# Verification report — AMR-096-0023 (Aldous: a spatial model of city growth, Conjecture 1 (a), (b), (c))

Verification date: 2026-10-14.

**Verdict.** The note decides each part of the conjecture; every theorem, lemma and corollary it states is proved,
and its numerical and simulation statements are labelled as tests. Checked by two independent AI-assisted
verification runs (A and B) of the first written version and by a further independent run of the final text
(independent run 2); no mathematical error was found.
- **Part (a): proved** for every c0 > 0, α > 1, β > 0 (Theorem B): M(∞) < ∞ and Σ_t p_t < ∞ almost surely.
  For all parameters, M(t) = O(t^(2/(β+2))) almost surely (Theorem A).
- **Part (b): disproved** for every c0 > 0, α > 0 and 2α/√3 < β < 2α (Theorem C for α > 1, Theorem D for every
  α > 0, in particular for the range 0 < α ≤ 1 of the companion paper): P(N_(1)(t)/t → 1) < 1.
- **Part (c): disproved** for every c0 > 0, α > 1 and 2α/√3 < β < 2α (Theorem C): P(N_(≥2)(∞) = ∞) > 0.
- **Explicit counterexample to (b) and (c):** α = 2, β = 3, any c0 > 0, cities near (1/4, 1/2) and (3/4, 1/2),
  δ = 1/20, T = max(10^5, ⌈100/c0⌉), with all constants (Section 6, Table 2).
- **Open:** whether (b) holds for 0 < β ≤ 2α/√3, and whether (c) holds for α > 1 and 0 < β ≤ 2α/√3.

The note is unrefereed.

## Statement checked
- **Primary source.** D. J. Aldous, untitled rough notes dated April 25, 2007 (41 pages), `cities-notes.pdf`, linked
  as "technical notes" from https://www.stat.berkeley.edu/~aldous/Research/OP/cities.html . Fetched on 2026-10-14
  (295,226 bytes, sha256 `9cfbe9da2660fcea7e44d7ef013cb7a8a1604ff54a9e7734e71ef76959c38b89`).
  - The model: notes, p. 2, equations (1)–(3): unit square [0,1]², parameters 0 < c0 < ∞, 0 < α < ∞, β > 0;
    I(y,t) = c0 max_i N_i(t)^α |y − x_i|^(−β); sphere of influence = where city i attains the maximum; one city of
    population 1 at a uniform point at time 1; at time t+1 a uniform immigrant founds a city with probability
    1/(1 + I(U,t)) and otherwise joins the city whose sphere contains U.
  - Conjecture 1 (notes, p. 8, Section 2): (a) α > 1 ⇒ M(∞) < ∞ a.s.; (b) β < 2α ⇒ N_(1)(t)/t → 1 a.s.;
    (c) α > 1 and β < 2α ⇒ N_(≥2)(∞) < ∞ a.s.; M(∞) and N_(≥2)(∞) are monotone limits.
  - Theorem 2 of the notes (p. 8): (a) α > 1 ⇒ P(N_(≥2)(∞) = 0) > 0; (b) β < 2α ⇒ P(N_(1)(t) > t^(1−ε)) → 1;
    (c) α < 1, β < 2α ⇒ P(N_(1)(t)/t → 1) > 0, with the rest of its proof marked as only sketched. Remark (ii)
    (criterion for M(∞) < ∞; the question whether convergence in probability can be improved to a.s.).
  - Lemma 3(b) (p. 9): a small city next to a large one has sphere area at most 8(N_B/N_A)^(2α/β).
  - Conjecture 32 (p. 40): the associated dynamic system.
  - The display of the founding probability in Section 1.3 of the notes carries an extra indicator
    1(y ∈ S(i,t)); the formula used in the note (∫ 1/(1+I)) is the one implied by the dynamics on p. 2.
- **Companion paper.** D. J. Aldous, B. Huang, "A spatial model of city growth and formation", arXiv:1209.5120v1
  (23 September 2012, 11 pages), read in full. Same model with 0 < α ≤ 1 (Section 2); heuristics, simulations and
  related models; no theorems. No journal version was found.
- **Corpus record.** ulamai/UnsolvedMath, AMR-096-0023 (dataset version 1.6.0; title "Unbalanced regimes of the
  spatial city-growth model"). Its statement is Conjecture 1 of the notes, complete and in the same form. Its
  upstream research summary describes the companion paper differently (year and content); the note relies on the
  documents themselves: only the 2012 preprint was found, it contains no theorems, and the partial results are
  those of the notes. The note does not use that summary.

## Readings
| Reading | Answer | Where |
|---|---|---|
| The notes' range 0 < α < ∞ (the record's reading) | (a) true; (b) false for 2α/√3 < β < 2α; (c) false for α > 1, 2α/√3 < β < 2α; open for β ≤ 2α/√3 | Theorems B, C, D; Corollary 1.1 |
| The companion paper's range 0 < α ≤ 1 | (a), (c) vacuous; (b) false for 2α/√3 < β < 2α | Theorem D |
| Torus or plane instead of the unit square | not the model of either source; not treated (the slope formula uses the square) | Section 4 |
| Proportional instead of maximal joining rule | not the model (the notes mention it only as an unstudied variant); not treated | — |
| "One or a few large cities" (the heuristic of the notes) instead of N_(1)(t)/t → 1 | the heuristic is consistent with the note; Conjecture 1(b) as written is refuted | Section 1.3 |
| The deterministic system of Conjecture 32 | a different conjecture; its part (a) fails for 2α/√3 < β < 2α (proved), for 0 < 2α/β < 1 only a sketch | Section 8 |

## Results in the paper
- **Lemma 2.2** (Ville's inequality for indicator sums): P(∃ t ≥ t0: Σ Y_s ≥ (e−1) Σ b_s + x | G_t0) ≤ e^(−x).
- **Lemma 2.3** (packing): at most (16/π)(λ/c0)^(2/β) cities are founded where the influence is below λ.
- **Lemma 2.4**: p_t ≤ κ N_(1)(t)^(−α), κ = 2^(β/2)/c0, and N_(1)(t) ≥ t/M(t).
- **Theorem A**: M(t) ≤ 1 + log2 t + (16/π + 2(e−1)/c0) t^(2/(β+2)) for all large t, a.s., for all parameters.
- **Theorem B** (part (a)): for α > 1, Σ p_t < ∞ and M(∞) < ∞ a.s., by lowering the exponent of Theorem A in
  finitely many steps (g ↦ αg).
- **Lemma 4.1**: for two cities at (1/2 ∓ a, 1/2): symmetry; the complement of city 1's region is a disc of
  Apollonius; under condition (V), f(1/2+h) = 1/2 + ∫ψ (formula (4)) and f(1/2+h) − 1/2 ≤ h q_a(h) (k ≥ 1);
  the slope f'(1/2) = k(a + 1/(12a)). Remark 4.2: a stable spacing exists iff k < √3.
- **Lemma 4.3, Corollary 4.4**: nearby positions; strip estimates with a margin η uniform on an open set G.
- **Theorem C**: α > 1, 2α/√3 < β < 2α: with positive probability, from a time T on, exactly two cities, with
  shares within δ of 1/2 (the event E_T, the founding bound (6), the Doob bound (8), the no-exit argument).
- **Section 6**: the explicit instance α = 2, β = 3, a = 1/4, δ = 1/20, r = 1/100, ρ = 10^(−5),
  T = max(10^5, ⌈100/c0⌉), with all constants (Table 2).
- **Theorem D**: every α > 0, 2α/√3 < β < 2α: with positive probability two cities keep shares within δ of 1/2 and
  at least 3/4 of the population, and all other cities stay below j0 inhabitants. The choice of T uses the T-free
  majorant B̄(t) of B_T(t), so it is not circular.
- **Section 8**: consistency with the notes; Conjecture 32(a) of the notes fails for 1 ≤ 2α/β < √3 (and the
  two clauses of Conjecture 32 are incompatible for α > 1, β > 2α); the open range.

## Computations (sanity checks; programs and outputs in reproducibility/)
No proof depends on a computation. The inequalities of Table 2 are proved in Section 6 with rational bounds.
- **Written with the note** (`writing_stage/check_note.py`, standard library, exact rational arithmetic, < 1 s;
  41 checks, TOTAL failures: 0): every entry of Table 2 and every inequality of Section 6 as printed, including
  (T1)–(T3) for all c0 > 0 at once and a proof of π < 355/113.
- **First version of the results** (`original/`): interval-arithmetic certificate of the instance for c0 = 1;
  formula (4) against a 6000×6000 grid (agreement to about 10^−5); simulations: two cities of 500 at
  (1/4,1/2), (3/4,1/2), α = 2: β = 3 shares within 0.013 of 1/2 at t ≈ 1.2·10^5 and within 0.006 at 10^7
  (4 of 4 runs), β = 2.4 shares 0.50–0.53, β = 2.2 and 2 slow drift up to 0.62, β = 1 monopoly (≥ 0.9996);
  basins (start at 0.54–0.64 returns towards 1/2, start at 0.72–0.78 goes towards 1); α = 1 and 0.8 with k = 4/3:
  shares within 0.003 of 1/2 at 2·10^6, 1 to 40 extra cities of at most 2 inhabitants; fresh starts α = 2, β = 3:
  second city above 20% at 10^6 in 0 of 40 runs (c0 = 1) and 7 of 40 runs (c0 = 0.01 and 0.001 each);
  α ∈ {1.2, 2}: M(10^6) ≤ 4; exploratory grid programs (zeros near 0.319, 0.5, 0.680; scan of 2463 two-city
  configurations per k: robustly stable interior zeros for k = 1.5 only).
- **Verification run A** (`verification_run_A/`): exact rational certificate (15 checks; η ≥ 0.0032439; its search
  for an admissible T stops at 51,229 for c0 = 1 and at 73,856 for c0 = 1/1000, while the smallest admissible T,
  with the tail bound 1/(T−1), is 51,223 and 73,828, by independent run 2); own area code (Lemma 4.1(ii) at 12 values of h,
  zeros 0.319066 / 0.5 / 0.680934, strip margins ±0.005498 on 40 positions in G, slopes for k = 1.7 and 1.8); own
  simulator (two cities of 500: deviation from 1/2 after 10^5 at most 0.0088 in 3 runs to 3·10^6 and at most 0.0145
  in 20 runs to 10^6; from (50,000, 50,000) at T = 10^5: at most 0.0018 in 10 runs; no founding; monopoly for
  β = 1; Theorem D's picture for α = 0.8; coexistence in 11 of 30 fresh runs with c0 = 0.001 at 2·10^5).
- **Verification run B** (`verification_run_B/`): independent exact share map (slope to within 3·10^−15 for six
  (k, a); Lemma 4.1(i) to 10^−47; Lemma 4.1(ii) at 299 + 975 points); exact rational certificate for c0 = 1,
  1/100, 1/1000 with five negative controls; strip margins about 0.0055 (fails for ρ = 10^−2); C simulator tested
  against the exact transition probabilities (|z| ≤ 2.14); campaign to 10^7 in agreement; fresh starts
  (100 runs each): second city above 20% in 1, 21, 37 runs for c0 = 1, 0.01, 0.001; with more runs the frequency
  for c0 = 0.001 is 34–39%, so the 7 of 40 of the first version is a low draw; Lemma 2.3 on 12 simulated paths.
- **Independent run 2** (`independent_run_2/`, written for the final text): `r2_constants.py` (exact rationals
  with other enclosures: artanh series with a tail bound, π by the Bailey–Borwein–Plouffe series, √2 by bisection;
  28 checks, 0 failures; η ≥ 0.003244197; (T1)–(T3) for every c0 > 0 through c0T ≥ 100, T ≥ 10^5,
  c0(T−1) ≥ 99, and at 67 sample values of c0); `r2_sharemap.py` (share map by Green's theorem on the boundary of
  disc ∩ square, 40–60 digits: formula (4) to 10^−40, Lemma 4.1(i) to 10^−38, Lemma 4.1(ii) at 590 points for ten
  pairs (k, a), slope at six pairs including k < 1, exactly three zeros 0.31907 / 1/2 / 0.68093, Figure 1, strip
  margin 0.0055 at x^0 and on 29 position pairs in G); `r2_symbolic.py` (44 algebraic steps with sympy).
- **Re-runs.** On 2026-10-14, when the note was written, every program of the package was run again from a copy
  of the package; every recorded output was reproduced byte for byte. Independent run 2 repeated this from an
  extracted copy of the archive `source.zip` (four lanes, wall time 6 min 15 s): 33 of 33 recorded outputs
  identical (`independent_run_2/outputs/rerun_comparison.out`). The recorded file
  `original/outputs/f_general_simE.out` lacks the `z=` lines of the program's output (it had been filtered); the
  complete output was regenerated and is included as `f_general_simE_complete.out`. Details and command lines:
  `reproducibility/README.md`.

## Independent verification runs
The results were first obtained, with proofs and programs, in a first written version. Two independent
verification runs, both AI-assisted, then examined that version (2026-10-14), each with its own programs and
without access to the other's report.

| Item | Run A | Run B |
|---|---|---|
| Statement, model and quantifiers against the notes and the companion paper | CONFIRMED | CONFIRMED |
| Lemmas 2.2–2.4, Theorem A | CONFIRMED (re-derived) | CONFIRMED (re-derived; packing tested on simulated paths) |
| Theorem B (part (a)) | CONFIRMED (re-derived; three attempts to break it failed) | CONFIRMED |
| Lemma 4.1 (symmetric configuration, condition (V), slope) | CONFIRMED (re-derived; own area code) | CONFIRMED (re-derived; independent exact method) |
| Lemma 4.3, Corollary 4.4 | CONFIRMED | CONFIRMED (margins tested) |
| Theorem C (parts (b), (c)) | CONFIRMED | CONFIRMED |
| Explicit instance | CONFIRMED (exact rational arithmetic) | CONFIRMED (exact rational arithmetic, negative controls) |
| Theorem D | CONFIRMED (optional wording fix: T-free majorant) | CONFIRMED (same wording fix required) |
| Conjecture 32 side remark | CONFIRMED for 1 ≤ k < √3; sketch for 0 < k < 1 | noted as a consequence of Corollary 4.4 for 1 < k < √3; for 0 < k < 1 not checked beyond numerics |
| Literature / novelty | no prior answer found | no prior answer found |
| Overall | CONFIRMED_WITH_FIXES | CONFIRMED_WITH_FIXES |

Neither run found a mathematical gap.

**Corrections required by runs A and B**, all applied in the note:
1. (A, B) State the per-part outcome everywhere ("(a) proved; (b) and (c) disproved in the stated parameter
   range"), never "disproved" alone; give the range 2α/√3 < β < 2α with every counterexample statement: abstract,
   Corollary 1.1, Table 1, Theorems C and D, Section 8, "Scope and priority", and the deposit metadata.
2. (A) The comparison with the heuristic exponent θ = (1−α)/(1−α+β/2) is stated for 0 < α < 1 only (it fails
   for α > 1 + β/2): Remark 3.1(iii).
3. (A) The case 0 < 2α/β < 1 of the remark on Conjecture 32 is marked as a sketch, and the observation that the
   dynamic system depends only on α/β is added: Section 8.
4. (A) The upstream summary of the corpus record ("Aldous–Huang 2014", "the paper proves several results") is
   not reused: Section 1.4 and "Scope and priority" describe the companion paper as a 2012 preprint with heuristics
   and simulations, the partial results being those of the notes.
5. (A optional, B required) The dependence of B(t) on T in the proof of Theorem D is resolved by the T-free
   majorant B̄(t), with a sentence saying that the choice of T is therefore not circular: Section 7.
6. (A optional) The range of simulated shares for β = 2.4 is 0.50–0.53: Section 9; the extra indicator in the
   notes' founding formula is mentioned after Lemma 2.1.
7. (B required) `f_general_simE.out` regenerated (complete output added as `f_general_simE_complete.out`) and
   the filter documented: Section 9, `reproducibility/README.md`.
8. (B optional) Corollary 4.4 says that only q_a(δ+ε) < 1 and η > 0 are used; the fresh-start frequency
   "7 of 40" for c0 = 0.001 is reported as a low draw, with the larger samples of run B: Sections 4 and 9.

**Changes made when the note was written, after runs A and B.**
- The explicit instance is given for every c0 > 0 with T = max(10^5, ⌈100/c0⌉) (the runs checked c0 = 1 with
  T = 10^5, and run B also c0 = 1/100, 1/1000 with T = 10^6). Checked by `check_note.py` in exact arithmetic.
- The proofs were rearranged and renumbered (see the numbering table in `reproducibility/README.md`); the index of
  the dyadic argument in the proof of Theorem A is called m (k is reserved for 2α/β).

**Second round: independent run 2 (final text).** One further independent verification run, also AI-assisted,
examined the final text of the note on 2026-10-14. It read the notes (pages 1–15 and 37–41 in full, the statements
of the other pages), the companion paper and the web page again and compared every statement about them with the
sources; re-derived every lemma, theorem and corollary line by line; wrote three programs of its own
(`independent_run_2/`); re-ran the whole package from an extracted copy of the archive (identical outputs); and
repeated the literature search (arXiv API, OpenAlex, Crossref, zbMATH Open, one web search).

| Item | Run 2 |
|---|---|
| Statement, model, quantifiers, statements about the sources | CONFIRMED (one misplaced description of a quotation corrected) |
| Lemmas 2.1–2.4, Theorems A and B (part (a)) | CONFIRMED (re-derived; algebra checked with sympy) |
| Lemma 4.1, Remark 4.2, Lemma 4.3, Corollary 4.4 | CONFIRMED (re-derived; share map by Green's theorem) |
| Theorem C (parts (b), (c)) and the explicit instance | CONFIRMED (re-derived; constants in exact arithmetic for every c0 > 0) |
| Lemma 7.1, Theorem D | CONFIRMED (re-derived; measurability condition in the use of Lemma 2.2 corrected) |
| Section 8 (consistency, Conjecture 32, open range) | CONFIRMED |
| Literature / novelty | no prior answer found; novelty not certified |
| Overall | CONFIRMED_WITH_FIXES; no mathematical error |

**Corrections required by run 2**, all applied in the note:
1. The bound (8) and the corresponding step of Theorem D use Doob's maximal inequality for the nonnegative
   submartingale Λ_t² (Kolmogorov's inequality), now named and stated in the form used (the usual "L² maximal
   inequality" carries a factor 4): Sections 1.4, 5, 7 and "Scope and priority".
2. In the proof of Theorem D ("Too many foundings"), Lemma 2.2 is applied with Y_s = 0 for s < T, so that its
   hypotheses (Y_s measurable with respect to G_{s+1}, P(Y_s = 1 | G_s) ≤ b_s) hold for every s.
3. The first sentence of Section 7 no longer asserts that new cities keep being founded for α ≤ 1 (not proved);
   the bound for a small city to reach j0 inhabitants is credited as a discrete form of Lemma 4 of the notes, with
   the same condition on j0 (Sections 1.4, 7, "Scope and priority").
4. Section 1.2: the sentence "What we can actually prove is somewhat weaker" follows the conjecture in the notes
   and introduces their Theorem 2 (it was described as introducing the conjecture).
5. Section 8: Theorem 2(c) of the notes is marked as proved there only in part.
6. Section 2: Lemma 2.2 is used three times (Theorems A, B, D), not twice.
7. Section 6: "≤" instead of "<" in κ(T/2)^−2 = 4·2^{3/2}/(c0T²) ≤ 4·2^{3/2}c0/10^4 (equality when c0T = 100).
8. Section 1.1: N_(≥2)(t) = t − N_(1)(t) is defined (it is used in the proof of Theorem C).
9. Section 9 and this report: run A's search for an admissible T stops at 51,229 and 73,856; the smallest values are
   51,223 and 73,828 (the earlier text called the former "the smallest").
10. The run is recorded in the Verification paragraph, in Section 9, in `reproducibility/README.md` and here.

## Relation to the literature, novelty and scope
- **Searches (14 October 2026, when the results were first obtained, again by verification runs A and B, and once
  more by independent run 2).** arXiv
  API (title and authors of the companion paper; city growth; sphere of influence; weighted Voronoi tessellations
  with urns or reinforcement; spatial Chinese restaurant processes; all later arXiv papers of Aldous), Crossref (no
  journal version of the companion paper), zbMATH Open (no document on the model), OpenAlex (no work citing
  arXiv:1209.5120, checked again at the writing and by run 2), Semantic Scholar (empty list of citing works), seven
  web searches in all.
  - Found: the notes, the companion paper, two Berkeley undergraduate simulation reports linked from Aldous's pages
    (they do not address Conjecture 1), and unrelated items.
  - No answer to Conjecture 1 was found. A search that finds nothing is not a proof of novelty.
- **What was read.** The notes (pages 1–14 and 38–41 in full, pages 15–37 for their statements; run A read all
  41 pages; run 2 read pages 1–15 and 37–41 again); the companion paper in full (v1, the only arXiv version); the
  web page. Hill–Lane–Sudderth (1980), Pemantle (2007), Ville
  (1939) and Williams (1991) were not read for the note; they are cited for standard facts and for orientation,
  and no result of theirs is used beyond those facts. All DOIs were checked with Crossref.
- **Credits.** The model, the heuristics and the conjecture are Aldous's; the constant κ and the small-city bound
  are taken from the notes, and the treatment of small cities in Theorem D follows the notes' sketch of their
  Theorem 2(c), with the comparison of their Lemma 4. The tools (Ville's inequality, Doob's maximal inequality,
  optional stopping, Lévy's extension of the
  Borel–Cantelli lemma, circles of Apollonius, trapping of an urn near a stable zero of its drift) are standard.
- **Scope.** New, to our knowledge and relative to the literature found: Theorems A to D, that is, the proof of
  (a) and the counterexamples to (b) and (c) for 2α/√3 < β < 2α. No priority is claimed. Open: (b) for
  β ≤ 2α/√3 and (c) for α > 1, β ≤ 2α/√3.

## 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/