# Attack 8: the second moment over a block, and whether it is allowed to work Block 1, v = 5 -> 25. Tile T_5 mod 30 (slots 11, 17, 29), folded at every prime in (5, 25]. Period W = 23# = 223,092,870; 22,309,287 T_5 slots; survivors form T_23. Everything below is exact enumeration or exact finite formula. No estimates, no fits except where labelled. Companion scratch scripts (not committed): `block8.js`, `block8b.js`, `block8c.js`, `block8d.js`, `block8e.js`, `block8f.js` in the session scratch directory. --- ## 0. Custody Reproduced before anything was computed: ``` survivors A = 7,952,175 = D_23 exactly OK combined L (longest deleted run) = 19 OK maxsum_20(T_5) = 204 OK G2(23#) = max survivor gap = 204 OK ``` The exact-variance code was separately checked against `paper/variance-note.md` §4, which it reproduces digit for digit at all six printed levels: ``` p_n L E[N] Var note says 13 289 14.3 2.4 14.3 / 2.4 19 529 20.7 4.0 20.7 / 4.0 31 1369 42.5 8.0 42.5 / 8.0 53 3481 85.4 19.1 85.4 / 19.1 71 5329 115.2 28.3 115.2 / 28.3 97 10201 195.3 48.1 195.3 / 48.1 ``` At block 1 the Theorem 2 formula and the brute-force walk over all 223,092,870 positions agree to five decimals at every H tested (204 through 5000). Two independent routes to the same variance. --- ## 1. THE LEGALITY BOUNDARY This is the deliverable. The brief asked which version of "the window is non-empty" is TPC-equivalent. Here is the line, stated four ways, because four different axes were candidates and only two of them are real. ### 1.1 The bridge, and the exact threshold Let w be the top prime of a block, W = w#, and let a "survivor" be a twin-admissible residue mod W. The only bridge from the tile to the integers is this: > If r and r+2 have no prime factor <= w, and r+2 < w^2, then r and r+2 are both > prime. So a survivor located inside the integer interval (w, w^2) is a twin pair. There is exactly one tile position that delivers such a survivor: the head, x = w. Every other position of Z/W is arithmetically inert. Consequently: - **G2(w#) < w'^2 - 2 for infinitely many w implies TPC** (`ZONE-POSTULATE.md` §3, the Gap Reformulation; the Ziller-Morack Prop 3.2/3.5 form is h_2(k) < p_k^2 - p_k, `covering-dive.md` line 47). Both are exponent 2 **with constant below 1**. - Exponent 2 with constant above 1 implies nothing. Since w'/w -> 1, no sparse choice of w with a large following prime gap rescues a constant above 1. **Refinement of the brief's Trap 2 statement.** "Exponent exactly 2 is TPC-equivalent" is the abbreviation. The precise line is: ``` G2(w#) = o( (log w#)^2 ) or <= (1-eps)(log w#)^2 i.o. ==> TPC. G2(w#) = O( (log w#)^2 ) with an unspecified constant ==> nothing. OPEN. ``` `covering-dive.md` line 141 already carries the qualifier ("Reaching exponent 2 **with the right constant**"); the brief dropped it. The distinction matters here because a second-moment argument never produces a clean constant, so it is worth knowing that landing on exponent 2 with constant 1.4 would be legal, publishable, and useless. ### 1.2 Axis A, tile versus integers: REAL, and the line is at H ~ w^2 A pure tile statement "every window of length H in Z/W contains a survivor" is G2(W) < H. It is TPC-implying exactly when H <= w^2 - w and it holds for infinitely many w. Above that it is legal, and it is a theorem: G2 << (log q)^{4.2665+eps} (`paper/beta2-note.md`) is a tile statement at H far above w^2. The open band (2, 4.2665] is entirely legal territory. ### 1.3 Axis B, all v versus almost all v versus infinitely many v: NOT REAL No escape here at all. "Infinitely many w" is already fatal, and almost-all and for-all are subsets of it. Averaged forms are fatal too: if the average of G2(w#)/w^2 over w <= V tends to 0 then infinitely many terms are below 1. Anyone hoping to buy legality by weakening the quantifier over w is buying nothing. This candidate distinction fails, and that is worth recording because it is the one the brief listed first among the plausible ones. ### 1.4 Axis C, all positions versus almost all positions: REAL, and it is the whole story Chebyshev bounds the **density** of empty positions. TPC needs **one** distinguished position, the head. A density bound eps reaches the head only if eps < 1/W, at which point the count of empty positions is below 1, hence 0, hence Axis A applies again. So the boundary is: ``` empty-density bound eps >= 1/W : LEGAL, says nothing about the head, provably safe. empty-density bound eps < 1/W : identical to the all-positions statement, TPC at H < w^2. ``` **The line is at eps = 1/W = e^{-(1+o(1))w}, not at any rate polynomial in w.** This is the single most important sentence in this report. It means a second-moment argument is legal for exactly the same reason it is useless: Chebyshev's bound is Var/E^2, and Var/E^2 is polynomially small in H while 1/W is exponentially small in w. ### 1.5 Axis D, bound on the count versus bound on the probability: REAL but trivial once separated Three "count" statements get confused: - `E[N] >= 1`, a bound on the **mean** count. True for H >= 29 at block 1. Vacuous. This is brief Trap 1 in another costume: it is capacity counting. - `P(N = 0) <= eps`, a bound on the **probability**. Axis C. Legal above 1/W. - `min_x N(x) >= 1`, a bound on the **minimum**. This is the target, and it is Axis A / TPC at H < w^2. Only the minimum matters, and no moment of N is a bound on its minimum without a maximal inequality. That is the whole content of §4 below. ### 1.6 What is left standing The legal-and-reachable set for a second moment at block scale is: 1. Almost-all occupancy at any H. Already done and published in the corpus (`paper/variance-note.md` §4). Attack 8 adds nothing there. 2. All-positions occupancy at H above the crossover measured in §3, which at block 1 is H* = 3,939,961 = W^0.79. That is a legal tile statement. It is also **13 times weaker than the already-proven bound**, since (ln 23#)^4.2665 = 3.00e5 at this level, and exponentially weaker asymptotically. 3. Lower bounds on G2, which run the safe direction. The second moment certifies these for free and they only make the Zone Postulate harder. Nothing in that list moves the programme. The second moment over a block is allowed to work, and it is allowed precisely because it cannot. --- ## 2. EXACT MOMENTS AT BLOCK 1 ### 2.1 Slot space: K consecutive T_5 slots, all 22,309,287 positions Slot survival density delta_s = 0.356451329 = prod_{7<=p<=23}(p-2)/p. ``` K E[N] Var Var/E Var/E^2 true empty frac minN Cheb/true 1 0.3565 0.2294 0.64355 1.8054e+0 6.4355e-1 0 2.8 5 1.7823 0.9558 0.53629 3.0090e-1 8.1229e-2 0 3.7 10 3.5645 1.6031 0.44973 1.2617e-1 4.7161e-3 0 26.8 14 4.9903 1.8277 0.36625 7.3391e-2 1.7804e-4 0 412.2 18 6.4161 1.8862 0.29398 4.5820e-2 1.7930e-6 0 25555.1 19 6.7726 1.9389 0.28628 4.2271e-2 2.6895e-7 0 157172.2 20 7.1290 1.9374 0.27176 3.8120e-2 0 1 inf 32 11.4064 3.2701 0.28669 2.5134e-2 0 3 inf 64 22.8129 4.3477 0.19058 8.3540e-3 0 13 inf 256 91.2515 8.5816 0.09404 1.0306e-3 0 78 inf 512 182.5031 12.8021 0.07015 3.8436e-4 0 166 inf ``` Smallest K with zero empty slot-windows is **20 = L + 1**, as it must be. The exceptional set at the critical K is tiny and exactly countable: ``` K=17: 186 empty slot-positions Chebyshev allows 1,129,144 K=18: 40 Chebyshev allows 1,022,203 K=19: 6 Chebyshev allows 943,033 K=20: 0 Chebyshev allows 850,437 ``` Six empty positions at K = 19 means six maximal deleted runs of length exactly 19. Chebyshev overstates that by a factor of 157,000, and at K = 20 it permits 850,437 empty positions where there are none. ### 2.2 Integer space: window (x, x+H], all 223,092,870 positions delta = 3.56451329e-2 = A/W exactly. ``` H E[N] Var(brute) Var(Thm2) Var/E Var/E^2 true empty minN 200 7.1290 1.97815 1.97815 0.27748 3.8922e-2 16 pos 0 201 7.1647 1.98555 1.98555 0.27714 3.8680e-2 12 0 202 7.2003 1.99040 1.99040 0.27643 3.8392e-2 8 0 203 7.2360 1.99272 1.99272 0.27539 3.8059e-2 4 0 204 7.2716 1.99250 1.99250 0.27401 3.7682e-2 0 1 289 10.3014 3.01875 3.01875 0.29304 2.8447e-2 0 2 529 18.8563 4.26485 4.26485 0.22618 1.1995e-2 0 8 841 29.9776 5.20460 5.20460 0.17362 5.7916e-3 0 18 5000 178.2257 12.39274 12.39274 0.06953 3.9015e-4 0 162 ``` The empty count falls by exactly 4 per unit of H from 200 to 204, so T_23 has exactly **four maximal gaps of length 204 and none longer**, which is where G2 = 204 lives. Full distribution of N at the critical H = 204, over all W positions: ``` N = 1 : 1,092 N = 6 : 42,174,000 N = 11 : 2,010,432 N = 2 : 43,920 N = 7 : 61,873,332 N = 12 : 205,440 N = 3 : 610,872 N = 8 : 55,046,928 N = 13 : 6,924 N = 4 : 4,284,006 N = 9 : 29,604,684 N = 5 : 17,354,244 N = 10 : 9,876,996 total 223,092,870 ``` The 1,092 positions with N = 1 alone force `Var >= 1092*(1-E)^2/W = 1.92528e-4`, which is already 812 times the 2.37015e-7 that Chebyshev would need. Nothing about the other 223 million positions is required to see that the method fails here. --- ## 3. THE GAP TO WHAT CHEBYSHEV NEEDS Chebyshev forbids emptiness exactly when `Var/E^2 < 1/W`, equivalently `Var < E^2/W`, equivalently `sum_x (N(x)-E)^2 < E^2`. ### 3.1 The measured shortfall ``` H measured Var needed Var shortfall 204 1.99250e+0 2.37015e-7 8.4066e+6 289 3.01875e+0 4.75675e-7 6.3462e+6 529 4.26485e+0 1.59377e-6 2.6759e+6 841 5.20460e+0 4.02816e-6 1.2921e+6 1000 5.99311e+0 5.69528e-6 1.0523e+6 10000 1.70516e+1 5.69528e-4 2.9940e+4 ``` In slot space at the critical K = 20: measured Var 1.93739, needed 2.27811e-6, shortfall 8.5044e5. **The shortfall is not a constant.** It equals W * Var/E^2 = W * Fano / E. At the operative window the Fano factor is about 0.27 and E is about 7.3, so the shortfall is of order W/w, which at block 1 is 2.23e8/23 = 9.7e6 against the measured 8.4e6. For a general block v -> v^2 with top prime w, that is **e^{(1+o(1))w}**. The gap grows superexponentially with the block index. ### 3.2 Where Chebyshev does start working, and why it does not help Bisecting for the first H with Var/E^2 < 1/W: ``` H* = 3,939,961 = 0.01766 * W = W^0.7900 at H*: E = 1.4044e5 Var = 88.215 Var/E^2 = 4.4726e-9 (1/W = 4.4824e-9) ``` So the strongest all-positions statement the second moment can produce at block 1 is `G2(23#) < 3.94e6`. The truth is 204. `(ln 23#)^2 = 369.5`. The already-proven bound is of order `(ln 23#)^{4.2665} = 3.00e5`. Chebyshev's legal output is 13 times worse than a theorem the corpus already has, and 19,000 times worse than the truth. ### 3.3 A tile-independent impossibility, which is the cleanest form of the answer N(x) is integer-valued, so every position contributes at least `dist(E, Z)^2` to `sum_x (N-E)^2`. Chebyshev's requirement `sum_x (N-E)^2 < E^2` therefore needs ``` dist(E, Z) < E / sqrt(W). ``` This has nothing to do with the tile. It is a statement about delta*H and W alone. At block 1: ``` H E = delta*H dist(E,Z) E/sqrt(W) verdict 204 7.2716 0.271607 0.000487 IMPOSSIBLE 289 10.3014 0.301443 0.000690 IMPOSSIBLE 529 18.8563 0.143725 0.001262 IMPOSSIBLE 841 29.9776 0.022443 0.002007 IMPOSSIBLE 1000 35.6451 0.354867 0.002386 IMPOSSIBLE 100000 3564.5133 0.486710 0.238648 IMPOSSIBLE 200000 7129.0266 0.026580 0.477296 possible ``` With a generic fractional part the constraint binds until `H > sqrt(W)/(4 delta) = 1.05e5`, and it binds in the worst case until `sqrt(W)/(2 delta) = 2.10e5 = W^0.637`. In general terms: > **The second moment cannot force a non-empty window shorter than about > sqrt(W)/(4 delta) = e^{(1/2 + o(1)) w}, for reasons that hold for any tile > whatsoever. The Gap Reformulation needs w^2.** That is a rigorous, one-line, measurement-free proof that this attack cannot reach its target, and it is stronger than the numerical shortfall because it does not depend on the tile being the one we have. The same floor applied to higher moments: `M_{2k} >= dist(E,Z)^{2k}`, so `M_{2k}/E^{2k} < 1/W` needs `dist(E,Z) < E / W^{1/2k}`. At H = 204 that fails for k = 1 and k = 2 and first becomes possible at k = 3. **The 2nd and 4th moment methods are dead on arithmetic grounds alone at the operative window; the 6th is the first that is not.** (Attack 3, `research/attack-03-higher-moments.js`, already computes M4 at these levels; this explains why it could never have been enough.) ### 3.4 The ceiling on all moment methods Even a perfect Gaussian lower tail, `P(N=0) ~ exp(-E^2/(2 Var))`, gives at H = 204: ``` E = 7.2716 Var = 1.99250 E^2/(2 Var) = 13.269 exp(-13.269) = 1.727e-6 needed: < 1/W = 4.4824e-9. Short by a factor of 385. ``` The count is simply too small at the operative window. At block 1 the Gaussian tail first suffices at H around 312 (E^2/2Var crosses ln W = 19.2231 between H = 300, where it is 18.413, and H = 350, where it is 23.319). --- ## 4. A RED FLAG WORTH MORE THAN THE COMPUTATION: the elementary maximal law is TPC-hard The corpus records the all-positions exponent under an unproven Gaussian maximal law as theta = 2.159 to 2.482 across z = 19 to 47 (`research/theta-ladder.md` §2), above 2 and rising, and reads that as "even granting the maximal law this route does not reach the finish line". That column is computed from the **sieve** mean square, the vector-sieve sawtooth R. `MORNING-2026-08-16.md` §2 established that the sieve mean square loses to the elementary second moment by about 1.4x. The consequence was not drawn. Running the same conditional on the **elementary** variance, using Theorem 2 of `paper/variance-note.md` exactly, gives the smallest L with E^2/(2 Var) > ln(p_n#): ``` p_n ln(p_n#) L_need theta = lnL/ln p_n L_need/p_n^2 13 10.310 120 1.8665 0.7101 19 16.088 191 1.7838 0.5291 23 19.223 312 1.8316 0.5898 31 26.024 462 1.7867 0.4807 47 40.960 985 1.7902 0.4459 71 61.586 1800 1.7584 0.3571 97 83.728 2935 1.7453 0.3119 199 188.564 9744 1.7351 0.2461 401 376.727 26378 1.6984 0.1640 ``` **Every value is below 2, and the column drifts down, not up.** Structurally L_need ~ 2 F ln(p_n#) / delta ~ c p_n ln^2 p_n, so theta -> 1. Read against §1, this is an alarm and not a result. A sub-Gaussian maximal law for the elementary two-class window count, at window length p_n^2, implies TPC. So: - it cannot be proved by any second-moment or soft-concentration argument; - the corpus is right to carry it as `[INFERRED heuristic arithmetic]` (`research/sift-limit-attack.md` §3, which already states the same conclusion in the form `G2 <~ (2F/0.41621) p_n ln^2 p_n` and immediately flags it as "the parity wall in concentration clothing, not a route"); - and the theta-ladder's above-2 column is not the general verdict on the maximal-law route. It is the verdict on the **sieve** version. The elementary version is below 2 at every computed level, which is exactly why it is unprovable. Anyone who later computes the elementary conditional and finds theta < 2 should read that as a soundness check firing, not as progress. That is the practical value of the Trap 2 discipline: it tells you in advance which of two nearly identical calculations is allowed to come out looking good. --- ## 5. THE `elementaryVariance` BUG, DIAGNOSED EXACTLY `research/sift-limit-lemmaV.js`:427 is bannered WRONG at line 395, "too large by roughly three orders", with the tell that Var/E^2 printed 0.500 across a decade in z and H. The banner is correct and the cause is now pinned to one line. The function does `if (d & 1) continue;` on odd lags. For odd d, J(d) = 0, so the correct contribution is `2(H-d)(0 - delta^2)`, which is negative. `continue` drops the `-delta^2` part along with the J part. The omitted mass is ``` sum_{d odd, 1<=d v^2. The first H at which Chebyshev works at all is W^0.79 = 3.94e6, whose conclusion is 13x weaker than an existing theorem. And §3.3 kills it without any measurement: the integer-valuedness floor makes Chebyshev impossible at every H below about sqrt(W)/(4 delta) = 1.05e5 = e^{(1/2+o(1))w}, against a target of w^2. **Anything legal that survives?** Three things, none of which move the programme: the almost-all corollary already published in `variance-note.md` §4; a vacuous all-positions bound at W^0.79; and lower bounds on G2, which run the wrong way. The one genuinely new item is §4's alarm and §5's bug diagnosis. **Trap 1 check.** This attack did not reduce to capacity counting at any point. Trap 2 fires on the all-positions branch and is the reason §1 is the deliverable. --- ## 8. COVERAGE **What I did not reach.** - Only block 1 was enumerated. Everything at v > 5 in §4 and §6 comes from the exact Theorem 2 formula and short-interval sieving, not from a full-period walk. The formula was validated against a full-period walk at block 1 and against `variance-note.md` §4 at six levels, so I trust it, but no second block has been enumerated end to end. - I did not compute the sixth moment at block 1. §3.3 shows k = 3 is the first order the arithmetic floor does not kill, and §3.4 shows a perfect Gaussian tail is still 385x short at H = 204, so I believe the sixth moment cannot close it either, but I did not compute M6 and the belief rests on the Gaussian reference, not on the true M6. - I did not check whether the H* = W^0.79 crossover is the same exponent at other blocks. The local exponent of Var/E^2 in H at block 1 wanders between -1.42 and -2.28 over the range H = 5e2 to 3e7 (the corpus's H^{-1.7} sits inside that), so W^0.79 is one measurement, not a law. - I did not read `paper/beta2-note.md`, so the 4.2665 comparison in §1.6 and §3.2 is taken from the brief and from `ZONE-POSTULATE.md` §3 without checking the implied constant. If that constant is large, the "13x weaker" claim in §3.2 could go either way at this small level. The asymptotic statement is unaffected. **What I suspect but could not prove.** - That the elementary conditional theta of §4 tends to exactly 1, not merely below 2. The derivation L_need ~ 2 F ln(p_n#)/delta gives 1 + 2 lnln p_n/ln p_n, which at p_n = 401 predicts 1.598 against a measured 1.698, so the Fano drift is doing something the formula does not capture over this range. - That the mild negative drift in the anchored z-scores of §6 at p_n = 127, 199, 401 is the beginning of reading 6's mechanism at zone scale, and that the column would cross -5 somewhere in the low thousands. Three points is not a trend and I am explicitly not claiming it. **Where I think I am wrong.** - §1.3 asserts that no quantifier over v buys legality. I checked all / almost all / infinitely many / average. I did not check quantifiers over a set of v defined by arithmetic conditions that could conceivably be finite in disguise, and I do not have a proof that no such gadget exists, only the observation that any construction producing infinitely many v is fatal. - §3.3's floor argument assumes the conclusion must come from Chebyshev's inequality literally. A cleverer inequality using Var plus one extra piece of information (the support of N, say, or its integer-valuedness used positively rather than as a floor) is not covered. I believe such an argument would have to be a maximal inequality in disguise, and §4 prices those, but that is an assertion. - The claim in §5 that the fix is one line assumes nothing else in `elementaryVariance` is wrong. I checked the odd-d omission accounts for the discrepancy to seven significant figures at three configurations, which is strong, but I did not audit the rest of the function against an independent derivation. **Calibration of searches.** Greps for `slot-space|slot window|slotVar` across `research/` and `paper/` returned only `a3-01-misalignment-ledger.js` and `a3-08-adjacent-pairs.js`, neither of which computes a slot-indexed variance; the same grep for `variance` returns 25+ files, so the pattern is live and the narrow result is a real absence, not a dead pattern. A grep for `Q5.4` in `covering-dive.md` returned nothing and the brief's citation is off by a label: the content is at `covering-dive.md` lines 47, 141 and 152 under "Realistic Target 6 / What NOT to attempt", which `research/history/staging/attack-block-05-exponent.md` line 300 cites the same way. A grep for `READING 6` in `natal5-variance.js` returned nothing because the readings block uses bare numbering; the content is at lines 200-240.