# Ten attacks on the wall The wall: prove the frontier zone (pₙ, p²ₙ₊₁) always (or infinitely often) contains a twin slot. Blocked by parity for pure counting. Each attack below is based on the audit (PRIOR-ART.md) and designed to produce a new learning even when it fails to breach. Status updated as executed; artifacts land as `attack-NN-*.js` with embedded output. | # | Attack | Idea | Status | |---|---|---|---| | 1 | **Gap cartography** | Where do the biggest twin-slot gaps live in the period? If worst gaps provably avoid [0, p²], that's a zone-safety mechanism. | ✅ **DONE.** Monster gaps migrate deep into the period (34% at 23#, 18.6% at 29#); the zone only ever holds the frozen actual-twin gap structure. G₂'s deep growth is IRRELEVANT to the zone — decoupled. | | 2 | **Head-bias profile** | Density of twin slots in [0, x] vs global average as x grows. | ✅ **DONE.** Head is ENRICHED, not starved, but the enrichment is capped at e^{2γ} = 3.172 and mortal; cumulative equidistribution locks at ~p³ — to within 0.006, tightening to the printed 1.000 by ~10p³, rescored against the script's own table 2026-08-20; the SCALE is the finding, not a three-decimal lock (our measurement, and **not a new object**: `attack-02-head-bias.js`'s own 2026-08-17 banner identifies it as the u ≥ 3 branch of the Unification Law, whose ω(u) is Buchstab's survival function per `GLOSSARY.md` and `PRIOR-ART.md`); universal dip ~0.90 near x~10³ is a finite-size blend, and the asymptotic dip constant is e^{2γ}/4 = 0.79305, not e^γ/2 (anchored-windows.md §4). | | 3 | **Higher moments** | 4th moment → quartic tail bounds; Gaussian test. | ✅ **DONE.** Kurtosis → 2.9 (Gaussian); quartic bound beats Chebyshev **30–37×** (34.8 / 30.4 / 37.4 at p = 13 / 17 / 19, recomputed from the script's own printed bounds; the "35-40×" this cell used to give contains neither endpoint); bonus: MIN window count over whole period = 9–12, nowhere near 0. | | 4 | **Fourier budget** | Spectral certification of window counts. | ✅ **DONE, and the per-prime union bound is dead.** Smooth conspiracies are dead (\|F(1)\| ~ 2ⁿ/P) and the dominant modes are the rigid mod-6 comb. The certification itself does not work: at x = 11 even an oracle for the exact per-prime cap fails (117.3 against N = 90), and the bound is structurally dead from there on, the nearest approach being 15% at x = 7 and moot. Source: `natal-cap-02-fourier-budget.js`, which supersedes `attack-04-fourier-budget.js`; the latter's "certified" column paired moduli with wrong kernel values and is not a valid certificate. | | 5 | **Annulus induction** | Twin in every prime-square annulus ⇒ TPC. | ✅ **DONE.** Zero empty annuli in all ~1,229 up to 10⁸; weakest are the tiny early crystallized ones. | | 6 | **Difference hierarchy** | G_d across even d; is 2 the hardest? | ✅ **DONE, and the surprise dissolved.** Slot counts follow H-L hierarchy exactly. The 19# split (G_8=G_16=198 vs G_2=G_4=150) looked like a law and is not: `attack-06b-difference-map.js` reading 2 shows the quadruple splitting four ways at 23# (204/186/210/264) and kills v2-monotonicity at every level, and its reading 3 is a stated NEGATIVE RESULT, **at its own scope, which this cell used to flatten**: it is titled "no simple statistic determines G_d WITHIN A DENSITY CLASS", and reading 1 of the same script says density DOES set the scale across all 105 differences. The 105-difference sweep runs at 13#, 17# and 19# only; at 23# the script computes d = 2, 4, 8, 16 alone. One faint signal survives — d ≡ ±2 (mod 5) outscores d ≡ ±1 (mod 5) at all three levels, a ~4-7% tilt the script flags as unconfirmed rather than a law. The map d ↦ G_d is therefore studied here and its easy structure refuted, not an open object. | | 7 | **LP adversary** | Optimal polynomial certificates against exact moments. | ✅ **DONE** (deg-4 grid). Best certificate 1.6e-4 at p=19 (57× better than Chebyshev); each +2 degrees buys ~1/(5μ); parity survives every degree at geometric cost. | | 8 | **Pigeonhole small-gap theorem** | Provable weak-twin theorem in every zone. | ✅ **DONE — THEOREM, for p ≥ 17.** For every prime **p ≥ 17** the zone (p,p²) contains two primes at distance ≤ 2(1+o(1))·ln p, unconditionally (crystallization+Chebyshev+pigeonhole). The hypothesis p ≥ 17 is where the Rosser–Schoenfeld input holds in the form used, and the o(1) is about the limit rather than any single p, which is why `paper/moire-primes.md` §7 states the finite form first and governs here. Certified constant already 2.00 at p=1009. TPC = removing one logarithm. | | 9 | **Chen-style almost-twins** | Price the parity step via spf(r+2) > p^θ. | ✅ **DONE** (empirical). The last step (θ=0.9→1) costs only 8%, stable across levels; Chen territory (θ=0.5) holds exactly ~2.0× the twins — the linear-sieve parity factor 2, visible in raw data. No cliff at θ=1: the wall exists only in certification, not in the statistics. | | 10 | **Anchored origin** | The zone is not a random window — the origin is pinned. | ✅ **DONE.** Three-phase life cycle of the head window measured (equidistributed → trough of asymptotic depth e^{2γ}/4 = 0.79305 at the zone edge → enrichment bounded by e^{2γ} = 3.172, then a crash to 0 as p_n → x). The zone sits at the phase boundary; anchored windows vs random windows is the framework's most original formalization target — **and it is no longer unexplored, which this cell used to say**. It now has `research/anchored-windows.md`, `research/anchored-calm.md`, the paper `paper/anchored-note.md` (ten exactly computed levels of β through @41), and the 38-script natal-cap campaign; `natal-cap-31-calm-vs-kill.md` records the fusion as PROVEN. What remains untouched is the LITERATURE side, which is `PRIOR-ART.md`'s wording: "The anchored half is untouched, and is the whole game." At window S = x'², measured later, the origin's density ratio tends to e^{2γ}/4, i.e. **21% below the tile mean asymptotically, and only after dividing out the window's own measured Hardy-Littlewood factor**. The raw measured ratios run 1.00 at x = 181 down to 0.912 at x = 6037, so the enumerated deficit is 0% to 8.8%, not 21%; the u = 2 argmin is INFERRED, not proven (origin-excess.md, its own boxed statement and reading 8). | Realistic goal: not a TPC proof (parity stands); the goal is new objects, new data laws, and possibly one or two small provable theorems (esp. #8, #3).