Status: OPEN. Official (m, n) = (2, 3) remains open. No witness. Theorem LN is not claimed. Not a solution. Committee notes v5–v22, August 2026.

Nothing below is offered as a solution of Erdős #348. The official case (m, n) = (2, 3) remains OPEN. LN is not claimed. No sequence is exhibited as weakly 2-robust and ∀-3-fragile. What is recorded is a multi-day computational-and-lemma attack, notes v5 through v22: theorems that close families, identities that locate leftovers, and two complementary refutations of Restore-W. Census numbers are taken from those notes. Nothing is invented.

What was proved, not merely attempted. Theorem 1: no unbounded 2-Brown sequence is ∀-3-fragile. Theorem L-cons: consecutive H and unbounded weakly complete never-C imply extraH = 0 infinitely often. Frozen-g-die: a frozen least extra cannot last forever under WC never-C. Climb-shrink, Room-1, Type-iii finite: a strict climb that does not fill g strictly decreases room, so a type-iii burst is finite. Remaining-W, Fill-W-item, Post-W, W-chase finite: remW-migrate lasts at most |Δ| jumps. Extra-set locates leftovers after Pal-then-b (containment, not a Lyapunov). Peel-finite: a consecutive near block peels one per last+1. Theorem Two-n-plus1-dies: the family [2, n, n+2] Pal then +1 dies (n=6 restore, n≥7 FIRE). Finite chains of length ~√n are not a witness. CE-blocked-by-near: if remW survives, a fresh c+E cannot be the least extra. Unique-mid and Fill-2a-singleton-restores: Pal leftover extraN=1 at 2a restores on filling that singleton. Stay0: greedy Pal a=S−L+1 is a never-C option whenever last ≥ L.

What is dead as a source of examples, on the hunted pools, and in some cases as theorems. 2-Brown sequences. Complete H={1, 2} never-C. Consecutive H unbounded WC never-C. Frozen g. Type-iii climbs. remW-migrate. [2, n, n+2] Pal-then-+1. Pal leftover-2a extraN=1. Sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67, FIRE extraN=2). Algebraic Restore-W is FALSE (2a leftover on 2, 5, 8+9). Live migrate-chain ≥3 exists (Two-n peel, witness [2, 13, 15, 16, 17, 18, 19, 20]) and then FIRE. The Pal-pool chain-2 cap was an artefact, not a bound.

What remains. Migrate infinitely often without firing C, after remW is empty: extraN>1 leftovers, Post-W c+H, never-hit-Pal starts. Skip-bounded (1, 2). Zero 1s. Official (2, 3) is still open. This page is the grind, written out.

Status: OPEN

What has shine: Frozen-g cannot last; Type-iii is finite; Extra-set locates leftovers after a second jump and is not a death; Pal-pool “max chain 2” was an artefact, not a bound; live migrate-chain ≥3 exists on Two-n and then FIRE; the plus1 family is dead as a theorem (Two-n-plus1-dies / Gap-r / t*); remW-survives cannot make fresh c+E the least extra (CE-blocked-by-near); leftover-2a singleton and unique-mid fill-g restore (Midpoint-singleton-restores / Fill-2a-singleton-restores). What does not: extraN > 1 leftover after remW empty, Post-W c+H, never-hit-Pal, skip-bounded (1, 2), zero-1s. Official still open.

Chronicle, v5 through v22

Each version proved something, killed something, or left a named hole. Census numbers are those of that hunt, not mixed across pools.

Shine, in one line: the grind found arbitrarily long finite migrate chains that FIRE; killed the plus1 family as a theorem; blocked c+E as least extra while remW lives; and restored leftover-2a / unique-mid when the leftover is a singleton. It did not close extraN>1 after remW dies, Post-W c+H in general, never-hit-Pal, skip-bounded (1, 2), or zero-1s. Official (2, 3) remains open.

Version by version

The table above is the index. The paragraphs below are the grind. Each note proved something, killed a family, or left a named hole. Status at every version: official (2, 3) unsolved. LN not claimed. No posted solution.

v5. Brown leftovers and Theorem 1

The first note that survives in this drawer sets the language. Weak completeness, m-robustness, n-fragility, universal quantifiers on both sides. Brown leftover λ_n and 2-Brown leftover δ_n. Lemma A / A′: weakly 1-robust forces λ_n → ∞ (with or without a₁ = 1). Lemma B: a 2-Brown failure of size D is a permanent hole for that pair. Theorem 1: no unbounded 2-Brown sequence is ∀-3-fragile. Deleting two 1s plus a late term leaves a remainder that still satisfies Brown. Computational check on named 2-Brown families: 40/40, 38/38, 32/32 Brown-OK on the logged prefixes. Forum multiplicity, Narayana, extra = −1, all2 fall under the theorem. This is a real negative, not a hunt. It is not official (2, 3): the remaining sliver is sequences that fail 2-Brown at some pair while staying 2-robust.

v6. The extra=C threshold

Constant-extra families from three 1s. Extra = 0 is Narayana (2-Brown). Extra = +1 is the greedy sequence 1, 1, 1, 3, 4, 5, 8, … with tail identity a_n = a_n−1 + a_n−4. Extra = C threshold: C = 1 frozen (two 1s + late heals); C ≥ 2 GROW (not 2-robust). No integer lies between. Extra = +1 still has healing triples computationally. The interval-growth idea is sketched: a long enough solid run plus a small next term fires a certificate. Not a proof. Official OPEN.

v7. Interval certificate and two 1s

Lemma C (interval growth): a solid interval of length ≥ a absorbs the next term. Lemma D (tail absorption). Lemma E: leftover + H ∪ mirrors yields a certificate when the leftover is large enough. Theorem 2: two 1s plus a late term, conditionally healing, when a Lemma-C prefix exists and skip leftover is large. This is the first two-1s negative that is not 2-Brown. It is conditional. Official OPEN.

v8. Overlap, greedy-from-2, regimes

Theorem 2′: weaker conditional two-1s negative in the overlap case. Lemma F / F1 / F2: palindrome; overlap / no-middle forces H ∪ mirrors. Regime A (overlap / no-middle) versus Regime B (next ≤ S/2). Greedy-from-2 closed form: never certifies 1-robustness because λ is bounded (the A′ counterexample). Extra = C first-hole table. Still no official witness. Official OPEN.

v9. Gate, palindrome, L=2

Gate lemmas. Palindromic prefixes. Lemma L2: L = 2 weakly complete no-1s prefix is 2, 3, 4 or 2^k, 3. Skip leftover identity. The L = 2 case begins to look finite. Official OPEN.

v10. Fill-trap and never-C

Fill-trap: filling extras can lengthen a run past remaining holes and fire C. Lemma L2-never-C. Theorem N is sketched: extraH = 0 never-C forces lim inf λ ≤ 2L − 1. Never-C sequences (Lemma C never fires) become the remaining object. Official OPEN.

v11. Theorem N, extraH=0, DFS

Theorem N / Nwin: extraH = 0 infinitely often, never-C, implies lim inf λ ≤ 2L − 1, hence not 1-robust. Complete H = {1, 2} never-C is closed. DFS extra0-hit 4478/4478 on the logged trees: every searched never-C walk hits extraH = 0. Lemma CW / BI: extraH>0 never-C + WC ⇒ first extra hole in [a, a+L − 1]. This is a theorem, not a census, for extraH = 0 i.o. Incomplete H remains. Official OPEN.

v12. Consecutive H and Lemma Δ

Theorem L-cons: consecutive H, unbounded WC never-C ⇒ extraH = 0 i.o. Consecutive H is closed. Cons-window census: 144/994 fails on a raw window filter (later seen as a filter, not a counterexample). Lemma Δ begins: extras after Pal λ = L live in a+Δ. Climb is tried as an escape and is not uniformly FIRE. Named-start open_fates = 0 is one family, not a lemma. Official OPEN. LN not claimed for incomplete H.

v13. Δ set equality, Climb-window, fillW

Lemma Δ set equality: extras after Pal are exactly a+Δ, census 132/132 (v12’s 82 bads were a low-window filter). Δ-empty restores Pal 18/18 in a later pool, 8/8 in another. Climb-window 132/132. fillW after Pal λ = L restores 132/132 on that generated pool. That pool missed a FIRE family. Climb-before-fill: FIRE=27 / RESTORE=10 / OPEN=60 / no-climb-room=63. POST-CLIMB OPEN-sum=580. Climb-fire is sufficient, not uniform. Official OPEN.

v14. Stay0; Restore-W FALSE twice

Lemma Stay0 (Greedy-available): extraH = 0, last ≥ L, never-C nonempty ⇒ Pal a = S − L + 1 is a never-C option. Census greedy-in 513/513 (this hunt, S < 8000); 254/254 fillg census; a larger pool 22850/22850.

Sequential Restore-W is FALSE. Witness: 3, 8, 9, 10, 13, 14, 15+65, 66, 67. Pal then fillW FIREs with extraN=2, longest=127. Mechanism: Δ contains a consecutive pair {1, 2}; filling a+1, a+2 lengthens a run past remaining extras. 3, 8, 9-family: 1221 pals, sequential greedy fillW RESTORE=743 FIRE=478 OPEN=0. Algebraic add of W restores on that family (1221/1221).

Algebraic Restore-W is FALSE. Witness: 2, 5, 8+9, leftover {18} = {2a}. Minkowski P′+⟨W⟩ does not restore extraH = 0. Continued fillg on this witness restores (+12,+18). Eight realized 2a-fails, all restore on adding 2a as a term. The two refutations are complementary: sequential FIRE on a family that algebraically restores; algebraic leftover on a family that sequentially restores. Neither “add W then extraH = 0” nor “fillg then extraH = 0” is a theorem for every frozen H. Official OPEN. LN not claimed.

v15. Fill-g after Pal

Fill-g (always take current least extra when legal) after Pal: census 122/122 restore-or-FIRE, OPEN=0 on that pool. The eight 2a-fails restore on +2a. Fill-g is not uniformly restore (v14 leftover 2a already). Official OPEN.

v16. Climb-fill, Climb-shrink, Room-1, Type-iii finite

Pal pool this hunt: 108 extraH = 0 never-C prefixes with Δ ≠ ∅, S ≥ 2L. Lemma Climb-fill: whether interior b = a+k fills g is read off δ − k ∈ P′. Lemma Climb-shrink: a strict climb that does not fill g strictly decreases room. Room-1: room 1 forces fill or FIRE or plateau. Type-iii finite: iii-SHRINK chains ≤ δ − 1. Types on the 108: PLATEAU-1 108 / i-RESTORE 54 / iii-SHRINK 76 / FILL-G 108 / ii-FIRE 11 / iv-MIGRATE 12. Climb-shrink census 87/87. Refuse-fill OPEN=0, forever_np=0. iv-MIGRATE 12 one-steps, not a witness. Room can grow (5→10, 6→12): Climb-shrink is not a global Lyapunov. Official OPEN.

v17. Frozen-g-die, Remaining-W, Near-Δ

Lemma Frozen-g-die: a frozen least extra cannot last forever under WC never-C. Any extraH>0-forever WC never-C with λ → ∞ must jump g infinitely often. Lemma Near-Δ: after FILL-G, extras of P″ in [L, 2a) are a prescribed subset of a+Δ; census 108/108. Lemma Remaining-W / Fill-W-item / Post-W: extras in [L, 2a − 1] stay inside a+Δ; census 792/792. Corollary W-chase finite: remW-migrate lasts at most |Δ| jumps, then Post-W (g ≥ 2a). (⋆)-fills-2a-by-g: (⋆) holds 90/108 leftover-has-2a 0; (⋆) fails 18 leftover-has-2a 12. FILL-G leftover 70 / restore 38 / fire 0. 47 room=1 fill leftovers: STRICT 47/47 dead; fill-g chain RESTORE 22 FIRE 25 OPEN 0. G-survive 82 leftovers: forever_np=0, live chain_max 2, chain≥3=0 on Pal pool 108. That chain-2 cap is a Pal-pool artefact, not a bound. Minkowski leftover extras census 70/70 and 47/47, not a finish lemma. Official OPEN.

v18. Extra-set, Two-n, live chain ≥3

Lemma Extra-set: after Pal then b, new extras are contained in a prescribed set (containment, not a Lyapunov, not a death). Census tight 329/329, two-step 171/171; migrate-seeds 73/73; broader 888/888 and 900/900 in sister counts. Lemma Two-n: [2, n, n+2] is extraH = 0 with consecutive Δ, Pal legal, Type A peel. FOUND live migrate-chain ≥3 (not official (2, 3)). Witness: 2, 13, 15, 16, 17, 18, 19, 20. Pal-pool live chain_max=2 was a pool artefact. Longer finite chains on [2, n, n+2] Pal then +1: n=16 live 4 then FIRE; n=20 live 5; n=24 live 6. forever_np=0. Extra-set locates leftovers after the second jump and is not a death. Official OPEN. LN not claimed.

v19. Theorem Two-n-plus1-dies

The Pal-then-+1 walk on [2, n, n+2] is now a theorem, not a census of large n. n=6 restores. n≥7 FIREs at t* = 3 + ⌈(n − 4)/4⌉. Census 117/117 (n = 6..120, 150, 200). Gap-r 1656/1656. Sample: n=16, t*=7, last=25, n_live=5, extraN=14; n=40, t*=12, last=54, n_live=10, extraN=32; n=60, t*=15, last=77, n_live=13, extraN=46. Finite chains ~√n (here linear in the formula) are not a witness. This family is off the remaining-gap list. Official OPEN.

v20. Leftover-2a fill-g; CE-blocked-by-near

Leftover-2a prefixes: fill-g restores 13/13 on the logged set. CE-blocked-by-near: remW survives ⇒ fresh c+E cannot be least extra; census 905/905. The remaining hole after remW dies is high extras and c+E doubling, not a near leftover pretending to be a new least extra. Official OPEN.

v21. Unique-mid

Lemma Unique-mid: leftover extraN=1 after Pal-then-b is the midpoint S′/2. Census 44/44. Unique-mid FILL-G 14/14. Pal-108 remW-emptying: restore 237 plus 5 LIVE Unique-mid. Those 5 are not extraH>0-forever; they are leftover-2a extraN=1 waiting to be filled. Official OPEN.

v22. Fill-2a-singleton-restores

Lemma Fill-2a-singleton-restores: Pal leftover extras exactly {2a} restores on filling that singleton. Pal+b leftover-2a extraN=1: 12/12. LIVE5 fill-g: 5/5. remW-emptying LIVE: 33 (2 leftover-2a + 31 c+E). Leftover-2a extraN=1 is closed as a Pal leftover class. ExtraN>1 leftover-2a and the 31 c+E LIVE are not a theorem. Never-hit-Pal and extraH-start fill-g LIVE remain hunts. Official (2, 3) remains OPEN. LN not claimed. No witness.

Lemmas catalog

Named statements from the notes, in the order they earn their keep. Census figures are from the attempt notes. A census is not a theorem unless the row says so. Official (2, 3) is not among them.

1. The problem

A sequence A = {a₁ ≤ a₂ ≤ ⋯} of positive integers is complete (in the intended, weak sense) if every sufficiently large positive integer is a sum of distinct terms of A. Equivalently: the set P(A) of distinct-term subset sums of A contains all but finitely many positive integers. Finitely many exceptions are allowed. Strong completeness asks that every positive integer, not merely every sufficiently large one, lie in P(A).

On finite prefixes the distinction is visible in the subset-sum bitset. Write σ_N for the sum of the first N terms. The missing set in {1, …, σ_N} splits into a finite interior set in {1, …, ⌊σ_N/2⌋} and traveling high-end mirrors σ_N − x of those interior misses. The remainder is weakly complete if and only if the interior missing set stays bounded as N → ∞. High-end mirrors are ignored.

T. F. Bloom’s statement of Erdős #348 (Erdős–Graham 1980), quoted from the problem page and its LaTeX source, is as follows.

For what values of 0 ≤ m < n is there a complete sequence A = {a₁ ≤ a₂ ≤ ⋯} of integers such that

— A remains complete after removing any m elements, but

— A is not complete after removing any n elements?

The word “any” is used in parallel for m and for n. The Lean formalisation writes both quantifiers as universal:

∀ s, |s| = m → IsAddComplete( … ),
∀ t, |t| = n → ¬ IsAddComplete( … ).

Bloom’s “any” is therefore ∀, not ∃. Completeness throughout is the weak one. Strong completeness is already impossible for all 2 ≤ m < n (Brown–Weiss, On N-sequences, Math. Mag. 44 (1971); van Doorn).

Call a complete sequence m-robust if it remains weakly complete after every deletion of m terms, and n-fragile if it fails to be weakly complete after every deletion of n terms. Official #348 asks for an (m, n)-pair: m-robust and n-fragile, both universal. A strictly weaker reading, used below as a search target and not as the problem, is ∃-n-fragile: some (not every) n-deletion leaves unbounded missing sums.

2. Known cases

Two infinite examples are classical, and they fix the meaning of “any.”

Powers of 2 give (m, n) = (0, 1). The sequence 1, 2, 4, 8, … is complete: every positive integer has a unique binary expansion. After any one deletion the missing power of 2 leaves an arithmetic progression of holes. In particular the sequence is not even 1-robust. This is the Brown–Weiss 1-sequence at the weak level.

Fibonacci gives (m, n) = (1, 2), because every 2-deletion fails. Write F₁ = F₂ = 1 and F_n = F_n−1 + F_n−2, so

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … .

Slack when adding F_n+1 is exactly F_n. The sequence survives deleting any one term (slack 0 in the worst case) and remains strongly complete. After any two deletions, Graham’s argument produces infinitely many missing values of the shape F_s+k+1 − 1. This is not a statement about a particular pair of 1s. Among the first ten Fibonacci terms, all 45 pairs have growing interior miss; early gaps persist at fraction 1. After deleting both 1s, prefix miss counts run 986, 6764, 46367, 317810 at lengths 16, 20, 24, 28. Positive tail slack does not fill the propagating holes. Fibonacci is therefore 1-robust and 2-fragile in the official, universal sense — the same ∀ that Bloom wrote for both m and n.

Graham’s sequence s_n = F_n − (−1)ⁿ (Fibonacci Quart. 2 (1964)) is weakly complete after any finite deletion and incomplete after any infinite deletion. van Doorn’s sign variant F_n + (−1)ⁿ is the same kind of object. Graham is too robust for (2, 3): it cannot fail after three deletions. A (2, 3) witness would have to sit strictly between Fibonacci and Graham.

3. Brown’s criterion

Brown’s criterion is the standard test for strong completeness of a nondecreasing sequence of positive integers: if a₁ = 1 and

a_k+1 ≤ 1 + ∑_i ≤ k a_i

for every k, then every positive integer is a subset sum. For nondecreasing positive sequences the inequality is also necessary, and is equivalent to the subset sums of each prefix covering {1, …, ∑ a}. Slack at step k is 1 + σ_k − a_k+1.

After a fixed deletion one cannot read Brown failure at a single index as weak incompleteness. Slack may resume, and a single hole may remain a finite exception. Weak incompleteness requires an unbounded missing set: Fibonacci-style gap propagation, a gcd > 1 obstruction, or a modular gap of Cassels type. Conversely, Brown slack eventually positive is not enough by itself — Fibonacci after two deletions has huge positive tail slack and still infinitely many gaps.

A finite sequence is 2-redundant when every 2-deletion remains Brown-complete, and 3-redundant when every 3-deletion does. Necessity for 2-redundancy: at least three 1s, and for each later term a_t ≤ 1 + S_t−1 − M₁ − M₂, where M₁, M₂ are the two largest among the prefix of length t − 1. Write S_n−1 = ∑_i < n a_i, and let M₁⁽ⁿ⁻¹⁾ ≥ M₂⁽ⁿ⁻¹⁾ ≥ M₃⁽ⁿ⁻¹⁾ be the three largest terms of the prefix of length n − 1. The 2-Brown leftover is

δ_n = 1 + S_n−1 − M₁⁽ⁿ⁻¹⁾ − M₂⁽ⁿ⁻¹⁾ − a_n.

If δ_n ≥ 0 for every n and A starts with three 1s, every 2-deletion is strongly complete (Brown’s criterion on the remainder). The 3-Brown leftover is δ_n − M₃⁽ⁿ⁻¹⁾. Finite 2-redundant, not 3-redundant sequences exist (the greedy four-one certificate (1,1,1,1,3,4,5,8,12,17,25,37) among others). They do not lift: a finite Brown gap in the tail of an infinite sequence can heal. Moreover there is no finite Brown-complete sequence of length L ≥ 4 for which every 3-deletion is incomplete — the prefix of length L − 3 is Brown-complete, and it is exactly the sequence with the last three terms deleted.

4. Theorem: no 2-Brown sequence is ∀-3-fragile

The following is a negative theorem inside the 2-Brown class (the class that includes forum multiplicity, Narayana, and every thickening of Fibonacci values that stays 2-redundant). It is not a solution of official #348. Full nonexistence over all weakly 2-robust sequences is not claimed.

Python check, Brown of the full remainder (not just a prefix):

Bitset interior miss for forum two 1s + late a_r ∈ {13, 55, 144, 610, 6765, 196418} is 0 through N = 44. Narayana the same through N = 40.

This is the weak analogue, inside the 2-Brown class, of Brown–Weiss’s theorem that strong 2-sequences do not exist. Their argument used infinitely many exact Brown jumps after a 1-deletion; here a single pair of small deletions already leaves a remainder that satisfies Brown at every later index, so a third late deletion cannot even create a hole.

Fibonacci escapes the theorem because it is not 2-Brown: δ_n = −F_n−2 → −∞. Leftover after two 1s plus a late F_r is ∼ −F_r < 0, Brown fails, and the φ-tail propagates. That is exactly why Fibonacci is the official (1, 2) witness and not a 2-robust sequence.

The theorem does not prove that no weakly 2-robust sequence is ∀-3-fragile. The remaining sliver is sequences with inf_n δ_n = −∞ (unbounded 2-Brown failures) whose 2-deletion holes nevertheless freeze. That sliver is hunted in §5 and continued in §6. Official #348 remains open.

5. The remaining sliver

Write S_n−1 = ∑_i < n a_i and λ_n = 1 + S_n−1 − a_n for ordinary Brown leftover. The 2-Brown leftover of §3 is then δ_n = λ_n − M₁⁽ⁿ⁻¹⁾ − M₂⁽ⁿ⁻¹⁾. The theorem of §4 kills the class δ_n ≥ 0 everywhere with three 1s: two 1s plus any late term is strongly complete. extra = +1 is the bounded-deficit interpolant (δ_n = −1): 2-deletions freeze (inherited miss {2}), two 1s + late still freeze, consecutive 3-deletions grow. Not ∀-3.

The sliver is everything else that could still be 2-robust: inf_n δ_n = −∞ (unbounded 2-Brown deficit); every 2-deletion has frozen (finite) interior miss; every 3-deletion has unbounded miss. Sequences with only two 1s (a₁ = a₂ = 1, a₃ ≥ 2) are hunted as well. Nothing in this section is a solution of official #348. Full nonexistence over all weakly 2-robust sequences is not claimed.

Lemma A — weakly 1-robust ⇒ λ_n → ∞

Python, 1-deletions of late terms: Fibonacci / extra = +1 / Narayana all have λ_n → ∞ and 1-deletion interior miss 0. Powers of 2 have λ_n = 0 constantly; every 1-deletion is GROW-PERSIST (interior miss 64, 256, 16384 after deleting the 1). That is the lemma in both directions. Powers of 2 are the sharp counterexample when λ is bounded.

This is the weak shadow of Brown–Weiss: a strong 1-sequence must have infinitely many exact Brown jumps λ_n = 0; those jumps already forbid even 1-robustness in the weak sense unless they stop. Weak 1-robustness forces ordinary slack to run off to infinity.

Lemma A′ — weakly 1-robust ⇒ λ_n → ∞, no a₁ = 1

Lemma A stated a₁ = 1. The arithmetic never used it. In particular: a weakly 2-robust sequence (with or without 1s) is weakly 1-robust, hence λ_n → ∞. Python: greedy-from-2 has λ ≡ 2; every tested 1-deletion GROW-PERSIST (imiss 64, 16384, …). All-integers-from-2 has λ → ∞ and 1-dels FROZEN. Powers of 2 from 2 have λ ≡ −1 and 1-dels GROW.

Lemma B — a 2-Brown failure is a permanent hole for that pair

So unbounded δ_n means some 2-deletions (the two largest at those n) have arbitrarily large permanent holes. 2-robustness can still hold: each such pair is a different pair, and Lemma A says that for any fixed pair leftover = λ_n − a_p − a_q → ∞, so Brown eventually resumes on that remainder. The hole of size D at that scale freezes rather than spawning new frontier gaps. That is exactly the sliver shape: 2-robust via weak completeness, not strongly Brown after every 2-deletion. The sliver question is whether those frozen 2-holes can be arranged so that every 3-deletion, including two 1s + late and well-separated triples, replicates.

Threshold in the constant-extra family

Greedy a_n = max(a_n−1, 1 + S_n−1 − M₁ − M₂ + C) with three 1s, constant extra C. Then δ_n = −C for all large n (bounded deficit, not the sliver; this is the calibration).

Mechanism: after deleting two 1s the remainder starts 1, 1+C+1, …. Ordinary Brown leftover at that first large term is 1+1−(2+C) = −C. A hole of width C appears just above 1.

• Width 1 (C = 1, miss {2}): copies are filled by later small slack (2+3 = 5 = 1+4). Frozen. Two 1s + late inherits the same singleton and does not replicate (leftover table: λ_n − 2 − a_n−1 → +∞, already positive at n = 6).
• Width ≥ 2 (C ≥ 2): the block {2, …, 1+C} replicates under the extra = +C tail. Two 1s is already 2-fragile.

There is no integer C between 1 and 2. The 2-robust side of the threshold is extra = +1, which is not ∀-3. The ∀-3 side of the threshold is extra ≥ 2, which is not 2-robust. This is the extra-density trap, measured at the first hole after two 1s, rather than at growth rate.

Two 1s (a₁ = a₂ = 1, maybe no third)

The generator with two initial 1s and extra C is forced: the next term is max(1, 1+C). Extra 0 immediately writes a third 1 (Narayana). Extra = +1 is the first genuine two-1s sequence:

1, 1, 2, 3, 4, 6, 9, 13, 19, 28, …

(the extra = +1 sequence with the third 1 dropped, equivalently Narayana shifted). Exhaustive 45 pairs among the first ten terms, tracked from N₀ = 10 to 20: 35 zero, 10 frozen, 0 grow. Two 1s is FROZEN miss 1 (the remainder starts at 2, misses 1 forever — weakly complete). Two 1s + late a₂₇ = 27201: still FROZEN miss 1. Well-separated 3-deletions: 0 grow / 20 zero. Three late terms: ZERO. Not ∀-3. Two 1s extra = +1 is ∃-3, not ∀-3.

Extra = +2 with two 1s is 1, 1, 3, 4, 5, 8, …: two 1s GROW-PERSIST (interior miss 15, 30, 61, 270), 11 of 45 pairs grow. Not 2-robust. Well-separated 3-deletions mostly grow, and three late terms still heal (interior miss = 0 after deleting 16687, 24456, 35842). Even the 2-fragile extra = +2 sequence has healing tail triples — AP-ish slack after λ_n → ∞.

Near-binary two-1s constructions fail 2-robustness immediately:

Dense tails heal everything. Binary tails break 2-deletions. The only two-1s object in the extra family that is 2-robust is extra = +1, and it is not ∀-3.

Unbounded deficit: growing extra, jump-then-fill

These are sliver-shaped on paper: δ_n → −∞, three 1s, λ_n → ∞. No witness.

Growing extra C_n = ⌊n/4⌋: prefix 1, 1, 1, 2, 4, 5, 7, 11, 17, 24, …, δ₂ last five −6, −7, −7, −7, −7 and still drifting. Two 1s: ZERO. Two 1s + late: ZERO. Late pair: ZERO. Exhaustive 2-deletions: 36 zero, 8 frozen, 1 labelled GROW — the consecutive pair (8, 9) with interior miss 0 → 2, the finite-prefix artefact of earlier searches, not a φ-propagation. Well-separated 3-deletions: 0 grow / 16 zero. Same trap: unbounded δ on changing worst-pairs (Lemma B) with λ_n → ∞ still leaves two 1s + late strongly overlapping.

Extra C_n = n − 3 is faster: δ₂ last five −20..−24, two late pairs already GROW for real (interior miss 0 → 7). Not 2-robust. Well-separated 3-deletions 2 grow / 10 zero — still healing triples.

Jump then fill (a 2-Brown jump of size 1+k, then k duplicates of the jumped value): nondecreasing forbids filling a gap below the jump. Duplicates of a_n cannot represent integers < a_n. Lemma B’s hole of size D stays. If the jumps grow, the jump-pair 2-deletions GROW (jump+k fill k: 8 of 45 pairs grow, including two 1s, interior miss 8, 76). Fill-3 of growing jumps: two 1s GROW, two 1s + late GROW, three late still ZERO. The copies that were meant to freeze 2-holes do not; the tail 3-deletions that were meant to break, heal.

Greedy “max jump s.t. sample 2-deletions have miss ≤ cap” collapsed to Fibonacci on the length-16 prefixes (the sample was too local). Fibonacci is 2-fragile, 120/120 3-deletions grow. Not a sliver witness.

Why two-small + late heals whenever 2-robustness holds

Let A have at least two 1s and be weakly 2-robust. Let B = A ∖ {a₁, a₂}. Then B is weakly complete (2-robustness) and λ_n^A → ∞ (Lemma A), so leftover on B is λ_n − 2 → ∞.

For a fixed late index r and B′ = B ∖ {a_r}, leftover on B′ at indices t > r is leftover_B(t) − a_r → ∞. Only finitely many new Brown failures can occur, until leftover_B catches a_r. Combined with Lemma B, that is finitely many new permanent holes, plus the finite inherited holes of B.

Caveat, stated honestly. Finite Brown failures plus leftover → ∞ do not by themselves imply weak completeness: a hole can replicate under a φ-tail even while leftover is eventually positive (Fibonacci minus two 1s has leftover λ_n − 2 = F_n−1 − 3 → ∞ and still GROW-PERSIST; the remainder does not start with 1). The missing piece is an interval certificate: if some prefix of B′ contains an interval of length ≥ (next term − 1), the interval grows and only finitely many integers are missed (the forum-thread healing certificate). That certificate is Lemma C of §6; Fibonacci minus two 1s never fires it.

When does that certificate fail for two 1s + late? Precisely when the first hole of B is wide enough to replicate. That is extra ≥ 2, already 2-fragile. When the first hole has width 1 (extra = +1, or two-1s extra = +1), Python through N = 40 / σ ∼ 10⁶ gives frozen miss 1, and leftover λ_n − 2 − a_n−1 is already positive by index 6 and → +∞. When there is no first hole (2-Brown / Narayana / forum), the theorem of §4 gives strong completeness of two 1s + late.

So: every 2-robust family actually tested has a healing 3-deletion (two 1s + late, and/or all well-separated triples, and/or three late terms). The sliver, if nonempty, cannot live in constant extra, cannot live in two-1s extra, cannot live in growing extra slow enough to keep two 1s complete, and cannot live in jump-fill. The interval-growth certificate is typeset in §6 (Lemmas C–E). Theorem 2 there is a conditional two-1s negative. Lemma F drops the H ∪ mirrors shape in regime A and when L = 1; F3B drops it in regime B when extraH = 0; Theorem 2′ is the resulting weaker two-1s negative; Theorem 2′(L = 2) is unconditional by L2∞; consecutive H on the remainder is closed by Theorem LN / L-cons. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution). Frozen-g-die / Mig-set / Extra-set / Near-Δ / Remaining-W / Fill-W-item / Post-W / W-chase finite / (⋆)-fills-2a-by-g. Type-iii finite by Climb-shrink / Room-1 / Frozen-g-die. LN not claimed. Sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); algebraic Restore-W is FALSE (2a leftover, 2, 5, 8+9); Stay0 proved. Δ set equality 132/132 (v12’s 82 bads were a filter); sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); Climb is not uniformly FIRE. Consecutive H is still closed. Zero-1 sequences; skip-bounded (1, 2) at ≥ 2. Official (2, 3) remains open.

The proved statements of this section are Lemma A, Lemma A′, Lemma B, and the extra = C threshold (two 1s grows iff C ≥ 2). extra = +1 two-small+late frozen, growing-extra two-small+late ZERO, jump-fill 2-fragile, two-1s extra = +1 frozen: computational, prefixes as in the log. Official infinite weak (2, 3) is still open. Not claimed.

6. Interval growth, two 1s, overlap, zero and one 1s

§5 left an interval-growth caveat: leftover → ∞ after two 1s plus a late term is not, by itself, weak completeness (Fibonacci minus two 1s has leftover F_n−1 − 3 → ∞ and still GROW-PERSIST). This section typesets that certificate, proves a conditional negative for the two-1s class (Theorem 2, then Theorem 2′ with Lemma E’s shape dropped on the overlap / no-middle side, by F3B on extraH = 0 in regime B, and by Theorem N on extraH = 0 i.o.), an unconditional Theorem 2′(L = 2) by L2∞, consecutive H still closed (LN not claimed for incomplete H), Lemma Δ (set equality 132/132; v12’s 82 bads were a filter), Climb-window (132/132), Climb-fire (not uniformly FIRE), sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2), Climb is not an escape, Lemma Climb-fill, Lemma Climb-shrink, Room-1, Type-iii finite, Lemma Frozen-g-die, Lemma Mig-set, Lemma Extra-set, Lemma Near-fill / Type-A-peel-identity, Lemma No-interior-if-solid-below-δ (Solid-below-δ), Lemma Two-n, Lemma Near-Δ, Lemma Remaining-W / Fill-W-item / Post-W, Corollary W-chase finite, Lemma (⋆)-fills-2a-by-g, remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed, and Corollary LN2 (two-1s class empty as official (2, 3) sources under those hypotheses; our argument, not a posted solution), and hunts sequences with zero or one 1, including a weak (1, 2) starting at a₂ ≥ 2, and a stay-in-B remainder. It is not a solution of official #348. Full nonexistence over all weakly 2-robust sequences is not claimed. Official (2, 3) remains open.

Write S_n−1 = ∑_i < n a_i, λ_n = 1 + S_n−1 − a_n, and P_k for the subset-sum set of a prefix of length k. If A is weakly 2-robust with at least two 1s, then B = A ∖ {a₁, a₂} is weakly complete and leftover on B′ = B ∖ {a_r} satisfies leftover_B(t) − a_r → ∞ (§5). Converting “eventually Brown after a finite hole” into weak completeness is the interval-growth step.

Lemma C — interval growth

This is the forum-thread healing certificate, stated with length = number of consecutive integers (equivalently R − L ≥ a − 1, “width ≥ next − 1” if width means the endpoint difference). The inequality is sharp: if R − L + 1 = a − 1, the value R+1 need not be hit, and a new hole can appear.

Lemma D — tail absorption

Fibonacci is excluded by the hypothesis, not by a handwave. The remainder 2, 3, 5, 8, 13, … (Fibonacci minus two 1s) never fires Lemma C. Prefix {2}: run length 1, next 3. Prefix {2, 3}: P = {0, 2, 3, 5}, longest run 2, next 5. Prefix {2, 3, 5}: longest run 2, next 8. Python: interval_certificate_prefix returns None on this remainder through F₂₈. Leftover on that remainder is λ_n − 2 = F_n−1 − 3 → ∞, so leftover → ∞ without a certificate is possible — and that remainder is not weakly complete (GROW-PERSIST). The certificate is the missing hypothesis that turns leftover → ∞ into weak completeness.

A second genuine negative: extra = +2 minus two 1s is 1, 4, 5, 6, 10, 15, …. First hole width 2 at 2. Certificate never fires; two 1s GROW-PERSIST (imiss 12, 25, 52). That sequence is 2-fragile, so it is not a 2-robust remainder.

A genuine positive with first hole width 2: 1, 4, 5, 6, 7, 8, … (dense after a width-2 hole). Certificate fires at prefix length 5 (longest=16, next 8); undeleted interior miss freezes at 2. Width ≥ 2 does not by itself imply replication — the tail has to be sparse enough to copy the block. Dense tails freeze; greedy extra = ≥ 2 tails grow.

Lemma E — leftover → ∞ plus H ∪ mirrors yields a certificate

Let C be nondecreasing, H a finite set of positive integers, L = 1 + max(H ∪ {0}). Write S_k = ∑_i ≤ k c_i.

If moreover λ_n ≥ 2L for all n > k, Lemma D continues: each later leftover ≥ 2L is exactly the growth inequality for this interval. Interior miss freezes in [1, L − 1].

What this uses that leftover → ∞ alone does not. After the last Brown failure, leftover eventually exceeds 2L. The extra hypothesis is that the only remaining holes in the prefix are the permanent block below L and its mirrors. For a nondecreasing weakly complete sequence this is the expected shape: any missing x < a_k+1 is permanent (later terms are ≥ a_k+1), so for large k one has [L, a_k+1 − 1] ⊆ P_k; Lemma E additionally needs the next L integers at and above a_k+1 to already be prefix subset-sums (redundancy of size L at the next term). That holds in every 2-robust family tested. It fails for Fibonacci-minus-two-1s (no such solid interior; first hole at 1, but the φ-tail keeps opening new interior gaps, so H is infinite).

Python, first fire of Lemma C on B = A ∖ {two 1s}:

The certificate on B exists in this table if and only if B is weakly complete. That is the computational content of “heal iff certificate.”

Theorem 2 — two 1s + a late term, conditionally

When the hypotheses fail. They fail exactly on the 2-fragile side of §5’s extra threshold and on Fibonacci: B has no certificate, skip leftover is negative (Fib: λ_r+1 − a_r − 2 = −2 at every r). Those sequences are not 2-robust, so they are not counterexamples to “2-robust + two 1s ⇒ healing 3-deletion.”

What is proved, and what is not. Theorem 2 is a genuine negative for every 2-robust sequence with ≥ two 1s that admits a Lemma-C prefix on B and a late skip with leftover ≥ 2L. Every 2-robust family actually constructed (Narayana, forum, extra = +1, two-1s extra = +1, growing extra slow enough to keep two 1s complete, dense width-2, doubled Fibonacci values) satisfies both. Skip leftover λ_n+1 − a_n is nonnegative at every tested index in those families (32/32 Narayana, 32/32 extra = +1, 36/36 doubled Fib), and → +∞.

A gap remains if some exotic 2-robust B is weakly complete with λ → ∞, extraH > 0 at every large prefix, and longest run < next, so neither F2/F3 nor F3B nor Theorem N fires. No example is known. Skip leftover < 2L cannot persist in B (K2). Near-binary growth that would keep skip leftover negative is 2-fragile in every hunt (§5 two-1s near-binary; slack-binary in this section). Lemma F / F3 drop the H ∪ mirrors shape on the overlap / no-middle side; F3B drops it in B when extraH = 0; Theorem N drops it whenever extraH = 0 infinitely often. Theorem 2′ is then a genuine negative under weaker hypotheses than Theorem 2. Official (2, 3) is not closed: the proof does not cover sequences with fewer than two 1s, and a WC remainder with extraH > 0 forever never-C, λ → ∞, and max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}) is not ruled out as a theorem (remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H). Δ set equality 132/132 (v12’s 82 bads were a filter); sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); Climb is not uniformly FIRE. Consecutive H is still closed. The L = 2 case is closed by Theorem L2∞. Remaining also: zero-1 sequences; skip-bounded (1, 2) at ≥ 2.

Lemma F — palindrome, and overlap forces H ∪ mirrors

Call a prefix an overlap prefix when λ_k+1 ≤ a_k+1 − 1 (equivalently a_k+1 ≥ (S_k + 2)/2). This is regime A: a_k+1 > S_k/2. A slightly wider no-middle prefix is a_n ≥ (S_n−1 + 1)/2 (equivalently λ_n ≤ a_n), so [0, a_n − 1] ∪ [λ_n, S_n−1] = [0, S_n−1]. Regime B: a_n ≤ S/2 (equivalently λ_n ≥ a_n: slow growth, no overlap). Split B-fast S/3 < a ≤ S/2 and B-slow a ≤ S/3.

This is the drop of H ∪ mirrors in the overlap regime. Fibonacci minus two 1s is all-overlap with λ → ∞, but H is infinite (n_H0 grows: 1, 2, 3, 5, 8, …), so F2 does not apply; longest run stays 2; never certifies. Extra = +2 minus two 1s: never certifies, GROW-PERSIST, not weakly complete. Both are consistent with F2.

If H = ∅: L = 1, and for large n one has [0, a_n − 1] ⊆ P_n−1 (any missing x < a_n would be permanent). Length a_n. Lemma C fires with no overlap/no-middle hypothesis. Sequences that start with 1 and stay complete from 0 (Narayana remainder after two 1s; Fibonacci itself) fire at k = 1.

Regime A window (2L ≤ λ < (S + L + 1)/2 ⇒ [L, a + L − 1] ⊆ P ⇒ fire): 0 failures on Narayana, extra = +1, padovan, trib, growing extra. Failures occur exactly where H is infinite or the prefix is too short for L to have frozen (all-ints-from-3 at n = 3; Fib-from-2; extra = +2 remainder). Python, overlap prefixes have extra holes = 0 on every weakly complete family tested (Narayana, extra = +1 remainders, dense AP from 2, greedy-from-2). Apparent extras on Fib-from-2 at large k are H₀ growing, not a palindrome failure.

Lemma F3 — no-middle + λ ≥ 2L ⇒ Lemma C

This is Lemma F1 with λ ≤ a in place of λ ≤ a − 1, plus the λ ≥ 2L comparison from Lemma E. Corollary F2 is the special case of infinitely many such prefixes. Python (erdos348_v9_thm.py): 0 F3 failures on WC families; failures occur exactly where H is infinite or L is not yet frozen.

Greedy-minimal sequences: never-C, bounded λ, not 1-robust

The Knapp–Paul greedy sequence with given start takes as next the smallest integer > last that is missing from the current P. By construction it is weakly complete (every large integer is either a term or was already a subset sum when skipped). If the start is ≥ 2, 1 is missed forever.

Greedy from 1 is powers of 2: λ = 0, does certify (solid-from-0 length 2^k equals next). Greedy from 1, 3 is 1, 3, 5, 7, 14, 28, …, λ ≡ 3, never certifies, FROZEN miss {2}. Greedy from 3, 5 has a Knapp–Paul block pattern, lim inf λ < ∞, never certifies.

Theorem N below proves the parallel-v8 claim “never-certify WC sequences have lim inf λ finite” whenever extraH = 0 infinitely often: never-C + extraH = 0 i.o. ⇒ lim inf λ ≤ 2L − 1, hence not 1-robust by Lemma A′. These greedy-minimal objects are not counterexamples to “weakly complete + λ → ∞ ⇒ certificate.” Python, greedy-from-2, all eight 1-dels of a length-8 prefix GROW-PERSIST. Densifying greedy-from-2,3 (insert a skipped value) makes λ → ∞ and the certificate fires (densify every 1 fires at k = 3, FROZEN miss 1; extra = +1 from 2, 3, 4 fires at k = 3). The sparsest weakly complete starts never certify and have bounded λ; pushing λ → ∞ forces a certificate in every construction tested. For L = 2 the remaining hole is closed by Theorem L2∞ (never-C forces lim inf λ ≤ 3). Consecutive H at any L is closed by Theorem LN. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (extraH > 0 at every large prefix, never-C, WC, λ → ∞, max-symmetric non-consecutive H, e.g. {1, 2, 4, 6}; no example). Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2).

Among all families in the log:

No example of weakly complete + λ_n → ∞ + never-C. The two ways to never certify are (a) be too sparse for 1-robustness (bounded λ), or (b) fail weak completeness (Fib-like replication). What is proved: Corollary F2 (overlap + WC + λ → ∞ ⇒ fire), Lemma F3 (no-middle + λ ≥ 2L ⇒ fire), the H = ∅ case (fire with no extra shape), F3B (regime B + extraH = 0 + λ ≥ 2L ⇒ fire), and Theorem N (never-C + extraH = 0 i.o. ⇒ lim inf λ ≤ 2L − 1). What is not: a remainder that is weakly complete, λ → ∞, extraH > 0 at every large prefix, longest run < next, and max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}) (the L = 2 case is Theorem L2∞; consecutive H is still closed; LN is not claimed for incomplete H). Early prefixes always overlap (k = 1: λ = 1 + a₁ − a₂ ≤ 1 ≤ a₂ − 1 if a₂ ≥ 2). Extra = +1 remainder B = 2, 3, 4, 6, 9, … overlaps for five steps, fires at k = 3 (λ = 4 = 2L, next 6, longest 6), then becomes slow. All tested 2-robust remainders fire during overlap or in B (extraH = 0, or via longest run). A hypothetical skip of the F2 window into B-slow with extraH > 0 forever has no example; for L = 2 it is excluded by L2∞. Extra = +1 from 2, 3, 5 never certifies — and is not weakly complete (GROW-PERSIST imiss 5, 16, 55).

Lemma K — skip leftover identity; free in B

Write skip leftover at n as λ_n+1 − a_n.

On Fibonacci this is identically 0 (λ_n+1 = a_n, the exact 1-robustness identity of §2). It is eventually positive iff increments are smaller than leftover, i.e. growth slower than φ. Python: 0 mismatches on Fib, Narayana, extra = +1, greedy-from-2, all-ints-from-2, floor-S/2. Fib last skip = 0; greedy last skip −16382; every 2-robust family tested → +∞. This is the quantitative difference that lets Lemma D finish along B′ = B ∖ {a_r} after a late skip. Extra = +2 (not 2-robust) has skip ≥ 0 in 16/18 and last skip 662 — positivity of skip leftover is not 2-robustness.

(The parallel-v8 bound ≥ 1 + S_n−1/4 additionally needs the current term in B. K2 as stated here does not.) Python: 0 failures on extra = +1 B, all-ints-from-2, Narayana B, floor-S/2. Growth slower than φ makes skip eventually positive on geometric tails (numerator −r² + r + 1 of skip/a_n vanishes at r = φ); this is a geometric illustration of K, not a separate theorem.

Theorem N / Lemma Nwin — extraH = 0 never-C next is near-Brown, any L

Write extraH = 0 at prefix k to mean holes of P_k in [0, S_k] are exactly H ∪ (S_k − H).

This is Lemma E’s length comparison, with extraH = 0 supplying the solid interval and WC supplying the first-miss bound, with no overlap hypothesis. Specialising to H = {1} recovers Lemma N2 (λ ∈ {2, 3}). Python: 214 extraH = 0 never-C prefixes on generated L ≥ 3 walks, 0 violations of the window. v12 re-check on non-consecutive Pal prefixes: extraH = 0 never-C hits=458, window-fail=33; all 33 are short-prefix artefacts ({2, 3}+5, {3, 3}+5, {2, 3, 5}+7/8 borderline, {3, 5, 7}+11). Large-S Pal prefixes obey the window. Short prefixes with S < 2L (e.g. {3}, {3, 5}) are outside the hypothesis.

Python: 0 violations of the 2L − 1 bound on greedy-from-2 (14 extraH = 0 never-C prefixes, λ ≤ 2), greedy-from-1,3, extra = +1 B before it fires, all-ints-from-2 before it fires. Fib-from-2 has only two extraH = 0 never-C prefixes, then extraH grows (not WC).

This is the parallel-v8 claim “never-certify WC sequences have lim inf λ finite,” proved whenever extraH = 0 infinitely often. For L = 2 the extraH > 0-forever hole is closed by Lemma N2, L2-cycle, and Theorem L2∞ below. Consecutive H at any L is Theorem LN. Remaining theoretical hole: extraH > 0 at every large prefix, never-C, WC, λ → ∞, max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}).

Lemma N2 — extraH = 0 never-C next is near-Brown

This is Theorem N specialised to L = 2, with the two explicit options. Python: after {2, 3, 4} extraH = 0, never-C options {7, 8}; after {2, 3, 4, 8} options {15, 16}; after {2, 3, 4, 7, 8} options {22, 23}. Always {S − 2, S − 1}.

Lemma CW — extraH > 0 + never-C pins the first extra hole

Python: CW holds on the WC-ish never-C prefix 2, 3, 4, 7 then greedy (cw_ok=1). Apparent failures on Fib-from-2 / extra = +2 B / extra = +1 from 2, 3, 5 have extra holes below a (permanent extra, H infinite): the WC hypothesis fails, so CW does not apply.

Lemma L2-cycle — the extraH = 0 / extraH = 1 oscillation

The closed cycle, as computed:

Along the S − 2 then S − 1 walk, leftovers at successive terms are 3, 24, 3, 69, 3, 204, 3, 609, …: lim inf λ = 3, not λ → ∞. Greedy (S − 1 always) has λ ≡ 2. Repeating the current last forever is bounded.

Theorem L2∞ / L2N — WC never-C L = 2 ⇒ lim inf λ ≤ 3

The only WC never-C L = 2 objects are greedy-minimal (λ ≡ 2, or lim inf λ = 3 along the S − 2 interpolant), all 1-fragile. Python: greedy-from-2, greedy 2, 2, 3, 6, …, greedy 2, 2, 3, 5, 6, 17, …, and the S − 2/S − 1 walk 2, 3, 4, 7, 8, 22, 23, 67, … all have every 1-del GROW-PERSIST.

Python (erdos348_v10.py). DFS of all never-C WC continuations (max length 11–13, 511 nodes from each of 2, 3, 4; 2, 3, 4, 7, 8; 2, 2, 3; 2, 2, 3, 5): 256 survivors each, 0 fire-forced leaves at the cap (the never-C window stays nonempty by always offering S − 2 or S − 1, or a repeat/fill of the CW hole). Every survivor is a finite prefix of greedy, of the S − 2 interpolant, or of a bounded repeat. Apparent λ ≥ 3 survivors (84 from 2, 3, 4) have lim inf λ ≤ 3 once continued (the large last λ is a single S − 2 step). Shape of extra holes is {1, S − 1} or {1, a+1, S − a − 1} except during repeats, where the AP ja+1 appears and is wiped by the next a+1. Shape violations (extras outside {m+1, S − m − 1}, m = solid-from-2) occur exactly on long repeats: 2, 3, 4, 7, 8, 22, 22, 22 has extras {23, 45, 67}; those walks are not WC (GROW-PERSIST).

This replaces the earlier L = 2 sketch after CW: filling a+1 returns extraH to 0 (L2-cycle (4)), so unbounded WC never-C cannot stay extraH > 0, and Theorem N supplies lim inf λ ≤ 3.

Lemma Inc / BI — extraH > 0 never-C has bounded increment

This does not apply at extraH = 0 prefixes: there never-C forces a near-Brown jump (Lemma N2 / Theorem N), which is how greedy and the S − 2 interpolant live. Python “violations” of a′ ≤ a+L − 1 on greedy-from-2 are exactly those extraH = 0 jumps, not failures of Inc.

For L ≥ 3, Inc says a never-C extraH > 0 remainder is syndetic with gap < L. Consecutive H = {1, …, L − 1} is still closed below. LN is not claimed for incomplete H. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (extraH > 0 forever with max-symmetric non-consecutive H, e.g. {1, 2, 4, 6}). No example. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). L = 2 is excluded by L2∞.

L ≥ 3 never-C walks from 3, 5, 7, 3, 4, 5, 4, 5, 6, 2, 3, 5: max-strategy is a greedy-like jump (lim inf λ finite, 2-fragile); min-strategy repeats and is bounded. Same two never-C mechanisms as v8, one L-level up. Consecutive H is still closed; mixed max-symmetric non-consecutive extraH > 0 forever (e.g. {1, 2, 4, 6}) is not a nonexistence proof. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H. Climb is not uniformly FIRE.

Lemma FT — fill-trap for general L

This is v9 Lemma Join without the fast-increment / a+H ⊆ P hypothesis: the excluded set is exactly a+H, which may remain unfilled. For H = {1} it is the v10 claim P′ ⊇ [2, 2a − 1] ∖ {a+1}. Python (erdos348_v11.py): 0/66 failures on listed prefixes (including L = 2 controls); 0/240 on generated L ≥ 3 walk prefixes (skipping steps that violate [L, a − 1] ⊆ P).

Lemma Res — extraH > 0 never-C residue

The “BI failures” on greedy-max L ≥ 3 walks (e.g. 6 → 16 after {3, 4, 5}) are exactly this h = 0 case: extraH becomes 0 at the filled prefix, then a Pal jump is legal. Python: 0 empty residues on extraH > 0 never-C steps with a ≥ 2L − 1 among the recorded walks; residue ⊆ a+H.

Lemma Seed — extras after extraH = 0 → > 0 sit in a window

Python: 0 far-low seed failures on extraH = 0 → > 0 transitions among the recorded walks. Palindromes at the high end of P′ are not a counterexample (they are S′ minus a low extra).

AP-clear — plateau extras are an arithmetic progression

Python, plateaus from {2, 3, 4, 7}, {3, 4, 5}, {4, 5, 6}, {2, 2, 3, 5}, {3, 5, 7}: extras after k = 1..5 repeats match the predicted AP (unexplained empty, up to palindromes). Filling the seed returns extraH = 0 on the consecutive starts; continuing the plateau forever is bounded and GROW-PERSIST, not WC.

Lemma Plateau — extraH > 0 forever cannot be only repeats

For H = {1, 2} at {3, 4, 5} (extras {6}) the only keep-extraH next is the repeat; unboundedness forces +6, which is Pal. That is the consecutive-H start of Theorem LN. For H = {1, 2, 4, 6} keep-extraH increments exist, so Plateau does not by itself close the case. That is the leftover max-symmetric non-consecutive H; Climb is not uniformly FIRE (Climb-fire is sufficient when δ ≤ (L − 1)/3; OPEN climbs exist). Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H.

Theorem LN — consecutive H WC never-C ⇒ extraH = 0 i.o. ⇒ lim inf λ ≤ 2L − 1

Call a prefix palindromic of consecutive type H = {1, …, L − 1} when P = [0, S] ∖ ({1, …, L − 1} ∪ {S − L + 1, …, S − 1}). Then extraH = 0 and [L, S − L] ⊆ P.

Taking a = S − L + 1 (λ = L). The first mirror hole is added as a term, so it is filled. Each remaining old mirror survives as a+j for j = 1, …, L − 2 (Seed). For L = 2 there are no remaining old mirrors: palindrome is restored in one step (v10 L2∞). For L ≥ 3 one has extraH = L − 2 immediately after, with extras in the CW window. The solid run is then [L, a]. First miss ≥ a is a+1. Never-C nexts are among {a, a+1}: a repeat, or fill of the least extra. Infinite repeats contradict unboundedness (AP-clear: extras accumulate as an AP of difference a, GROW-PERSIST). Filling a+1, then a+2, …, through a+(L − 2) clears the old mirrors (AP-clear). After those L − 2 fills the holes in [0, S″] are again H ∪ (S″ − H): palindrome is restored. During the finite fill, leftover may be large; each restoring step itself has next > longest (never-C), and the next never-C choice at the restored palindrome again has leftover ≤ 2L − 1.

Taking a = S − 2L + 2 (λ = 2L − 1), or any interior option. Some old mirrors remain unfilled. Lemma FT puts them in the CW window (or as an AP of difference a after repeats). Unboundedness forces a later fill of each generator; filling the window extras returns to palindrome, as in the λ = L case. Python, from {3, 4, 5, 6} (H = {1, 2}, S = 18, never-C {14, 15, 16}):

The same trichotomy at every later Pal prefix {3, 4, 5, 6, 16, 17} (S = 51, never-C {47, 48, 49}).

For L = 2 this is v10 Theorem L2∞ / L2N. For L = 3 (H = {1, 2}): start-at-3, and any two-1s remainder that misses both 1 and 2. Unbounded WC never-C ⇒ extraH = 0 i.o. is not proved for arbitrary frozen H. Non-consecutive H (e.g. {1, 4}, {1, 2, 4, 6}) still has extraH = 0 i.o. in every unbounded WC never-C walk tested, but that is not a proof. DFS of WC never-C continuations: from {3, 4, 5} (max_n = 9) 266 seen, 151 survive, 0 extraH-never-0 at the cap; from {3, 4, 5, 6} 150/85/0; from {2, 3, 5} 2970/2276/0. Survivors with large last λ have hit extraH = 0. Apparent extraH-forever walks are min/repeat plateaus (GROW-PERSIST) or FIRE-FORCED finite prefixes. v14 still does not claim Theorem LN for incomplete H. Greedy-available (Stay0). Sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Algebraic Restore-W is FALSE (2a leftover, 2, 5, 8+9). v13’s 132/132 missed the FIRE family. Δ set equality is now 132/132 (v12’s 82 bads were a filter). Climb is not uniformly FIRE. Consecutive H remains closed. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (e.g. {1, 2, 4, 6} from 3, 5, 7).

Lemma Δ — extras after Pal λ = L are exactly a+Δ

Write M = L − 1 and Δ = H ∩ ((L − 1) − H) ∖ {0} (max-symmetric holes). This is set equality, not v12’s coded “low” window.

The set a+Δ is palindrome-closed in P′. v12 reported 82 “bads.” Those were not equality failures: e.g. 3, 5, 7+9 has Δ = {2, 4}, pred = {11, 13}, newE = {11, 13}, but only {11} sat in a coded “low” window. A filter, not a counterexample. Python (erdos348_v13.py), generated Δ-positive Pal prefixes (132 prefixes including later Pal-cycle restorations): ok=132, fail=0, skip=0.

Δ-empty (max-asymmetric H): Δ = ∅, so Pal λ = L restores extraH = 0 in one step (v12, re-checked). Log: 2, 3, 5+6; 2, 4, 5+8; 2, 3, 6+7; 2, 4, 7+8; 3, 4, 6+8; 2, 3, 5, 7+11; 2, 4, 6, 7+14; 2, 3, 4+8, extraN′ = 0. ok=8 fail=0. Those walks are not extraH > 0-forever sources.

Climb-window

Python: ok=132, fail=0. Examples: 3, 5, 7+9 window {9, 10, 11}, extras {11, 13}; 3, 5, 7, 9, 11, 13+42 window {42, 43, 44}, extras {44, 46}; 3, 3, 5+7 window {7, 8, 9}, extras {9}. If δ = 1 there is no climb room (consecutive-H / L = 2 shape). Δ-positive with climb is δ ≥ 2. This subsumes the v12 named-start check (ok=15, bad=0).

Lemma Finite-burst

So a never-C WC path cannot climb forever without filling g. Climb is not an unbounded extraH > 0 escape from a single Pal-add window. Sequential fillW may fail to complete the burst (FIRE witness below): extras stay in a finite palindromic W-block, but the never-C walk is not obligated to finish it.

Lemma Climb-fire (sufficient, not necessary)

After Pal λ = L, add a climb c with a < c < g. Let S″ = S′+c. If g remains a hole, palindrome of P″ makes S″ − g a hole, and first miss ≥ c is still g.

Model equality: H = {1, 2, 4, 6}, L = 7, δ = 2 = (L − 1)/3. Climb 3, 5, 7, 9, 10 does not fill g = 11 and is FIRE-FORCED (longest = 11). Same at later Pal-cycle greedies (+42 climb 43, +174 climb 175, +702 climb 703).

Climb is not uniformly FIRE. Counterexample: H = {1, 2, 4}, L = 5, δ = 2 > (5 − 1)/3. Climb 3, 3, 5, 7, 8 is OPEN (extras {9, 17}, opts {8, 9}, g = 9 unfilled). Then fill 9 PAL-RESTORE. Finite-burst still applies. The proposed test “climb-fire if min Δ ≤ (L − 1)/3” without the solid-run hypothesis is refuted (v14): 3, 6, 8 has L = 8, min Δ = 2 ≤ 7/3, HAS-OPEN. Do not use δ ≤ (L − 1)/3 as a lemma.

Python, climb-before-fill on 132 Pal prefixes: FIRE=27, RESTORE=10, OPEN=60, no-climb-room=63. OPEN examples are the δ > (L − 1)/3 family (335-cycle). No OPEN sample is extraH > 0-forever: the window still ends at g.

POST-CLIMB FIRE-FORCED is not general. v12 had open_fates = 0 on the one named family with post-fill climb room (3, 5, 7). Generated Pal prefixes have post-fill climb room on 124/132 and OPEN-sum = 580 (e.g. 3, 3, 5 after fill then climb 19 stays extraH > 0). Not a lemma. Those OPEN post-fill climbs still restore when g is later filled.

Lemma Stay0 — Greedy-available

Pal-L already says opts ⊆ {S − 2L + 2, …, S − L + 1}. Stay0 is the missing endpoint: the greedy top is attained whenever last ≥ L and never-C is nonempty. Python (erdos348_v14.py): generated Pal prefixes greedy-in 513/513 (this hunt, S < 8000); 254/254 (fillg census). A larger pool reported 22850/22850. Not claimed for last < L (no such extraH = 0 never-C prefix appeared). Stay0 says greedy is legal, not that it is taken. Interior Pal is a different next in the same Pal-L window; on generated pals, only-greedy count = 0. Never-λ = L is the typical choice, not a rare escape.

Restore-W / fillW after Pal λ = L

Sequential Restore-W is FALSE. Pal λ = L then fillW (always take the current least extra g when it is a never-C option) need not restore extraH = 0: it can FIRE with extraH still > 0. Witness: 3, 8, 9, 10, 13, 14, 15, extraH = 0, H = {1, 2, 4, 5, 6, 7}, L = 8, Δ = {1, 2, 5, 6}, S = 72, never-C {58, …, 65}, greedy a = 65 (λ = 8 = L) legal by Stay0. W = {66, 67, 70, 71}.

FIRE-FORCED: wc nexts {67, …, 71}, longest = 127 ≥ 71 = g. extraH still 2. fillW did not restore. Mechanism: Δ contains a consecutive pair {1, 2}. Filling a+1, a+2 lengthens a run past the remaining extras a+5, a+6. Palindrome copies {132, 133, 136} are twins of leftover W, not a middle migrate. The run, not a migrating residue, fires C.

Algebraic contrast on this same prefix. Adding the set W = {66, 67, 70, 71} as summands (not never-C constrained) yields extraH = 0 at S_f = 411. The Minkowski join of W restores here; the never-C walk cannot finish W.

After filling W = a+Δ: [L, 2a − 1] solid; leftover ≥ 2a

Write S′ = S+a, so S′ − L = 2a − 1. Let Q be the subset-sum set after adding every w ∈ W = a+Δ as a term (order-independent Minkowski: Q = P′+⟨W⟩), S_W = S′+∑W.

Thus any leftover extra of Q in [L, S_W − L] is ≥ 2a. The v13 high-interval obligation is exactly [2a, S_W − L] ⊆ Q.

If S ≥ 3L − 3 then a − d ≥ a − (L − 2) = S − 2L+3 ≥ L, so a − d ∉ H automatically and 2a ∈ Q. The interesting leftovers are small S.

Algebraic Restore-W is FALSE on a realized frozen prefix. Witness: 2, 5, 8, extraH = 0, H = {1, 3, 4, 6}, L = 7, Δ = {3}, S = 15, a = 9, never-C {8, 9}. W = {12}. Minkowski Q = P′+⟨W⟩ at S_W = 36 has extraN=1, leftover {18} = {2a}. Criterion: a − d = 6 ∈ H, so 2a ∉ Q. Lemma Δ still holds: extras after Pal are exactly a+Δ, so the Pal-add window [L, S′ − L] is filled by appending a+Δ; the expanded extra range after those generators is not. v13’s “high interval filled in the log, not proved” is not an omission of a true lemma: the high interval can fail. Not LN. Not a solution.

Continued fillg on this witness restores. After +9, extras {12}, window {9, …, 12}. After +12, extras {18}, window {12, …, 18}, 2a = 18 legal. After +18, extraH = 0. Seq 2, 5, 8, 9, 12, 18, extraH 0,0,0,1,1,0. lim inf λ = 7, FROZEN, 1-fragile (Nio+A′). Length-3 2a-fails: only 2, 5, 8 among a ≤ 9. Length-4: seven more (2, 2, 7, 12; 2, 3, 8, 14; 2, 4, 7, 12; 2, 4, 9, 14; 2, 5, 5, 13; 2, 5, 7, 15; 3, 4, 7, 13). On all eight, Minkowski-W leftover includes 2a as least extra, 2a is a legal never-C next, and adding 2a as a term restores extraH = 0. fillg-after-lamL on a 254-prefix pool that includes these: RESTORE=254, FIRE=0, OPEN=0. That pool missed the FIRE family above.

After adding the term 2a: FT-L plus the identities 2a+h = a+(a+h) for h ∈ H put 2a+H ⊆ Q already before the new term. The new term fills 2a+[1, L − 1] ∖ H. Shift [L, 2a − 1]+2a = [2a+L, 4a − 1]. Thus [L, 4a − 1] is solid after +2a. The remaining tail [4a, S_new − L] of length ∑W is not proved in general; it is empty after +2a on the eight realized 2a-fails. Abstract palindromic P leftovers after Minkowski-W: 244 failures, all with S − 2L ≤ 4. Realizable High-fails: none in the hunt.

So the two refutations are complementary:

Neither “add W then extraH = 0” nor “fillg then extraH = 0” is a theorem for every frozen H. A 3, 8, 9-family DFS (pool): 1221 pals, sequential greedy fillW RESTORE=743 FIRE=478 OPEN=0; algebraic add W: 1221/1221 restore. Small named pools that miss this H = {1, 2, 4, 5, 6, 7} report 100% sequential restore (v13’s 132/132; this hunt’s 254/254) — those pools are not general.

fillW from extraH > 0 Δ-positive starts (no Pal λ = L first): restore or fire, never OPEN at cap (3, 5, 6 / 4, 6, 7 / 3, 6, 7 / 3, 5, 7, 9 / 3, 3, 5, 7 restore; 5, 7, 9 / 4, 7, 9 / 5, 6, 8 / 5, 7, 8 fire). Fire is not never-C. Restore hits extraH = 0. Neither is extraH > 0-forever WC never-C.

Interior Pal then fillW is not a uniform restore. Restore or FIRE; 0 OPEN leftover on searched pools (FIRE is C-death, not extraH > 0-forever). Named interiors: 3, 5, 7+8 RESTORE; +9 RESTORE; +7 (min) FIRE at 3, 5, 7, 7, 9, 11; several options on the 3, 8, 9 seed FIRE. Immediate extraN′ = 0 interiors exist (e.g. 3, 5, 7, 9, 11, 13+37). Refusing greedy does not escape Theorem N if extraH returns to 0: never-greedy named walks Pal-cycle with extraH = 0 i.o. (lim inf λ = L or an interior ≤ 2L − 1) or FIRE. Example: 3, 3, 5 interior-fill hist 0,0,0,2,2,0,1,0,1,…, nλL=0, still extraH = 0 i.o., 1-fragile. Same v12 min-plateau FIRE on incomplete H = {1, 2, 4, 6}. Contrast v11 complete H = {1, 2}: k fives then 6 restores extraH = 0 for every k. Job (2)’s implication (ever Pal-greedy ⇒ extraH = 0 i.o.) does not fire as a theorem: fillW does not always restore, and the sequence need not Pal-greedy.

Climb is not an escape

Climb is not uniformly FIRE. After Pal λ = L, a climb either fires (Climb-fire when the solid-run hypothesis holds), restores without FIRE, or leaves g unfilled with the window still ending at g (finite burst toward g). OPEN climbs exist (v13: 60 on 132 prefixes; v14 after any Pal option: CLIMB 33 restore / 78 OPEN / 21 FIRE). They are not extraH > 0-forever. Climb OPEN then fillW: RESTORE 75, FIRE 3. Sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Algebraic Restore-W is FALSE (2a leftover, 2, 5, 8+9). Algebraic add of W restores on that FIRE family; consecutive pair in Δ lengthens the run.

Pal-cycle walks (greedy_fill / interior_fill / climb_first / max, max_n=14, freeze L): all FROZEN or FIRE; λ bounded; extraH = 0 i.o. on survivors (e.g. 3, 5, 7 greedy_fill extraH=0 i.o., λ≡7; climb_first FIRE at 9, 10). Escape-DFS “nonplat-extra-cap” samples all have extraH = 0 in the history (mid-cycle, e.g. 3, 5, 7, 9, 11, 13, 42, 44, 46, 174 about to fillW). Witness extraH > 0-forever WC never-C λ → ∞: none. The cycle picture is not Theorem LN.

On the named start 3, 5, 7 (v12, kept): after λ = L extras {11, 13}, opts {9, 10, 11}. Climb 10 is FIRE-FORCED. Named-start post-fill open_fates = 0 was that family only, not a lemma for every incomplete H.

Adversarial climb-prefer (keep extraH > 0 if possible): every named Δ-positive start restores Pal cyclically or fires. 3, 5, 7: fill 11,13, Pal-cycle, lim inf λ = 7, FROZEN. 3, 3, 5: climb is OPEN, used, then fillg restore, lim inf λ = 5. 5, 7, 9: FIRE-FORCED at 5, 7, 9, 10, 11. No extraH > 0-forever path in the log.

v12 named-start traces (kept). Every OPEN extraH > 0-at-cap sample is a plateau or a truncated fillg/climb at the length cap — not WC (plateau-fail). Deeper DFS (max_n = 11–14): extraH-never-0 non-plateau =0 on 3, 5, 7 (16127/16127 extra0-hit), 3, 3, 5, 2, 5, 6, 3, 4, 7, 3, 5, 8, 3, 5, 6, 4, 5, 7, 3, 5, 7, 9, 3, 5, 7, 9, 11, 13. Leftover never0_nonplat samples (4, 6, 7; 3, 7, 8; 5, 7, 9) are truncated plateau-then-one-increment at the cap, not unbounded WC λ → ∞. Zero-1 official (2, 3) candidates 0. Skip-bounded (1, 2) from a₂ ≥ 2 candidates []. No witness. Not LN. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); algebraic Restore-W is FALSE (2a leftover, 2, 5, 8+9); Stay0 proved; interleaved climb/plateau restore is computational). Consecutive H is still closed. Official (2, 3) remains open.

Lemma Climb-fill, Climb-shrink, Room-1 — type-iii finite

Refuse-fill after Pal λ = L: take a never-C next strictly below g = a+min Δ whenever such a climb exists; fill g only if forced. Must-fill is false (Climb-window width δ; only-greedy 0/108; plateau always legal). Python (erdos348_v16.py / erdos348_v16b.py / erdos348_v17.py / erdos348_v18.py / erdos348_v18_plus1.py). Pal pool this hunt: 108 extraH = 0 never-C prefixes with Δ ≠ ∅, S ≥ 2L. Official (2, 3) remains OPEN. LN is not claimed.

Census, Pal pool 108, interior climbs with g not filled: shrink 87/87. Predicted window [b, g] when g not filled: ok=141, bad=12 (11 ii-FIRE with opts empty, longest ≥ g; one additional 2, 6, 9+10+12, longest = b = 12, opts = {13} = {g} — still a shrink, next is forced fill). Climb-shrink is not a Lyapunov: type-iv fills g and room can grow.

Named iii-SHRINK: 2, 5, 8+9+11 extras {12, 23}, opts {11, 12}, room 3→1; 3, 6, 8+10+11 room 2→1; 3, 3, 5+7+8 room 2→1. Then Room-1 forced fill or FIRE/restore. Strict walks on the Pal pool: RESTORE0 225, FIRE-FORCED 207, OPEN-EXTRA 0; forced-fill 322 (restore-after-forced 155, fire-after-forced 167).

Python (erdos348_v17.py / erdos348_v18.py). Pal pool this hunt: 108 extraH = 0 never-C prefixes with Δ ≠ ∅, S ≥ 2L (same as v16). Official (2, 3) remains OPEN. LN is not claimed.

Lemma Frozen-g-die — frozen least extra cannot last

This closes frozen-g / Type-iii refuse-fill forever. Climb-shrink is the one-step room drop used above; Frozen-g-die is why that drop does kill the frozen subclass, even though it is not a Lyapunov for migrate. The remaining sliver is (extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution)

Lemma Mig-set — leftovers after Pal then b

Leftover 2a ∈ S′ − H always, because M ∈ H. It becomes an extra iff 2a − b ∉ P′. By 2a-crit this is a finite check, not an unbounded family. Scope. Mig-set locates leftovers after one Pal-then-b. After a second migrate the prefix is no longer extraH = 0, so Δ / Pal-add no longer locate extras. Not a Lyapunov for migrate i.o. Census Pal+b: ok=369 bad=0.

Lemma Extra-set — leftovers after +c on an extraH > 0 prefix

Census (v18 tight): Extra-set on migrate+FILL-G nexts ok=329 bad=0, two-step ok=171 bad=0. Earlier iv-MIGRATE-seed scan ok=73 bad=0; broader Pal then extraH > 0 then any next (opts[:12]): ok=900 bad=0. After a second migrate extras need not sit in Mig-set; Extra-set still locates leftovers (containment, not a death; c+E can sit farther out). v18 tight 329/329 and two-step 171/171. Not a Lyapunov for unbounded chains. extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution.

Lemma No-interior-if-solid-below-δ

Census: solid-below-δ prefixes 69/108 and 138/505. Interior steps on those: 21/21 and 61/61, all g-unfilled, BAD-filled=0. Corollary: Type A interior migrate requires some j ∈ [1, δ) with j ∉ H. On this Pal pool the realised case is δ = 3, 2∉H, k = 1. Gapped Δ is common; gapped-Δ interior migrate is rare because most gapped Δ still have [1, δ) ⊆ H.

Lemma Near-fill / Type-A-peel-identity

Census (Pal pool 108): the 9 Type A iv-MIGRATE events all have this shape. Identity 9/9. Each subsequent last+1 peels one more from the consecutive block and keeps room 3. Finite block ⇒ pure near-fill is finite. Not a theorem that the walk dies as never-C: high extras (≥ 2a) and Extra-set c+E can remain, and FIRE can occur before the block is empty.

Lemma Two-n family

FOUND live migrate-chain ≥3 (not official (2, 3)). Pal pool 108 live chain_max=2 was a pool artefact. Witness: 2, 13, 15, 16, 17, 18, 19, 20 (never-C, extraH > 0). Longer finite chains on [2, n, n+2] Pal then +1: n=16 live 4 then FIRE; n=20 live 5; n=40 live 9; n=60: 13 then FIRE. forever_np=0. Each walk is finite C. No uniform bound on chain length, and no extraH > 0-forever example. Extra-set locates leftovers after the second jump and is not a death. Official (2, 3) remains OPEN. LN not claimed.

Lemma Near-Δ — FILL-G extras in [L, 2a)

Census (this Pal pool): ok=108 bad=0.

Lemma Remaining-W / Fill-W-item / Post-W

Near-Δ is the k = δ case. Census, 185-prefix pool, every Pal-then-next: Remaining-W 792/792, Fill-W-item 792/792.

Post-W one-steps on the 185 pool: 11/11 have g ≥ 2a, extras exactly {2a, S″ − 2a} (including the three that look like {2a, 2a+2}). The three “high” leftovers {26, 28}, {30, 33}, {30, 33} are those twins (S″ = 54, 63, 63). Fillg taking 2a restores all 11. HAS-2a-PLUS is not a new high-tail family.

Lemma (⋆)-fills-2a-by-g

Pal+FILL-G on this pool: (⋆) holds 90/108, leftover-has-2a among those 0. (⋆) fails 18, leftover-has-2a 12 (the 8 LEFTOVER-2a plus 4 HAS-2a-PLUS).

FILL-G is one per prefix (Fill-available). It is not uniformly restore (v14 leftover 2a; v16 tagged FILL-G 108). Filling only min Δ, not all of W = a+Δ, is why leftover is common — distinct from v15 algebraic-W leftover (rare 2a).

Among the 70 leftovers: near extras nonempty 62, only-high 8. Kinds: LEFTOVER-2a 8, HAS-2a-PLUS 4, NEAR-PLUS 58. Room after FILL-G can grow 24, stay 37, or shrink 9 — not a Lyapunov. Examples: 2, 5, 8+9+12 extras {18} room 3→6 (only-high 2a); 3, 5, 7+9+11 extras {13, 22} room 2→2 (near 13=a+4 + high 22); 2, 6, 8+9+12 extras {13, 24} room 3→1; 2, 2, 7, 12+13+18 extras {26, 28} room 5→8 (only-high); 4, 6, 9+11+12 extras {14, 28} room 1→2. Common, finite, not extraH > 0-forever. Prototype leftover-2a 2, 5, 8, 9, 12: G-survive seen 21, fire 9, restore 16, plat 4, mig 0, fnp 0.

iv-MIGRATE: interior climb 1 ≤ k < δ, old g filled, extraN > 0, new g′ > g. Distinct from FILL-G (k = δ) and from i-RESTORE (extraN=0). One-step 12, all DFS-die, not a theorem. Room grow 3, stay 9, drop 0. Type A peel 9 / Type B leftover-2a 3. Type A: interval Δ starting at 3, 2∉H, k = 1, room 3→3, consecutive leftover |Δ| − 1. Type B: singleton Δ, leftover 2a, room grows, further mig = 0 on those seeds.

That is why Climb-shrink is not a global potential. Deeper DFS ALL never-C from these 12 (n ≤ 16, 12000 nodes): every fnp=0. Sum seen 309, fire 195, restore 132, plat 30, later-mig 77. Max migrate-chain along a path on this Pal pool: 2 (only 2, 10, 12, 13, 14; chain≥3 = 0) — a Pal-pool artefact, not a bound. Sample of a second jump: 2, 10, 12, 13, 14+15 g 17→18; then +17+18 g 18→21 extraN 7→2 extras {21, 65}, then FIRE/plat. v18 Two-n peel: live migrate-chain ≥3 exists (witness [2, 13, 15, 16, 17, 18, 19, 20]); longer finite chains on [2, n, n+2] Pal then +1 (n=60: 13 then FIRE). None of the 12 Pal-pool seeds is extraH > 0-forever. Not a theorem for every incomplete H. No infinite walk.

Unique leftover prefixes after a g-filling step: 82 (70 FILL-G leftover + 12 iv-MIGRATE). G-survive DFS all never-C (n ≤ 13, 6000 nodes): seen 1260, fire 762, restore 591, plat 135, later-mig 198, forever_np=0. Max chain 2, chain≥3 0 on this Pal pool (v17 artefact; v18 Two-n live ≥4 then FIRE). fillg-continue (always take current g when legal): RESTORE0 49, FIRE 33, OPEN 0. Later-mig-on-fillg 41. No OPEN leftover under greedy fill-g. Deeper FILL-G leftover: seen 951, fire 567, restore 459, plat 105, mig 121, fnp=0. G-survive forever on this Pal-pool leftover class is empty at cap. Closed as a source of examples. Not a theorem.

Later-mig samples (room can grow again after leftover): 4, 6, 9, 11, 12+14 g 14→28 room 2→14; 2, 2, 7, 12, 13, 18+22 g 26→28 room 8→6; 2, 8, 11, 12, 15+16 g 16→32 room →16. Second jumps exist on Pal pool 108; a third on a live never-C extraH > 0 path was not seen there. v18 Two-n peel: live migrate-chain ≥3 exists (witness [2, 13, 15, 16, 17, 18, 19, 20]); longer finite chains on [2, n, n+2] Pal then +1 (n=60: 13 then FIRE). Not a death. No infinite walk. Finite-tail min λ > 2L − 1 occurred once (4, 6, 9, 11, 12, 14, 28): Pal term λ = L = 9 ≤ 17 = 2L − 1, later fill-g terms λ = 19, 29, 29. Not a Pal-L counterexample (liminf of the whole walk includes the Pal term). Pal-L stands.

The v16b 47 room=1 fill leftovers: from Pal, strict-climb until room=1 (or no strict), then fill g: restore 52, fire 5, still-extra 47. Room after those fills: grow 25, stay 22, shrink 0. STRICT / all-opts DFS from the 47: forever_np = 0, 47/47 dead in the hunt (fire / restore / forced-then-die). fill-g chain from the 47: RESTORE 22, FIRE 25, OPEN 0; chain length min/max 2/3 (greedy 3 then FIRE). Rechecked, still empty as extraH > 0-forever. Not a theorem for every incomplete H.

No Minkowski finish lemma. Algebraic Restore-W of W = a+Δ is already false (v14 leftover 2a). Minkowski-add of leftover extras after filling g restored extraH = 0 on the 70 Pal+FILL-G leftovers and on the 47 room=1-fill leftovers (one-shot census). Sequential never-C fillW can FIRE before the leftover is finished (v14). After a second migrate, extras are not located by Δ. Tail after adjoining leftover extras is census-empty, not an identity. Leftover extras look Minkowski-finite on this pool; greedy fill-g from leftovers is restore-or-FIRE with OPEN=0. That is a census, not a finish lemma, and not LN.

3, 7, 9+11, 13, 15 is FIRE, not extraH > 0-forever. Pal then FILL-G leftover then a second fill that FIREs. Prefix 3, 7, 9: extraH = 0, H = {1, 2, 4, 5, 6, 8}, L = 9, Δ = {2, 4, 6}, S = 19, Pal a = 11, W = {13, 15, 17}.

Same kill as 3, 8, 9, 10, 13, 14, 15+65, 66, 67: filling Pal-add extras lengthens a run past remaining W. DFS from 3, 7, 9, 11, 13: seen=9, fire=6, restore=0, forever=1 (plateau), fnp=0. Not extraH > 0-forever WC. Not a migrate-forever seed.

Greedy migrate-seeking walks from the 12 seeds: RESTORE0 8, FIRE-FORCED 28, OPEN-EXTRA 0; max migrates on a greedy walk 3 then FIRE (seed 2, 10, 12, 13, 14+15+16+17). Live DFS chain_max = 2 on Pal pool 108 (v17 artefact). Expanded iv-strict chain_max = 3, ge4 = 0 (census on 22 expanded interior seeds; not a bound). Pal-pool/expanded DFS ge4=0 does not cover Two-n plus1. v18 Two-n: live migrate-chain ≥3 exists (witness [2, 13, 15, 16, 17, 18, 19, 20]); longer finite chains on [2, n, n+2] Pal then +1 (n=60: 13 then FIRE). Extra-set locates leftovers after the second jump (73/73, v18 tight 329/329) but is not a death.

Remaining (v22). extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution). By Frozen-g-die, any extraH > 0-forever WC never-C λ → ∞ walk must jump g infinitely often. Remaining-W / Fill-W-item / Post-W / W-chase finite close remW-migrate (at most |Δ| jumps, then g ≥ 2a); high extras after Post-W remain. Near-Δ / Mig-set describe one leftover after Pal-then-b. After a second migrate the prefix is no longer extraH = 0. Hunt: Pal-pool live chain_max 2 was an artefact, not a bound. Expanded iv-strict chain_max 3, ge4 = 0 (census, not a theorem). v18 FOUND live migrate-chain ≥3 exists (witness [2, 13, 15, 16, 17, 18, 19, 20]); longer finite chains on [2, n, n+2] Pal then +1 (n=60: 13 then FIRE). Extra-set locates leftovers after the second jump but is not a death. 47/47 room=1 leftovers dead. forever_np = 0 on hunted leftovers. No infinite walk. Not a theorem. Climb-kill is false; Restore-W is false; Climb-shrink is not a Lyapunov; Frozen-g-die / Remaining-W / Extra-set are proved (Extra-set is containment, not a death); Frozen-g-die does not apply once g jumps; W-chase finite does not kill high extras after Post-W; Near-Δ does not apply after the second jump; Minkowski of leftover extras is not a finish lemma. LN is not claimed. Official (2, 3) remains OPEN. No witness.

Lemma Peel-finite — near migrates ≤ |Δ|

Type-A-peel-identity is the consecutive-Δ, 2∉H, k = 1 case: room stays 3, leftover |Δ| − 1 consecutive. Two-n FIREs with remW still nonempty — Peel-finite does not say the walk restores; it says a pure near-peel cannot be extraH > 0-forever. Forever needs high-jumps. Census inherited: Remaining-W / Fill-W-item 792/792; Near-Δ-k 369/369 and 76/76; Room-3 peel on Two-n t=2..t* n=60 ok=60 (v19).

Theorem Two-n-plus1-dies / Lemma Gap-r / t*

Census Gap-r, n=6..40, t=1..t*+2, r=1..min(t, 11): n=1656 ok=1656 bad=0. Pal-then-+1 vs t*, n=6..120 plus 150, 200: 117/117 exact match (reach t*, +1 legal until then, death as predicted). FIRE=116, RESTORE=1, illegal=0, early=0. Named: n=16 t*=7 last=25 n_live=5 extraN=14 FIRE; n=40 t*=12 last=54 n_live=10 FIRE; n=60 t*=15 last=77 n_live=13 FIRE. Room-3 on t=2..t*: n=60 ok=60. Off the remaining-gap list as a forever source.

Kill “max chain 2” as a remaining-gap bound. v17 Pal-pool live chain_max=2, chain≥3=0 was a pool artefact (n≤12). v18 FOUND live ≥4 then FIRE, growing with n. v19 proves the plus1 walk dies at t*(n) for every n≥6. Finite chains getting longer in n are the opposite of extraH > 0-forever, and also the opposite of a uniform chain bound. Not a witness. Not official (2, 3).

all-opts DFS from Pal+k=1 seeds (census, not a theorem): n=8,10,13,16 STRICT/ALL fnp=0, live_cm=1,2,3,4. n=20 STRICT fnp=0 live_cm=5; ALL fnp=1 is a DFS cap, greedy FIREs at t*=8 — not a certified forever walk. Neighbour plus1 [2, n, n+1] / [2, n, n+3] / [3, n, n+3] OPEN=0 at listed n (census, not a theorem).

Lemma CE-blocked-by-near

Census: Pal-108 leftover nexts remW-survives ⇒ g∉c+E 905/905 (v21 recensus). leftovers ineq 1105/1105. TOTAL live g∈c+E = 0 on Pal-108 / expanded / Two-n leftovers (v20). v22 remW-survives leftover nexts 3701, all CE-blocked-by-near; fresh c+E as live new g after remW empty: 0. Extra-set still locates high c+E; they are not least until remW dies, and Two-n FIREs first. Containment, not a Lyapunov.

Lemma Palindrome-midpoint / Unique-extra-is-mid

Same statement as Lemma Unique-extra-is-mid (v21). Census unique-extra states (leftovers + leftover nexts + extra_starts_small): 44/44. Of those, x≥2L−1 on 19, x<2L−1 on 25. The inequality for fill-g restore needs the large side.

Lemma Midpoint-singleton-restores / Fill-2a-singleton-restores

Same Unique-mid-FILL-G-restores (v21) for x≥2L−1, census 14/14 on those. Skip 25: x not in opts or x<2L−1 — the inequality is not claimed for small unique-mids (e.g. extra_starts_small [2, 4, 4] g=5). FILL-BELOW of Unique-mid restored 106/106 on 21 emptying-LIVE seeds (census, not a theorem).

LIVE5 / remW-emptying 2g-jump singleton (v21 seeds, v22 recensus): filling old g leaves extras {2·last} = S′/2. Identities 5/5. fill-g RESTORE 5/5. DFS fnp=0 cm=0. Greedy slow/fast/grow: RESTORE0 9, FIRE-FORCED 6. Frozen-g-die applies to the fire-forced branches.

Pal+b leftover-2a, extraN>1, and the h* lemma

v20 algebraic legs (2a-leftover-location, h*-not-in-Δ, 2aH-in-P″) are theorems, census 13/13 on union 543. extras-are-pair and SH-preimage were 13/13 on that hunt, not identities for every incomplete H. Fill-2a-restores is a theorem given pair+SH. v22 Pal+b leftover-2a unique 12 (union 378): extraN hist {1:8, 2:4}, extraN>2 0, extras==pair 12/12, 2a+H⊂P 12/12, fill-g RESTORE 12/12. extraN=1 ⇒ h*=0: 8/8. extraN>1 ⇒ h*∉H: 4/4. Off Pal+b, extras-are-pair fails (extraN=3 exists).

Not “fill leftover-2a always restores, every incomplete H.” extraN>1 leftover after remW empty is the remaining leftover-2a hole.

remW-emptying LIVE / Post-W / never-hit-Pal (v21–v22 hunts)

Pal-legal union 378 (pool 108 + expanded 185 + Two-n n≤36 + other Pal-legal 230). Pal-then-any leftovers 1274 (remW>0 1262, Post-W 12). Leftover nexts 4636. remW-survives 3701 (CE-blocked-by-near). remW-emptied 799: RESTORE 766, FIRE 0, LIVE 33. already-Post-W 136.

Unique remW-emptying LIVE: 33. Shape and fill-g:

extraN=1 21 (all midpoint, g≥2L−1 21/21, pred empty 21/21 — theorem applies). extraN>1 12 (extraN=3 twice: [2, 4, 14, 20, 21, 27, 28] extras [56, 58, 60]; [2, 4, 16, 22, 23, 30, 31] extras [62, 64, 66]). fill-g RESTORE 33/33. fill-g LIVE: none. DFS all 33: seen 6706, fire 5229, restore 2239, plat 422, mig 613, fnp=0, chain_max=1 (only extraN>1 seeds migrate once then die). Greedy OPEN-EXTRA 0. extraN>1 remW-emptying is census restore, not a theorem: pair identity relative to Pal 2a fails off Pal+b; extraN=3 exists.

v21 Pal-108 slice (subset): remW-emptying 242 = RESTORE 237 + LIVE 5 Unique-mid + FIRE 0. The five LIVE all land Unique-mid extras={2g}=S′/2, g is not fresh c+E. ALL-OPTS DFS: seen 95, fire 154, restore 133, plat 30, mig 0, fnp=0, chain_max 0. Those five die by the Unique-mid theorem + Frozen-g-die + FILL-BELOW census.

Post-W leftovers 12 (v22 union; v21 Pal-108 had 11), all g=2a. fill-g RESTORE 12/12. Post-W live g=2a 72 (frozen; Frozen-g-die), g>2a 4. Post-W live migrates 4, all primary tag c+H, extras a palindrome pair with high partner (not least):

• [2, 2, 7, 12, 13, 18]+22: g 26→28, extras [28, 48], fill-g RESTORE
• [2, 3, 8, 14, 15, 21]+27: g 30→33, extras [33, 57], fill-g RESTORE
• [3, 4, 7, 13, 15, 21]+27: g 30→33, extras [33, 57], fill-g RESTORE
• [3, 4, 10, 16, 18, 24]+33: g 36→39, extras [39, 69], fill-g RESTORE

fill-g after migrate RESTORE 4/4. DFS fnp=0 cm=0. S−H as live new g (not also 2a/c+H): 0. Not a theorem that every Post-W c+H migrate restores.

Never-hit-Pal. extra_starts_small 81. g=2·last: 0. fill-g RESTORE 7, LIVE 74. These are never-hit-Pal incomplete starts (typically g=last+1), not remW-empty leftover-2a. Midpoint hypothesis g≥2L−1 fails (e.g. [2, 5, 5] g=6). true-never-at-cap 0 (v16). Unchanged as a theorem. v20 oldE-high fill-g LIVE 17, DFS fnp=0 chain_max 3 — not leftover-2a, not extraH>0-forever on that hunt.

Families killed, v16–v22 (never-C λ→∞ WC 1-robust / official (2, 3))

Witness (official (2, 3) or extraH>0-forever WC λ→∞): none.

Honest remaining gap (v22)

Not a solution of official #348. LN not claimed. Two-1s not negatively settled. The sliver that would still feed official (2, 3) as a two-1s remainder is a WC never-C incomplete-H walk that never returns extraH = 0 (S≥2L) and has λ→∞. FIRE branches are dead as never-C. Pal-cycle branches are extraH = 0 i.o. (1-fragile). Plateau branches are not WC. Frozen-g / Type-iii / Two-n plus1 / remW-survives-c+E-as-g / leftover-2a singleton are empty.

1. extraN>1 leftover after remW empty. Singleton leftover-2a is closed. extraN>1 remW-emptying fill-g restores on 12/12 here (not a theorem; extraN=3 exists; pair identity fails off Pal+b). DFS chain_max=1 then die. No example of regenerating forever.
2. Post-W c+H. Four Post-W c+H migrates fill-g restore 4/4 here. Not a theorem for every H. S−H as live new g (not 2a/c+H): 0 on this hunt.
3. Never-hit-Pal. 81 extraH>0 incomplete starts; fill-g LIVE 74. true-never-at-cap 0 (v16). Midpoint hypothesis g≥2L−1 fails. Unchanged as a theorem.
4. Skip-bounded weak (1, 2) from a₂≥2. Unchanged.
5. Zero-1 official (2, 3). Theorem 2′ never covers these. Unchanged.

Pal-pool “max chain 2” is not a remaining-gap bound. Live migrate-chain ≥3 exists, then FIRE (Two-n). plus1 is a theorem-death, not a remaining family. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. No fake solve.

Lemma F3B — Regime B + extraH = 0 ⇒ Lemma C

This is Lemma E evaluated in B: the leftover is large, so the H ∪ mirrors shape (which palindrome plus frozen H supplies whenever extraH = 0) is enough. Together with Theorem N: a B-prefix that doesn’t fire and has extraH = 0 must have λ ≤ 2L − 1, which for large S contradicts λ ≥ S/2 + 1. So in B, extraH = 0 and never-C are incompatible for large prefixes. Python: 0 F3B failures on every family that entered B with extraH = 0 (Narayana B, extra = +1 remainders, all-ints-from-2/3, padovan B, floor-S/2, floor-S/3). Families that skip the check (dense width-2 with the 1 included; odds-from-3 with L mis-set as 2) certify on the A side or have extra holes from a larger true L.

Lemma F5 — B-ultrafast never-C cannot persist for WC

Call a prefix B-ultrafast if it is in B and a > (S − L)/2. Then the extra zone [a, S − a] has length S − 2a + 1 < L + 1.

Python: never-C WC sequences in the log are all-A (greedy-from-2, λ ≡ 2), not perpetual ultrafast. Sequences that are perpetual B-fast (floor-S/2 from 2, 3; extra = +1 remainder) have extraH = 0 and fire at k = 3 or 4. Ultrafast-and-never-C occurs only for non-WC families (Fib, extra = +2, extra = +1 from 2, 3, 5), where extraH grows.

Lemma L2 — the L = 2 prefix, including extra 2s

Every WC λ → ∞ continuation constructed after 2, 3, 4, 7 fires (floor-S/2 at k = 5; floor-S/3 at k = 7 via longest run; pick-hole-B at k = 5). The only never-C WC L = 2 objects found are greedy-minimal (possibly after a finite 2^k or 2, 2, 3, 5 prefix) and the S − 2 interpolant, all with lim inf λ ≤ 3. Theorem L2∞ now proves there is no unbounded WC never-C L = 2 sequence with λ → ∞, so none of these is 1-robust.

Adversarial stay-in-B: no never-C WC counterexample

Generators aimed at skipping the F2 window then remaining in B: floor-qS after {2, 3, 4, 7} or {2, 3, 4, 8}; pick-a-hole-in-[last, S/2]; greedy for k = 3..6 steps then floor-S/2 or S/3; densify-greedy every 2/3/4; mixed extra; floor-S/2 from 2, 3, 5; dynamic-L never-C walks from 2, 3, 4 (exactly two WC never-C choices at each extraH = 0 step: next S − 2 or S − 1, leftover 3 or 2).

No example of WC + λ → ∞ + never-C, including inside B. For L = 2 this is Theorem L2∞, not a hunt. The two never-C mechanisms remain exactly v8’s: (a) bounded λ, not 1-robust; (b) not WC, extraH growing, Fib-like. Consecutive H extraH > 0 forever is empty. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (max-symmetric non-consecutive extraH > 0 forever, e.g. {1, 2, 4, 6}; no example).

A genuine extraH > 0 fire in B-slow: 2, 3, 4, 7 then floor-S/3 has extraH = 2 from n = 5 onward, longest run catches next at n = 7 (long=13, next 13). extraH = 0 is sufficient, not necessary. Middle-run scan: 20 B-slow extraH≤ 2 (shape-ok) prefixes, 4 fire, 16 never-C, and those 16 are all the same 2, 3, 4, 7, 7 step (S = 23, λ = 17, extras {8, 15}, longest = 6). After +8, extraH = 0 and F3B/N apply. extraH > 0-forever is empty for L = 2 by L2∞.

Tribonacci minus two 1s: never-C, all-A, skip leftover → −∞, GROW-PERSIST. Growth faster than φ (trib constant ≈ 1.84). Not a 2-robust remainder.

Brown-equality a_n+1 = 1 + S_n−1 from 2, 3 produces 2, 3, 3, 6, 9, 15, … (Fib-like without two 1s): never-C, λ → ∞, GROW-PERSIST, 1-fragile. φ-growth starting at ≥ 2 does not even give WC.

Theorem 2′ — two 1s, overlap, extraH = 0 in B, Theorem N, L2∞, and LN

Python, two 1s of A, remainder B:

The two-1s class is empty as a source of official witnesses on overlap, on extraH = 0-in-B, on extraH = 0 i.o. (Theorem N), on L(B) = 2 (L2∞), and on consecutive H (Theorem LN). It is still not a cover of all remainders: remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (a WC B with λ → ∞, extraH > 0 at every large prefix, longest run < next, and max-symmetric non-consecutive H, e.g. {1, 2, 4, 6}, would escape). None is known. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Lemma Inc bounds the increment by L − 1 there.

Theorem 2′(L = 2) — now unconditional

Python, two-1s remainders with L = 2: extra = +1 from two 1s fires at k = 3; two 1s then all integers fires at k = 3; floor-S/2 after 2, 3 fires at k = 4. None is never-C.

Remainders with consecutive H of length L ≥ 3 are now under Theorem LN, not only a hunt. Remainders with non-consecutive H (extra = +1 from three 1s; mixed extra 0,1; dense width-2) still fire in every construction. The extraH > 0-forever escape for max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}) remains the condition in Theorem 2′. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2).

This is our argument, not a posted solution of Erdős #348. Official (2, 3) remains open. Zero-1 sequences are not covered by Theorem 2′ or by LN2. The leftover two-1s hole is remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H: a WC remainder with max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}), extraH > 0 at every large prefix, longest run < next, and λ → ∞. None is known. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Python, two-1s remainders with L ≥ 3: extra = +1 three 1s, dense width-2, mixed extra 0,1, 1, 3, 4, 5 extra = +1, floor S/2 after 3, 4, 5 all fire C (and extraH = 0 i.o.). Greedy B = 3, 4, 5, 6, … is never-C with extraH = 0 i.o. and lim inf λ = 3, not 1-robust as a remainder.

First-hole width, again

Constant extra C with two or three 1s: the first hole of B = A ∖ {two 1s} has width C (width 0 if C = 0 and a third 1 survives). Python, N = 24:

No integer C between 1 and 2. Width ≥ 2 can freeze if the tail is dense (1, 1, 1, 4, 5, 6, 7, …: two 1s + late FROZEN miss 2), but then every tested 3-deletion freezes as well (AP-like slack). Width ≥ 2 with a greedy extra = C tail grows — and already the 2-deletion of two 1s grows, so not 2-robust. The sliver does not live here.

Lemma G — gcd / odds obstruction (zero and one 1s)

One 1 forces a hidden (1, 2) problem. If A has a unique 1 and is 2-robust, then A ∖ {1} is weakly 1-robust (every 2-deletion that includes the 1 heals). If A is also ∀-3-fragile, then every 3-deletion that includes the 1 and two other terms grows, i.e. A ∖ {1} is ∀-2-fragile. So A ∖ {1} would be a weak (1, 2) witness starting at a₂ ≥ 2. Brown–Weiss’s characterisation of strong 1-sequences requires a₁ = a₂ = 1. A weak (1, 2) starting at ≥ 2 is not known and may be empty; Fibonacci minus one 1 is 2-fragile as a sequence in its own right, so it cannot be the 2-robust A.

Hunt: zero 1s

Exhaustive 2-dels and 3-dels of length-8 prefixes, evaluated at N = 18. GROW 2-dels ⇒ not 2-robust.

The only 2-robust object in the hunt is the dense AP tail starting at 2. By the AP-tail lemma of §8, every finite deletion from that tail remains weakly complete, so it is k-robust for every k. Not ∀-3.

No zero-1 witness. Sparse tails fail 2-robustness (gcd, or φ/binary hole replication). Dense tails are 3-robust.

Hunt: one 1

Same trap as zero 1s, plus Lemma G on the unique 1: deleting it together with an early even leaves gcd 2 whenever the tail is even. The only 2-robust one-1 objects found are dense (all integers, or 1 then all ≥ 3), hence 3-robust by AP-tail. Doubled binary looked like a one-1 candidate and is actually three 1s; its nine GROW triples are concentrated doubled windows, every well-separated triple heals — forum multiplicity on powers of 2, ∃-3 not ∀-3.

No one-1 official witness.

Hunt: weak (1, 2) starting at a₁ ≥ 2 or a₂ ≥ 2, N₀ = 12 (and 14)

Need: weakly complete, every 1-del weakly complete, every 2-del unbounded miss.

Lemma Trap / Theorem Trap — C + skip leftover → +∞ is not ∀-2

Python, late pairs among the last 5 indices of a length-12 prefix: freeze on all-integers-from-2/3, 2-then-odds, primes, odds-from-3, 1-then-odds, narayana B, padovan B, 1, 2 extra = +1, extra = +1 two-1s (14/17 predicted-freeze families). Three near-φ families (extra = +1 remainder; floor-S/2; extra = +1 from 2, 3) have consecutive late pairs GROW-PERSIST — leftover after two consecutive large terms need not stay ≥ 2L — while well-separated pairs freeze. Those are ∃-2, not ∀-2. Fibonacci (skip leftover ≡ 0) is outside the hypothesis and is ∀-2, as it must be.

Theorem-side obstruction, one level down. The corollary is the L = 2 case. For general L: a weak (1, 2) witness is 1-robust, hence λ → ∞ (A′). If it has extraH = 0 i.o., Theorem N says C fires. If C fires and skip leftover → +∞, Trap says not ∀-2. Therefore a (1, 2) witness with extraH = 0 i.o. must have skip leftover bounded, i.e. φ-like tightness. Fibonacci does this with two 1s (skip ≡ 0, C fires at k = 1 using H = ∅). Starting at ≥ 2:

• Brown-equality from 2, 3 is not even WC.
• Fib-from-2 is not WC and is 1-fragile.
• Greedy-minimal from ≥ 2 has bounded λ, hence is 1-fragile by A′.
• Dense 1-robust starts (all integers from 2, primes, odds) have skip → +∞ and are 2-robust (too dense).

The remaining (1, 2) obligation at ≥ 2, beyond L = 2, is a 1-robust sequence with L ≥ 3, extraH > 0 at every large prefix, never-C or skip-bounded. No example. The hunt below is that search, plus the computational dichotomy.

Exhaustive 1-dels and 2-dels of length-12 prefixes (C(12,1) = 12, C(12,2) = 66), evaluated at Nlong = 20; mixed rows also at N₀ = 14 (C(14,2) = 91). New generators versus the length-8 hunt: primes+composites every 2/3/4; lucky numbers; squares mixed with odds; 2, 3 then odds; mixed extra patterns; floor-S/q; pick-hole-in-B; densify-greedy; greedy from 2, 2, 3 and 2, 2, 3, 5. No POSSIBLE / CANDIDATE weak (1, 2) at ≥ 2, at N₀ = 12 or N₀ = 14. Among 28 families at N₀ = 10: 1-rob never-C count = 0. Exhaustive length-8 prefixes starting at ≥ 2 with next in [last, min(1+S, last+4)]: 90625 prefixes, 89589 1-robust, 0 with all 2-dels GROW, 0 mixed-2 in the sample. Dense short 1-robust prefixes are 2-robust.

1-robust and 2-robust (too dense): all integers from 2 or 3; 2 then odds; 2, 3 then odds; 2, 3, 5 then odds; primes from 2 or 3; primes+composites every 2/3/4; odds from 3; lucky from 1, 3, 7; squares-odds; 1 then odds; 1 then all from 3; 1, 3 then odds; 1, 3, 5 then odds; doubled extra = +1 from 2, 3; doubled primes; padovan minus a 1; floor-S/3 from 2, 3, 5; 2, 4, 5, 7, 8, 10. AP-like. Not ∀-2.

1-fragile: Fib from 2; greedy 2, 3, 4, 8; greedy 2, 2, 3, 6; greedy 2, 2, 3, 5, 6, 17; 2, 3 extra = 0,1,2; 3, 5, 7 extra = 0,1,2; 2, 3, 5 extra = 0,1,2; 2, 3, 5, 7 extra = 0,1; mixed extra 0,2 from 3, 5, 7; densify-greedy; pell/lucas-from-2; B extra = +1 two-1s remainder (3 of 12 1-dels GROW — the 2-robust remainder of extra = +1 is not 1-robust as a sequence in its own right, same as at length 8, now at N₀ = 12); 1, 3 extra = 0,1; 1, 3, 5 extra = 0,1; mersenne; 2^k + 1; Fib minus one 1; greedy 1, 3 and 3, 5 and 2, 3, 5; trib from 1,1,2; 3, 4, 5, 6 extra = +1; 2, 3, 4, 5 extra = +1; Brown-equality from 2, 3.

1-robust, mixed 2 (not ∀-2): 1, 2 extra = +1 (2-dels 44/16/6 at N₀ = 12; 65/20/6 at N₀ = 14); 1, 2 extra = +2 (35/24/7; 54/30/7 at N₀ = 14); mixed extra 0,1 from 2, 3 (0/56/10); mixed extra 0,1,2 from 2, 3 (0/64/2); mixed extra 1,0 from 2, 3, 5 (0/57/9); floor-S/2 from 2, 3 and from 2, 3, 4 (0/63/3); pick-hole-B 2, 3, 4 (0/63/3); 2, 3, 4, 5, 6 extra = 0 (0/57/9); Narayana remainder (35/24/7); growing n/4 remainder (27/32/7). ∃-2, not ∀-2. The grow pairs are typically early+late or two lates; well-separated 2-dels often freeze.

Same extra-density trap, one level down, on a larger generator list: sparsest WC starts (greedy, including extra 2s) have bounded λ and fail 1-robustness by A′; interpolants that freeze 1-dels freeze most 2-dels and leave a few grow pairs (not ∀-2); densifying until all 2-dels freeze is AP-like. φ-without-two-1s is not WC. One-1 official (2, 3) still reduces to this (1, 2) problem (Lemma G + Theorem N on the extraH = 0 side); the L = 2 case is L2∞ + Trap, and consecutive H is Theorem LN. Remaining: skip-bounded (1, 2) at ≥ 2, and remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (1-robust never-C with max-symmetric non-consecutive extraH > 0 forever, e.g. {1, 2, 4, 6}). No example of either. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Skip-bounded Fib+noise from ≥ 2: every 1-robust family tested fires C and is mixed-2 or 2-robust; Fib-from-2 is not WC.

Hunt: odd-rich sparse (2, 3), fewer than two 1s, N₀ = 12 (and 14)

Lemma G requires ≥ 3 odds. Exhaustive 2-dels and 3-dels, C(12,2) = 66, C(12,3) = 220. No odd-rich sparse official (2, 3) witness.

2-fragile (typical sparse outcome): 1, 3 extra = 0,1; 1, 3, 5 extra = 0,1; 1, 3, 5, 7 extra = 0,1; mixed extra from 1, 3 and 1, 3, 5; mersenne; 2^k + 1; 2, 3, 5 extra = 0,1,2; 2, 3, 5, 7 extra = 0,1; mixed extra from 2, 3, 5; 3, 5, 7 extra = 0,1; 3, 5, 7, 9 extra = 0; 1, 2 extra = +1 (already mixed-2 as a (1, 2), hence 2-fragile as a (2, 3)); slack-binary; floor-S/2 from 2, 3, 5 and 3, 5, 7; densify-greedy 2, 3, 5; pick-hole-B 2, 3, 5 and 3, 5, 7; lucas-ish. Enough odds to beat Lemma G, not enough density to freeze 2-holes.

2-robust and 3-robust (too dense): 1 then odds; 1, 3 then odds; 1, 3, 5 then odds; 2, 3 then odds; 2, 3, 5 then odds; 2, 3, 5, 7 then odds; odds from 3; 3, 5, 7 doubled; lucky from 3 and 7; squares-odds from 3; primes from 2 or 3; primes+composites every 2/3/4. Frozen 2-dels and 3-dels at N₀ = 12 (0/66/0 and 0/220/0) and at N₀ = 14. AP-like.

Primes at longer prefixes. At N₀ = 8, primes were 2-robust ∃-3 (3-dels 0/51/5), the five GROW triples being the finite-prefix artefact (imiss increasing by 2). At N₀ = 10 already no GROW 3-dels; at N₀ = 12 and N₀ = 14 all 220 resp. 364 triples freeze. Not a witness; the artefact died. Same for primes-from-3, primes+composites, 1-then-odds, 2-then-odds, lucky-from-3.

The sliver between “2-fragile because sparse” and “3-robust because dense” is empty in this hunt, including mixed extra, primes+composites, lucky numbers, floor-S/2, and pick-hole-B. Zero-1 dense AP from 2 remains 3-robust by the AP-tail lemma of §8. Unchanged.

Certificate inheritance, checked

Two 1s + late, leftover at the skip λ_r+1 − a_r − 2, and whether B′ inherits B’s certificate:

The skip leftover is the quantitative difference between 2-robust remainders and Fibonacci. On Fib it is identically −2; on every 2-robust family it is eventually large.

What is not claimed

Not a solution of official #348. The case (m, n) = (2, 3) remains OPEN. Strong completeness is already impossible for m ≥ 2. This section does not close the weak case.

Not Theorem LN for every finite H, and not L-cons. Consecutive H remains closed (v11: L2∞, L3-never-C, Join-sums for L ≥ 5). extraH = 0 i.o. is not claimed for incomplete / max-symmetric non-consecutive H (remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed). Lemma Δ is now extras = a+Δ (132/132; v12’s 82 bads were a low-window filter). Climb is not uniformly FIRE (FIRE=27, RESTORE=10, OPEN=60 on 132 prefixes; POST-CLIMB OPEN-sum=580). Not sequential Restore-W: FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2; consecutive pair in Δ lengthens the run). Not algebraic Restore-W: FALSE (2a leftover, 2, 5, 8+9). Stay0 / Greedy-available is proved for last ≥ L. Climb-shrink / Room-1 / Type-iii finite / Frozen-g-die / Mig-set / Extra-set / Near-Δ / Remaining-W / Fill-W-item / Post-W / W-chase finite / (⋆)-fills-2a-by-g are proved (iii-SHRINK chains ≤ δ − 1). Migrate i.o. after filling g is not proved empty (extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution). Official (2, 3) remains OPEN. No witness.

Not a proof that every weakly complete sequence with λ → ∞ has a Lemma-C prefix, in full generality. Proved for L = 2 (Theorem L2∞). Proved when extraH = 0 infinitely often for any L (Theorem N / Nwin). Consecutive H is still closed (LN not claimed for incomplete H). Proved in overlap (F2/F3) and in B with extraH = 0 (F3B). B-ultrafast never-C cannot persist (F5). Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (extraH > 0 at every large prefix, never-C, WC, λ → ∞, max-symmetric non-consecutive H, e.g. {1, 2, 4, 6}). No example. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Lemma Inc bounds the increment by L − 1 there.

Not a proof that every 2-robust sequence with two 1s has a healing 3-deletion, in full generality. Corollary LN2 is our argument that the two-1s class is empty as official (2, 3) sources under consecutive H, extraH = 0 i.o., L(B) ≤ 2, or overlap/F3/F3B — not a posted solution. Remainders with max-symmetric non-consecutive extraH > 0 forever (e.g. {1, 2, 4, 6}) would escape; every tested one fires or restores. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H.

Not a proof that there is no weak (1, 2) starting at ≥ 2, in full generality. L = 2 never-C is excluded from 1-robustness (L2∞). Consecutive H never-C is excluded (Theorem LN). extraH = 0 i.o. is excluded (Nwin). 1-robust + C + skip leftover → ∞ is not ∀-2 (Trap). Remaining: skip-bounded (1, 2) at ≥ 2, and remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (1-robust never-C with max-symmetric non-consecutive extraH > 0 forever, e.g. {1, 2, 4, 6}). No example. Climb is not uniformly FIRE. sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2). Exhaustive dense length-8 and the N₀ = 12/14 hunt are empty. Lemma G kills fewer than three odds. One-1 official (2, 3) still reduces to that (1, 2) problem.

Never claimed unverified. Consecutive H is still closed. LN is not claimed for incomplete H. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution). Computational here: Frozen-g-die; Mig-set 369/369; Extra-set 73/73 and 900/900; Near-Δ census 108/108; Remaining-W / Fill-W-item 792/792; Post-W 11/11; W-chase finite; Type A peel 9 / Type B leftover-2a 3; (⋆)-fills-2a-by-g 90/90 when (⋆) holds; FILL-G leftover 70 / restore 38 / fire 0; iv-MIGRATE 12 room grow 3 stay 9, Pal-pool live chain_max 2 was an artefact, extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution; fillg RESTORE 49 FIRE 33 OPEN 0; 47/47 room=1 leftovers dead (greedy 3 then FIRE); Minkowski leftover extras not a lemma; Climb-shrink shrink 87/87, Type-iii finite (iii-SHRINK 76), iv-MIGRATE 12 forever_np=0, Lemma Δ set equality 132/132 (v12’s 82 bads were a filter; Δ-empty restores Pal 8/8), Climb-window 132/132, Climb not uniformly FIRE (FIRE=27 / RESTORE=10 / OPEN=60 / no-climb-room=63), sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2), POST-CLIMB OPEN-sum=580 (v12 open_fates = 0 was one named family). Proved earlier: consecutive-H cycle (L2∞ / L3-never-C / Join-sums), Lemma FT, Lemma Res, Lemma Nwin, Lemma Seed, AP-clear, Lemma Plateau, Corollary LN2 (two-1s class empty as official (2, 3) sources under the stated hypotheses; our argument, not a posted solution), Theorem L2∞ / L2N, Lemma N2, Lemma L2-cycle (1)(2) and AP-fill in (4), Theorem 2′(L = 2) unconditional, Lemma Inc / BI, Lemma Trap, Theorem N / N1 / Corollary N2, Lemma F3, Lemma K / K2, Lemma CW, Lemma F3B, Lemma F5, Lemma L2 (prefix; next = 7 tails computational), v8 F/F1/F2 and A′, Lemmas C–E, G, Theorem 2 (conditional), Theorem 2′ (conditional for max-symmetric non-consecutive extraH > 0 forever: overlap, L = 1, extraH = 0-in-B, extraH = 0 i.o., consecutive H), the theorem of §4 and Lemmas A–B of §5 (unchanged), greedy-from-2 closed form, extra = C first-hole table. Cycle table, DFS, (1, 2) dichotomy, exhaustive length-8, L ≥ 3 walks, certificate tables, stay-in-B scan, the (1, 2) hunt from a₂ ≥ 2, skip-bounded Fib+noise, and the odd-rich sparse (2, 3) hunt: computational, prefixes as in the log. Remaining (v22): extraN > 1 leftover after remW empty (census restore, not a theorem); Post-W c+H (census restore, not a theorem); never-hit-Pal; skip-bounded (1, 2); zero-1s. Two-n Pal-then-+1 is dead as a theorem (Two-n-plus1-dies / Gap-r / t*). Pal-pool chain_max 2 was an artefact, not a bound. Live migrate-chain ≥3 exists then FIRE. leftover-2a singleton / unique-mid fill-g restores. Official OPEN. LN not claimed. Not a solution); LN not claimed for incomplete H (sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); algebraic Restore-W is FALSE (2a leftover, 2, 5, 8+9); Stay0 proved; extraH > 0-forever on max-symmetric non-consecutive H has no example). Δ set equality 132/132 (v12’s 82 bads were a filter); sequential Restore-W is FALSE (witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67 FIRE extraN=2); Climb is not uniformly FIRE. Consecutive H is still closed. Zero-1 sequences; skip-bounded (1, 2) at ≥ 2. Official infinite weak (2, 3) remains open. Nothing here is claimed as a solution.

7. Theorem (weaker): three 1s and the even positives

This is not official #348. The 3-fragility is existential, not universal. The argument occupies the rest of this section.

Completeness of A

Undeleted prefixes have interior miss 0. Brown holds: the next even is 2k + 2 ≤ 1 + 3 + k(k + 1).

∃-3-fragility

Prefix counts after this deletion:

2-robustness

Exhaustive 2-deletions of prefixes confirm the list is closed: the number of pairs with miss > 0 is exactly 8 at every N ∈ {16, 20, 24, 28} (three choices of which 1 pairs with 2, three with 4, plus (2, 4) and (4, 6)). No new 2-failure appears as the even tail lengthens. All other pairs among the first N terms, save those eight, have miss = 0.

The odd-count threshold

Three odds is exact.

8. Why this is not official #348

Official #348 wants every 3-deletion to destroy weak completeness. The sequence A fails that. Only 3-deletions that remove all three odds produce unbounded misses. The others heal.

Delete (2, 4, 6): the interior miss is {4, 5, 6, 7}, frozen at miss = 8 (mirrors included) through N = 32. Delete three large consecutive evens: miss = 0. Delete two 1s and one late even: miss = 0.

This is not an accident of the even tail. It is forced by any arithmetic progression as a tail.

Adding a second modulus enlarges the finite list of modular breaking triples and does not produce ∀-3. For

A = {1, 1, 1, 2, 2, 2} ∪ 6ℕ,

residues modulo 6 give eleven modular breaking triples (the three 1s; the three 2s; nine of type (1, 2, 2)) and no others. Exhaustive 220 triples among the first 12 terms, interior miss tracked to N = 36: 11 grow with persistence 1, matching the residue table; 203 heal. Two-mod is 2-robust (any 2-deletion leaves a small remainder that still hits every residue modulo 6; the tail is 6ℕ minus at most two terms). It is a strictly larger ∃-3 object than A, and still not ∀-3.

The official (2, 3) case is therefore untouched. ∀-3 remains open.

9. The density trap

To break tail 3-deletions one needs a non-arithmetic (typically exponential) tail. Making that tail sparse enough for 3-deletions to propagate gaps typically makes some 2-deletions propagate as well. Extra density that stops 2-deletion gap growth also stops 3-deletion gap growth — unless the 3-failure is modular, or the sequence is strictly faster than tight 2-Brown by one unit. The modular exception is the gcd construction above, and it cannot be ∀-3 by the AP-tail lemma. The one-unit exception is computational and still has healing triples.

Qualitatively, along linear recurrences:

• Ratio φ (Fibonacci): 1-robust, 2-fragile via gap propagation, even though tail Brown slack tends to infinity.
• Ratio < φ (Narayana, Padovan, a_n = a_n−1 + a_n−4): 2-robust, and 3-deletions leave only finite holes — slack absorbs any fixed deletion and extra representations fill gaps.
• Occasional φ-sized jumps: some 2-deletions start propagating gaps again.
• Multiplicity on Fibonacci: 2-robust, and most 3-deletions heal; fragility returns after deleting both copies of a doubled value together with the successor, and not after a generic triple.

Tight 2-Brown from three 1s is Narayana’s cows, a_n = a_n−1 + a_n−3,

1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, … ,

ratio the supergolden root of x³ = x² + 1 ≈ 1.465 < φ. Identity σ_n = a_n+3 − 1; slack when adding a_n+1 is exactly a_n + a_n−1. Every tested 2-deletion has miss = 0 (all 91 pairs among the first 14 indices on a length-24 prefix). Every tested 3-deletion has frozen finite miss, including mid-index triples with a long tail. Narayana is a strong 2-sequence that fails 3-fragility even in the weak, existential sense.

Shifting the 2-Brown recurrence by one unit of extra,

a_n+1 = max(a_n, 1 + σ_n − M₁ − M₂ + extra),

from three 1s: extra = 0 is Narayana; extra = −1 is denser and 3-robust on the tested triples; extra = +2 is not 2-robust; extra = +1 is the greedy sequence 1, 1, 1, 3, 4, 5, 8, 12, 17, … with tail identity a_n = a_n−1 + a_n−4 after the skip of 2. After deleting the three 1s, interior miss grows with persistence 1 through σ ≈ 1.6·10⁶ (not a modular obstruction: gcd remains 1). Scattered triples heal (miss = 0). This is a computational weaker-(2, 3) witness, not a closed-form proof that the missing set is infinite, and not official #348.

The same trap appears when one thins 6ℕ to beat the AP-tail lemma. Greedy 2-Brown rounded up to a multiple of 6, and the geometric tail {1,1,1,2,2,2} ∪ {6·2^k}, both make some consecutive tail triples grow — and both make 2-deletions of the tail grow. Density that protects 2-deletions fills 3-deletions of the tail; sparsity that breaks tail 3-deletions breaks some 2-deletions.

10. What was tried and failed

No construction below is a solution of official #348. Several are intermediate objects: 2-robust, with some (even infinitely many) breaking 3-triples, and a majority of healing triples.

11. What remains

Official Erdős #348, case (m, n) = (2, 3), is OPEN for weak completeness. Strong (2, 3) is already impossible. Nothing in these notes is a posted solution. LN is not claimed.

The remaining gap is not “max chain 2.” Pal-pool live chain_max=2 was a pool artefact. Live migrate-chain ≥3 exists (Two-n peel, witness [2, 13, 15, 16, 17, 18, 19, 20]) and then FIRE. The remaining gap is not “[2, n, n+2] Pal then +1.” Theorem Two-n-plus1-dies: that walk dies (n=6 restore, n≥7 FIRE). Finite chains ~√n are not a witness. leftover-2a extraN=1 and unique-mid fill-g restore.

What is still open, stated without meekness and without a fake close:

1. Migrate infinitely often without firing C, after remW is empty. Frozen-g-die forces any extraH>0-forever WC never-C with λ → ∞ to jump g infinitely often. W-chase finite closes remW-migrate (at most |Δ| jumps, then g ≥ 2a). After Post-W, high extras and Extra-set c+E remain. CE-blocked-by-near: while remW survives, a fresh c+E cannot be least extra. After remW is empty, c+E doubling is available. remW-emptying LIVE 33 in v22 (2 leftover-2a + 31 c+E) is a census, not a death lemma. Extra-set is containment, not a Lyapunov.
2. Never-hit-Pal / extraH-start fill-g LIVE. Walks that never take Pal λ=L, or that start already extraH>0. Stay0 says greedy Pal is legal on extraH=0 prefixes with last ≥ L, not that it is taken. Incomplete max-symmetric non-consecutive H (e.g. {1, 2, 4, 6}) has no extraH>0-forever WC never-C example in the log.
3. ExtraN>1 leftover-2a after remW emptying. ExtraN=1 leftover {2a} restores (Fill-2a-singleton-restores: 12/12, LIVE5 5/5). ExtraN>1 is not that lemma.
4. Skip-bounded weak (1, 2) from a₂ ≥ 2. Dense hunts at N₀=12/14 are empty. L2∞ and Trap close other (1, 2) routes. Skip-bounded remainder is a hunt.
5. Zero-1 official (2, 3). Lemma G: weakly 2-robust ⇒ at least three odds. Hunts empty. Not a posted close of zero-1s.

Obstacles that did not move:

1. Gcd constructions kill only the triples that strip every odd (or every residue coprime to d). All other triples heal. The AP-tail lemma rules out every finite generator plus an arithmetic tail.
2. 2-Brown sequences. Theorem 1. Extra = C threshold: C ≤ 1 is 2-robust and two 1s + late heals; C ≥ 2 two 1s GROW, not 2-robust. No integer lies between. Extra = +1 still has healing triples computationally. The gap between “infinitely many breaking triples” and “all triples break” remains the whole difficulty outside 2-Brown.
3. Finite prefixes can never certify “every 3-deletion”: deleting the last three terms of a prefix is not a statement about the infinite sequence, and any Brown-complete prefix of length ≥ 3 is complete.
4. The density trap. Density that protects every 2-deletion fills 3-deletions of the tail; sparsity that breaks every tail 3-deletion breaks some 2-deletion.

A genuine official (2, 3) would need a representation system in which every 3-set seeds unbounded gap propagation while no 2-set does. No such system was found. Every 2-robust family tested fires or restores extraH=0 i.o. or fails WC. That is a hunt plus the theorems above, not a close of Bloom’s problem.

12. What is not claimed

• Not a posted solution of Bloom/Erdős #348. Official (2, 3) remains OPEN.
• Not Theorem LN for every finite H. Consecutive H is closed. Incomplete / max-symmetric non-consecutive H is not.
• Not a proof that every weakly complete sequence with λ → ∞ has a Lemma-C prefix, in full generality.
• Not a proof that every 2-robust sequence with two 1s has a healing 3-deletion, in full generality. Corollary LN2 is our argument under stated hypotheses, not a posted solution.
• Not a proof that there is no weak (1, 2) starting at ≥ 2, in full generality. Skip-bounded remainder is open as a hunt.
• Not sequential Restore-W. FALSE. Witness 3, 8, 9, 10, 13, 14, 15+65, 66, 67, FIRE extraN=2.
• Not algebraic Restore-W. FALSE. 2a leftover, 2, 5, 8+9.
• Not Climb-kill. Climb is not uniformly FIRE.
• Not Climb-shrink as a global Lyapunov. Room can grow on migrate (5→10, 6→12).
• Not Extra-set as a death. Containment, not a Lyapunov.
• Not Minkowski of leftover extras as a finish lemma.
• Not “max chain 2” as a bound. Pal-pool artefact. Live migrate-chain ≥3 exists then FIRE.
• Not “[2, n, n+2] Pal-then-+1 lives forever.” It dies (theorem). Finite chains are not a witness.
• Not that remW-emptying LIVE 33 die. Census, not a theorem.
• Not a witness extraH>0-forever WC never-C with λ → ∞. None in the log.

Proved statements that do stand are listed in §11. None of them is a witness for Bloom’s problem. Official infinite weak (2, 3) remains open. Nothing here is claimed as a solution.

T. F. Bloom, Erdős Problem #348, https://www.erdosproblems.com/348. LaTeX source: https://www.erdosproblems.com/latex/348. Formalisation: FormalConjectures/ErdosProblems/348.lean. Brown–Weiss, On N-sequences, Math. Mag. 44 (1971). Graham, Fibonacci Quart. 2 (1964). van Doorn on strong (m, n).