# L and κ(m): the proven bounds, and why L is not the target *Parent: `research/U-FRAME.md` §10, which carries these claims in summary. Three proven theorems on L, the lemma that makes them, and the two reasons the bound cannot reach the branch the route needs.* L, the longest adjacent-kill run, is the parameter in U-FRAME §5a step 3's upper bound; κ(m), the maximum kill count in a window of m consecutive new gaps, is its generalisation to the whole family (U-FRAME §5a step 3a), with κ(1) = L. Both are governed by the criterion that two adjacent slots at distance g can both be deleted by the fold at p only if g ≡ 0 or ±2 (mod p), PROVEN in one line from the strike classes {0, −2}, hence by f, the fraction of gaps meeting it, which `research/f-decays.md` measures. This note is the status of both: what is proven, where the proofs stop, and why the object to aim at is κ(m). ### The qualifying gaps, in closed form (PROVEN and exact) Grain gaps are multiples of 6, so d = 6k qualifies iff k ≡ 0, ±3⁻¹ (mod p). The qualifying set is therefore three arithmetic progressions of modulus 6p carrying weights 1, 1, 2: > **{2p−2, 4p+2, 6p} when p ≡ 1 (mod 6), and {2p+2, 4p−2, 6p} when p ≡ 5 (mod 6)** so the smallest qualifying gap is exactly **2p ∓ 2**, not "about 2p". The weight 2 on the p-divisible size is a multiplicity: d ≡ 0 admits both kill patterns (0, 0) and (−2, −2), so two of the p copies realise it, while d ≡ ±2 admits one. Checked against brute force at all 302 primes from 5 to 1999, derived twice by routes that share nothing (A3's census argument and A8's pair count), and it reproduces U-FRAME §5's qualifying-gap table at all eight folds. **It also explains the fold-11 anomaly, which is a census accident and not an exclusion.** The law predicts a single qualifying value 24 below G₂(T₇) = 30, while U-FRAME §5 records fold 11 as having no qualifying gap at all. Both are right: 24 is legal, and T₇'s gap set is {6, 12, 18, 30}, which contains no 24. ### The Alternation Lemma (PROVEN, and it is the prize) A run is a walk on two states: from residue a only gaps ≡ 0 or −2 are legal, from a−2 only 0 or +2. So the class word is a two-state walk rather than merely a word avoiding three residues, and **along a run the non-zero class gaps must strictly alternate** between class +2 and class −2. One class is small (2p∓2) and the other large (4p±2), and they sum to exactly 6p. Hence L ≥ 3 forces a gap of at least 4p−2; L ≥ 2k+1 forces k of them; and the run's internal gaps average at least 3p − p/(L−1). **The operative threshold is 3p on a window, not 2p on a single gap.** In quantitative form, with no independence assumption: #(3-windows) ≤ 2(min(N_P, N_M) + N_Z). **ATTRIBUTION (2026-08-19): the LEMMA is ours, the LANGUAGE FAMILY is printed.** The two-state walk this lemma defines — non-zero marks strictly alternating in sign — is a textbook constrained-coding object: it is the **B = 1 charge constraint**, equivalently **alternate-mark-inversion**, in Marcus–Roth–Siegel **§2.3 p. 47**, with its capacity in their **§3.2 p. 75** table. So the strict-soficity of the constraint graph and its capacity ln 2 are REPRODUCTIONS of printed results and may not be presented here as new structure. What stays ours is the lemma's arithmetic content: that this is the constraint the fold imposes at all, the 3/p → 2/p rate correction, the weight-(1,1,2) multiplicity that puts the small class at 2p ∓ 2 and the large one at 4p ± 2 summing to exactly 6p, and the wall address below (`IMPORT-MAP.md` row 2, `history/staging/import-sofic.md`). **The kill graph makes all of it computable on the old gap word alone.** Take 2D nodes (i, σ), one per kill, with an edge (i, σ) → (i+1, σ′) iff σ′ − σ ≡ g_i (mod p). Both degrees are at most 1, so the graph is a disjoint union of paths: edges are the adjacent-kill pairs, components are the maximal runs, and L is the largest component. All of it is O(D), with no fold in memory. **And it explains the fold-29 dip, which nothing else did.** At fold 29 the classes are +2 → 60 and −2 → 114, and every one of the 288 adjacent qualifying-gap pairs in T₂₃ is (60, 60), both in the SAME class. Alternation forbids all of them, so no run of three exists and L = 2. The dip is forced by the lemma, not by scarcity. VERIFIED: every long run in the data is a strict alternation, 48+90 and 90+48 at fold 23, 60+126 and 126+60 and 60+126+60 at fold 31, never 60+60. `research/operator-and-pair-count.md` carries the same check at fold 37, the deepest we have. ### The proven bounds on L, and their transfer to κ(m) **Theorem A (Run Cost, PROVEN; VERIFIED at ten primes).** Any two ADJACENT gaps of a run sum to at least 6p, because min(class +2) + min(class −2) = 6p exactly. Hence the span of a run of length L is at least c_min(L−1) ≈ 3p(L−1). **Attained with equality at every fold with L ≥ 2, all seven.** At fold 31 the extremal run is 60, 126, 60 = 2p−2, 4p+2, 2p−2, whose adjacent sums are 186 = 6p to the unit. **Theorem B (PROVEN, the first unconditional bound on L).** L ≤ 1 + max{m : maxsum_m(T_x) ≥ c_min(m)}. It gives 2, 2, 2, 4, 4, 4, 5, 6 against the true 2, 1, 2, 2, 2, 3, 2, 4. The run-consistency condition alone would give 3, 2, 8, 5, 11, 8, 10, 13, so the alternation condition roughly halves the bound. **Theorem C (PROVEN).** The argument never used adjacency, so it transfers to κ(m): κ(m) ≤ max{k : maxsum_{m+k−2} ≥ c_min(k−1)}. Verified in all 56 cases. ### The wall, located precisely The alternation condition is worth exactly 3/2 and no more: it raises the per-gap floor from about 2p to exactly 3p, and the extremal runs attain it. That changes the constant, not the branch. **Theorem B is structurally capped**, since maxsum_m ≥ G₂ always, so it can never prove L below G₂/(3p) ≈ 0.18x, which is LINEAR, the failing branch; its projection clears x² only to about x = 100. Theorem C inherits the ceiling, so redirecting L to κ does not move the wall. What remains is a decay hypothesis on runs of large gaps, and there are two sharp reasons it is hard. It is a two-dimensional lower-bound sieve at sifting parameter 1+o(1) against the dimension-2 sifting limit β₂ = 4.26645 (`research/dhr-verification.md`, rigorous, Booker and Browning). And **no polynomial moment of any fixed order can give it**, because Markov at order k bounds the window fraction below by (m̄/3p)^k independently of m, threshold and mean both scaling linearly in m. Decay needs exponential moments, that is quasi-independence. Measured, Markov sits flat at 0.32 to 0.48 while the true fraction falls one to two orders per step. *(CORRECTED 2026-08-18: this read "0.32 to 0.44", which drops fold 7's 0.4762. The band came from `a3-05-bound-L.js` reading 6 and so did the identical band in `research/a3-05-bound-L.md` section 7, so the two sites agreeing was descent, not corroboration, and a prior wave marked both clean on the strength of it.)* **The cap is more general than A5, and that is the bad news.** `research/gate-multiplies.md` §§2, 5 shows it is exactly the linear cost feedback of the no-fixed-point argument, so Theorem B is the supplier of the hypothesis that closes every chain fed by it, and the Overshoot Budget then closes the whole accumulating-index family on the tile. ### The lower bound never becomes exact, so L cannot be made irrelevant Exactness of G₂(new) = maxsum₂(old) holds at 3 of the 9 ladder folds (11, 13, 19) and over a grid of 329 (tile, prime) cells at rate 0.927. **It does trend, but not along the ladder.** Binned by 2p/m̄ the exactness rate is 0.44, 0.39, 0.81, 0.91, 0.96, 1.000, clean and monotone. But the fold ladder is pinned near 2p/m̄ ≈ 2, the worst band, moving only 1.40 to 2.30 across folds 7 to 37, and on a Mertens fit it reaches the safe band 2p/m̄ ≥ 4 only near x ≈ 130, a tile of width 10⁵⁷. The excess over the lower bound runs 6, 0, 0, 12, 0, 18, 24, 18, 120: it does not shrink, and the deepest reachable fold is by far the worst. **Half the failure has a closed form (PROVEN, VERIFIED).** A miss whose record carries a qualifying gap happens **iff p divides one of G₂(old), G₂(old)−2, G₂(old)+2**: the record gap is then itself deletable at both ends, so the merge takes both its neighbours. Predicted tile by tile with no errors and no false alarms: T₅{7}, T₁₃{17}, T₁₇{53}, T₁₉{37}, T₂₃{101, 103}, empty for T₇ and T₁₁. It is the sufficient half only, catching 6 of 24 misses; L ≥ 2 is necessary but over-flags 45 cells. The real criterion is a size question rather than a residue one: whether the best three-gap sum anchored at a qualifying gap beats the best two-gap sum anywhere. **Why that closes the route outright.** The divisibility event has chance about 3/p per fold, so it recurs forever, about 3·lnln x times. **Exactness can therefore never hold from some level on**, and the hope of settling the growth law by making L irrelevant is dead. The second mechanism is statistical and does fade, since exactness is essentially certain once 2p/m̄ ≥ 4, but the arithmetic one does not, which is why the fading half cannot rescue it. **And no constant can replace the effective run length either.** The effective run length of U-FRAME §5a step 3 is the better coordinate, never worse than L+1 and strictly better at three folds, but a constant value for it cannot exist, for two independent reasons. The telescoping argument that would make it useful, Σ_{p≤x} m_eff·m̄(p), is U-FRAME §5a step 4, and step 4 is false. And even taken at face value it self-destructs: Σ_{p≤x} 3·m̄(p) ~ 7.2·x·(ln x − 1) falls **below** the measured G₂ from about x = 45, so it would certify a bound smaller than the truth. The data agrees directly, since the effective run length leaves 3 at fold 37, where G₂(37#) = 528 overshoots maxsum₃(T₃₁) = 510 and a run of three wins. ### Where the L question stands Theorem B is unconditional and structurally too weak, linear rather than polylog, and it names exactly why. `research/f-decays.md`'s census supplies strong evidence for the polylog branch and `research/operator-and-pair-count.md`'s operator makes the tail computable exactly at any level in reach, but the step from L to G₂ went through U-FRAME §5a step 4 and is missing. **The interval difficulty is now concentrated in one scalar, P(gap ≥ 2p), and the missing piece is no longer measurement but the valid replacement for step 4.** **So the target is κ(m), and inside κ(m) the rarer of the two ±2 classes at 4p**, a quantity 38 times smaller than the 2p count at fold 31 and 553 times smaller at fold 29. That is a narrower hole in a better coordinate, and there is about L + 2 of headroom to spend in it. **Honest limit:** the independence model is exact at L = 2, within 20% at L = 3, and over-predicts L = 4 by a factor of 70, and nothing here shows min(N_P, N_M) decays, so A5's hole remains open. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.* **Closed 2026-08-19:** L has no law of its own — it is G₂ divided by the block's own spacing, and it is super-additive on disjoint sets, so splitting gives floors and merging gives G₂ back (`research/sift-limit-attack.md` §7a-bis, §7a-ter); `L ≤ 62` supersedes the adjudicated `L ≤ 111` (`research/OUTCOMES.md`).