# Glossary — the shared vocabulary of primeoire **Scope.** This document owns the vocabulary and nothing else: one term per object, with the canonical alias attached. It is a lookup document, read one entry at a time, and it is **not the first thing to read**. It does not own the status of any route. The reading path for a fresh reader is in [README.md](README.md); the state of the central object is in [G2-STATE.md](G2-STATE.md); why these objects are the right ones is in [THE-LENS.md](THE-LENS.md); what to try next is in [../TODO.md](../TODO.md). Adopted 2026-08-14. One metaphor family (moiré/tiling), one term per object. First use in any public text states the canonical alias once, then uses our word. ## Core objects - **Tile (Tₚ)** — the repeating unit of the moiré after stacking all primes ≤ p: one full period, width p# (the primorial). *The 31-tile* T₃₁ has width 200,560,490,130. Canonical aliases: the primorial wheel mod p#; Holt and Rudd's *cycle of gaps* G(p#) among the generators of Z mod p#. - **Fold** — what a new prime does to the tile: p **folds** the tile p times (lays p copies end to end), then strikes its residues through every copy. "Adding 11 folds the 7-tile eleven times and strikes each twin site twice." Canonical alias: Holt and Rudd's R1/R2/R3 recursion (identify the next prime, concatenate p copies, close adjacent gaps); a kill is their *fusion*. - **Hole / candidate** — a position no stacked prime hits; where all further primes must live. - **Twin slot** — a position r with both r and r+2 holes; a *possible* twin pair. - **Twin opener** — the lower member a of a realized twin pair (a and a+2 both prime): the twin slot that survived every fold. Consecutive openers define the twin gaps g_i, and those gaps are the process the head statistics of TODO Z4 run on (history/staging/head-residual-factor.md §1). - **Census (Dₚ)** — the number of twin slots per tile: Dₚ = ∏₃≤q≤p (q−2). The 31-tile's census is 6,226,553,025. (Counted exactly; Copying Theorem.) Canonical aliases: OEIS A059861; Schemmel's totient (1869) at the primorial — the pair-analogue of Euler's φ, which is the SINGLE-hole census ∏(q−1); Holt and Rudd's N2(p#), "Twin Generators". - **Width (|Tₚ|)** — the tile's size: |Tₚ| = p# (the primorial; classically the pattern's period). The two growth laws per fold: |Tₚ| = p·|T_prev| and Dₚ = (p−2)·D_prev. ## Places in the tile - **Seam** — where adjacent copies meet after a fold: the multiples k·(width of the previous tile). Every seam carries the pair (kP−1, kP+1) — twin slots by the mirror. **Seam Lemma:** each fold kills exactly 2 seam pairs; p−2 survive. **The seam's enrichment is slot-level and stops there.** Against a random integer a seam point is 10–20× more likely to carry a twin prime, with the Hardy-Littlewood constant E(P) = 2·∏_{odd p|P} p/(p−2) VERIFIED to 0.1–0.6% (10×/14×/17×/20× at P = 30..30030, sieved to 3·10⁹). That whole factor is slot-hood: a seam position is roughly 20× to 28× more likely than average to be a slot, level by level (20.2, 22.9, 25.6, 28.1 at T₁₃, T₁₇, T₁₉, T₂₃, recomputed from `attack2-01-06-seam-census.js`; no script prints a single figure, and the round "25×" that stood here is the T₁₉ value), and **conditional on being a slot it is not more likely than average to be a twin prime**. Seam neighbourhoods at half-widths 300 to 3·10⁵ measure twin-prime survival at 1.016, 0.980, 1.007, 0.986 times the tile mean at T₂₃, against one control offset each, and the seam points themselves give 1 genuine twin pair from 20 slots against 2.25 expected (`research/fold-profile-12-anatomy-survival.js`). The 400-control test is a different run, `research/fold-profile-13-hotspot-sweep.js`, which puts the seams at z = −1.42 to +0.26 across T₁₉ and T₂₃, every cell consistent with the null; those two extremes are different tiles at different half-widths. Nor is the enrichment a neighbourhood effect even at slot level: seam-anchored windows measure 1.002 ± 0.005 (`attack2-01-06-seam-census.js`). - **Mirror** — the palindrome: the tile reads the same backward (r ↔ width−r); center at width/2, edges (±1 from the seams) always survive. - **Zone** — the quiet stretch (p, p′²), where p′ is the next prime after p: every hole in it is a genuine prime and every twin slot lying wholly inside it is a genuine twin pair. p's own only strike below p′² is at p itself. Everything the conjecture needs happens here. Nearest prior-art object: Holt's *interval of survival* Δ-H(p_k) = [p_k², p_{k+1}²], which shares the upper endpoint and starts higher. - **Frontier / crystallization** — at p′² the tile below is *finished forever*: holes there are real primes, twin slots real twin primes. The frontier moves outward at speed p′²; settled territory is never touched again. Canonical alias: Holt's *horizon of survival*. - **Stratum (fossil)** — the permanent dent each prime digs at [p², 2p²] in its own tile, copied into every later tile unchanged. **Exact Invariance Lemma:** a stratum's depth is frozen at birth (every band scales by exactly (p−2) per fold). Depth at birth given by the Unification Law's band average. - **Kill image** — the Redundancy Lemma's object: p's new strikes are exactly p × (previous holes) — a p-times magnified copy of the tile's own hole pattern, igniting at p². Its dense head digs the stratum. - **Head** — one word, three objects, and every use states which. (i) The **scour head**: the scour primes with q³ ≤ W+1, where the heaviest caps sit and where the twin question lives; the certificate engine's proven share is the head share only (research/certificate-engine.md §1). (ii) The **head window** [0, x] at the start of a tile period, the twin-poor start whose ratio is capped at e^{2γ} by the Unification Law (research/attack-02-head-bias.js). (iii) The **zone head** F(p): the distance from a prime p to the first twin opener strictly above it, the object of TODO Z4, of history/staging/destroyer-census-01.md §6 and of zonegap-01.js's a_first − p. Senses (i) and (ii) are regions and collide only on the word; sense (iii) is a distance. - **Tail** — the scour primes with q³ > W+1, the complement of the scour head. There every fresh victim v = qm has m prime, so cap₂ counts primes rather than rough integers and the tail sum becomes a prime-counting statement: on the Natal@5 comb the elementary 2 ln 2 drops to (ln 2)/2 at modulus 30 (paper/staircase-note.md §4, §6; research/bv-import-survey.md §3.1). Distinct from the distributional tail of a statistic, which the corpus also calls a tail. ## The ledger - **Remover window** — for a slot at position r in the tile at level x, the only primes that can strike it are q with x < q ≤ √r (q strikes only at positions ≥ q²). Empty window ⟺ crystallized ⟺ real twins. Just 34 removers for all of T₁₃. - **Overlap credit** — kill events landing on already-dead slots (divisible by two removers ⇒ dies once, absorbs two strikes). Removers' total capacity EXCEEDS the census (~2 ln x factor), so the family survives on this forced inefficiency alone: survivors = census·∏(1−2/q) ~ 2C₂W/ln²W → ∞. Removers act exactly like folds not yet performed. Resolves folder-17 steps 6–7: the hoped "capacity < census" inequality reverses; salvation is overlap. Read as a fluctuating quantity rather than a total, the overlap credit is the **X-channel** (see the anchored layer below), which is where the anchored survivor deficit was localized on 2026-08-15. - **Self-strike = twin found (NOT a kill)** — when prime q removes the slot that contains q itself (position q or q−2), it is not destroying a candidate: it is q graduating to wheel-prime status. PROVEN: a self-strike occurs iff the pair (q, q±2) is a genuine twin prime (a composite partner would have been bulk-killed by a smaller prime first). So the self-strike sequence of a tile ENUMERATES its twin primes; honest destruction count = kills − self-strikes. The ancestor (5,7) is the first self-strike. Contrast: genuine kills (bulk + the hit at q²) unmask COMPOSITE candidates. - **Effective Scour / √W tooth-loss** — inside a FIXED tile of width W, a prime q kills a slot only via a self-strike (at position q — a twin FOUND, see above) or a genuine strike at q² and beyond (smaller multiples are already dead). So if q² > W the prime is nearly toothless — it can only self-strike (find a twin), destroying nothing. The effective *destructive* Scour on a width-W tile is just the primes ≤ √W; the rest is boundary noise. Per-prime kills follow kills(q) ≈ 2·(alive before q)/q — the (p−2)/p law as a kill sequence. Marching all primes ≤ √W reconstructs exactly the real twins in the tile (crystallization from the kill side). See scour-into-fixed-tile.js. - **Onset shell** — the Effective Scour's origin-local dual, on the integer line rather than in a fixed tile (Chris's hand derivation, 2026-08-21, machine-verified over 1..10⁴: `natal-onset-01.js`). The first FRESH kill of scour prime q is exactly q² — any earlier multiple q·m with m > 1 coprime to the wheel has lpf(m) < q and was already struck — so at position n the destructive influencer set is exactly {q : q² ≤ n}: a √n staircase, one new tread per q². The onset shell of q is the stretch [q², q′²) on which that set is frozen at the primes ≤ q. The zone (p, p′²) sits entirely below p′'s tread, which is the certification fact in onset coordinates: inside a zone the only active destroyers are the tile's own primes, so holes are primes and twin slots are twin pairs. Per prime, three onsets lag in order: first fresh kill (q²), first kill landing in a twin channel, first destruction of a LIVE pair — the instrument measures all three. The tile-global statement is the Effective Scour entry above (`scour-into-fixed-tile.js`). - **The Scour** — the removers as a collective, with their strike pattern: ⋃_q q × (the tile's own holes). The tile's crystallized output, re-scaled and turned against it; deterministic, periodic per remover, misses = survivors. "The scour of T₁₃ is 34 primes firing 1,699 strikes; its misses are the 456 real twins." - **Two-Moiré Argument** — the conjecture in final native form: Grain and Scour come from disjoint prime alphabets, so they are EXACTLY independent (CRT; provable — aggregate alignment impossible, miss classes guaranteed in the joint tile). TPC = "the Scour never achieves perfect local alignment with the Grain" — the guaranteed misses live in a joint tile ~10⁷³ wide while our window is a ~10⁻⁶⁹ sliver of it. See two-moire-argument.md. - **Origin Excess Lemma** (PROVEN, VERIFIED 14/14) — the origin carries more slots than the mean over the ensemble of all x# translates, for windows S ≤ y′². It has exactly **one** hypothesis, and three further statements travel with it that are easy to miscount as hypotheses: the **non-vacuity remark** (the conclusion is empty unless y′² > x), the **Origin Excess Corollary** (the advantage and its expiry at x < x\*(y)) and the **Scale Collision Proposition** (y′² < x′² strictly, a statement about the parameters rather than about the construction, so no better lemma repairs it). The canonical enumeration by logical type is `maier-matrix.md` §4a; cite it rather than keeping a private count. The advantage is capped: at most a factor of about 2.2, measured maximum 1.372 over the fourteen cells, and at the zone's own width S = x′² it reverses to ρ(2) = e^{2γ}/4. - **Two-class discrepancy (ΔΦ₂), and the ×3 ceiling** (PROVEN) — the signed discrepancy of the two-class sieve against its main term. Its ceiling is 3^{π(x)}, which is exactly the Möbius term count, and the identity R_k(p) → k+1 goes with it. The two-class object does not appear in the literature; the one-class version is Holt's (`discrepancy-two-class.md`, `G2-STATE.md` §8). - **Driving term** — Holt's one-class mechanism adapted to two classes. The **two-class driving-term lemma is PROVEN**, and cleaner than the one-class original because Fact A rules out the second kill. **The driving-term ROUTE to a bound on G₂ is REFUTED**: it certifies gaps of size O(x) and cannot reach x² (`two-class-lower-bounds.md` §7). Keep the two statuses apart; the lemma being proven says nothing about the route. ## The dynamics - **The family** — all twin slots in every tile descend from the single ancestral slot (5,7) in T₃; no new lineage is ever born (zero orphans, verified). Each slot has p−2 children per fold. - **Edge lineage** — the family branch that keeps the seam address kP±1; the Seam Lemma is the Copying Theorem restricted to this line. - **The three houses** — the @5 tile's slots, the family's complete and final aristocracy: House 11 and House 17 (mirror images of each other; the fold-5 graduates) and House 29 (self-mirror; the edge house, owner of all seams). Each carries exactly one third of every census forever. @5 is the unique tile fully crystallized at birth (width 30 < frontier 49): all three founders are real twin primes, immortal by theorem. - **Natal set @p** (the *born-at-p* cohort) — the twin opportunities that first appear as *interior* slots at fold p, born at the **seam** (the mirror protects the edge: position 1 and width−1 are never struck). Size at birth = **p−3** (and 2 at @5). Mechanism: the Seam Lemma — the edge lineage bears p−2 children each fold, 1 stays the eternal edge, **p−3 graduate inward** as the natal set. Carried forward, a natal set scales by (q−2) per later fold. Honest note: "born" = newly-*interior*, not new-lineage (zero orphans — every natal slot descends from the ancestor; the seam is its birth canal). Paired with: - **Carried set** (legacy / inheritance) — the surviving copies of earlier-born opportunities; each existing slot keeps **(p−2) of its p copies** per fold (coprimality forbids more than 2 struck). - **Eternal edge** — the single slot (House 29 / ancestral seam pair, ≡ −1 mod width) that is *always* the edge, never born; descends straight from the ancestor. The "1 / width−1 never touched" slot. Fold law: D_new = 1 (edge) + carried·(p−2)-per-slot + natal(p−3). Every tile decomposes as edge + Σ_q (natal@q, carried to the present). Verified T₅..T₁₃ (e.g. @13 = 1 + 990 + 396 + 88 + 10). See [[birth cohorts]] below. - **Birth cohorts** — the exact decomposition of the census by creation fold: each fold p the edge bears p−2 children; one remains the edge, p−3 graduate into the interior as that fold's newborns (cohort(5) = 2, the special case). D_x = 1 + Σ cohort(p)·∏_{p 0 implies twins are infinite, because E → ∞ is Mertens and survivors-are-twins is proven (paper/anchored-note.md §7). No converse is proven or claimed, so β's positivity is at least as strong as TPC and may be strictly stronger; do not write "TPC *is* the positivity of β". The weakest sufficient statement the corpus has is weaker still and is the one to aim at: **Non-Annihilation Sufficiency** (PROVEN, paper/anchored-note.md §8 Proposition 2) — if the anchored survivor count S(x) ≥ 1 for infinitely many x, twins are infinite. Conjectured limit of β is e^{2γ}/4 = 0.793055, the Unification Law's zone-edge value. Do not call β "the anchored ratio" or "R"; older files use both. - **Assumption A** — the positivity of β, stated as an assumption and priced honestly at Hardy-Littlewood strength. **"Assumption A" alone is not a statement: every occurrence carries a form.** Its *sharp* form (β → e^{2γ}/4) is algebraically equivalent to HL on the 11/17 comb; its *weak* form (β ≥ c) is blocked by the parity floor 2 at every known input including GEH. The name is used across the papers, so keep it. - **Fused window** — the anchor's two strike classes {0, −2} mod q are mirror-adjacent, so they glue into a SINGLE cyclic window of length L_q ≈ 2W/q in the tile's own dilated natal pattern, rather than two independent ones. PROVEN — this is the **Fusion Identity** plus the **Mirror-Phase Doubling Lemma** (natal-cap-19-calm-lemma.md). It is the mechanism behind the anchored calm, and it is why W/2, the unique mirror-fixed phase, duplicates instead and is provably variance-doubled. - **The anchored calm** — the measured fact that the anchor sits in the extreme low tail of the ensemble's strike-variance statistics (rank 2 of 510,510 at @17 on equal-weight Z2). One sighting, not four: three of the original four dissolved under exact control. **"The anchored calm" names a phenomenon and is never a claim**: it decomposes into separately-named sub-claims at separate statuses, from PROVEN through CERTIFIED at six levels to one that is REFUTED in its uniform-in-q form and one that is merely MEASURED, and the status table is [anchored-calm.md](anchored-calm.md). There is no "Fused-Window Calm Lemma"; the name is retired. WARNING, on the record: the calm does NOT concentrate survivors (corr(VR, S) ≈ 0). Calm is about strikes; survival is about the X-channel. - **The X-channel** — the overlap credit read as a fluctuating quantity. 97% of the anchored survivor fluctuation lives here, with corr(X, S) = 0.99, and it is invisible to every strike statistic. **X-limitation Theorem** (PROVEN PER LEVEL, at the enumerated levels @11, @13, @17 and @19): strikes alone cannot annihilate the natal set at any loudness; annihilation requires an overlap collapse of 8 to 15 σ_X at @13 and @17. The all-x form is OPEN, because no level-uniform bound on max VR exists, so "from x = 13 upward" remains an extrapolation rather than a theorem. Of the two measured trends it rested on, one broke and the other held decisively. Max VR was enumerated at a fourth level and does not fall: 2.78, 2.35, 2.14, 2.293 at @11, @13, @17, @19. The driver S̄/√(K·V̄) does keep rising, 3.12, 4.88, 9.75, 24.96, and since the per-level hypothesis is exactly √(K·V̄·maxVR) < S̄, the @19 driver is what promoted @19 to a theorem (margin ×271.7 on the squared form; natal-cap-38-loudness-driver.js). So Assumption A restated in its true channel: the anchored overlap-credit deficit stays below (1−ε)·S̄. Measured X(0)/X̄ = 1.4541, 0.9908, 0.9482, **0.9558** at @11/@13/@17/@19 — **down for two levels, then UP at the fourth, so no monotone descent holds** (`natal-cap-35-x-multiplicity.js`, whose @19 row is a u-form means row). That is the same shape, at the same level, as max VR (2.78, 2.35, 2.14, 2.293): @19 breaks both monotone claims and in the same direction. The deficit lives in multiplicity m ≥ 3, which pair-based instruments track poorly (natal-cap-31-calm-vs-kill.md). - **The skeleton** — the mod-30 kernel that survives when the anchored ledger's 2ⁿ branch algebra collapses. Two objects, at two calibrations. **Skeleton Collapse Theorem** (PROVEN, all x and all q): the whole ledger reduces to one kernel K = 15C − 2P (natal-cap-30-skeleton-bound.md). **Aggregate 30-Skeleton Bound** (CERTIFIED at six levels, @11 through @29): G30_agg < 1/2 in exact BigInt. Its all-x form is the **Skeleton Equidistribution Conjecture** (OPEN, door named and analytic: equidistribution of ⌈W/q⌉ mod 30 over the scour primes); the door as named is CLOSED as a route (2026-08-30, `OUTCOMES.md`): over every scour prime it governs 9.2%, −0.9%, 0.2%, 0.5% of the mass at @13..@23, at most 0.0102 of G30_agg, and the open side is the inequality itself at moduli exceeding the window. - **Staircase cap (cap₁, cap₂, cap_K)** — the per-prime, HISTORY-BLIND hard upper bounds on a scour prime's fresh kills. cap₁ is the divisibility skeleton via Legendre-Buchstab counting; cap₂ folds in the deterministic natal residue system on the cofactor; cap_K adds, for each of the first K scour primes q′ < q, the **one** forbidden freshness residue that can fire (v ≢ −2 mod q′ on the A side, v ≢ +2 on the B side). Its companion v ≢ 0 (mod q′) is void under P⁻(m) ≥ q, so the density factor is ∏(1 − 1/(q′−1)), one class per prime, and the sifting dimension is 1. In a formulation that *sifts* for P⁻(m) ≥ q rather than assuming it, m ≢ 0 is a real second class and the dimension is 2 (history/staging/thm-buchstab-transfer-shallow.md §2(c)). The caps form a monotone ladder fresh ≤ cap_K ≤ … ≤ cap₂ ≤ cap₁ (paper/staircase-note.md). - **Ladder depth K, and K*** — K is how many freshness moduli a cap folds in; **K\*** is the least K at which the pigeonhole closes, Σ cap_K < N. K* is the exact price of history-blindness. Measured 0, 0, 2, 10, 27, 69 at @11..@29, which is SUB-LINEAR in the scour and tracks the quarter-power phi-band. Certificate efficiency at fixed relative depth improves with level; read only at the crossing point it appears to collapse, and that appearance is an artifact of the crossing point rather than a property of the certificate (natal-cap-24-boundK-curve.js). - **Sub-Poisson window counts (Var/E)** — the natal field's suppressed large-scale density fluctuation: window counts are sub-Poisson, Var/E drifting 0.152 → 0.396 across @7..@37 rather than sitting at 1. This is NOT hyperuniformity; see that entry below. It is the reason the extremal window excess obeys E = 0.97·σ(ℓ)·√(2ln(W/ℓ)) with no free multiplier (natal-cap-25-excess-law.js). Var/E is not a constant in either the level or the window, and the two ladders that carry it are different objects. For the **full** twin process the stable structure is a scaling law in the window exponent u, ln(Var/E) ≈ −(0.24u² + 0.13u), whose u = 2 value drifts 0.251 → 0.321 over y = 97 to 2003 with the limit unknown (paper/variance-note.md §6). For the **comb-restricted** Natal@5 process the diagonal u = 2 ladder above (0.152 → 0.396) is the one that reaches deep levels, and its limit is Paper III's last open question. The earlier reading "≈ 0.611, favoured 10:1" is refuted as an inference: the note's own fit protocol, run on a model whose limit is 0.455 by construction, returns 0.615 and wins the frozen forecast too, so no fit on this lever arm can see the limit. The standing HEURISTIC closed form is lim Var/E = Pr[GD(2) > u] = 1 − e^{−2γ}(9/2 − 4 ln 2) = 0.45546 at u = 2, zero parameters, residuals ≤ 0.002 from x = 13 against the Monte-Carlo model (+0.0029 at x = 13 against the exact model), exact at one excluded class, with the model's limit theorem PROVEN (`history/staging/varE-limit-theorem.md`) and one step open, the θ = 2 identification (`varE-theta2-proof.md`; `varE-spectral.md`, red-teamed 2026-08-28; paper/variance-note.md §7 still carries the older reading pending its edit). - **Hyperuniformity** — Torquato's word, and not a synonym for the entry above. A point configuration is hyperuniform when its structure factor vanishes at zero wavenumber, and the taxonomy sorts configurations by how the local number variance Var[N_L] grows with the window: class I as the window surface area, which in one dimension means bounded; class II with an extra ln L; class III as L^{1−α}, α ∈ (0,1). Torquato, *Physics Reports* **745** (2018) 1–95, read at the author's arXiv full text and cited by ITS pages, not the journal's: arXiv:1801.06924 eq. (14) p. 10 (the definition), §5.3 p. 23 (the three classes), §5.5 p. 28 (every periodic configuration is class I). The published article runs 1–95 on different pagination and those are not its page numbers. **The tile is class I, and vacuously so.** It is periodic with period W, so Var[N_L] is W-periodic and Var[N_W] = 0 exactly by the sum rule; periodicity alone puts every level in class I, and no window any live document reasons about exceeds one period, so the classification carries no arithmetic (PROVEN, `history/staging/import-repulsive.md` §3). **At the window scale the programme actually reads, L = y^u with u fixed, the twin-slot count is sub-Poisson and not hyperuniform**: Var[N_L] grows linearly in L with Var/E between 0.152 and 0.396, which is the Poisson scaling with a reduced prefactor rather than a suppressed one, MEASURED. That reading is a statement about the diagonal family of configurations, one per level with L and the sieve level moving together, not a hyperuniformity class for any one tile; the standing values and the open limit sit under **Sub-Poisson window counts (Var/E)** above. Torquato's practical fallback fails here too: the effective hyperuniformity criterion S(0)/S(k_peak) ≲ 10⁻⁴ (§11.1.6 eq. (252), arXiv p. 78) is of order 10⁻¹ to 10⁰ on this object. What is genuinely small-k here is the comb's structure factor S(ν) = δŴ(ν), and natal-cap-29-sigma-plateau.js has been computing it under another name. - **Lemma V** — the two-dimensional analog of Iwaniec's 1980 linear-sieve error term: signed cancellation of the bilinear interval sawtooth remainder, uniform in position. NEEDED, NOT PROVEN in the supremum form the route needs; its MEAN-SQUARE form is PROVEN and unconditional (G2-STATE.md §0; re-derived adversarially 2026-08-29, redteam-0829-theorem1.md §§2, 9), a period mean rather than a supremum, so Lemma V proper is untouched. It is what the Brüdern-Fouvry vector sieve needs to decouple, and full decoupling gives 1 + √e ≈ 2.649, with any partial decoupling past θ_total = 1.2090 beating our proven 4.2665 (corrected 2026-08-18 from 1.2417, which was the wrong asymmetry; research/sift-limit-attack.md §4.5). **But Lemma V is not the operative assumption at the measured working point.** There s/u ≈ 1.2, outside the range s ≤ u that Lemma V is stated in, so the ladder rests instead on the unproven Gaussian Maximal Law below, which is the whole price of the route (research/sift-limit-attack.md §§3, 4.5; research/theta-ladder.md). Below 2 is TPC outright. That sidelining is TPC-specific (2026-08-20 night): at the β₂ target — the ρ maximal law's F2 form — the working point s/u = 3/u sits inside Lemma V's stated range for every delivered exponent u ∈ (3, β₂), so the sub-question is un-mooted there in its weakest form only; what re-enters is the ask, not an import route (research/history/staging/rho-maximal-law.md §5). - **Gaussian Maximal Law for the interval sawtooth** — the operative unproven input on the exponent road, and the one to price the route against: that the bilinear interval sawtooth remainder obeys a Gaussian maximal inequality at the measured working point s/u ≈ 1.2 to 1.25. NEEDED, NOT PROVEN, and every number in theta-ladder's conditional column rests on it. Its mean-square form is done and verified; the uniform-in-position form is not (research/sift-limit-attack.md §3, research/theta-ladder.md §5b). - **The Gap Reformulation** — TPC as a statement about the two-class Jacobsthal exponent: an exponent below 2 proves it. The only route in the programme with a defined finish line, which is why the exponent road is TODO item 0. ## The pane (adopted 2026-08-16) Vocabulary for Chris's square-window target. Kept separate from the zone because the two windows are different objects and confusing them wastes time. - **Pane** — the window (n², (n+2)²), of width 4n+4. Contrast the **zone** (p, p′²), of width ≈ p². The zone is quadratic and its consecutive instances overlap almost entirely, so a handful of twins satisfies vast stretches of it. The pane is LINEAR, that redundancy is gone, and the statement develops genuine exceptions. Canonical alias: the interval between squares two apart. - **Pane slots** — the Natal@5 positions in a pane, meaning r ≡ 11 or 17 (mod 30) with the whole pair inside. About (4n+4)/15 of them. - **Pane Postulate** — every pane contains a twin prime pair. MEASURED: one exception, n = 26, for n up to 1e5. The tighter (n², (n+1)²) version has twelve exceptions ending at 122, and is OEIS A091592, conjectured complete, tested to 1e7, keyword `hard`; its companion A091591 records that proving it would prove the twin prime conjecture. - **The Pane Bound** — a proven upper bound on the number of pane slots destroyed by the primes 7 ≤ q ≤ n, lying strictly below the pane's slot count. By pigeonhole a Pane Bound forces a survivor, and a survivor in a pane is a genuine twin pair, so establishing it for all large n proves TPC. **VERDICT (attack A7, research/a3-07-pane-overlap.js): the pane is STRICTLY HARDER than the zone route it was proposed as an alternative to.** Scanning the sieve cutoff, the first z at which any provable bound enters the corridor has lnW/ln z decreasing toward **1**. Classical TPC sits at sifting parameter 2, the parity floor; dimension-2 sieves are proven only above β₂ = 4.2665. In the repo's own currency the pane's ledger entry reads **"4.2665 proven, 1 needed"**, which is HALF the parity floor. The best provable family falls short by 159 corridor widths at n = 3e6 and the miss grows like ln²n. This refutes a family of counting arguments, not the Pane Postulate itself, and does not touch bounds using structure other than counting. - **The corridor** — the band a valid Pane Bound must land in: at or above the true removal count, strictly below the slot count. Its width is exactly the twin count, **5·C₂/ln²n = 3.301/ln²n** of the slots (VERIFIED at 3.348, 3.285, 3.386, 3.326, 3.325 for n = 3e4 to 3e6). MEASURED at 25% at n = 100, 8.2% at n = 1000, 3.2% at n = 30,000, 0.52% at n = 3e6. **The corridor closes.** That is the one-line statement of the difficulty: a valid Pane Bound is not merely a better bound, it is an asymptotically exact one. The factor 5 rather than 15/2 is the point to get right: the slot set sees only two of the three twin residues mod 30, so counting all twins against it overstates the corridor by exactly 3/2. - **The overshoot** — why the naive bound misses. Counting strikes ON COMB SLOTS, total capacity is Σ_q 2·slots/q, so **capacity/slots = 2·Σ_{7≤q≤n} 1/q ≈ 2(lnln n − 0.772)**, independent of the pane's width. VERIFIED against exact counts: 1.32, 1.94, 2.33, 2.61, 2.89, 3.13 at n = 100, 300, 1000, 3000, 10⁴, 3·10⁴, matching the closed form to three digits. It exceeds 1, so the naive pigeonhole fails, but it grows only like a double logarithm: about 3.7 at n = 10⁶ and still only about 9 at n = 10¹⁰⁰. Capacity EXCEEDS the census, so twins survive on [[overlap credit]] rather than scarcity, and a Pane Bound must bound OVERLAP rather than capacity. Both sides of the ratio must be counted on the comb: strikes on ALL integers against comb-only slots is a factor-15 mismatch and inflates the overshoot to 25 to 60. - **Width invariance (PROVEN, one line)** — slots and capacity both scale linearly in the pane's width, so their ratio does not depend on it. The overshoot is a function of the sieve depth alone, and the sieve depth is set by the pane's HEIGHT, not its width. Widening a pane therefore cannot close the pigeonhole; it only suppresses the O(π(n)) boundary term from the per-prime ceilings, which is a real but bounded gain (about 26% of capacity at n = 3·10⁴). ## Deprecated / historical terms - "Copy set" → **tile**. "Pattern/wheel" in older research files → tile. - "Kill shadow" → absorbed into the Unification Law (kept only historically). - Files written before 2026-08-14 use the older words; readings stand. ## The problem, in this vocabulary Every fold multiplies the family by p−2 and moves the frontier to p². The Twin Prime Conjecture: *the family never stops sending at least one child into the zone before it crystallizes.* Measured: the zone's supply grows like p²/ln²p with safety margin ~p. Provable: two primes within 2·ln p of each other in every zone. The gap between those two sentences is the parity wall. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*