[zero-aska]

This is a follow-up to #33. It makes the four wording fixes that #33's reviews left as non-blocking:

• three from the re-review of 607265a;
• one from the review of e1cd891.

No number, tolerance or check changes.

Changes

• Newman and Ziff's p_c is a measurement. 0.59274621(13) is a Monte Carlo result, not an exact value.
  • README.md (two places), critical/README.md's opening paragraph and the header of tests/critical.mjs now say "exact or high-precision".
  • The tests' header names the threshold and says its error is negligible here (1.3e-7).
  • The two p_c checks print it as reference, where they used to print exact. near() already took a label argument.
• Salas and Sokal pinpoint. The Sources entry now cites secs. 1, 4.1 and 4.3 and Tables 2 and 5:
  • sec. 1 holds the Binder-cumulant definition the README credits to them;
  • sec. 4.3 and Table 5 hold the free-exponent fit the tests quote (Δ = 2.01 ± 0.22).
• Section 1 header in the tests. It now says what the section checks: the closed forms against the published formulas, the open-edge shift also against Ziff and Newman's measured crossings (Table 1), and the random-number generator.
• The site's turbulence entry (index.html #studies) now adds that where the cascade ends depends on the medium, as the root README and turbulence/README.md say.

Edited paragraphs are rewrapped only as far as the change reaches. To keep the tests' header rewrap from reaching further lines, four words were also dropped ("it is", "those", "The"). The meaning is unchanged.

Checked

• node tests/run.mjs passes all 8 check files on Node 18.20.8, 20.20.0 and 22.22.0.
• python3 scripts/check_site.py is clean.
• No open PR touches the edited lines. Four open PRs (Bound gravity-only witnesses for dark matter (issue 3) #12, Cosmic homogeneity: reproduce where galaxy clustering stops looking fractal, and fix the README's range #26, Halos within halos: how deep the nesting goes, and what the dark matter particle sets as the floor #35, Orbits: is a galaxy a big solar system? Where Kepler's law holds, where rotation flattens, and what ends a Kepler region #37) change other parts of README.md, and Critical-point study: wording fixes left over from #33's reviews #44 adds no conflict for any of them.

Corrections after merge

• This PR was merged before the independent review it promised had finished.
• That post-merge review said merge, with no blocking findings.
• It found the relabelling incomplete: the results table and critical/table.mjs still called p_c exact, Salas and Sokal's V4 is high-precision too, and the turbulence entry still implied viscosity always ends the cascade. Critical-point study: finish the exact/high-precision relabelling from #44's review #45 makes those changes.

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[zero-aska]

This review of 02f9fe6 was written by agent instances that did not draft the branch. Two reviewers covered it: one checked sources and claims, the other checked tests, prose and the site. Each finding was then given to a refuter (blocking ones would have had two). The PR was merged as 5fa2448 at 05:21Z, after ahrazzle approved it, so the fixes below need a follow-up PR.

What was verified

• Newman and Ziff 2000 (arXiv cond-mat/0005264). p_c = 0.59274621 ± 0.00000013 is a Monte Carlo estimate (their R^(h)), not an exact value. "Negligible here" holds:
  • the error is 0.06% of the p_c error bars and 1.4e-4 of the smallest p_c tolerance;
  • in the most sensitive check it is 0.2% of the L = 256 shift, against a 10% tolerance.
• The Salas and Sokal pinpoint (arXiv cond-mat/9904038) matches the LaTeX. Sec. 1 has U = 1 - V4/3 and V4 = 1.1679229(47). Sec. 4.1 has Delta = 7/4. Sec. 4.3 has Delta = 2.01 ± 0.22 and B4 = -0.48 ± 0.23. Table 2 has V4(16) = 1.16586(38). Table 5 has 2.007 ± 0.223 and -0.477 ± 0.228.
  • The pinpoint covers every place the repository uses the paper.
  • The 1.0e-4 shift at tests/critical.mjs:142 recomputes. So does 1.28e-4 for Delta = 7/4.
• Test headers.
  • The new section 1 header (tests/critical.mjs:73-74) lists exactly what lines 75-124 do, with no Monte Carlo.
  • The header's "two checks" against Ziff and Newman's measurements are the section-1 crossings check and the section-7 p_0.5(32) check.
• Test output.
  • Base and head output of tests/critical.mjs differ only in "exact" becoming "reference" on the two p_c lines (one near() call in the pool.wide loop) and in the runtime. Every value, tolerance, sigma and margin is byte-identical: 33 PASS, 0 FAIL.
  • node tests/run.mjs passes on Node 18, 20 and 22.
  • The CI runs test (18), (20) and (22) succeeded on 02f9fe6.
• Prose.
  • In README.md:338 and :361, every value called "exact or high-precision" is one or the other. The largest deviation is 1.43 sigma (xi at T = 2.4), so "1.5 standard deviations" still holds.
  • The new clause in index.html:238 matches README.md:32 word for word.
• Wrapping and site checks.
  • scripts/check_site.py is clean, and so is git diff --check.
  • Edited lines are within the wrap width. The only lines over 119 columns already existed on main.
  • Word diffs of the rewrapped paragraphs contain only the intended edits, apart from the tests header (refuted item below).
• PR body. Its account of the four changed files matches the diff.

Blocking findings

None.

Non-blocking findings

1. critical/README.md:109, :121, :28. The results table still calls Newman and Ziff's p_c "Exact". Both reviewers raised this; it is the same issue as finding 1 of #33's re-review, in places that review did not name.
   • The "What it reproduces" table heads its reference column | Check | Exact | Model | (:109) and lists p_c, 0.59275, under it (:121).
   • The opening paragraph (:9-10), edited by this PR, says the same table is compared with "an exact or high-precision value".
   • The short answer (:28) still says "lands on or near the exact numbers", and its bullet quotes 0.592746 (Newman and Ziff 2000).
   • No number is affected, and the wording dates from before #33.
   • Fix: rename the column "Exact or high-precision" (or "Reference"). At :28 say "exact or high-precision numbers", to match README.md:338.
2. critical/table.mjs:1, :19, :26, :89-91, :96-98. table.mjs still prints p_c as "exact". Both reviewers raised this. critical/README.md:4 names this script as the one that prints the study's numbers.
   • It prints PERC.pc under "Exact values (infinite lattice)".
   • The rows "p_c, L = 32/128", "p_c, L = 64/256" and "p(R = 1/2), L = 256" print "exact 0.59275, mean - exact ...".
   • The header comment (:1) says "the exact values beside them".
   • tests/critical.mjs now prints "reference" for the same two means, so the repository labels one comparison two ways.
   • Fix: label p_c "reference" or "high-precision" in the heading and in the :98 printout. For example, give the p_c columns their own label, as near() now does.
3. critical/README.md:9-10 and tests/critical.mjs:7, :147, :170-171. V4 is also a numerical result. Salas and Sokal's V4 = 1.1679229(47) is partly a Monte Carlo integration ("we performed a Monte Carlo integration using 10^9 measurements", sec. 2). Their Table 2 caption says "exact except for a numerical integration".
   • The new parenthetical in critical/README.md names only Newman and Ziff's threshold as not exact, yet the measurement-trap cumulant on the same list is compared with 1 - V4/3.
   • tests/critical.mjs:7 groups V4 under "exact", and both Binder checks print "exact 0.6107".
   • The error in U is 1.6e-6, against tolerances of 0.0035 to 0.005, so no result changes.
   • Fix: write "(Newman and Ziff's measured threshold and Salas and Sokal's V4, whose errors are negligible here)", and pass 'reference' to the two Binder near() calls.
4. index.html:238. The turbulence clause does not fully fix #33's review finding 2. The entry still names only "the size where viscosity takes over".
   • The added "where the cascade ends depends on the medium" still reads as true if viscosity is the only cutoff, because the viscous size itself varies between media.
   • In README.md:32-36 the same phrase is followed by the solar-wind ion scales and the sonic scale in molecular clouds, which settle it. The site entry has no such examples.
   • Fix: use the review's "what ends the range depends on the medium", or add a short example (for example, "ions in the solar wind").
5. PR body, "Checked", bullet 3. "No open PR touches these files." is false. Open PRs #12, #26, #35 and #37 change README.md, and all their heads predate #44.
   • None of them touches the edited lines (335-363).
   • git merge-tree gives identical conflict sets for each of them against 4600f90 and against 5fa2448, so #44 added no conflict.
   • Fix: none needed in the repository. Check open PRs before making this claim next time.
6. PR body, "Checked", bullet 4. The promised independent review did not happen before merge. The bullet says such a review "will follow before merge".
   • The PR was opened at 05:17:59Z. ahrazzle approved it at 05:20:39Z and merged it at 05:21:21Z.
   • Fix: put items 1-4 in a follow-up PR.

Refuted:

• tests/critical.mjs:7-8, dropped words. The rewrap of the header dropped three words ("it is", "those", "The"). These are copyedits inside the one sentence the PR openly rewrites. Lines 4-6 still say which two checks use Ziff and Newman's measurements, and the count is still right.

Verdict: merge

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