# Attack 3: is there a block-level alternation law? *(2026-08-18. Object and rules from the block brief. Every number below was recomputed here from scratch; nothing is quoted from another document without being rederived. Calibration is marked on every claim: PROVEN, VERIFIED by exact computation, MEASURED, REFUTED.)* ## Verdict **A block-level inequality exists and it is a real theorem. It is not an alternation law.** Over a block, the alternation law of `research/a3-05-bound-L.md` section 2 is strictly weaker than plain slot counting, by the exact factor `mbar(T_v)/6`, and that factor is at least 10/6 and grows like `ln^2 v`. The strongest per-prime statement is the exact two-class count `K_p`, which subsumes both. The resulting Block Partition Inequality closes at v = 5 and v = 7 and is vacuous from v = 11 onward, because its slope is `2 * sum_{v **sum over p in (v, v^2] of k_p >= L.** (the partition constraint) Three upper bounds on k_p are available, in increasing strength. **(a) The alternation bound.** The k_p slots killed by p lie in one set {a, a-2} mod p and are all congruent to 5 mod 6, so their consecutive separations obey Lemma 1 and Lemma 2 of a3-05 verbatim. Adjacency was never used in those lemmas, which is the same observation that gives Theorem C. Hence Theorem A applies to the separations and > **k_p <= 1 + m*_p(X)**, where m*_p(X) = max{m : c_min^{(p)}(m) <= X} and > c_min^{(p)}(m) = 3pm for m even, 3pm - p - 2*eta for m odd. Since c_min(m) >= 3pm - p - 2 this is k_p <= 1 + (X + p + 2)/(3p), so about 1 + X/(3p). PROVEN, given a3-05 sections 2 to 4. **(b) The free-slot lemma.** If 2p - 2*eta > X then k_p <= 1, because two run slots in one 2-set are separated by a positive multiple of 6 lying in classes 0 or +-2 mod p, and the least such is 2p - 2*eta by Lemma 2. This is the one place where the alternation law says something counting cannot: it halves the naive "2 free slots per large prime" to 1. PROVEN. It never fires; see section 5. **(c) The exact two-class count.** Since s_i = s_0 + d_i, the kill condition s_i = 0 or -2 mod p reads d_i = a or a-2 mod p with a = -s_0, so > **k_p <= K_p(L, f) := max over a of #{i < L : d_i = a or a-2 (mod p)}**, a computable function of the run length L and the phase f alone, independent of where the run sits. K_p is exact: it is the true maximum, so it dominates (a) and (b) automatically. PROVEN. Combining (c) with the partition constraint gives the only theorem this attack produces: > **Theorem D (Block Partition Inequality, PROVEN).** If a run of L cyclically > consecutive T_v slots at phase f is entirely deleted by the block (v, v^2], > then > > L <= sum over p in (v, v^2] of K_p(L, f). > > Since any run of length L contains runs of every shorter length, the combined > L of the block satisfies > > **combined L <= L0 - 1, where L0 = min{ l : max over f of sum_p K_p(l, f) < l }.** The alternation form of the same theorem, which is all that is available when the gap word of T_v is not in hand, replaces K_p by 1 + m*_p(X) with X = maxsum_{L-1}(T_v). ## 3. Evaluated at block 1, against the truth and against capacity All five columns are per-prime caps on k_p for the measured run, L = 19, X = 180. VERIFIED by exact computation. | p | true k_p | K_p exact | 1 + m*_p(180), alternation | mean 2L/p | integer capacity 2X/p | |---|---|---|---|---|---| | 7 | 6 | 6 | 10 | 5.43 | 51.4 | | 11 | 4 | 4 | 6 | 3.45 | 32.7 | | 13 | 3 | 4 | 6 | 2.92 | 27.7 | | 17 | 3 | 3 | 4 | 2.24 | 21.2 | | 19 | 3 | 3 | 4 | 2.00 | 18.9 | | 23 | 3 | 3 | 3 | 1.65 | 15.7 | | **sum** | **22** | **23** | **33** | **17.69** | **167.6** | Read against L = 19: - **The exact cap is attained at five of six primes.** Sum of K_p is 23 against a true sum of 22 and a need of 19. The counting side of the block problem is as tight as the residue side is at a single fold. - **The alternation cap is 33, which is 43 percent worse than counting**, and it loses at every prime except p = 23 where it ties. Over a block, the alternation law is not an improvement on counting. It is a degradation of it. - **The 189 is reproduced as an integer count, not a slot count.** `2 * G2(23#) * sum_p 1/p = 2 * 204 * 0.465623 = 189.97`, and the floor of that is 189. The span version, 2X/p summed, is 167.6. Both count integers in the two classes. Only one integer in ten is a T_5 slot, and 189.97/10 = 19.0 against the mean slot capacity 17.69. That factor of 10 is mbar(T_5), and it is the whole of the "vacuous by 10x". GREP CALIBRATION, per the rules: 189 does not appear anywhere in `research/` or `paper/` in a capacity context. The pattern was calibrated on a known positive, `grep "204" research/history/staging/` returns the G2 sequence in applied-Q.md. So the 189 is the adjudicator's own arithmetic, made today, and the reconstruction above matches it to the unit but is a reconstruction. ## 4. Why it dies: two slopes, one of them fatal Both forms of Theorem D are linear in L with a slope, and both are useless the moment the slope reaches 1. **Counting form.** K_p is 2L/p on the mean, so > slope_count(v) = 2 * sum_{v < p <= v^2} 1/p, limit 2 ln 2 = 1.386294. **Alternation form.** 1 + m*_p is 1 + X/(3p) with X = mbar * (L-1), so > slope_alt(v) = mbar(T_v) * sum_{v < p <= v^2} 1/p / 3, and therefore, exactly, > **slope_alt / slope_count = mbar(T_v) / 6.** That ratio is 10/6 at v = 5, 14/6 at v = 7, 20.2/6 at v = 13, and it grows like `ln^2 v` forever. The alternation law prices a killed slot at 3p of span; plain counting prices it at mbar/2 slots, that is 6p/mbar of the same span per unit of 2/p. The two agree at mbar = 6 and mbar is never below 10. **This is the clean reason attack 3 has nothing to add: Theorem A wins at a single fold because there the run is every slot and counting says nothing, and it loses over a block because there the run is a sparse subset and counting says everything.** The crossover, VERIFIED: | v | 2 * sum_{v X. The trivial construction "one prime per slot" gives combined L >= pi(v^2) - pi(v) (set a_p = s_i for distinct primes), hence X >= mbar(T_v) * (pi(v^2) - pi(v) - 1). Comparing against the largest possible 2p - 2 = 2v^2 - 2: | v | mbar(T_v) | pi(v^2)-pi(v) | X at least | 2v^2 - 2 | free-slot primes | |---|---|---|---|---|---| | 5 | 10.000 | 6 | 50 | 48 | none | | 7 | 14.000 | 11 | 140 | 96 | none | | 11 | 17.111 | 25 | 411 | 240 | none | | 23 | 28.054 | 90 | 2497 | 1056 | none | | 101 | 53.278 | 1226 | 65266 | 20400 | none | | 1009 | 115.752 | 79661 | 9220807 | 2036160 | none | The margin is `mbar * pi(v^2) / (2v^2)`, about `C ln v / 4`, and it widens. At block 1 directly: the largest 2p - 2*eta in the block is 48, against X = 180. VERIFIED at every v listed; PROVEN for all v >= 5 by the displayed inequality plus the growth of mbar. So the free capacity in the block is not 2 per prime. It is the fluctuation of K_p above its mean 2L/p, MEASURED at 0.9 per prime at v = 5 and 1.9 per prime at v = 7. The death is entirely the slope, not the intercept. ## 6. The numbers, in full Theorem D in counting form, evaluated exactly by sweeping l and all phases. | block | truth | Theorem D bound | capacity as quoted | trivial lower bound | |---|---|---|---|---| | v = 5 | **19** | **L <= 62** (first dead l = 63, sums 62, 62, 62) | 189 | 6 | | v = 7 | >= 49 | **L <= 659** (first dead l = 660, sum 659) | | 11 | | v >= 11 | | **none, at any L** | | 25 and up | At v = 5 the raw inequality is satisfied again at some l above 63, last at l = 111, and fails at every l from 112 to 200, the limit of the sweep. The monotone reading, that a run of 64 contains a run of 63 which must also be covered, is what makes 62 the answer rather than 111. So the honest scoreboard at block 1, all four quantities in the same units: > **truth 19, sharp per-prime sum 23, Theorem D bound 62, alternation form > infinite, quoted capacity 189 (in integer units, 19.0 in slot units).** Theorem D beats the quoted capacity by 3.0x and sits 3.3x above the truth. It is the first finite block-level bound on combined L in this repository. It is also worthless, for the reason in section 4: it exists only at the first two blocks. **It just misses being interesting.** L <= 62 gives G2(23#) <= maxsum_63(T_5) = 630. The p^2 rule threshold at x = 23 is 23^2 = 529. The bound overshoots by a factor 1.19. Had it read L <= 52 it would have proved "there is a twin prime in (23, 529]" by pure counting, which is true but finite and so no contradiction; the point is only that the block-1 arithmetic sits within 20 percent of a p^2-rule conclusion. ## 7. Trap tests **Trap 1, capacity. HIT, but not where expected.** Theorem D is capacity counting with an exact per-prime maximum instead of a mean. It dies at the Mertens constant 2 ln 2 > 1. The correction to the brief is that at block 1 capacity is not vacuous by 10x; in slot units it is 17.69 against 19 on the mean and 23 against 19 at the maximum, and the 10x was mbar(T_5). **Trap 2, TPC. NOT HIT.** This needs saying carefully, because the temptation is to claim the trap and stop. Suppose Theorem D were subcritical for infinitely many v, giving L <= B(v) = O(pi(v^2)) = O(v^2 / ln v). Then G2(v^2 #) <= maxsum_{B+1}(T_v) <= (B+1) * G2(v #). To reach the p^2 rule at x = v^2 one needs G2(v^2 #) < v^4, hence B * G2(v #) < v^4, hence G2(v #) < v^2 * ln v roughly. **With the currently proven exponent that fails**: G2(v #) <= v^{4.2665+eps} gives v^{6.27}, nowhere near v^4. With the measured exponent 1.54 it succeeds, at v^{3.54} < v^4. So a subcritical block law implies TPC only when fed an input at least as strong as the Zone Postulate itself, which is already known to be TPC-equivalent by the p^2 rule. **Subcriticality is not TPC-hard on its own.** Attack 3 therefore dies of trap 1 and only trap 1, and the honest statement is that the Mertens constant, not the twin prime conjecture, is what kills it. That distinction matters for the programme. It means the block law is not protected by a hardness barrier; it is defeated by an arithmetic constant. A different constraint on the same object is not ruled out by trap 2. ## 8. The unique-killer refinement (MEASURED, no force) Requested as the fallback if the main line died. It does die, so here it is. MEASURED at block 1: **17 of the 19 run slots have exactly one killer.** The distribution of killer counts along the run is 1,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1, and by prime the unique-killer counts are 7:5, 11:3, 13:2, 17:2, 19:3, 23:2. Over-coverage is 3 slots' worth out of 22. So the sub-run of uniquely-killed slots is 17 of 19, and a constraint on it would be nearly a constraint on the whole run. **It has no force, and the reason is structural rather than numerical.** Write U_p for the slots killed only by p. Then U_p is a subset of the kill set of p, so every bound of section 2 applies to |U_p| verbatim and gives nothing new; and the only available aggregate statement is sum_p |U_p| = #{uniquely killed slots} <= L, which points the wrong way. To get force one would need a proven lower bound on the number of uniquely killed slots, that is a proven upper bound on the forced overlap. There is none: the residues a_p are free by CRT, and pigeonhole forces an overlap between p and q only when k_p + k_q > L, which at block 1 fails for every pair (the largest is k_7 + k_11 = 10 against 19). The adversary is never compelled to overlap, so it is never compelled to be unique either. REFUTED as a source of force. The measurement is still worth keeping for a different reason. It says the block covering at v = 5 is almost exactly a partition, 22 kills for 19 slots, which is why the sharp counting bound is attained at five of six primes. The block problem is not loose; it is a tight packing that counting can see and that no residue argument improves on. ## 9. Corrections to claims elsewhere Both are made from computations in this file and should be checked before being propagated. **`research/history/staging/attack-block-10-target.md` section 5, trap 1.** It states "a prime p kills at most 2(L/p + 1) of them" and concludes "capacity exceeds need for every L, at every v, forever". Two issues. - The per-prime bound 2(L/p + 1) is not valid. The exact K_p exceeds it in 100 or more of the (L, phase, p) triples tested at block 1 for L <= 300, first at L = 27, p = 7 where K_p = 10 against 9.71, and at L = 20, p = 23 where K_p = 4 against 3.74. The violations are always under one unit, so the formula is a good approximation and its aggregate conclusion is unaffected, but it is not a bound. The valid object is K_p. - "For every L, at every v, forever" is false at the first two blocks. Even that file's own formula, 0.9312 L + 12 at v = 5, closes at L = 175. The exact form closes at L = 62. Capacity at v = 5 and v = 7 gives finite bounds. It gives none from v = 11 on, which is the part that is right and is the part that matters. **The brief's "coverage available 189 against 19 needed, vacuous by 10x".** The two numbers are in different units. In slot units the reading is 17.69 on the mean, 23 at the sharp maximum, against 19. The conclusion (any attack reducing to capacity is dead) survives, because it is decided by 2 ln 2 and not by the block-1 margin, but the margin as quoted overstates the case by mbar(T_5) = 10 and would mislead anyone calibrating a new attack against it. ## 10. Reproduction Scripts are in the session scratch directory, not in `research/`, per the rules: `blk.js` (block-1 object, 22.3M slots, 20 s), `cap.js` (the five per-prime caps and the self-consistency sweep), `cross.js` (the Mertens crossover), `bv.js` and `bv7.js` (Theorem D at v = 5, 7, 11, 13), `freeslot.js` (section 5), `truth7c.js` (block-2 covering optimum by branch and bound), `chk.js` (the 2(L/p+1) violations). ## COVERAGE, blunt **What I did not reach.** - **Block-2 truth is not pinned.** Combined L at v = 7 is at least 49, by an explicit covering found with a complete branch and bound over all 15 phases, and at least 35 independently because G2(37#) = 528 and the smallest m with maxsum_m(T_7) >= 528 is 36. The upper end is unknown. The search was still running at l = 50 when I stopped. So "Theorem D at v = 7 is 13x loose" is a lower bound on its looseness, not the figure. - **The LP relaxation of the block covering problem was not computed.** Theorem D uses one inequality, sum_p k_p >= L, and per-prime caps. The full linear program (choose a fractional a_p per prime, cover every slot) is strictly stronger, its value at block 1 sits somewhere in [19, 62], and it is the natural next object. I did not build it. If anyone does: the only content beyond Theorem D lives in the interaction between primes, since K_p is exact and cannot be improved. - **G2 for the block-2 tile.** I used G2(37#) = 528 from the brief and did not recompute it. - **Nothing was tested above v = 13 for Theorem D itself**, only the slope. The slope is what decides it, but the exact bound at v = 11 and 13 was inferred from supercriticality rather than swept. **Where I was wrong, caught here.** My first branch and bound reported combined L = 33 at v = 7 and called l = 34 infeasible. It was incomplete: it branched on "prime p_i covers the first uncovered slot, or p_i is unused", which discards every solution where p_i covers some other slot and a later prime covers the target. The error was caught by an independent consistency check, not by inspection: G2 is monotone in x, so G2(47#) >= G2(37#) = 528, and maxsum_36(T_7) = 528 forces combined L >= 35 at that block. The corrected complete search then found coverings up to 49. Any number from the first search is void. **Where I still suspect I am wrong.** - The identification of 189 as `floor(2 * G2(23#) * sum 1/p)` is a reconstruction of someone else's arithmetic that matches to the unit. It could be a different computation that coincides. The substantive point, that the quantity is in integer units while 19 is in slot units, holds for both candidates I could construct (167.6 over the span, 189.97 over G2). - Theorem D's monotone step, "a run of 63 contains a run of 63 at some phase", is sound, but the bound 62 depends on my having swept all three phases at every l. The three phases give identical sums at l = 19, 63 and most other l, which is either a genuine symmetry of the 6, 12, 12 word or a bug I did not find. The numbers differ at other l (l = 15 gives 21, 19, 21), so it is not a constant function, which is the only check I ran on it. - I have assumed throughout that a block prime deletes exactly the residues 0 and -2 mod p, per the brief. If the fold's 2-set at block level were ever {a, a-2} with a not equal to 0 for reasons other than CRT copy choice, the measured k_p profile of section 1 would change, though Theorem D would not, since it maximises over a.