 A point to write honestly——“$5\%$ per rung” is an assumption to display the structure, not the actual error of each rung. And if systematics correlate across rungs, a sum of squares is not enough——correlated errors add at first order.
Observation 3.6 Paper 255 separated two things called “accuracy,” noting that calibration removes only one. Each rung of the ladder is calibrated against the one below——so a systematic error in the first rung vanishes at no rung at all.
4. At the Top Rung, Two Routes Disagree by $4.85\sigma$
Setting: The Hubble constant $H_0$ is measured by two entirely different routes.
Check 4.1:
| Route | $H_0$ [km/s/Mpc] | What it rests on |
|---|---|---|
| Ladder (SH0ES) | $\mathbf{73.04\pm1.04}$ | stacking from the local |
| Ruler (Planck) | $\mathbf{67.36\pm0.54}$ | the acoustic horizon of the CMB |
Check 4.2:
| Quantity | Value |
|---|---|
| Difference | $5.6800$ |
| Combined $\sigma=\sqrt{1.04^2+0.54^2}$ | $1.1718$ |
| Discrepancy | $\mathbf{4.8471\sigma}$ |
| Two-sided $p$ | $1.2528\times10^{-6}$ |
| By chance | once in $\mathbf{798{,}192}$ |
Observation 4.3 It is $\mathbf{4.85\sigma}$. Particle physics conventionally calls $5\sigma$ a “discovery,” so this is one step short——awkward to explain by chance, insufficient to call settled.
Check 4.4: We recast the difference into something felt.
| Quantity | Value |
|---|---|
| Relative difference | $\mathbf{8.09\%}$ |
| $1/H_0$ (SH0ES) | $13.3872$ Gyr |
| $1/H_0$ (Planck) | $14.5160$ Gyr |
| Difference | $\mathbf{1.1288}$ Gyr |
Observation 4.5 Recast as an estimate of the age of the universe, the disagreement is $\mathbf{1.1}$ billion years. In percentage it is only $8\%$, but both sides claim $1$--$2\%$ precision——it is because the precision rose that the disagreement became visible.
Observation 4.6 A point to write honestly——this paper says neither which is right nor what the cause is. The number $4.85\sigma$ rests on the premise that the systematics have been correctly estimated——break that premise and the number loses its meaning.
5. Separator: Ladder or Ruler
Observation 5.1 Separator——how do the two measurements differ?
| Ladder side | Ruler side | |
|---|---|---|
| Starting point | nearby stars (geometry) | the CMB acoustic horizon (model) |
| Direction | stacked from near to far | extrapolated from far to near |
| Governing assumption | calibration of each rung | the cosmological model ($\Lambda$CDM) |
Observation 5.2 The two share not a single rung. The ladder’s errors and the ruler’s errors are independent——which is why the combined $\sigma$ may be taken as a sum of squares.
Observation 5.3 And they measure the same quantity. This is what makes the disagreement serious——had they measured different quantities, disagreement would be expected.
Observation 5.4 Paper 250 treated a case where measuring one question with two instruments reverses the sign. That concerned the coding of the corpus’s own text; this concerns a physical constant——the same figure with a different object. Both ask what is at stake when two instruments fail to agree.
Observation 5.5 Finally we join Sections 2 and 4. The ladder has seven rungs because direct measurement falls $6.4$ orders short——and at the tip of those seven rungs it disagrees with the ruler side by $4.85\sigma$. That there are many rungs and that there is a disagreement cannot be unrelated——though this paper does not assert that it is the cause either.
6. Roll Call and Closing
- Parallax at $10\%$ reaches $5000$ pc (Check 2.1): $500$ pc at $1\%$.
- $200\%$ error at $10^{5}$ pc (Check 2.1): Larger than the distance.
- Ratio $2.8\times10^{6}$ (Check 2.3): $6.4$ orders short.
- Direct covers $3.6\times10^{-7}$ (Observation 2.4): The rest is ladder.
- Seven rungs (Check 3.1): Only the first is geometry.
- Accumulated $12.41\%$ (Check 3.3): A sum of squares if independent.
- $73.04$ against $67.36$ (Check 4.1): Ladder and ruler.
- $4.8471\sigma$ (Check 4.2): Once in $798{,}192$.
- Relative difference $8.09\%$ (Check 4.4): $1.1288$ Gyr.
- Not one shared rung (Observation 5.2): Hence independent.
Closing: Cosmic distance is measured by a ladder of seven rungs, and only the first is measured directly. Trigonometric parallax reaches $10\%$ precision only to $\mathbf{5000}$ pc, while the observable universe is $1.4\times10^{10}$ pc——$\mathbf{6.4}$ orders short, so direct measurement covers a mere $\mathbf{3.6\times10^{-7}}$ of the whole. The rest is stacked by spectroscopic parallax, main-sequence fitting, Cepheids, TRGB, type Ia supernovae and Hubble’s law, and $5\%$ per rung already brings the top rung to $\mathbf{12.41\%}$. And at that top rung, a value measured by an entirely different route disagrees by $\mathbf{4.8471\sigma}$——$73.04\pm1.04$ against $67.36\pm0.54$, once in $\mathbf{798{,}192}$ by chance, a difference of $\mathbf{1.1}$ billion years in the age of the universe. One thing separates them——whether one stacked up from nearby or extrapolated in from afar. The two share not a single rung, which is why their errors are independent, and why the difference means something. And both are measuring one and the same quantity.
References
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- Yamagishi, Y. (2026). Sciama’s Famous Coincidence Is an Identity [Paper 169]. Zenodo. 10.5281/zenodo.22093563
- Yamagishi, Y. (2026). “Rare” Is Measured on a Logarithmic Scale [Paper 190]. Zenodo. 10.5281/zenodo.22106182
- Yamagishi, Y. (2026). Two Instruments, Opposite Signs [Paper 250]. Zenodo. 10.5281/zenodo.22160733
- Yamagishi, Y. (2026). “Accuracy” Is Two Things, and Calibration Removes Only One [Paper 255]. Zenodo. 10.5281/zenodo.22168883
Reference
Yamagishi, Y. (2026). Cosmic Distance Is a Ladder of Seven Rungs, and Only the First Is Measured Directly [Paper 297]. YOI: https://www.yamagishiyuuki.com/yoi/297
DOI: 10.5281/zenodo.22171788
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 297. This paper is a preprint and has not been peer reviewed.