# Displaced prose: `research/README.md` as it stood, 2026-08-13 to 2026-08-17 *(Written by partition D of the 2026-08-17 consistency campaign. `research/README.md` was repurposed into the research directory's router, per Chris's ruling recorded in `qc-CAMPAIGN.md`. The prose below is what the router displaced. It is an **accurate dated summary of the work as of 2026-08-13**, which is why it was moved here rather than deleted: every claim in it was true when written, and the sections it duplicated are each owned by a document written later. The parent session files this into `research/history/` at the location it judges right — a dated summary document beside `CHRONICLE.md`, or an appendix to it.* **Where each section's content now lives.** | displaced section | current owner | |---|---| | Definitions used throughout | `research/GLOSSARY.md` | | The proven spine | `research/THE-LENS.md` §§1-2 for the mathematics, `research/G2-STATE.md` §8 for the attribution | | The wall, honestly | `research/THE-LENS.md` §5, `research/ZONE-POSTULATE.md` §5, `paper/moire-primes.md` §7A | | The ten attacks | `research/ATTACKS.md` | | Prior art & novelty | `research/PRIOR-ART.md` | | Provenance | `research/history/CHRONICLE.md` (the 26 folders), `paper/moire-primes.md` §9 (the rediscovery list) | | The measurements (six-script table) | **kept in the router**, with the script filenames added; it is the only index of scripts 01-06 | --- ## Definitions used throughout **Full adopted vocabulary: [GLOSSARY.md](GLOSSARY.md)** (tile, fold, seam, stratum, zone, frontier, family — adopted 2026-08-14; files written earlier use older words, readings stand). The essentials: - **Tile Tₚ** — the repeating unit at prime p, width P# = 2·3·…·p (canonical alias: the primorial wheel). Each new prime **folds** the tile p times, then strikes its residues. - **Hole / candidate** — a position untouched by every stacked prime's beats. - **Twin slot** — r with both r and r+2 holes; **census Dₚ = ∏(q−2)** per tile. - **Zone / frontier** — the quiet stretch (p, p²); below p² the tile is crystallized — holes are real primes, twin slots real twin primes, final forever. - **G₂(n)** — largest gap between consecutive twin slots in the tile (the "twin Jacobsthal function"). ## The proven spine (each provable in a paragraph; proofs to appear in the paper) 1. **Redundancy Lemma.** A new prime p's *new* kills are exactly p × (old survivor set) — a p-magnified copy of the hole pattern itself, starting at p². Per new period: φ(old period) new kills (measured 2020–2025 in the original research corpus as 1, 1, 2, 8, …). 2. **Crystallization.** Below p², possible = actual, permanently (script 04: 8/8). 3. **Copying Theorem.** Adding p copies every twin slot p times and kills exactly 2 copies: D_{n+1} = D_n·(p−2). Created/destroyed = (p−1)/2 → ∞. Slot counts ∏(p−2) verified exactly up to 6.5·10⁹ (script 05b). 4. **Euclid-in-moiré.** The mirror edges P#±1 survive every stacking; any survivor's smallest prime factor is new. Primes are infinite. (Euclid's construction, rediscovered as the edge of the mirror.) 5. **Zone Equivalence.** Twin Prime Conjecture ⟺ infinitely many guaranteed zones contain a twin slot. 6. **Gap Reformulation.** If G₂(n) < p_{n+1}² − p_n infinitely often, TPC follows. 7. **Variance Theorem** (script 06; full note in paper/variance-note.md). The pattern is a CRT product, so window-count variance is *exactly* computable. Verified against brute force; yields per-level certified bounds: ≥ 99.87% of all length-p_{next}² windows mod P_97# contain a twin slot. The pattern is provably **sub-Poisson** at every computed level and window exponent. Var/E is not a constant: it drifts with the level (0.25 → 0.32 at zone scale, limit unknown), and the stable empirical structure is ln(Var/E) ≈ −(0.24u² + 0.13u) in the window exponent u, with the limit posed as an open question. *(Campaign note, for whoever reads this later: items 1-4 are classical and the router now says so at the point of use — Holt and Rudd arXiv:1408.6002 Thm 2.3 for 1 and 3, Pritchard 1982 for 2, Euclid IX.20 for 4. Item 5 is a framing biconditional rather than a spine theorem, per `PRIOR-ART.md`.)* ## The wall, honestly All lower-bound routes to "the zone contains a twin" run through the **parity problem** (Selberg): sieve counting cannot distinguish numbers with an odd vs even number of prime factors. History's breaches all imported outside structure: Chen (almost-primes), Friedlander–Iwaniec (algebraic bilinear structure), Zhang/Maynard (equidistribution). In this framework the wall has two sharp faces: - **Gap face:** extend Iwaniec's g(q) ≪ ln²q (one forbidden residue per prime — exactly the critical exponent) to two forbidden residues with exponent < 2. - **Window face:** certify in *the* zone the √-cancellation that scripts 03/04 measure everywhere. The Twin Prime Conjecture, in this language: *the moiré contains no conspiracy.* Measured out to 10⁸: no conspiracy ever strays past one standard deviation. ## The ten attacks (see ATTACKS.md + attack-NN-*.js) Ten systematic assaults on the wall, all executed 2026-08-13. What they yielded: - **A theorem (attack 8, unconditional):** every zone (p, p²) contains two primes at distance ≤ 2(1+o(1))·ln p — crystallization + Chebyshev + pigeonhole; certified constant already 2.00 at p=1009. TPC compressed into: *remove one logarithm from an elementary bound.* - **Decoupling (attack 1):** the pattern's monster gaps migrate deep into the period; the zone only inherits the frozen actual-twin gap structure. Deep G₂ growth cannot threaten the zone. - **The anchored origin (attacks 2, 10):** head windows live a three-phase life — exactly fair until p³ reaches them (new empirical law), a trough with asymptotic depth e^{2γ}/4 = 0.79305 at the zone edge, then enrichment that is **bounded by e^{2γ} = 3.172 and mortal**, crashing to 0 as p_n → x (`anchored-windows.md` §3). "Anchored vs random windows" is this framework's most original open formalization. What the origin does not have is a general advantage: at S = x'² it carries 21% LESS than mean density. - **A new object (attack 6):** the map d ↦ G_d. Equal-density differences split (G_8 = 198 > G_2 = 150 at 19#): extremal gaps depend on offset arithmetic beyond the Hardy–Littlewood density. Unstudied per our audit. - **The wall, priced (attacks 3, 4, 7, 9):** Gaussian moments (kurt 2.9); Fourier budget grows ~2ⁿ vs Legendre's 3ⁿ and misses certifying the p=11 zone by 18%; optimal quartic certificates beat Chebyshev 57× with parity surviving every polynomial degree at geometric cost; the parity step itself skims only 8% of candidates (spf > p^0.9 vs prime), and Chen territory holds exactly 2.0× the twins — the linear sieve's factor of 2, visible raw. The statistics show no cliff at "prime": the wall exists only in certification. - **Zero exceptions (attack 5):** all ~1,229 prime-square annuli below 10⁸ contain twin primes; the annulus-induction ladder is uncracked. *(Campaign note, and it is a retraction rather than a caveat: the attack-4 Fourier-budget numbers in the fifth bullet — the 2ⁿ growth and the 18% near-miss at p = 11 — are outputs of the column that `research/attack-04-fourier-budget.js`'s own banner calls "NOT a valid certificate". **The corrected artifact `natal-cap-02-fourier-budget.js` reverses the conclusion**: at x = 11 even an oracle fails, cap 117.3 against N = 90, and the per-prime union bound is structurally dead from there on. The real near-miss is 15% at x = 7 and it is moot. That is finding P-1 of the campaign, the reason this block is history rather than body, and the reason nothing carrying an 18% survives into the router.)* ## Prior art & novelty A three-agent literature audit (2026-08-13) checked every claim above against the published record — see **[PRIOR-ART.md](PRIOR-ART.md)** for the full verdict table with sources. Summary: the classical spine is exactly that (classical, now properly cited, from H.J.S. Smith 1857 to Iwaniec 1978 to Táfula 2015); the wave/superposition framing is published up to PRL 2019 and must be cited; three things survived as candidate novelties — the difference-2 twin Jacobsthal function G₂ (sequence not in OEIS; reduction refines Ziller–Morack 2017), the exact two-class window-variance formula with its sub-Poisson scaling law, and the moiré/palindrome pedagogical synthesis. The open problem behind the Gap Reformulation is sharp: **no Iwaniec-type upper bound is published for two omitted residue classes per prime** — Ziller–Morack's bound is conjectural, and the 2021 FKMPT "long gaps in sieved sets" machinery does not cover this case. The lower side is not open: G₂(x#) ≥ g(x#) pointwise imports the whole Erdős–Rankin literature for free (`two-class-lower-bounds.md` §§1, 3). A large part of the fold apparatus is prior art: Holt and Rudd own the cycle of gaps, the fold recursion, fusions, the closure theorem, the transfer operator, the population models, the interval of survival and the one-class discrepancy; Maier owns the tile-as-matrix; Buchstab owns the survival curve ω(u). See **[PRIOR-ART.md](PRIOR-ART.md)**. ## Provenance Built on a 2020–2026 corpus of experiments (26 folders) that independently rediscovered: wheel factorization, φ(P#) = ∏(p−1), the Euler product density, the wheel palindrome, ∏(p−2) twin-slot counts (the Hardy–Littlewood local factors), Fermat's factorization (three ways), and CRT residue lifting. The framing throughout: independent rediscovery as evidence the moiré lens works, with canonical names attached wherever they exist.