# Verification report — AMR-096-0008 (Aldous: Metropolis on Cayley graphs)

Verification date: 2026-10-09.

**Verdict.** The note gives a **partial answer**; every statement it makes is proved, and the scope is as follows.
- **Settled, for the relaxation time τ = 1/(1 − β₂) and for the items as literally posed.** Item (i) of the
  problem page ("τ(p) is monotone decreasing in p") fails on every finite Cayley graph: τ(p) > τ(0) for all small
  p > 0 (Theorem 1.6). Item (ii) (τ(p) ≤ C τ(∞) for a universal constant C) fails under both readings of the undefined
  symbol τ(∞), namely τ(0) and lim_{p→1} τ(p) = d, on hypercubes and on Cayley graphs of bounded degree with a
  two-sided spectral gap (Corollaries 1.4 and 1.5). With the orientation of p reversed, (i) fails on the cycles
  C_n, n ≥ 7, and on the hypercubes of dimension d ≥ 3.
- **Not new mathematics.** The complete-graph formula τ(p) = (n−1)(1+(n−2)p)/(n−p) (Proposition 1.1) is a
  corollary of a known result: on K_n the Metropolis chain with any target π has relaxation time (n−1)·max π
  (Liu 1996; Diaconis–Saloff-Coste 1998, Theorem 2.2; Aldous–Fill, Section 11.4.2, Proposition 11.2(b)). The
  methods of the note are standard.
- **Caveat.** Item (i), and item (ii) under the reading τ(∞) = τ(0), are thus already contradicted on complete
  graphs by a consequence of a proposition in the Aldous–Fill monograph. The question that was intended on the
  page may differ from the literal one. This cannot be decided from the sources, and the note does not say that
  the problem is solved.
- **Not answered.** The third item of the page (a decreasing bound on τ(p) in terms of τ(∞), p, the number of
  states and graph parameters) and the open-ended opening question; the shape of τ on (0,1) in general.

The note is unrefereed. Two independent verification runs, both AI-assisted, examined it (see below).

## Statement checked
- **Primary source.** D. Aldous, open-problem page "Metropolis on Cayley graphs",
  https://www.stat.berkeley.edu/~aldous/Research/OP/cayley.html , entry "(1.9)" among the open research problems
  of the list https://www.stat.berkeley.edu/~aldous/Research/OP/index.html .
  - Fetched on 2026-10-09 in the second stage, in the first independent verification run, when the note was
    written and in the second independent verification run (1480 bytes, sha256
    `9989446bf73a5d08579242e70a89cb6d09cdc4b60032ff9cbf29f09064eb8f31` each time; index: 6517 bytes, sha256
    `69de0a61207bac04b9452263afbc75321409adb75e673d0f6dfb29a2441adb40`). A Wayback Machine copy of 2015-12-03 has
    the same text, except that the range 0 < p < 1 in the first sentence is missing there and one word is
    misspelt ("deceasing bound"); a copy of the index of 2010-07-18 already lists the entry (both copies were
    fetched again in the second run). The 2003 list `problems_old.pdf` linked from the index does not contain the
    problem.
  - Content (paraphrase). On a finite Cayley graph, X is the random walk started at the identity and T_p has the
    geometric distribution with parameter p. For 0 < p < 1, μ(p) is the distribution of X(T_p − 1); μ(0) is taken
    to be uniform. τ(p) is the relaxation time of the Metropolis chain, based on the random walk, with stationary
    distribution μ(p). Question: can one prove anything about τ(p) in this general setting; for instance (i) is
    τ(p) monotone decreasing in p; (ii) is τ(p) ≤ C τ(∞) for some universal C; and, in an item labelled (ii) a
    second time, is there some decreasing bound on τ(p) in terms of τ(∞), p, the number of states and familiar
    parameters of the graph. History line: first written March 2009.
  - Not defined on the page: the symbol τ(∞) (p ranges over [0,1)); whether the generating set is symmetric and
    whether it contains the identity; which geometric distribution; which definition of the relaxation time.
- **Conventions fixed in the note (Section 1.2).** Symmetric generating set without the identity, simple random
  walk; P(T_p = k) = p(1−p)^{k−1}, k ≥ 1, which is forced by "X(T_p − 1)" and "μ(0) uniform"; the Metropolis chain
  for a symmetric proposal as in Aldous–Fill, Section 11.2.1; the relaxation time 1/(1 − β₂) as in Aldous–Fill,
  (3.38)–(3.39). A lazy walk, a generating set containing the identity, or continuous time only reparametrise the
  family (Remark 2.2).
- **Corpus record.** ulamai/UnsolvedMath, AMR-096-0008 (dataset version 1.6.0; upstream status `open`). Its
  statement asks to analyse τ(p), whether τ(p) is decreasing in p, whether it is universally bounded by a constant
  times "its endpoint value", and for decreasing bounds. It replaces τ(∞) by "its endpoint value" and omits that
  μ(0) is uniform and the shift in X(T_p − 1).

## Readings
| Item | Reading | Answer | Where |
|---|---|---|---|
| (i) τ is non-increasing in p | as posed | no, on every finite Cayley graph | Theorem 1.6 (and Proposition 1.1 on K_n, known) |
| (i) | orientation of p reversed (not what the page says): τ is non-decreasing | no: cycles C_n, n ≥ 7; hypercubes Q_d, d ≥ 3 | Corollary 4.1; Corollary 1.4(c) |
| (ii) τ(p) ≤ C τ(∞) | (a) τ(∞) = τ(0) | no: K_n (ratio n, known); Q_d (ratio ≥ 2(d−1)/15); bounded degree with a two-sided spectral gap (ratio ≥ c (log n)²) | Proposition 1.1; Corollaries 1.4, 1.5 |
| (ii) | (b) τ(∞) = lim_{p→1} τ(p) = d | no: cycles (τ(0)/2 → ∞); Q_d (ratio ≥ (d−1)/15); bounded degree with a two-sided spectral gap | Corollaries 4.1, 1.4, 1.5 |
| (iii) a decreasing bound | — | not answered; necessary conditions only | Remark 7.1 |
| opening question | — | not answered; three statements valid on every finite Cayley graph | Theorems 1.2, 1.3, 1.6 |
| relaxation time by the absolute gap | not the definition of the note | Theorems 1.2 and 1.3 hold; (i) holds on K_2 and, weakly, on K_3; Theorem 1.6 holds if \|λ_n\| < λ₂ and can fail otherwise | Remark 7.2 |

## Results in the paper
- **Proposition 1.1 (known).** On K_n: μ_p(e) = (1+(n−2)p)/(n−p), τ(p) = (n−1) μ_p(e), strictly increasing,
  τ(0) = (n−1)/n, τ(1−) = n−1, sup τ(p)/τ(0) = n. **Proposition 3.1 (known):** any target on K_n has relaxation
  time (n−1)·max π; a short proof is included.
- **Lemma 2.1.** μ_p = p (I − (1−p)K)^{-1} δ_e; μ_p > 0 with a strict maximum at e; spectral formula and
  μ_p → uniform as p → 0 (also on bipartite graphs); μ_p(x) = p q^{|x|}(N(x) d^{−|x|} + ε), 0 ≤ ε ≤ q/p.
- **Theorem 1.2.** lim_{p→1} τ(p) = d. The limit matrix is block lower triangular for the word length; its
  diagonal blocks have row sums ≤ 1 − 1/d, with equality on level 1, and are symmetric for the weights N.
- **Theorem 1.3.** τ(p) ≥ (1−p) m (1−m)/(m−p), m = μ_p(e) (test function 1_{e}; equality on K_n), and
  τ(p) ≥ 2 Var|X(T_p − 1)| (test function |x|).
- **Corollary 1.4 (hypercube).** τ(0) = d/2, τ(1−) = d, τ(p) ≥ H_d(p); (a) τ(1/d) ≥ d(d−1)/15; (b)
  τ(p) ≥ (1−p)d/(3p) for pd ≥ 4; (c) H_d(p) > d iff p > 4/(d²−2d+4). **Proposition 5.1:** μ_p(x) = w(|x|) with w
  a moment sequence; mean and variance of |x|; equality in the Dirichlet form. **Proposition 5.2:** β₂ has an
  eigenfunction depending only on |x| (monotone coupling), so τ is the relaxation time of a birth-and-death chain.
- **Corollary 1.5.** If n ≥ 32 d⁶, then sup_p τ(p) ≥ (κ/40)(log_d n − 6)², κ = min{1, log(1/ρ)/log d}, ρ the
  largest absolute value of an eigenvalue of K other than ±1. The existence of Cayley graphs of fixed degree with
  ρ ≤ ρ₀ < 1 (Lubotzky–Phillips–Sarnak) is quoted and was not checked.
- **Theorem 1.6.** 1/τ(p) = 1/τ(0) − s* p + O(p²) with s* = max of an explicit quadratic form Q on the λ₂-eigenspace,
  and s* ≥ Q(f₀) > (m₂ − 1)/2 ≥ 0. Proof: expansion of the symmetrised matrix S(p) (Lemma 6.2) and an elementary
  first-order perturbation lemma (Lemma 6.1).
- **Remark 4.2 (prior instances).** For the targets θ^{|x|} of Diaconis and Hanlon: on the hypercube the chain of
  the Hamming weight has the eigenvalues 1 − (j/d)(1+θ) (their Section 3, Theorem 1), and so has the Metropolis
  chain itself, which is a product chain; on S_N with all transpositions the chain lumped to conjugacy classes has
  the second eigenvalue 1 − 2θ/N − 2/(N(N−1)) (their Section 4), and the relaxation time of the Metropolis chain on
  S_N is at least that of the lumped chain (equal in the computations for N ≤ 7).
- **Section 7.** Table 1 (the readings); Remark 7.1 (item (iii): necessary conditions; the comparison bound (14),
  an instance of the comparison lemma of Diaconis and Hanlon); Remark 7.2 (absolute gap); Remark 7.3 and Figure 1
  (shapes of τ: computations only).

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation.
- **The exact program of the note** (`writing_stage/check_note.py`, 60 checks, less than one
  minute; TOTAL failures: 0). Exact rational targets and kernels, eigenvalues with 30 digits, 34 graphs
  (33 Cayley graphs and the Petersen graph). It tests Lemma 2.1 and Remark 2.2 exactly; the example of Remark 2.3
  exactly; Proposition 3.1 on 42 random rational targets and Proposition 1.1 for n ≤ 12 exactly, with the full
  spectrum on K_n; the structure of the limit matrix exactly and the limit τ(1−ε) → d; the two formulas of
  Diaconis and Hanlon quoted in Remark 4.2; Theorem 1.3 at 25 values of p on every graph; Proposition 5.1 exactly
  for d ≤ 5, Corollary 1.4(a)–(c) and identity (11) in rational arithmetic for d up to 1000 and symbolically,
  Proposition 5.2 (full chain against level chain); the arithmetic of the proof of Corollary 1.5 and its Steps 1
  and 2 on ten graphs; Lemma 6.2 ((a), (b), (d) exactly), the formula for s* against difference quotients of 1/τ at
  p = 10⁻⁴, 10⁻⁶, 10⁻⁸, and Lemma 6.1 with its explicit constant on 300 random symmetric matrices; the comparison
  bound (14), Remark 7.2 and the shapes of Figure 1.
- **The programs with which the results were first obtained** (`original/`): Proposition 1.1 symbolically for
  n ≤ 8; the exact limit matrix on 38 graphs; the bounds of Theorem 1.3 on 40 graphs and the slope at p = 0 on 41
  graphs (double precision); both limits on six graphs with 60 digits; the hypercube with the level chain up to
  d = 1024; the steps of Corollary 1.5 on eleven graphs; cycles up to n = 256, SL₂(Z/q) for q ≤ 13 and symmetric
  groups up to S₆.
- **The first independent verification run** (`independent_run/`, own programs, exact rational targets,
  eigenvalues with 40 to 60 digits): Proposition 1.1 symbolically for n ≤ 8 and in 280 exact rational cases;
  Theorems 1.2, 1.3, 1.6 on 46 graphs (all flags OK); all 1204 Cayley graphs of the 23 groups of order ≤ 12 and
  2000 random Cayley graphs of 29 groups of orders 15 to 120 in double precision (no violation of any statement;
  six flags of the program's own tolerance for a difference quotient, resolved with 60 digits); the hypercube
  (exact identities for d ≤ 6, full chain against level chain for d ≤ 10, level chain up to d = 400); all steps of
  the proof of Corollary 1.5 on SL₂(Z/53) (n = 148,824 ≥ 32 d⁶); the formulas of Diaconis and Hanlon with 50
  digits; the absolute gap; the shapes.
- **The second independent verification run** (`independent_run_2/`, own programs, written from the text of the
  note before the other programs were opened; 106 checks, 0 failures): Lemma 2.1, Remarks 2.2 and 2.3 and
  Lemma 6.2 exactly on 40 graphs; Proposition 3.1 on 56 random rational targets and Proposition 1.1 symbolically
  for n ≤ 9; the limit matrix with exact counts of the real roots of its blocks, and the limit of τ with 40 digits;
  Theorem 1.3, Propositions 5.1 and 5.2, identity (11) and Corollary 1.4 (d up to 10⁶ in rational arithmetic);
  all steps of the proof of Corollary 1.5 on PGL₂(Z/31) with three generators (d = 3, n = 29,760 ≥ 32 d⁶:
  ρ = 0.9748, κ = 0.0233, t = 431, τ(1/t) = 100.8 ≥ 2 Var = 29.2 ≥ κr²/40; sup_p τ(p) ≥ 135.9 = 4.3 τ(0));
  Lemma 6.1 with its constant in 7200 random cases with multiplicities up to N, and on the matrices S(p);
  Theorem 1.6 with 50 digits on 40 graphs; the 1204 Cayley graphs of the groups of order ≤ 12, built anew, on
  three grids; the two families of Remark 4.2 for the lumped chains and for the Metropolis chains (S_N for N ≤ 7,
  Q_d for d ≤ 8); the numbers of Figure 1 and Remark 7.3.
  Two findings on Remark 7.3: the numbers 776 / 428 (increasing / up-then-down among the 1204 graphs) are those of
  the grid of 23 values; on a grid of 209 values reaching p = 1 − 10⁻⁶ they are 764 / 440, because on twelve
  Cayley graphs of the dihedral group of order 12 the maximum of τ lies near p = 0.99 (τ − d ≤ 0.0005 there). And
  the six random graphs for which the first run recorded no shape are all up-then-down, so that 21 is the number
  of graphs with a third shape among all 2000.
- **Re-runs.** On 2026-10-09, when the note was written, all programs of the package that have a recorded output
  were run again, one process at a time. `reproducibility/run_quick.sh` was run from an extracted copy of the
  archive (344 seconds): of 33 outputs, 28 are identical to the recorded files and 5 are identical up to fields
  which record running times. The one slow program (`stress_followup.py`, 686 seconds) was run separately; its
  output is identical. The second verification run repeated both from an extracted copy of the archive
  (410 seconds: 24 outputs identical, 9 identical up to running times; `stress_followup.py`, 798 seconds:
  identical), and its own programs were run again from an extracted copy of the archive
  (`independent_run_2/run_all.sh`: 13 outputs identical, 106 checks passed). Details: `reproducibility/README.md` and `reproducibility/RERUN_LOG.txt`.

## Independent verification runs
The results were obtained in three stages, all AI-assisted: (1) the complete-graph formula; (2) a second,
independent derivation of that formula, and Theorems 1.2 and 1.3, Corollaries 1.4(a), (b) and 1.5, and Theorem 1.6
in the form 1/τ(p) ≤ 1/τ(0) − Q(f₀) p + O(p²), with proofs; (3) a first independent verification run on the written
results of stage 2, with its own programs. After the note was written, a second independent verification run,
also AI-assisted, examined the final text and the package.

| Item | First run (written results of stage 2) | Second run (final text) |
|---|---|---|
| Conventions (reading of the source) | CONFIRMED_WITH_FIXES (caveat on the literal reading; K_2 and K_3 under the absolute gap) | CONFIRMED (page and index fetched again and compared sentence by sentence; the description of what the page leaves undefined is fair) |
| Proposition 1.1, Proposition 3.1 | CONFIRMED (re-derived; exact); not new: attribution required | CONFIRMED; the reduction to Aldous–Fill, Proposition 11.2(b), checked against the HTML edition |
| Lemma 2.1, Remark 2.2, Lemma 2.4, Corollary 2.5 | CONFIRMED | CONFIRMED (text of the note) |
| Theorem 1.2 (limit p → 1) | CONFIRMED (re-derived; a second argument for the real spectrum of the limit matrix) | CONFIRMED (block structure, row sums of the level-1 block, convergence of the second eigenvalue) |
| Theorem 1.3 (a), (b) | CONFIRMED | CONFIRMED |
| Corollary 1.4 (a), (b), H_d, the constants | CONFIRMED; the inequality for the Dirichlet form is an equality; Proposition 5.2 proved | CONFIRMED |
| Corollary 1.4(c), identity (11) | not in the material of this run | CONFIRMED |
| Corollary 1.5 (all steps and constants) | CONFIRMED; the existence of the expander families is quoted, NOT CHECKED | CONFIRMED, including ρ > 0; existence still quoted, NOT CHECKED |
| Theorem 1.6 (τ(p) > τ(0) for small p) | CONFIRMED on every finite Cayley graph (and vertex-transitive graph), for τ = 1/(1 − β₂) | CONFIRMED |
| Exact derivative of 1/τ at 0 | proved by the run, through Rellich's theorem | CONFIRMED in the form of Section 6: Lemma 6.1 (min–max, the constant 2a²/γ + C₀), Lemma 6.2 (a)–(d), proof of Theorem 1.6 |
| Remark 2.3 (example on Z/16), s* ≥ (d+1)/2 | not in the material of this run | CONFIRMED (exact values; symbolic expansion) |
| Citations of Aldous–Fill, Diaconis–Saloff-Coste, Diaconis–Hanlon | read by the run | Aldous–Fill and Diaconis–Saloff-Coste: exact. Diaconis–Hanlon: CONFIRMED_WITH_FIXES (the report gives the eigenvalues of the lumped chains) |
| Conclusion on (i) | CONFIRMED_WITH_FIXES (wording "as literally posed"; convention) | CONFIRMED |
| Conclusion on (ii) | CONFIRMED (both readings) | CONFIRMED |
| Novelty | CONFIRMED_WITH_FIXES (Proposition 1.1 is known; the two families of Diaconis and Hanlon are prior instances) | CONFIRMED_WITH_FIXES (nothing found; one credit added) |

Neither run found a gap in a proof or a counterexample.

**Corrections required by the first run**, all applied in the note:
1. Attribution of the complete-graph formula: Proposition 3.1 with Liu 1996, Diaconis–Saloff-Coste 1998
   (Theorem 2.2), Aldous–Fill (Proposition 11.2(b)); Proposition 1.1 is labelled "known": Sections 1.3, 1.4, 3.
2. Literature: the 1998 survey and the 1992 paper of Diaconis and Hanlon (in its technical-report version) were
   read; the hypercube family and the symmetric-group family with targets θ^{|x|} are presented as prior instances
   of the phenomenon behind Theorems 1.2 and 1.6: Section 1.4, Remark 4.2, "Scope and priority".
3. Wording: "the items (i) and (ii) as literally posed have negative answers", with the caveat stated without a
   judgement of the source: abstract, Section 1.4, Section 7.
4. Convention: all statements are for τ = 1/(1 − β₂); Remark 7.2 describes the absolute gap (K_2: τ_abs = 1/p;
   K_3: constant; odd cycles; bipartite graphs).
5. Shapes: Remark 7.3 reports three shapes as computations and suggests no dichotomy.
6. Upgraded statements: equality in the hypercube Dirichlet form (Proposition 5.1(c)); β₂ attained on a function
   of the Hamming weight (Proposition 5.2); the exact one-sided derivative (Theorem 1.6).

**Added or changed when the note was written, after the first run; examined by the second run.**
- Section 6 is organised around the symmetric matrices S(p) and the elementary Lemma 6.1 (first-order perturbation
  of a multiple eigenvalue, one-sided, with an explicit constant), which takes the place of Rellich's theorem;
  Lemma 6.2(c), (d) are the corresponding form of the expansion. Rellich's theorem is cited (Remark 6.3(a)) but no
  proof depends on it.
- Corollary 1.4(c) with identity (11): H_d(p) > d iff p > 4/(d²−2d+4). The first run had the reversed orientation
  on hypercubes for d ≥ 17 only (from τ(1/d) ≥ d(d−1)/15).
- Remark 2.3 with the example on Z/16; the bound s* ≥ (d+1)/2 on Q_d in Remark 6.3(b); the observation that ρ > 0
  in the proof of Corollary 1.5.

**Corrections asked for by the second run**, all applied:
1. Section 1.4 and Remark 4.2: the eigenvalues given in the report of Diaconis and Hanlon are those of the
   lumped chains (Hamming weight; conjugacy classes). The note now says so, adds that the Metropolis chain on the
   cube is a product chain with the same eigenvalues, and that on S_N the relaxation time of the Metropolis chain is
   at least that of the lumped chain, with equality in the computations for N ≤ 7. The sentence on the sign in the
   general eigenvalue formula of the report says what the scan shows.
2. Remark 7.1: the comparison bound (14) is credited to the comparison lemma in Section 2 of that report.
3. Remark 7.3: the numbers of shapes on the finer grid (764 / 440) and the twelve graphs which change their class.
4. Section 8: four parts; the programs of the second run; "the first independent verification run" where the
   earlier run is meant.
5. "Verification" and "Scope and priority": final state; what the second run read and searched, including the
   requests that were refused.
6. Abstract and Section 1.4: the complete graphs contradict item (i) and item (ii) under reading (a), not "the
   literal items" without qualification.
7. Lemma 6.1: the condition on p for a = 0.
8. (Recommended.) One sentence of context in Section 1.4, with the reference to Kargin's note on the mean-field
   Ising model (abstract read).
9. Package: this report; `reproducibility/independent_run_2/` with its README; `README.md` and `RERUN_LOG.txt`.
10. "Scope and priority": the archived copy of the page of 2015 has the same text as the present page except for
    the range 0 < p < 1 in the first sentence, which is missing there, and one misspelt word (the note said "the
    same mathematical text").

## Relation to the literature, novelty and scope
- **Searches (all on 9 October 2026, in stage 2 and in the two verification runs).** arXiv API: 13, 34 and 38
  completed queries; OpenAlex: 21 and 17 queries in stage 2 and in the first run, including the lists of the works
  citing the 1998 survey (154, and 31 for its conference version) and the 1992 paper (41), of which the titles were
  read; Crossref and zbMATH (in the second run 14 searches and 10 queries); five web searches. In the second run
  the search queries to OpenAlex (17) and to Semantic Scholar (6) were refused because of the limits for requests
  without a key, and requests to dblp were answered with a bot check; none was repeated or worked around. No
  treatment of the family μ_p, no answer to the question and no citation of the problem page was found. The
  nearest titles concern the targets θ^{|x|} on the symmetric group and related models, the effect of the energy
  gap on the mixing time of the Metropolis algorithm (Nakade and Biswas, 2012), and Metropolis chains with uniform
  stationary distribution on graphs with symmetries (Boyd, Diaconis, Parrilo and Xiao, 2005); these papers were
  not read. The arXiv API returned only 415 records with "Metropolis" in the title, so this part of the search is
  weak evidence.
- **What was read.** The problem page and the index; of the Aldous–Fill monograph, in the HTML edition of 2014:
  Section 3.4 around (3.36)–(3.41), Sections 3.6.1 and 3.6.3, Sections 11.2 and 11.4 (the rest was not read; a
  text search of the PDF edition found no statement on Metropolis chains on Cayley graphs); Diaconis–Saloff-Coste
  1998, all 17 pages, in the copy on the web page of its second author (the publisher's site refused automated
  access and was not pursued), and Sections 2.1 and 2.2 again in the second run; Diaconis–Hanlon 1992 in the
  technical-report version (Stanford, Technical Report No. 392, March 1992; 29 scanned pages), all pages in the
  first run and Sections 1–4 again in the second run; of Kargin 2011 the abstract. All bibliographic data with a
  DOI were checked with Crossref (eleven DOIs), and two entries also with zbMATH.
- **Not read.** The published version of Diaconis–Hanlon (Contemp. Math. 138), which may differ from the report
  (the publisher's site refused automated access in both runs); Liu 1996 (no open copy was found in the second
  run; used only as quoted in the survey and in the monograph); Lubotzky–Phillips–Sarnak 1988; Miclo 2002 beyond
  the abstract; Metropolis et al. 1953, Hastings 1970, Rellich 1937, Kato's and Horn–Johnson's books (not
  consulted for this note; from them only the min–max principle is used in a proof).
- **Caveats.** In the scan of the 1992 report the general eigenvalue formula of Section 4 shows a plus sign where
  the report's own example N = 3 and its formula for the second eigenvalue require a minus sign; the note uses
  only the second eigenvalue, which the exact lumped chain confirms for N ≤ 7. That the second eigenvalue of the
  Metropolis chain on S_N (not lumped) equals that of the lumped chain is a computation for N ≤ 7, not a quoted
  theorem. The existence of expander families of Cayley graphs is quoted, not checked. What the proposer intended
  is not known. A search that finds nothing is not a proof of novelty.
- **Scope.** Negative answers to the two literally posed items; the open-ended parts of the question are not
  answered. No priority is claimed.

## Public release and license
This paper, its source files, and this verification report are licensed under the Creative Commons Attribution
4.0 International License (CC BY 4.0): https://creativecommons.org/licenses/by/4.0/