Extinction sets of the wheel graphs in a graph-based model of HIV infection
Abstract
Mukwembi's graph-based model of HIV infection is a deterministic three-state cellular automaton on a graph G, driven by one integer parameter R, the rate at which depleted cells are replenished. Its extinction set E(G) is the set of those R for which the infection dies out from every admissible initial state. Espinosa-García, Figueroa, Fresán-Figueroa, Maldonado and Sánchez-Solís computed E(Wₙ) exhaustively for the wheels Wₙ=K₁vee Cₙ₋₁ with 4 ≤ n ≤ 10, sampled it for 11 ≤ n ≤ 26, and conjectured that E(Wₙ)={3} ∪ {R:R ≥ n-1} for every even n ≥ 12 and E(Wₙ)={4} ∪ {R:R ≥ n-1} for every odd n ≥ 17. We determine E(Wₙ) exhaustively for 4 ≤ n ≤ 31 and for every positive R, and refute the conjecture in both of its halves: the least counterexamples are n=18, where 4 ∈ E(W₁₈), and n=23, where 5 ∈ E(W₂₃), the conjectured description is exactly right at every smaller n that it covers, and from n=27 on it fails at every n we can reach. Seven of the sixteen sampled rows of the published table are incomplete, although the two extinction parameters read off that table are unaffected. On the whole verified range the data obey one linear law: for 4 ≤ R ≤ n-2 one has R ∈ E(Wₙ) if and only if n ≥ 5R-3. Five thresholds sit on that line, namely R=3,…,7 at n=12,17,22,27,32, and the last two were predicted before they were computed. Where the slope 5 comes from we do not know, and we pose that as the question this leaves open. Every theorem below is machine-checked in Lean 4. What carries the search past n=26 is an orbit quotient that is itself part of the formal development: the dihedral group Dₙ₋₁ acts on Wₙ by automorphisms, the transition map is equivariant for it, and a descent argument — using neither a canonical-form theory nor the order of the group — restricts each sweep to the orbit minima, worth a measured factor 32 to 38 here.
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Archived files
- Version 2 · current (opens in a new tab)
Source snapshot 2026-08-30 15:34 UTC
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0aa9e0deb99150baa74455a3477c02bb606fa53a69af631a6a93f8dabda1fc5e
Claim ledger
Stated results
HW1candidate2026-08-28
Conjecture 1 of arXiv:2608.00340, even half, is FALSE: 4 is in E(W₁8) while the conjecture says E(Wₙ) = 3 u R >= n-1 for even n >= 12; and n = 18 is the least even counterexample
HW2candidate2026-08-28
Conjecture 1, odd half, is FALSE: 5 is in E(W₂3) while the conjecture says E(Wₙ) = 4 u R >= n-1 for odd n >= 17; and n = 23 is the least odd counterexample
HW3known data2026-08-28
E(Wₙ) determined exhaustively for 11 <= n <= 16, for EVERY positive replacement parameter; the values confirm the source's sampled Table 3
HW4correction2026-08-28
E(Wₙ) determined exhaustively for 17 <= n <= 21: Table 3 is confirmed at n = 17, 19, 21 and is INCOMPLETE at n = 18 and n = 20, each missing R = 4
HW5correction2026-08-28
E(Wₙ) determined exhaustively for 22 <= n <= 26: all five rows of Table 3 are INCOMPLETE, missing R = 4 and/or R = 5
HW6candidate2026-08-28
The corrected description: E(Wₙ) = 3: n even, n >= 12 u R: 4 <= R <= floor((n+3)/5) u R: R >= n-1, verified exhaustively for 11 <= n <= 26; the source's hiv and HIV formulas survive on that range
HW7known2026-08-28
The source's own exhaustive range reproduced: E(Wₙ) = R: R >= n-1 for 4 <= n <= 10, so hiv(Wₙ) = HIV(Wₙ) = n-1 there
HW8routine2026-08-28
Negative controls: E(W n) is not an up-set (3 in E(W₁8), 5 not in E(W₁8)); it is strictly larger than its tail from n = 12 on; and some admissible state of W₁8 is latent at R = 5, so the quantifier inside InE is not vacuous
HW9routine2026-08-28
The two certificate formats and the R >= n reduction: a bounded sweep proves membership, a forward-invariant orbit prefix avoiding the all-healthy state proves non-membership, and on a graph of order n the transition map is independent of R once R >= n
HW10candidate2026-08-29
𝓔(W n) determined exhaustively for 27 ≤ n ≤ 29, for every positive replacement parameter: 4,5,6 ∪ R ≥ 26, 3,4,5,6 ∪ R ≥ 27, 4,5,6 ∪ R ≥ 28
HW11candidate2026-08-29
𝓔(W n) determined exhaustively for 30 ≤ n ≤ 31: 3,4,5,6 ∪ R ≥ 29 and 4,5,6 ∪ R ≥ 30
HW12routine2026-08-29
The dihedral orbit quotient, proved in Lean: step is equivariant for the Dₙ₋₁ action on UInt64 masks, and a descent on the mask value reduces the 2 ^ n sweep to the dihedral-canonical masks – with no canonical-form completeness lemma and no use of the group order
HW13routine2026-08-29
Measured leverage of the quotient: the same proposition proved both ways at n = 24, 25, 26 costs 105/250/594 s by the full 2 ^ n sweep and 3.26/6.63/15.7 s by the quotient – 32x to 38x against a theoretical 2(n-1) of 46/48/50
HW14known data2026-08-29
The 64-bit quotient stack reproduces 𝓔(W n) for 4 ≤ n ≤ 20 value for value: the source's own exhaustive range 4 ≤ n ≤ 10 and Table 3's experimental range 11 ≤ n ≤ 20, including the two corrections 4 ∈ 𝓔(W 18) and 4 ∈ 𝓔(W 20) that Table 3 misses
HW15candidate2026-08-29
The corrected description R ∈ 𝓔(W n) ⟺ n ≥ 5R - 3 (for 4 ≤ R ≤ n-2, i.e. below the tail R ≥ n-1) holds on 27 ≤ n ≤ 31; the R = 6 threshold is exactly n = 27 and the R = 7 threshold is exactly n = 32, both predicted before they were computed; Conjecture 1 of the source fails at every n in 27 ≤ n ≤ 31; and hiv(W n), HIV(W n) are determined in Lean on 27 ≤ n ≤ 30 (hivₑq, HIVₑq), the case n = 31 being the same arithmetic applied to the verified row extinctionSet₃1
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- S. Mukwembi's graph-based model of HIV infection (2008) is a deterministic three-state cellular automaton on a graph G. A state is a map f: V(G) → 0,1,2, read as *healthy / infected / dead*. An admissible initial state is a map f₀: V(G) → 0,1 — no cell starts dead — so a graph of order n has exactly 2ⁿ of them. Given a positive integer R, the *replacement parameter*, and writing d(v) for the number of infected neighbours of v, one time step is
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7