# Attack block 7: the block version of kappa(m) *(2026-08-18. Block attack 7. Object: kappa_block(m) over a block v -> v^2, the aggregated form of Theorem C of `research/a3-05-bound-L.md` section 8a. Calibration marked on every claim: PROVEN, VERIFIED by exact computation, MEASURED, REFUTED. Scripts in the attack's scratch directory, listed in section 7; nothing in `research/` was edited.)* ## 0. Verdict first **kappa_block(1) = 19 = combined L at block 1, and the two are equal by construction, not by accident. The tightness that kappa buys per fold does not survive aggregation: the minimal shift j*(m) equals kappa_block(m) at every m from 1 to 12, so there is zero slack anywhere in the block object.** **And the block form of Theorem C is not merely weak, it is EMPTY.** Aggregating the alternation law over the six primes of the block excludes no kill count at all, at any m, because the aggregation replaces a per-prime SUM by a per-prime MAX and what remains is capacity counting. That is trap 1 of the brief, reached by a different road. The vacuity coefficient at block 1 is 1.552 against a requirement of 1, and it grows like m_bar(v) ~ ln^2 v, so it is worse at every later block and permanently so. Worse still, it is dominated: plain counting of T_v slots, using no residue law at all, is smaller by the exact factor m_bar(v)/6 and does return a finite bound at block 1 (158, against the true 19). **The alternation law adds nothing to a block.** So a3-05's sentence "moving the target from L to kappa buys tightness and buys nothing against the obstruction" transfers to the block with one word changed: at the block it does not even buy tightness. This closes the kappa route at the block level. ## 1. The definition Fix v and the block v -> v^2. Let T_v be the twin slot tile mod v#, let Q = {p prime : v < p <= v^2}, and let P = v# * prod(Q) = (v^2)# be the block period. Composite folds in (v, v^2] are the identity (established, 136 composite folds and 0 live deletions), so Q is the whole block. A T_v slot r is **killed by the block** if r = 0 or -2 (mod p) for at least one p in Q. The **survivors** are exactly the T_{v^2} slots. Read the T_v slots cyclically around P and let > r_t = the number of killed T_v slots strictly between survivor t and > survivor t+1, for t = 1..D_{v^2} cyclically. Then, for m >= 1, > **kappa_block(m) = max_t ( r_t + r_{t+1} + ... + r_{t+m-1} ),** the largest number of T_v slots the block kills inside a stretch that becomes m consecutive new gaps of T_{v^2}. Equivalently and identically: take m+1 consecutive survivors s_0 < ... < s_m and count the killed T_v slots strictly between s_0 and s_m; kappa_block(m) is the maximum over position. **kappa_block(1) = combined L**, since a stretch that becomes one new gap is exactly a maximal run of consecutively killed slots. The consequence that makes it the right object, PROVEN in one line, is the block form of the A4 recursion. A stretch that becomes m new gaps spans exactly m + K old gaps when it contains K kills, so > **maxsum_m(T_{v^2}) <= maxsum_{m + kappa_block(m)}(T_v).** VERIFIED at block 1 for all m from 1 to 12; see the table in section 3. ## 2. The degeneration check **A block containing one prime must reproduce the per-fold kappa(m) of a3-05 section 8a.** The block code was run with Q = {p}, over the full period v#*p, at six single-prime blocks. This is the correctness test, so it is reported before any new number. | fold p over T_v | kappa_block(m), m = 1..8 | a3-05 reading 9 | agree | |---|---|---|---| | 7 over T_5 | 2 2 4 4 5 5 6 6 | 2 2 2 1 0 0 0 0 | m >= 3 differ | | 11 over T_7 | 1 2 2 2 3 4 4 4 | 1 2 2 2 3 4 4 4 | all 8 | | 13 over T_11 | 2 2 2 2 3 3 3 4 | 2 2 2 2 3 3 3 4 | all 8 | | 17 over T_13 | 2 2 2 3 3 3 3 4 | 2 2 2 3 3 3 3 4 | all 8 | | 19 over T_17 | 2 2 3 3 3 3 3 4 | 2 2 3 3 3 3 3 4 | all 8 | | 23 over T_19 | 3 4 4 4 5 5 5 5 | 3 4 4 4 5 5 5 5 | all 8 | **Five of six folds agree in all eight entries, and kappa_block(1) reproduces the published L diagonal 2, 1, 2, 2, 2, 3 at folds 7 through 23 exactly.** The degeneration is therefore established. **The sixth is a defect in a3-05's reading 9, not in the definition, and it is visible without any computation: the published fold-7 row DECREASES in m.** kappa(m) is a maximum of window sums over a fixed window family that grows with m, so it is nondecreasing in m by definition; 2, 2, 2, 1, 0, 0, 0, 0 cannot be a kappa profile. The cause is that T_5 carries three slots, `kappaProfile` extends it by EXT = min(N, 64) = 3, and its window loop is guarded by `t >= m - 1`, so for m larger than the number of runs seen it never fires and the entry stays 0. Over the true period 210 there are 21 T_5 slots, 15 survivors and 6 kills, so kappa is capped at 6 and reaches it at m = 7, which is what the block code returns. **The same short-tile fault corrupts a3-05's maxsum table, and therefore its Theorem C row at fold 7.** Reading 5 prints maxsum_m(T_5) = 12, 24, 30, 42, 48, 0, 0, 0. The truth, from the exact 6, 12, 12 gap word, is 12, 24, 30, 42, **54, 60, 72, 84**. Recomputing Theorem C at fold 7 with the correct maxsums gives 2, 2, 4, 4, 5, 6, 6, 8 against the published 2, 2, 3, 3, 2, 1, 1, 1, and the corrected bound holds over the corrected truth at every m, tightly at five of eight. **Scope of the defect, checked rather than assumed.** Only the T_5 row is affected: T_7 already carries 15 slots and its published maxsums are correct. **Theorem B is unaffected at fold 7**, because it only ever compares maxsum_1 = 12 against c_min(1) = 12 and maxsum_2 = 24 against c_min(2) = 42, both below the corruption. The L diagonal is unaffected. So this is a reading-9 and reading-5 presentation error at one fold, not a load-bearing one, and it is reported here because the degeneration check is what found it. ## 3. Block 1, measured v = 5 -> 25. Full period 23# = 223,092,870. T_5 slots at 11, 17, 29 mod 30. Folds at 7, 11, 13, 17, 19, 23. Exact, no sampling. Ground truth reproduced, all four: | quantity | measured | expected | |---|---|---| | T_5 slots in the period | 22,309,287 | 22,309,287 | | survivors | 7,952,175 | D_23 = 7,952,175 | | G2(23#) | 204 | 204 | | maxsum_20(T_5) | 204 | 204 | killed = 14,357,112, so the mean kills per new gap is 1.8054 and maxsum_19(T_5) = 192 < 204 <= 204 = maxsum_20(T_5). **The table.** j*(m) is the minimal shift, j*(m) = min{ j : maxsum_{m+j}(T_5) >= maxsum_m(T_23) }, which is the quantity the recursion actually consumes; kappa_block(m) is an upper bound for it. | m | kappa_block(m) | m*L | kappa_block/(m*L) | maxsum_m(T_23) | j*(m) | maxsum_{m+kappa}(T_5) | recursion holds | |---|---|---|---|---|---|---|---| | 1 | **19** | 19 | 1.000 | 204 | **19** | 204 | OK | | 2 | 21 | 38 | 0.553 | 234 | 21 | 234 | OK | | 3 | 27 | 57 | 0.474 | 300 | 27 | 300 | OK | | 4 | 31 | 76 | 0.408 | 348 | 31 | 354 | OK | | 5 | 34 | 95 | 0.358 | 390 | 34 | 390 | OK | | 6 | 40 | 114 | 0.351 | 462 | 40 | 462 | OK | | 7 | 43 | 133 | 0.323 | 498 | 43 | 504 | OK | | 8 | 45 | 152 | 0.296 | 528 | 45 | 534 | OK | | 9 | 45 | 171 | 0.263 | 540 | 45 | 540 | OK | | 10 | 47 | 190 | 0.247 | 570 | 47 | 570 | OK | | 11 | 47 | 209 | 0.225 | 582 | 47 | 582 | OK | | 12 | 49 | 228 | 0.215 | 612 | 49 | 612 | OK | Increments of kappa_block: 2, 6, 4, 3, 6, 3, 2, 0, 2, 0, 2. A least squares fit over m = 1..12 gives kappa_block(m) = 18.9 + 2.83m, and the cleaner reading is **kappa_block(m) = 19 + 2.73(m-1) to within 2 over the measured range**: the first new gap costs 19 kills and every further new gap costs about 2.73, which is 1.51 times the mean 1.8054. ## 4. Is kappa_block tighter than combined L? Two readings of the question, and they answer differently. **Against m*L, yes, and the margin grows.** m*L - kappa_block(m) reads 0, 17, 30, 45, 61, 74, 90, 107, 126, 143, 162, 179 and the ratio falls monotonically from 1.000 to 0.215 over m = 1..12. Aggregating over m is worth a lot: paying L per new gap overstates the cost by a factor approaching 7 at m = 12. This is the same qualitative gain the per-fold kappa profiles show, and it is much larger here because L itself is large. **Against combined L at the point that matters, m = 1, no, and this is the finding.** kappa_block(1) = combined L identically, by the definition, exactly as kappa(1) = L per fold. The substantive question is whether the object the recursion consumes, j*, sits strictly below it, which is what A4 found per fold. It does not. **j*(m) = kappa_block(m) at every one of the twelve measured m.** For contrast, the per-fold picture computed the same way from a3-05's maxsum table: j*(1) against L reads 2 vs 2, 1 vs 1, 1 vs 2, 2 vs 2, 1 vs 2, 2 vs 3, 2 vs 2, 2 vs 4 at folds 7, 11, 13, 17, 19, 23, 29, 31, so j* is strictly below L at four of eight folds and by as much as 2 at fold 31. That slack is what motivated A4's redirection. At block 1 the slack is zero at every m. **Why, and it is not luck.** Two conditions must both hold for j* = kappa. First the window realising maxsum_m(T_{v^2}) must also be a window of maximal kill count, which at block 1 it is: the record new gap 204 spans 20 T_5 gaps and therefore contains exactly 19 kills, the maximum. Second the granularity of maxsum_k(T_v) must be fine enough that the previous step falls short, and at block 1 maxsum_k(T_5) advances by 6 or 12 per step against a target of 204, a resolution of 3 to 6 percent, where the per-fold ladder advances by about m_bar(T_old) = 20 to 30 against targets of 100 to 350, a resolution of 10 to 25 percent. **The per-fold slack is a rounding artefact of a coarse ladder, and aggregation removes it.** That is a discouraging finding rather than an encouraging one: it means the block bound is attained, so there is nothing left to win by sharpening the target, and the relation maxsum_m(T_23) <= maxsum_{m + kappa_block(m)}(T_5) is a re-description of the truth rather than an inequality with room in it. ## 5. Does Theorem C's mechanism survive aggregation? **The lemma survives per prime. The theorem does not survive the sum. Committed answer: NO.** ### 5a. What survives, VERIFIED Theorem C works because the alternation lemma never uses adjacency: if one 2-set deletes slots j_1 < ... < j_k anywhere, the separations between consecutive deleted slots are all 0, +2 or -2 mod p and the two-state walk governs them, so the span is at least c_min(k-1). That argument is per prime and it is untouched by having other primes present. Other primes delete other slots; they do not change which residues mod p the p-killed slots occupy. So inside any stretch, for every p in Q, > span of p's own kills >= c_min(k_p - 1), where k_p = the number of slots p kills > in the stretch. PROVEN, unchanged from a3-05 section 8a. VERIFIED inside the block, and attained. The extremal stretches at block 1 decompose as follows, with each prime's own first-to-last-kill span against its own floor: | m | K | kills by 7, 11, 13, 17, 19, 23 | sum | stretch span | binding prime | |---|---|---|---|---|---| | 1 | 19 | 6, 4, 3, 3, 3, 3 | 22 | 198 | 23: span 138 = c_min(2) = 138, EQUALITY | | 2 | 21 | 6, 4, 4, 3, 3, 4 | 24 | 234 | 23: span 186 = c_min(3) = 186, EQUALITY | | 4 | 31 | 12, 6, 6, 5, 4, 5 | 38 | 348 | 23: span 276 = c_min(4) = 276, EQUALITY | | 8 | 45 | 17, 10, 9, 7, 6, 6 | 55 | 528 | 23: span 366 vs c_min(5) = 324 | Every per-prime floor is respected in all 24 cells, and at three of the four stretches the largest prime attains its floor exactly. **So the alternation law is live and sharp inside the block**, exactly as a3-05 section 4 found per fold. The sum column exceeds K because slots with two killers are counted twice: 22 against 19 distinct at m = 1, an overlap factor of 1.16, so 3 of the 19 killed slots carry two killers. ### 5b. What breaks The per-fold Theorem C converts "span >= c_min(k-1)" into a bound on k because there is one prime and therefore one constraint on one unknown. With |Q| primes the constraints are > span >= c_min(k_p - 1) for every p, and sum_p k_p >= K, and the adversary chooses the allocation. The binding statement is therefore **a maximum over p, not a sum over p**, and the adversary defeats it by spreading: inverting each constraint gives k_p <= span/(3p) + 1, so > **K <= span * sum_{p in Q} 1/(3p) + |Q|.** **That is capacity counting.** It is the same shape as trap 1, with the naive 2/p per prime replaced by 1/(3p), so the entire content of Theorem A and the alternation law, aggregated over a block, is a factor 6 improvement on the brief's integer-unit capacity count and nothing else. The brief's number at block 1 is 189 available against 19 needed, a factor 10 short; a factor 6 improvement leaves 31.5, plus |Q| = 6, so 37.7 against 19 needed, still short by a factor of 2. Section 5e shows that this comparison flatters the alternation law, and that against the correct capacity count it is not an improvement at all. ### 5c. It is empty, not weak Worse than short: the bound has no fixed point, so it excludes nothing. Feeding span <= maxsum_{m+K}(T_v) back in, with maxsum_k(T_5) = 10.2k at the record and 10k on average, > K <= 1.552 * (m + K) + 6, whose coefficient on K exceeds 1, so it holds for every K. Solved exactly as an integer allocation rather than through the linear approximation, with S = maxsum_{m+K-2}(T_5) and kmax_p(S) = max{k : c_min(k-1) <= S}: | K | S = maxsum_{1+K-2}(T_5) | kmax at 7, 11, 13, 17, 19, 23 | capacity | excludes K? | |---|---|---|---|---| | 5 | 42 | 3, 2, 2, 2, 2, 1 | 12 | no | | 10 | 90 | 5, 4, 3, 2, 2, 2 | 18 | no | | **19** | 180 | 10, 6, 6, 4, 4, 3 | **33** | no | | 30 | 294 | 15, 10, 8, 6, 6, 5 | 50 | no | | 60 | 594 | 29, 19, 16, 12, 11, 9 | 96 | no | | 200 | 1992 | 96, 61, 52, 40, 36, 30 | 315 | no | | 1000 | 9990 | 476, 304, 257, 196, 176, 146 | 1555 | no | The capacity/K ratio starts at 2.4 and settles at 1.552, never crossing 1. Same at m = 8. **The block form of Theorem C returns no finite bound on kappa_block(m) at block 1, for any m.** **CALIBRATION of that null, since a null result proves nothing until the test is shown able to fire.** The identical exclusion routine, run with Q = {p} a single prime, reproduces a3-05's published Theorem C row exactly at folds 11, 13, 17, 19, 23 and 29, all eight entries each, and at fold 7 reproduces the corrected row of section 2. It returns finite bounds there because the single-fold coefficient is m_bar(T_old)/(3p), which reads 0.476, 0.424, 0.439, 0.397, 0.402, 0.371, 0.323 at folds 7 to 29, all below 1. **The routine fires when it should, so the "no exclusion" verdict at block 1 is a finding.** ### 5d. Why it can never be repaired, and where the crossover is The single-fold Theorem C terminates iff m_bar(v)/(3p) < 1. The block replaces the one prime by all of Q, so the block Theorem C terminates iff > **C(v) = m_bar(v) * sum_{v < p <= v^2} 1/(3p) < 1.** Mertens gives sum_{v ln 2 = 0.693147, a constant, so C(v) -> 0.2310 * m_bar(v), and m_bar(v) = 6*prod_{5<=q<=v} q/(q-2) increases without bound, of order ln^2 v. Measured: | v | primes in block | m_bar(v) | sum 1/(3p) | C(v) | |---|---|---|---|---| | 5 | 6 | 10.000 | 0.15521 | **1.552** | | 7 | 11 | 14.000 | 0.16182 | 2.265 | | 11 | 25 | 17.111 | 0.19423 | 3.324 | | 23 | 90 | 28.054 | 0.20185 | 5.663 | | 37 | 207 | 34.051 | 0.21636 | 7.367 | | 101 | 1226 | 53.278 | 0.22421 | 11.945 | | 1009 | 79661 | 115.752 | 0.22986 | 26.607 | **C(v) > 1 already at block 1, and it rises monotonically over every block we can evaluate.** So the block Theorem C is vacuous at the first block and more vacuous at every later one. It never had a good regime and it never will. The requirement is quantitative and it says exactly what a repair would have to supply. For C(v) < 1 the per-kill floor would have to be at least 0.693 * m_bar(v) * p in place of 3p, that is, it must grow like p * ln^2 v rather than being a constant multiple of p. **The alternation law gives exactly 3p and cannot give more, because 3p is attained with equality**, per fold at all seven folds with L >= 2 (a3-05 section 4) and, as section 5a shows, inside the block as well at p = 23 in three of four extremal stretches. The residue side is exhausted at 3p, and 3p is a factor ln^2 v below what aggregation needs. ### 5e. The alternation law is not merely insufficient, it is dominated There is a second per-prime cap that uses no residue law at all. A stretch with K kills and m new gaps holds n = m + K + 1 T_v slots, and the number of those that lie in {0, -2} mod p is bounded by the exact maximum over windows of n consecutive T_v slots, which is computable from the 3p-slot cycle of T_5 modulo p. Call it maxKill(p, n). Both caps are valid, so the truth obeys the smaller. | n = m+K+1 | S = maxsum_{m+K}(T_5) | (a) alternation, sum S/(3p)+1 | (b) slot count, sum maxKill(p,n) | ratio a/b | |---|---|---|---|---| | 21 | 204 | 37.66 | **25** | 1.507 | | 31 | 312 | 52.56 | 35 | 1.502 | | 61 | 612 | 99.12 | 61 | 1.625 | | 101 | 1014 | 161.52 | 100 | 1.615 | | 1001 | 10014 | 1558.39 | 940 | 1.658 | **Cap (b) is smaller at every n, and the ratio converges to exactly m_bar(v)/6 = 1.6667.** The reason is arithmetic and not statistical: the alternation cap is denominated in LENGTH and converts to slots at 6 units per slot, because T_v slots are 5 mod 6; the slot cap is denominated in slots directly and converts at m_bar(v) = 10 units per slot. Every factor of m_bar(v)/6 that separates "5 mod 6" from "actually a T_v slot" is thrown away by any bound that reasons about span. So **the aggregated alternation law is strictly weaker than plain slot counting, by the exact factor m_bar(v)/6, which is 1.667 at block 1 and grows like ln^2 v.** This also corrects the brief's capacity figure at the point where it matters. 189 is 2 * 204/p summed, an INTEGER count over the record span; the same count in T_5 slots is 2 * 21/p summed = 19.56, against the true 19. **Capacity counting in the right units is not vacuous at block 1, it is nearly exact.** Its slope in n is 2 * sum_{v **The block redirection from combined L to kappa_block buys nothing at all. > It does not buy tightness, because j* = kappa_block at every measured m. And > its bound does not merely inherit the wall, it evaporates before reaching it, > because aggregating a per-prime constraint over a block turns a max into a sum > and lands on capacity counting, which is vacuous by a factor 1.552 at block 1 > and by ln^2 v asymptotically, and which is in any case denominated in the > wrong units and so sits a further factor m_bar(v)/6 above the counting bound > it was supposed to beat.** The kappa route is therefore closed at the block level, and closed for a reason that is structural rather than quantitative: there is no allocation of the alternation law over many killers that survives, because the constraint is a maximum over primes and the adversary has v^2/(2 ln v) primes to spread across while needing only about G2/m_bar kills. **Trap 2 was tested and is not triggered.** Nothing above asserts that any window (u, u^2) contains a twin prime, and no statement here is an existence claim about primes; the results are a definition, an exact census over one period, and a vacuity proof about a counting bound. The block object kappa_block is a statement about the tile, not about the primes in the zone. ## 7. Reproduction Three scripts, in the attack's scratch directory `.../43d455bc-.../scratchpad/`, not in `research/`: ``` node --max-old-space-size=6144 kblock.js # 3.1 s: degeneration, block 1 table, composition node kblock2.js # <1 s: exact aggregated Theorem C, C(v) ladder node --max-old-space-size=6144 kblock3.js # 5.8 s: calibration against a3-05 reading 9 node kblock4.js ; node k5.js # <1 s: alternation cap vs exact slot cap ``` `kblock.js` streams the 22,309,287 T_5 slots of one 23# period, storing a one-byte killer bitmask per slot, then derives runs, survivor gaps, kappa_block, j* and the composition of the extremal stretches in one pass over the mask. Its tile builder is independent of `research/Lgrowth.js`, so the reproduction of D_23 = 7,952,175, G2 = 204 and the L diagonal 2, 1, 2, 2, 2, 3 is a genuine cross-check rather than a shared routine. ## 8. COVERAGE: what this did not reach **Only block 1 is measured.** Every kappa_block number above is v = 5 -> 25. Block 2 is v = 23 -> 529, period 529#, and it is not computable here: T_23 has 7,952,175 slots but the block has 90 primes and the period is astronomically beyond a full pass. So "the margin grows with m" and "j* = kappa_block" are one-block measurements. The vacuity argument of section 5 is not; it is an inequality plus Mertens and it holds at every v. **j* = kappa_block is measured, not proven.** Section 4 gives a mechanism for it (the record gap is also a record kill window, plus fine granularity) but the first half of that is an empirical coincidence at block 1 and could fail at block 2. If it does fail, kappa_block would regain some slack over j*, and section 4's discouraging reading would soften. I do not expect it: the granularity argument alone forces j* to within one of kappa whenever m_bar(v) is small against maxsum_m(T_{v^2}), and that ratio improves with v. **The multiply-killed slots are not exploited.** Section 5a measures 22 killer incidences against 19 distinct kills, so sum_p k_p exceeds K by about 14 percent at block 1. That overlap makes the capacity constraint sum_p k_p <= capacity slightly stronger than K <= capacity, and I did not chase it. It cannot rescue anything: the deficit is a factor 1.552 and rising, and overlap is worth 1.16 at block 1 and falls as the primes get larger and their kill sets sparser. Recording it because it is the one term I dropped. **I did not attempt the correlated version.** The bound of section 5b treats the per-prime constraints as independent and takes the worst allocation. A real allocation must also be realisable by CRT, and the two-class lower bound result (`research/two-class-lower-bounds.md` section 1) says every adversarial choice does occur somewhere in the tile, which is why I did not expect a gain there and did not test it. **This is the place I am most likely to be wrong**: the CRT argument makes each prime's choice free, but it does not obviously make the joint allocation free at a single position, and a joint obstruction is the only door left in this direction. I did not find one and I did not prove there is none. **I did not push the surviving object, cap (b) of section 5e.** Its exclusion set is not monotone in K: the smallest excluded K at block 1, m = 1, is 121, but some larger K are not excluded, so the honest bound is the largest non-excluded value, 158 up to K = 20,000. That non-monotonicity comes from the exact maxKill counts jumping in steps, and I did not smooth it or chase where it settles. Block attack 3 owns that object and has taken it further. **I did not re-derive combined L = 19 by a second method.** It is reproduced here from an independent tile builder and agrees with the brief, and G2 = 204, D_23 = 7,952,175 and maxsum_20 = 204 all reproduce, so the four ground-truth anchors agree. That is agreement of two computations, not a proof. **The a3-05 defect found in section 2 is reported, not fixed.** Per the brief no existing repo file was edited. Reading 5's T_5 maxsum row and reading 9's fold-7 kappa and Theorem C rows need the corrections stated there. Theorem B, the L diagonal, and every load-bearing claim in that file are unaffected, which I checked rather than assumed.