Cosmic homogeneity: reproduce where galaxy clustering stops looking fractal, and fix the README's range

CORRECTIONS.md

@@ -175,3 +175,15 @@ Claims made during the exploration that did not survive it, kept so the record s
     not 1.36. Whether the clouds' 1.36 matches the laboratory's 1.3 to 1.4 for a shared reason is not shown. The
     laboratory measures sections of turbulent surfaces (7/3 in 3D for the interface), the clouds a projection of
     supersonic gas, and the two only meet near 4/3.
+
+## 18 October 2026: cosmic homogeneity (issue #17)
+
+21. **"Beyond about 80 to 300 million parsecs (h⁻¹ Mpc, 80 to 300)."** Two errors, and a missing caveat. The 80
+    to 300 h⁻¹ Mpc in Scrimgeour et al. (2012) is the range over which they rule out a fractal with D₂ below
+    2.97, and 300 is the largest scale their survey reached. Their homogeneity scale is 70 to 81 h⁻¹ Mpc: 71 ± 8,
+    70 ± 5, 81 ± 5 and 75 ± 4 at z ≈ 0.2, 0.4, 0.6 and 0.8. And h⁻¹ Mpc is not Mpc: with h = 0.674, 70 to
+    81 h⁻¹ Mpc is 104 to 120 Mpc. The caveat: the scale is a convention, not a fixed length. It depends on the
+    tracer, the epoch and the threshold; a 0.1% threshold instead of 1% puts it at 185 h⁻¹ Mpc for matter today.
+    See [cosmic-homogeneity](cosmic-homogeneity/).
+22. **"The nesting stops, the way a coastline's roughness stops at the size of molecules."** Points the wrong
+    way. A coastline is rough down to a smallest scale; galaxy clustering is clumpy up to a largest one.

README.md

@@ -24,8 +24,11 @@ with Claude (Anthropic), for [Foundation0](https://github.com/Foundation-0/f0).
   that roughness only if they kick it harder than its own momentum: a cold atom in a room-temperature vacuum,
   yes; nothing as warm as, and heavier than, the gas around it ([quantum-paths](quantum-paths/)).
 - **Cosmic structure is fractal only up to a point.** Galaxies cluster in a fractal-like way on small
-  scales, but beyond about 80 to 300 million parsecs (h⁻¹ Mpc, 80 to 300) they are spread almost evenly
-  (Scrimgeour et al., 2012). The nesting stops, the way a coastline's roughness stops at the size of molecules.
+  scales: well inside 5 h⁻¹ Mpc (the surveys' unit, about 1.5 million parsecs), the number of neighbours
+  within a distance r grows as r^1.2 rather than r³. Beyond a radius of about 70 to 80 h⁻¹ Mpc, roughly 100 to
+  120 million parsecs, the galaxies of the WiggleZ survey are spread evenly to within 1% (Scrimgeour et al.,
+  2012). The nesting stops at a largest scale, as a coastline's roughness stops at a smallest one, and where
+  depends on the galaxies and on how even counts as even ([cosmic-homogeneity](cosmic-homogeneity/)).
 - **Turbulence repeats, but not exactly.** Between the size where a flow is stirred and the size where viscosity
   takes over, the same energy flux passes through every scale. In a cascade model it holds to 0.2% over the first
   nine of 21 shells (2.4 decades), and a third-order law that follows from it to 0.1%. The bursts do not scale

cosmic-homogeneity/README.md

@@ -0,0 +1,200 @@
+# Cosmic homogeneity: where galaxy clustering stops looking fractal
+
+Status: **exploration, not peer reviewed, no new result claimed.** Issue
+[#17](https://github.com/Foundation-0/fractals/issues/17).
+
+Galaxies cluster: around any galaxy there are more neighbours nearby than an even spread would give. On large
+enough scales the excess fades. This study computes where, from the standard cosmological model, checks it
+against the surveys, and shows how much the answer depends on the definition.
+
+## The measure
+
+Count the galaxies within a distance r of a galaxy, divided by the count an even spread would give:
+N(<r) = 1 + ξ̄(r), with ξ̄ the correlation function averaged over the sphere. For a fractal of dimension D the
+raw count grows as r^D; for an even spread, as r³. The correlation dimension is
+
+```
+D₂(r) = 3 + d ln N / d ln r = 3 + 3 (ξ − ξ̄) / (1 + ξ̄)
+```
+
+Scrimgeour et al. (2012) define the homogeneity scale R_H as the radius beyond which D₂ stays within 1% of 3
+(D₂ = 2.97). A second definition, where N comes within 1% of 1, gives R_H a few h⁻¹ Mpc larger in ΛCDM (in the
+WiggleZ data not always: 78 against 81 at z ≈ 0.6); every number here uses D₂ unless it says N. Lengths are
+comoving and in h⁻¹ Mpc, the surveys' unit (see Units below).
+
+## Small scales: about 1.2, not 2
+
+Galaxy clustering on small scales follows a power law, ξ = (r/r₀)^−γ with r₀ ≈ 5 h⁻¹ Mpc and γ ≈ 1.8
+(Zehavi et al. 2005). Then:
+
+| r (h⁻¹ Mpc) | 0.1 | 1 | 3 | 5 | 10 | 20 |
+|---|---|---|---|---|---|---|
+| D₂ | 1.20 | 1.24 | 1.45 | 1.71 | 2.25 | 2.69 |
+
+Well inside r₀ the neighbour counts grow as r^1.2, a fractal-like dimension of 3 − γ. Beyond about
+1 h⁻¹ Mpc, D₂ never holds still: it climbs steadily toward 3. Between 1 and 30 h⁻¹ Mpc the counts grow on
+average as r^1.9 (endpoint slope; a least-squares fit to log-spaced points gives about 1.86), an average through
+the climb, and one reading of the "D ≈ 2" of Sylos Labini, Pietronero and colleagues, who argue the distribution
+stays fractal to the largest scales. Their 2009 value, D = 2.1 ± 0.1 up to about 30 h⁻¹ Mpc, is about 2σ above
+this average. Their estimator and sample differ, and the power law here is fitted only below about 10 to
+20 h⁻¹ Mpc, so the comparison is rough.
+
+## Large scales: the homogeneity scale
+
+Linear theory: the matter power spectrum of Eisenstein and Hu (1998) with baryon wiggles, normalised to σ₈,
+gives ξ and ξ̄. Galaxies trace matter with a bias b (ξ_g = b²ξ), and redshift-space distortion adds the Kaiser
+factor 1 + 2β/3 + β²/5.
+
+**WiggleZ.** With Scrimgeour et al.'s fiducial cosmology and each slice's measured bias and β, the model
+gives their own ΛCDM homogeneity scales to within about 1 h⁻¹ Mpc. The paper fits the bias separately for each
+statistic (Table 4); the b² here is the fit for D₂:
+
+| z | b² | Model | Their ΛCDM | Measured |
+|---|---|---|---|---|
+| 0.2 | 0.70 | 77.1 | 76 | 71 ± 8 |
+| 0.4 | 0.87 | 78.6 | 78 | 70 ± 5 |
+| 0.6 | 1.25 | 81.5 | 81 | 81 ± 5 |
+| 0.8 | 1.46 | 78.7 | 78 | 75 ± 4 |
+
+The model runs 0.5 to 1.1 above their values: they use a nonlinear spectrum and damp the redshift-space signal
+(see What the model leaves out). R_H barely moves with redshift because the galaxies' bias grows as the
+clustering of matter weakens, as the paper says. The other definition, with the bias fitted to the counts
+(b² = 0.68, 0.87, 1.23, 1.43), gives 80.7, 83.0, 85.3 and 82.5, against their ΛCDM counts of 80, 83, 85 and 82
+(Table 4).
+
+Other surveys use different criteria and tracers. Hogg et al. (2005) found SDSS luminous red galaxy counts
+consistent with r³ beyond about 70 h⁻¹ Mpc, within their errors; for such strongly biased galaxies (b ≈ 2) the
+model's 1% scale is about 130. Yadav et al. (2005) and Sarkar et al. (2009) found 60 to 70 h⁻¹ Mpc from
+multifractal dimensions. Ntelis et al. (2017) used D₂ = 2.97 on BOSS CMASS galaxies, corrected the counts for bias
+and redshift-space distortion, and found 63.3 ± 0.7 h⁻¹ Mpc for the matter distribution, against a ΛCDM 62.0
+(abstract; their Conclusions give 61.9 ± 0.8 for 0.538 < z < 0.592). The 62.0 is a real-space matter value: the
+paper's matter values (Sec. 5.5, Table 5) run from 59 to 67, its galaxy values from 109 to 147. In their
+fiducial cosmology, Planck 2015, the model gives 61.2 for matter at z = 0.57, near the middle of that slice (the
+paper attaches no redshift to 62.0). With Planck 2018 it gives 59.9, mostly because σ₈ is lower (0.811 against 0.831).
+For galaxies of their bias in that slice (b ≈ 1.94) in redshift space the model gives 124; they measure 114 ± 6.
+
+## The scale is a convention
+
+R_H is where clustering falls below a chosen threshold, not a length nature sets.
+
+| How close to even (D₂ = 3(1 − ε)) | ε = 1% | 0.5% | 0.2% | 0.1% |
+|---|---|---|---|---|
+| R_H, matter today | 77 | 95 | 149 | 185 |
+
+| z | Matter | Galaxies, b = 1 | Galaxies, b = 2 |
+|---|---|---|---|
+| 0 | 77 | 87 | 147 |
+| 0.5 | 62 | 76 | 127 |
+| 1 | 50 | 64 | 95 |
+| 2.4 | 29 | 41 | 69 |
+
+(Galaxy columns are in redshift space.) A ten times stricter threshold more than doubles R_H. Matter becomes
+even on smaller scales in the past, because its clustering has had less time to grow. Strongly biased tracers
+look clumpy out to twice the scale. Other definitions give other numbers: Yadav, Bagla and Khandai (2010) put
+an upper limit of about 260 h⁻¹ Mpc on ΛCDM by asking where the deviation from 3 drops below its own
+statistical scatter, and an entropy measure gives about 140 h⁻¹ Mpc for SDSS (Pandey and Sarkar 2015).
+
+The baryon acoustic peak shows up too. The linear ξ has a bump at 99.6 h⁻¹ Mpc (Eisenstein et al. 2005 found
+the peak near 100), and it holds D₂ almost flat between about 100 and 120 h⁻¹ Mpc. In linear theory D₂ even dips
+slightly near 113, and then R_H jumps: for a b = 2 tracer at z = 0.5 in real space (R_H = 99), half a percent
+more clustering amplitude, near ×1.055, moves R_H by 13 h⁻¹ Mpc. Nonlinear motions smooth the peak: fits to BOSS
+galaxies before reconstruction give Σ ≈ 8 h⁻¹ Mpc (Beutler et al. 2017). With Σ = 7 to 10 h⁻¹ Mpc the dip and
+the jump are gone, the peak moves to about 96, and the 1% scale for matter today is unchanged (77).
+
+## Where the README's "80 to 300" came from
+
+From the last sentence of Scrimgeour et al.'s abstract: they exclude a fractal with D₂ below 2.97 "on scales
+from ~80 h⁻¹ Mpc up to the largest scales probed by our measurement, ~300 h⁻¹ Mpc". That is the range they
+could rule a fractal out over, not the scale where evenness starts, and 300 is where their survey ran out. It
+is also not Yadav et al.'s 260.
+
+## Units
+
+h⁻¹ Mpc is a megaparsec divided by h, the Hubble constant in units of 100 km/s/Mpc. With h = 0.674 (Planck
+2018), 70 to 80 h⁻¹ Mpc is 104 to 119 Mpc, about 340 to 390 million light-years. The README read
+"80 to 300 h⁻¹ Mpc" as "80 to 300 million parsecs".
+
+## The coastline
+
+A coastline is rough down to some smallest scale (in the end, grains and molecules); the galaxy distribution
+is clumpy up to some largest scale. Both are fractal only between two limits, but the cosmic limit that
+"stops the nesting" is the upper one.
+
+## Bigger structures
+
+Some structures are claimed to exceed the homogeneity scale: the Huge Large Quasar Group (Clowes et al. 2013,
+about 1.2 Gpc long), the Giant Arc (Lopez et al. 2022, about 1 Gpc) and the Big Ring (Lopez et al. 2024, about
+400 Mpc). Nadathur (2013) showed that the same group finder turns up Gpc "structures" in random points, and
+Sawala et al. (2025) find Giant Arc-like patterns common in ΛCDM simulations; Lopez and Clowes dispute that.
+Homogeneity is a statement about averages, so single large structures do not by themselves contradict it.
+
+## Run it
+
+```
+node cosmic-homogeneity/table.mjs    # D₂(r), R_H against threshold, tracer and redshift, and the WiggleZ comparison
+npm test                             # includes tests/cosmic-homogeneity.mjs
+```
+
+`homogeneity.js` is the whole model. It loads in Node (`require`) and in a browser (`window.HOMOGENEITY`).
+
+## Checks
+
+`tests/cosmic-homogeneity.mjs`:
+
+| Check | Published | Model |
+|---|---|---|
+| WiggleZ ΛCDM R_H, four slices (Scrimgeour et al. 2012) | 76, 78, 81, 78 | 77.1, 78.6, 81.5, 78.7 |
+| Measured WiggleZ R_H within 2σ of the model | 71±8, 70±5, 81±5, 75±4 | yes |
+| WiggleZ ΛCDM R_H by the counts definition (their Table 4) | 80, 83, 85, 82 | 80.7, 83.0, 85.3, 82.5 |
+| Matter R_H at z = 0.57, Planck 2015 (Ntelis et al. 2017, ΛCDM, matter) | 62.0 | 61.2 |
+| Sound horizon at Planck's drag redshift (Planck 2018) | 147.09 Mpc | 147.25 Mpc |
+| Baryon acoustic peak in ξ, sanity check (Eisenstein et al. 2005) | ~100 | 99.6 |
+| D₂ deep inside r₀ (model; γ = 1.8 from Zehavi et al. 2005) | 3 − γ = 1.2 | 1.20 |
+| Endpoint count slope over 1–30 h⁻¹ Mpc (model) | between 1.2 and 3 | 1.90 |
+| Threshold and redshift move R_H (model) | | 77 → 185 for 1% → 0.1%; 77, 62, 50, 29 at z = 0, 0.5, 1, 2.4 |
+| Acoustic plateau, b = 2, z = 0.5: largest R_H change for a 0.5% amplitude step (model) | | 12.8 (linear), 0.8 (smoothed) |
+
+## What the model leaves out
+
+- Linear theory: no nonlinear clustering, which matters below about 20 h⁻¹ Mpc, apart from the optional
+  smoothing of the acoustic peak, and no velocity damping of the redshift-space signal. Scrimgeour et al.'s
+  ΛCDM uses a nonlinear spectrum (CAMB with HALOFIT) and a Lorentzian damping term; with linear Eisenstein–Hu
+  and the Kaiser factor alone the model comes out 0.5 to 1.1 h⁻¹ Mpc above their D₂ values and 0 to 0.7 above
+  their counts values.
+- The Eisenstein–Hu equations, constants and equation numbers were checked against the paper's arXiv LaTeX
+  source (astro-ph/9709112v1) in review on #26: an independent transcription gives the same T(k) to within
+  7 × 10⁻¹⁶. Hu's own code (tf_fit.c) differs slightly from the paper text, moving T by at most 0.06% and R_H by
+  at most 0.02 h⁻¹ Mpc; this model follows the paper. Their fitted drag redshift (1021) is lower than Planck's
+  (1060), by construction (the paper reduces z_d by a factor of 0.96), so their sound horizon is 2.6% larger;
+  at Planck's drag redshift the same formula gives 147.25 Mpc against Planck's 147.09.
+- The WiggleZ bias, β and R_H values and the Ntelis et al. values were checked against the papers' arXiv
+  LaTeX sources in review on #26.
+- The galaxy power law is a fit below about 10 to 20 h⁻¹ Mpc: r₀ and γ change with galaxy luminosity, and γ
+  steepens for the brightest. The 1-to-30 average extrapolates it.
+- Neutrinos are left out: Planck's Ωm includes about 0.06 eV of neutrino mass, which the Eisenstein–Hu fit
+  does not model.
+- Clustering lengths are comoving and in h⁻¹ Mpc. The sound horizon and the large-structure sizes are in Mpc,
+  as their sources give them. The grid resolves ξ out to 300 h⁻¹ Mpc, so R_H is not searched beyond that.
+
+## Sources
+
+- M. I. Scrimgeour et al., *The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity*, MNRAS 425, 116 (2012). https://arxiv.org/abs/1205.6812
+- D. J. Eisenstein and W. Hu, *Baryonic features in the matter transfer function*, ApJ 496, 605 (1998). https://arxiv.org/abs/astro-ph/9709112
+- Planck Collaboration, *Planck 2018 results. VI. Cosmological parameters*, A&A 641, A6 (2020). https://arxiv.org/abs/1807.06209
+- Planck Collaboration, *Planck 2015 results. XIII. Cosmological parameters*, A&A 594, A13 (2016). https://arxiv.org/abs/1502.01589
+- D. W. Hogg et al., *Cosmic homogeneity demonstrated with luminous red galaxies*, ApJ 624, 54 (2005). https://arxiv.org/abs/astro-ph/0411197
+- J. Yadav, S. Bharadwaj, B. Pandey and T. R. Seshadri, MNRAS 364, 601 (2005). https://arxiv.org/abs/astro-ph/0504315
+- P. Sarkar, J. Yadav, B. Pandey and S. Bharadwaj, MNRAS 399, L128 (2009). https://arxiv.org/abs/0906.3431
+- J. K. Yadav, J. S. Bagla and N. Khandai, *Fractal dimension as a measure of the scale of homogeneity*, MNRAS 405, 2009 (2010). https://arxiv.org/abs/1001.0617
+- P. Ntelis et al., *Exploring cosmic homogeneity with the BOSS DR12 galaxy sample*, JCAP 06 (2017) 019. https://arxiv.org/abs/1702.02159
+- B. Pandey and S. Sarkar, MNRAS 454, 2647 (2015). https://arxiv.org/abs/1507.03124
+- I. Zehavi et al., *The luminosity and color dependence of the galaxy correlation function*, ApJ 630, 1 (2005). https://arxiv.org/abs/astro-ph/0408569
+- D. J. Eisenstein et al., *Detection of the baryon acoustic peak in the large-scale correlation function of SDSS luminous red galaxies*, ApJ 633, 560 (2005). https://arxiv.org/abs/astro-ph/0501171
+- F. Sylos Labini, M. Montuori and L. Pietronero, *Scale-invariance of galaxy clustering*, Phys. Rep. 293, 61 (1998). https://arxiv.org/abs/astro-ph/9711073
+- F. Sylos Labini, N. L. Vasilyev, L. Pietronero and Y. V. Baryshev, EPL 86, 49001 (2009). https://arxiv.org/abs/0805.1132
+- R. G. Clowes et al., MNRAS 429, 2910 (2013). https://arxiv.org/abs/1211.6256
+- S. Nadathur, *Seeing patterns in noise: gigaparsec-scale 'structures' that do not violate homogeneity*, MNRAS 434, 398 (2013). https://arxiv.org/abs/1306.1700
+- F. Beutler et al., *The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Fourier space*, MNRAS 464, 3409 (2017). https://arxiv.org/abs/1607.03149
+- A. M. Lopez, R. G. Clowes and G. M. Williger, MNRAS 516, 1557 (2022). https://arxiv.org/abs/2201.06875
+- A. M. Lopez et al., *A Big Ring on the sky*, JCAP (2024). https://arxiv.org/abs/2402.07591
+- T. Sawala et al., MNRAS Letters 541, L22 (2025). https://arxiv.org/abs/2502.03515