# The bridge between Z2 and G2: every object on the road, and what each reduction discards > **RIDER 2026-09-04 (orchestrator, from `redteam-0904-r0-extension.md`).** > The zone property quoted near line 471, "coprime to x#, and x < l₁ < x′²", > must bound both coordinates, `x < l₁, l₂ < x′²`, for its extension to be > the pairs of primes in the zone and for H5 to hold; with one coordinate > bounded it is Tao's Example 2 and H5 fails. Prefer "pairs of primes l, l+2 > in (x, x′²)" over "twin prime pairs", which borrows the conjecture for its > (−1, −1) witness. Conditional on Claim 1 throughout. *(Staging note, HELD under the publication moratorium. Computes nothing new; proposes no route. Every claim carries a calibration marker and a pointer. Written 2026-08-29 as one of four parallel reads.)* ## 0. What is open, first - **The band on G₂ is open and unmoved.** `G₂(x#) ≪_ε x^{4.26645+ε}` is PROVEN (`paper/beta2-note.md`, via `research/G2-STATE.md` §0); the Gap Reformulation needs exponent 2 with constant strictly below 1 (`history/staging/attack-block-08-secondmoment.md` §1.1 as quoted in `history/staging/attack-wrongdirection-audit.md` §1); the open band is (2, 4.26645] and every route in this corpus has left it where it was (`README.md` §Status). - **There is no band on Z₂ at all.** Below the zone's own width, every statement the corpus can make about Z₂(p) is either a measurement or is TPC-strength. `Z₂ + head + tail < width` for all p is the strong Zone Postulate; i.o. it is TPC; and any a-priori bound on any one of the three pieces at the window scale is TPC-strength, because the three objects are only DEFINED when the zone is occupied (`history/staging/zonegap-02-reduction.md` §2, (R0) and (R1); PROVEN, with the circularity stated there). The one non-circular sufficient form, (R2), is a first-slot bound, and its one proven instrument is the Euclid anchor, which is G₂ again. - **The headroom the reduction spends is a factor of about p, not p².** The postulate needs F < x′², and F ≥ x′ always, so window/F can never exceed x′ and is saturated within 1.0000004 of that ceiling at x = 10⁹; the genuine advantage of the origin over the worst case is G₂/F, MEASURED at roughly 7 and growing only polylogarithmically (`research/ZONE-POSTULATE.md` §5). - **What this note does not do.** It computes nothing new: every number below is cited to the artifact that produced it, per the standing compute rule. It proposes no route, and it is not an attack. Three of this corpus's closed routes were bridges of exactly the kind a map like this invites (`research/REFUTED.md`: the localized merge chain, the maxsum bridge as a 0c→L converter, the L = 1 residue count), and a fourth, fifth and sixth were sub-targets that turned out TPC-strength. - **One thing in the brief's own framing needed correcting before the map could be drawn**, and it is carried in §5: the separation at issue cannot be "tile versus zone", because the tile statement `G₂(x#) < x′² − 2` is about the tile alone and does reach the zone statement (PROVEN, `research/G2-STATE.md` §1b). §5 states what it is between instead. ## 1. The chain of objects, zone to tile **Read the caveats in the columns, not around them.** "Expires" is the largest window or level at which anything is proven, certified or exhaustively verified about the object, not the range over which it has been sampled; several rows expire at a level far below where they have been measured. "Rot-inv" asks whether the object is invariant under translation of the tile (equivalently, whether it is a statement about residues alone rather than about a distinguished position). TPC-strength labels are the audit's: **(i)** strictly weaker than TPC in the corpus's witnessed sense, **(ii)** implies TPC or more, **(iii)** undetermined (`history/staging/attack-wrongdirection-audit.md` §2). | object | definition pointer | rot-inv | scale | expires at | grade | TPC-strength | |---|---|---|---|---|---|---| | **Z₂(p)**, max gap between consecutive twin openers in the zone (p, p′²) | `history/staging/zonegap-01.md` §0 | **no** (interval, anchored at p) | ln³p family, band constants c₃ = 3.426, 3.681, 4.022, 3.930 at 10², 10³, 10⁴, top band (`zonegap-01.md` §4) | nothing proven at any scale; VERIFIED exhaustively to p = 10¹¹ (27,292 zones), envelope record-exact against A113274/TOS | MEASURED | **(ii)** in every form that bounds it below the width (`zonegap-02-reduction.md` §2 R1) | | **head F(p)**, distance from p to the first twin opener above it | `research/GLOSSARY.md` §head, sense (iii); `zonegap-01.md` §6 | **no** | ln²p: mean head/ln²p = 0.7229 against HL's 0.7574 (`zonegap-01.md` §6) | one proven bound only: F(p) ≤ G₂(p#) − p − 1, i.e. p^{4.2665+ε} (`zonegap-02-reduction.md` §3.1) | MEASURED + one PROVEN bound | **(ii)**: F(p) < p′² − p − 2 i.o. ⟹ weak ZP ⟺ TPC (`zonegap-02-reduction.md` §2 R2) | | **tail**, p′² − a_last | `zonegap-01.md` §0 | **no** | ln²(p′²): c_tail = 0.7771 at [3163, 10⁴), against the head's 0.6693 ln²p on the same zones; the factor 4 is the unit and nothing else (`history/staging/zone-tail-01.md` via `QUESTIONS.md` `Q-zone-tail`) | nothing proven; field measured at 1,225 zones, band drift does not settle | MEASURED, PARTIAL | **(ii)** as a summand of R0 | | **M(x, x′²)**, largest twin-slot gap inside the zone | `research/maxgap-law.md` §8 | **no** | M/ln³x flat at 3.2 to 3.7 over a 47-fold range in x | measured at seven levels x = 211..9973; reproduced 7 of 7 EXACT by the zone sweep, the two windows (x, x²) and (x, x′²) binding the same gap at all seven (`zonegap-01.md` §1) | MEASURED | **(ii)**, same object as Z₂ under the Zone Restriction Lemma (see §6) | | **stretch S_Q = [Q², Q′²), and T(Q)** its twin count | `history/staging/stretch-01.md` §0, §1 | **no** | width ~ 2Q ln Q | occupancy VERIFIED at all 5,484,596 stretches with Q′² < 2⁵³ ≈ 9.0·10¹⁵, one exception cleared directly (`stretch-01.md` §3) | VERIFIED (occupancy), the postulate itself OPEN | SP ⟹ strong ZP ⟹ weak ZP ⟺ TPC, so **(ii)**, and per level strictly above zone occupancy (`stretch-01.md` §2) | | **X(K)**, both-composite pairs in S_Q rough beyond p_K; **floor_K**; **capU_K(r)** | `history/staging/quadpoint-identity-01.md` §1 | **no** | stretch-local | identity VERIFIED at every depth of all 1,227 anchors to Q = 10007 | PROVEN (the identity), the certificate OPEN | certificate **(ii)** (audit §3.1); an upper bound on X alone is the legal half | | **y\***, the crossing depth, candidate h^{1/(2e^γ)} | `quadpoint-identity-01.md` §2, §3 | **no** | ln y\*/ln h ≈ 0.2807 | y\*(Q) exists ⟺ T(Q) ≥ 1, so the law is measured only where it is defined (all 1,227 anchors have T ≥ 1) | HEURISTIC derivation, MEASURED-consistent | **(ii)** in the all-Q form, and stronger; legal only stated conditionally on {T ≥ 1} (audit §3.2) | | **S(x)**, anchored survivor count | `paper/anchored-note.md` §1, §3 | **no** (one phase of W) | whole tile, W = x# | ten levels, to x = 41, W = 3.04·10¹⁴, "the last level this arithmetic reaches" (`paper/wall-note.md` §2) | computed exactly at ten levels | **(ii)**: S(x) ≥ 1 i.o. ⟹ TPC (Proposition 2, PROVEN, `anchored-note.md` §8) | | **E(x)**, exact rotation-ensemble mean | `anchored-note.md` §1, §3 | **yes** | whole tile | exactly computable at every computed level; E → ∞ is Mertens | PROVEN + computed | **(i)**: an ensemble mean decides nothing about one member (Proposition 1, `anchored-note.md` §3) | | **β(x) = S(x)/E(x)** | `anchored-note.md` §3; `GLOSSARY.md` §anchored bias | **no** | descends 1.156 → 0.846 over @7..@41 | ten levels | MEASURED; liminf β > 0 ⟹ TPC is PROVEN | **(ii)**, and possibly strictly stronger; no converse is proven | | **anchored calm δ, ρ, F**, with (1+δ)(1+F) = 1+ρ | `history/staging/import-suen.md` §7 and its correction list | **no** (δ is anchored) | per-level percentages: \|δ\| = 0.29% to 2.02% | seven levels; ρ/F = 2.001 at @11 and 1.266 at @19, so δ exceeds the forced scale at 2 of 7 | the identity is exact, F exact, ρ measured | **(ii)**: any \|δ\| ≤ c < 1 is TPC-strength through the identity (`REFUTED.md`, the fourth wrong-direction arrival) | | **anchored caps cap₁, cap₂, cap_K, capU_K; floors; K\*** | `paper/staircase-note.md`; `GLOSSARY.md` §staircase cap | **no** | per-level integer counts | floors at @11, @13, @17, @19; K\* measured 0, 0, 2, 10, 27, 69 at @11..@29 | PROVEN caps, CERTIFIED floors, finite | **(i)** while at named levels; any all-x family with floor(x) → ∞ is **(ii)** (audit §3.4) | | **advmin@11 = 16**, the class-uniform ceiling | `history/staging/attack-advmin-1113.md` | **yes** (quantified over classes) | one level | exact at @11, two disjoint proof stacks | PROVEN at that level | **(i)**, and it CLOSES the class-uniform family: 16 sits 18 below the anchored floor 34 | | **maxsum_m**, largest sum of m cyclically consecutive tile gaps | `research/a3-05-bound-L.md` §5; `GLOSSARY.md` (scan statistic) | **yes** | maxsum_m ≥ G₂ always | exact at T₃₇ over all 217,929,355,875 gaps (`history/staging/scanstat-t37.md`) | PROVEN inequality, MEASURED law | **(i)** as an upper-bound instrument; the Bridge Floor caps what it can deliver at ≈ 0.183x | | **θ ladder, all-positions exponent; R_H** | `research/theta-ladder.md` §5b; `history/staging/phase1-T4-maximal-law.md` | **yes** (all positions) | need_true/z² = 0.5485, 0.4877, 0.4637 at z = 19, 23, 29, self-consistent | exact full-period walks to z = 31; prefix bounds above | MEASURED exactly, the law itself unproven | **(ii)** for the sharp maximal law on R_H at the operative window (audit §3.6a); the H-free ρ̃ form is **(i)** | | **G₂(x#)** | `research/G2-STATE.md` §1a | **yes** | needs < x′² − 2; proven ≪ x^{4.26645+ε}; measured exponent 1.50 (control 1.57) | exact ladder to x = 37 with an exhaustive maximality certificate (G₂(37#) = 528) | PROVEN upper and lower bounds; the band OPEN | **(i)** across the whole band (2, 4.26645]; **(ii)** only at exponent 2 with constant < 1 | | **g(x#)**, one-class Jacobsthal at primorials | `G2-STATE.md` §0; `research/two-class-lower-bounds.md` §9 | **yes** | G₂ ≥ g pointwise | Iwaniec 1978 gives g ≪ x² with an inexplicit constant | PROVEN | **(i)** above exponent 2; at exponent 2 with an explicit constant it is prize-adjacent (Erdős #687) | | **Ĝ(s) = G₂(P(s)#)**, the doubling target's object | `history/staging/attack-doubling-01.md` §1 | **yes** | base-2 chain ratios 3.0000, 5.0000, 2.2000, 5.2727, 3.1034 at s = 2..32 | the last chain datum rests on Wang 2024's A144311 a(18), literature grade | MEASURED | **(i)** for the all-s form; **(ii)** for the eventual form at every C₂ ∈ (3.1034, 4) (audit §3.3) | **The one exception to "rot-inv means free" is worth stating here rather than in §5**: E(x), maxsum_m, the θ ladder and G₂ are all rotation-invariant, and three of the four are still hard. Rotation invariance is what makes a statement expressible in residues; it is not what makes it cheap. `research/THE-LENS.md` §5 owns the rule and states it as a triage heuristic, not a theorem. ## 2. Every proven inequality between them, with direction **What is wrong here first.** Not one of these inequalities is a bridge in the useful direction. Every arrow that runs zone-to-tile is either vacuous when read backward (the floors it puts under G₂ are polylog against a proven floor of x ln x) or is the postulate again. That is the finding of this section, and it is stated before the list so the list is not read as an inventory of options. **(P1) Z₂(p) ≤ G₂(p#), when the zone holds ≥ 2 pairs [PROVEN, elementary; stated in the corpus at `zonegap-01.md` §5 as "a one-line theorem"; VERIFIED at the shared exact levels there].** The step the statement turns on is that consecutive in-zone openers are consecutive twin-slot openers of the tile, so the two gap sets agree inside the zone and no summing of slot gaps is needed. Written out: *Proof.* Fix p ≥ 7, W = p#, and note p′² < W (`stretch-01.md` §0), so the zone (p, p′²) sits inside one period. Let A = {a : p < a, a+2 < p′², gcd(a(a+2), W) = 1} be the in-zone twin-slot openers, which by the Zone Restriction Lemma is exactly the set of in-zone twin-prime openers (`zonegap-02-reduction.md` §1, PROVEN). Suppose a < b are consecutive in A, and suppose some twin-slot opener c of T_p satisfies a < c < b. Then c > a > p, and c + 2 < b + 2 < p′², so c ∈ A, contradicting consecutiveness in A. Hence no tile opener lies strictly between a and b: a and b are **consecutive** twin slots of T_p, and b − a is one of the tile's consecutive-opener gaps, so b − a ≤ G₂(p#). Taking the maximum over consecutive pairs in A gives Z₂(p) ≤ G₂(p#). For p = 2, 3, 5 check directly. ∎ *Direction of use.* Forward it is used nowhere as an instrument, because it runs the wrong way: it bounds the easy object by the hard one. Backward it is **vacuous**: Z₂(p) is MEASURED at 3.43 to 4.02 ln³p (`zonegap-01.md` §4), so the floor it puts under G₂(x#) is polylog in x, against the PROVEN floor G₂(P(y)) ≫ y (ln y)³ (lnlnln y)²/(lnln y)⁴ (`G2-STATE.md` §0). It is outclassed by a factor x. What the inequality does carry is the measured slack: EQUALITY at p = 2, 3, 5, 7, then G₂/Z₂ = 1.40, 2.20, 3.00, 2.08, 1.36, 1.72, 2.32, 3.52, 3.64, 4.12 at p = 11..43, so **past level 7 the tile's extremal gap lives beyond p′² and Z₂ is not G₂ restricted** (`zonegap-01.md` §5, MEASURED). That ratio is the reduction's loss, measured in the only place it can be measured exactly. **(P2) F(p) ≤ G₂(p#) − p − 1 [PROVEN; VERIFIED at p = 7..17, `zonegap-02-reduction.md` §3.1 SEC C1].** F(p) is the distance from p to the tile's first twin-slot opener above p. The −p−1 is the Euclid edge anchor: the tile carries the twin slot (W−1, W+1), so cyclically an opener sits at a₀ = −1; the next opener a₁ satisfies a₁ = p + F(p), and the cyclic gap a₁ − a₀ = F(p) + p + 1 is one of the tile's consecutive-opener gaps, hence at most G₂(p#). The verification cited reads anchor gaps 12, 18, 18, 30 against G₂ = 30, 42, 66, 108, so the inequality is loose at every checked level. *Direction.* Forward this is the corpus's ONE proven head bound, and its shortfall is exactly the open band: p^{4.2665+ε} against a needed p². Backward it says G₂(p#) ≥ p + F(p) + 1 ≈ p + 0.72 ln²p, again far below the proven x ln x floor: vacuous in that direction too, and `zonegap-02-reduction.md` §3.2 notes it is the same fact as the mirror's edge isolation, not a second one. **(P3) head + Z₂ + tail ≤ width, equality iff k = 2 [PROVEN; VERIFIED digit-exact at five levels, `zonegap-02-reduction.md` §2 (R0)].** With k the number of in-zone pairs, width = head + Σ gaps + tail exactly, and for k ≥ 2 the maximum gap is at most the sum of the gaps, with equality iff there is exactly one gap. The identity is the decomposition the whole zone side rests on. Its measured shares at band B4 are **1.82% head, 89.72% Z₂, 8.46% tail, with the tail's share falling** (`zone-tail-01.md`, `QUESTIONS.md` `Q-zone-tail`, MEASURED at 1,225 zones), so filling in the two small pieces moves the difficulty nowhere: Z₂ carries nine tenths of the width and carried it before the tail was measured. *Direction.* Forward it gives the chain (R1): Z₂ + head + tail < width i.o. ⟹ infinitely many occupied zones ⟺ TPC. Backward, and this is the honest reading the corpus insists on: **the premise's three objects are defined only when the zone holds ≥ 2 pairs, so the premise already contains occupancy**, and by (R0) the strict inequality is exactly "≥ 3 pairs". The implication is trivially PROVEN and the whole difficulty is producing the premise (`zonegap-02-reduction.md` §2). **(P4) The Gap Reformulation: G₂(x#) < x′² − 2 ⟹ Zone Postulate at x ⟹ (i.o.) TPC [PROVEN, `G2-STATE.md` §1b, `ZONE-POSTULATE.md` §3].** Same edge anchor as (P2): reading forward from the origin the first twin slot lies within G₂ of the edge slot, so it lands below x′² − 2 and is a genuine pair. Since ln(x#) = θ(x) ~ x, the window x² is (ln x#)² and the sufficient condition is a two-class Jacobsthal exponent below 2. *Direction.* Forward it is the programme's one route with a defined finish line (`GLOSSARY.md` §Gap Reformulation). **Backward it fails, and by a measured factor:** the Zone Postulate does not imply G₂ < x′² − 2, because the tile's extremal gap lives beyond the frontier past level 7 (P1's ratio table). The implication is strictly one-way, and §3 quantifies the loss. **(P5) The stretch chain: A091592-complete ⟹ SP ⟹ strong ZP ⟹ weak ZP ⟺ TPC ⟺ weak SP [PROVEN arrow by arrow, `stretch-01.md` §2].** SP ⟹ strong ZP is containment (S_p ⊆ zone p). The converse containment FAILS, PROVEN: no zone fits inside any stretch, so the strong Zone Postulate delivers SP by no interval argument, and whether some non-containment derivation closes the gap is OPEN. Weak SP ⟺ TPC is elementary in both directions. *Direction.* Forward, A091592-completeness is an OEIS conjecture with keyword `hard` whose companion A091591 records that proving it proves TPC (`ZONE-POSTULATE.md` §5a), so the top of the chain is not a lever. Backward, the chain says SP is per level strictly more demanding than zone occupancy, so descending it costs and never pays. **(P6) The capture identity: Σ_r capU_K(r) = (C − T) + X(K), equivalently floor_K = T − X(K) [PROVEN, elementary; VERIFIED at every depth of all 1,227 anchors, `quadpoint-identity-01.md` §1].** The half-open window convention is load-bearing: the loose convention breaks the identity by exactly one at every anchor tested. *Direction.* Forward, the certificate floor_K ≥ 1 is T ≥ X(K) + 1 ≥ 1, a twin pair in the stretch, hence **(ii)** (audit §3.1). Backward it is the useful reading and the corpus already carries it: the cap machinery needs no caps, the certificate at (Q, y) is exactly "X(y) < T", and the transplant instrument was measuring a classical object in disguise. **(P7) (1+δ)(1+F) = 1+ρ [exact identity, F exact, ρ measured; `import-suen.md` §7 as corrected].** δ is the anchored dependence defect, F the forced scale, ρ the measured ratio. *Direction.* Forward it converts a bound on δ into a statement about the anchored survivor count. Backward it kills the target: δ > −1 is equivalent to S(0) > 0, so ANY bound |δ| ≤ c < 1 is TPC-strength, which is the fourth recorded wrong-direction arrival (`REFUTED.md`). And δ in fact exceeds the forced scale at 2 of 7 levels, ρ/F = 2.001 at @11 and 1.266 at @19, so the bound is not merely TPC-strength but MEASURED false in the shape it was wanted. **(P8) G₂(x#) ≥ g(x#) pointwise [PROVEN, elementary; VERIFIED at all ten shared terms, ratios 2.00 to 8.00, `ZONE-POSTULATE.md` §3].** Twin slots are a subset of holes. *Direction.* Forward it imports the Rankin–Pintz–FGKMT lower bounds free. **Backward it is a warning and the corpus reads it as one** (`ZONE-POSTULATE.md` §6, route A): G₂(x#) < x′² − 2 implies g(x#) < x′² − 2, an explicit-constant Jacobsthal bound at primorials, where Erdős problem #687 pays $1000 for merely g(x#) = o(x²). The floor was checked and holds: no explicit-constant route to g(x#) < x′² exists in the named literature. **(P9) maxsum_m ≥ G₂ always, the Bridge Floor [PROVEN, `a3-05-bound-L.md` §7].** Hence Theorem B (L ≤ 1 + m\*, PROVEN, §5 there) can never prove L ≤ 1 + m for any m below G₂/(3p), which on the measured law is ≈ 0.183x and is a positive power of x on any law consistent with the data. *Direction.* Forward, maxsum bounds L. Backward it is what CLOSED the maxsum bridge as a 0c→L converter (`REFUTED.md`): any maxsum bound whatsoever floors the bridge output at ≈ 0.183x. **(P10) The complementary-window duality maxsum_m + minsum_{D−m} = W [PROVEN; VERIFIED at all 1484 m on T₁₃; and NOT ours, it is Cressie 1977's complement identity for the circular scan statistic, `IMPORT-MAP.md` row 1].** Direction: it converts a max statement into a min statement on the same tile and stays inside the rotation-invariant layer, so it crosses no bridge. **(P11) Two more, recorded for completeness because they are the ones that close routes rather than open them.** The L = 1 residue count is PROVEN EQUIVALENT to the Zone Postulate, the residue condition being the kill condition with zero slack and the chain sum vanishing iff G₂ < θ (`attack-l1-residue.md`); and Shearer's exact criterion on a complete dependency graph IS the union bound (`import-shearer.md` §4), which is why the whole local-lemma family closes. Both run only in the closing direction. ## 3. What the Gap Reformulation discards, quantified **The caveat first: none of these ratios is evidence.** They are flat or divergent over short ladders, and `ZONE-POSTULATE.md` §5a's square-window lesson applies to all of them: a margin that grows because the window grows faster than the gap scale is a scale mismatch, not the primes becoming better behaved. What follows is a ledger of the loss, not an argument about it. **The four ratios, at every level where they are computed.** | ratio | what it measures | values | source | |---|---|---|---| | x′²/G₂(x#) | the whole-tile margin against the frontier | 4.50, 4.17, 4.08, 4.03, 4.02, 4.38, 3.34, 3.53, 4.12, 3.72, 3.93, 3.18, 3.39, 3.57 at x = 2..43 | `G2-STATE.md` §2, the fourteen exact terms | | x²/G₂(x#) | the same in the exponent frame | 2.08, 1.63, 2.88, 2.56, 2.68, 2.41, 2.59, 3.26, 2.76, 2.59 at x = 5..37 | `ZONE-POSTULATE.md` §5 | | G₂/F | the origin's genuine advantage over the worst case | ≈ 7, growing only polylogarithmically | `ZONE-POSTULATE.md` §5 | | window/F | what the postulate actually has | can never exceed x′ since F ≥ x′; saturated within a factor 1.0000004 of that ceiling at x = 10⁹ | `ZONE-POSTULATE.md` §5 | | x²/certificate | the best construction's share of the window | 3.8, 7.8, 18.1, 44.9 at x = 37, 229, 1009, 4001 | `two-class-lower-bounds.md` §10, cited via `ZONE-POSTULATE.md` §5 | | M(x, x′²)/ln³x | the localized margin | flat at 3.2 to 3.7 over a 47-fold range in x, so the localized margin is x²/(3.5 ln³x) and diverges | `maxgap-law.md` §8 | | G₂/Z₂ | the reduction's loss, measured exactly | 1.00, 1.00, 1.00, 1.00 at p = 2, 3, 5, 7, then 1.40, 2.20, 3.00, 2.08, 1.36, 1.72, 2.32, 3.52, 3.64, 4.12 at p = 11..43; extended 2026-08-29 to all 22 trusted terms, reaching 8.1429 at x = 79 with the maximum 8.3214 at x = 71 (`measure-g2z2-0829.md` §3, red-team reproduced) | `zonegap-01.md` §5; `measure-g2z2-0829.md` | | head : Z₂ : tail | where the zone's width sits | 1.82% : 89.72% : 8.46% at band B4, tail share falling | `zone-tail-01.md` via `QUESTIONS.md` `Q-zone-tail` | **The one number that is not flat is the one that runs the wrong way for the route and the right way for the postulate.** x²/certificate climbs monotonically 3.8 → 44.9 over a 108-fold range in x: the best construction's share of the window collapses. That is an upper bound on x²/G₂ moving, not a lower bound, and `ZONE-POSTULATE.md` §8 item 4 insists the direction of inference be stated every time the number is used. It is used here only to say that the flatness of x²/G₂ over fourteen terms is a small-number effect. **The structural statement, and it is what the reformulation actually buys and sells.** Passing from Z₂(p) to G₂(p#) discards the origin. The zone is an interval anchored at a distinguished point; the tile's maximum gap is taken over all positions, so the resulting statement is invariant under translation of the tile and is a statement about residues alone. That is the whole trade: **rotation invariance is restored, and the price is the factor measured in the G₂/Z₂ row, which reaches 4.12 by p = 43 and has no proven ceiling.** **Is it true that every origin-keeping object either expires by x′² or is TPC-strength?** Checked row by row against §1, and the answer is yes with one row needing a third word rather than an exception. - **Expire at or before the zone's width.** The Origin Excess Lemma's advantage is positive only below a threshold with ln x\*/ln y ≈ 1.44 at every computed level, so the advertised (ln x/ln y)² factor is capped at about 2.2 with a measured maximum of 1.372 across the 14-cell table, and at S = x′² the advantage does not merely vanish, it **reverses to 0.79305 = e^{2γ}/4**, the minimum of the survival curve (`origin-excess.md` §0, §2, PROVEN identity plus MEASURED 0.79303 at x = 1487). The anchored caps expire even more sharply: inside the zone every strike by a prime q > p on an in-zone slot is a self-strike, so the whole cap ledger reduces to the certification identity and **adds zero information there** (`zonegap-02-reduction.md` §3.3, PROVEN; VERIFIED at five levels). The mirror expires too: certification does not survive σ, with composite-membered slots in the image window at every level, 1 of 16 at p = 7 rising to 21 of 34 at p = 17, and the mechanism is structural because the frontier is not a mirror-covariant object (`zonegap-02-reduction.md` §3.2, REFUTED as a transfer channel). - **TPC-strength.** Z₂, head, tail, M, T(Q), floor_K, β, S(x), and any bound on the anchored δ: §1's last column, each with its audit pointer. - **The row needing a third word: X(y) alone.** An upper bound on X with no comparison to T is a sieve statement, is legal, and does not expire; it is simply never sharp enough. Certifying Σ capU below C − 1 needs the count bounded above with relative precision at twin scale, (T − X)/C ~ 1/ln²h, on intervals as short as the stretch, down to width 4Q + 4, and **no sieve upper bound delivers a constant sharp to 1 + O(1/ln²)**: that precision class is the parity wall's (`quadpoint-identity-01.md` §4). So the dichotomy's honest form is: expires, or is TPC-strength, or is available at every scale and never at the precision the comparison needs. - **One further qualification, and it is the sharpest fact in this section.** The proven bound on the ANCHORED gap is not a consequence of the proven bound on G₂; it is the identical theorem, because the DHR sieve is position-uniform (`QUESTIONS.md` `Q-anchored-vs-global-gap`, CLOSED, `attack-block-09-anchored.md`). The separation between anchored and global was measured, is real, and is worth exactly one unit of exponent against a deficit of 2.2665. So the information the Gap Reformulation discards is information **the proven bound never used in the first place**: the reduction gives up an advantage the instrument on the far side cannot exploit. ## 4. The three killers, placed on the chain **What is wrong with treating these as three separate walls:** they are not independent, and two of the three are not theorems. The corpus states them together because three days of attacks moved neither exponent and said why from independent directions (`README.md` §Status), which is a convergence of evidence, not a proof of impossibility. Graded lowest-rung-first: **Killer 1, the class-blind cap at 4.26645: HEURISTIC, with a PROVEN number inside it.** What is proven is `G₂(x#) ≪_ε x^{4.26645+ε}` (`paper/beta2-note.md`) and that β₂ has been unimproved since Diamond–Halberstam 2008, everything after being worse at κ = 2 (Blight 4.45, Franze 4.516, Opera de Cribro 4.83; `REFUTED.md`). What is **not** proven is that a two-class-only argument cannot go below it: the exact value of β(κ) is unknown for κ > 1/2 except κ = 1, and Ford's notes confirm no κ = 2 extremal example exists in print (`research/sift-limit-attack.md` §2, `ZONE-POSTULATE.md` §3). So the cap is a property of a construction class plus a closed search, not a barrier theorem. *Where it bites.* At the input to (P4): the Gap Reformulation is a PROVEN link from tile to zone, and the only instruments that feed it are dimension-2 sieve statements. It bites again at (P8) read backward, where the target silently contains an explicit-constant one-class bound, and once more at Z2's route (b), where the sieve half needs a κ = 2 lower bound at s = u\*/2 = 1.7829 against β₂ = 4.26645, short by 2.393× (`attack-wrongdirection-audit.md` §3.8, hand arithmetic, unstamped, no adversarial pass). **Killer 2, the almost-all quantifier: this is the one with a THEOREM in it.** The core is exact counting and needs no measurement: the ensemble has W members and the anchor is one of them, so a bound admitting an exceptional fraction ε decides the anchor only if εW < 1 (`anchored-note.md` §3, Proposition 1 part (i), exact). The audit states the same line as Axis C: the boundary sits at empty-density ε = 1/W = e^{−(1+o(1))w}, **not at any rate polynomial in w**, and Axis B records that no quantifier over w (infinitely-many, almost-all, for-all, averaged) buys anything, since infinitely-many is already fatal (`attack-wrongdirection-audit.md` §1). The measured half is how far short the available bounds sit: the second-moment bound gives ε ≈ (Var/E)/E ≍ ln²W/W, measured 8.38·10⁻⁶ at @19 against the decision threshold 1.03·10⁻⁷, a factor 81 short and growing like ln²W. *Where it bites.* Squarely on the link E(x) → S(x), which is the only place the ensemble layer touches the origin. It bites again on the θ ladder wherever an all-positions statement is weakened to almost-all, and on every "almost all intervals" import, which is the whole Maier/BV/EH family. It does **not** bite on maxsum_m or on G₂ itself, both of which are already all-positions objects. **Killer 3, the equivariance wall: a MEASURED pattern over the attack fleet, with two proven components.** Operationally, "fails to localise" is this claim about the ensemble-to-origin step: > No statement quantified over the rotation ensemble, a mean, a variance, a > tail bound, a Fourier bound or a correlation inequality, determines the value > of the statistic at the single phase t = 0, because the ensemble carries no > structure that distinguishes that phase, and the anchor is measurably > atypical at the depth any such bound would have to be stated at. Two components of that are proven. **(a)** The tile's Fourier transform on ℤ/W **vanishes nowhere**: every local factor is 1 + e(2v/q) up to a unit phase, zero only when 4v ≡ q mod 2q, which parity forbids for odd q; so the support is maximal and the whole uncertainty-principle family (Donoho–Stark and its sharpenings) holds with maximum slack and constrains nothing (PROVEN in one line, checked numerically at twelve primes, `recon-0828-farfields.md` §2a). **(b)** The dependency graph is complete, on which Shearer's exact criterion IS the union bound, which closes the entire local-lemma family (`import-shearer.md` §4, PROVEN); and generic chaining cannot beat that union bound either, its entropy integral already exceeding it by 1.054 to 1.099 at the lower branch, flat across z = 13..23 (`import-chaining.md`, CLOSED). The remaining component, that the one non-generic structure available, the constant shift making twin slots S₁ ∩ (S₁ − 2), buys nothing at the anchor, is measured, and the three tests the brief names are the three that were run: 1. **Mirror symmetrisation gains exactly 0**, and the zero is structural rather than empirical: every cap family in play is itself mirror-covariant, the covariance σ(K_q(a)) = K_q(w−a) with the comb fixed being a one-line theorem (`attack-anchored-01.md` §3, `redteam-0820-night-proofs.md` §1; REFUTED as an improvement channel). 2. **The QR-immune classes deliver exactly the guardrail's r/(r−2) and nothing past it**: per-prime Z from −0.48 to +1.66 over 3.4 million twin openers with the matched control reading the same, and as a certificate the immune subset is dominated by simply naming the two classes each small prime kills, 1219/1219 anchors against 1126 (`attack-z3-immune-01.md` §3, §4; CLOSED). The pre-registered matched control is what voided a 6.5σ surplus the control reproduced (`README.md` §Status). 3. **The anchored δ exceeds the forced scale at 2 of 7 levels**, ρ/F = 2.001 at @11 and 1.266 at @19, so the room-for-dependence reading cannot be quoted as a bound and only the identity (1+δ)(1+F) = 1+ρ survives (`import-suen.md` §7, as corrected). *Where it bites.* On the same link as killer 2, E(x) → S(x), and on every arrow in §1 that would carry a rotation-invariant statement to an origin-keeping one. The general form the corpus has settled on is not about Fourier at all: an interval of length L is a diagonal segment of L points in a CRT box of x# cells, and any theorem that counts, covers or bounds over the box sees the segment only once L approaches x# (`recon-0828-farfields.md` §0, item 3, which disposes of the polynomial method, the sumset family and the density-to-interval bridge at once). **Summary of the grades, since the brief asks for it explicitly.** Killer 2 is the theorem (its core is exact counting; its distance-from-available is measured). Killer 3 is the measured pattern over the attack fleet, with two proven components and a third that is three tests deep. Killer 1 is the heuristic: the number is a theorem, the cap is not. ## 5. Is the discarded information provably needed? **The short answer, and it is a correction to the question as posed.** No. As literally asked, is the origin information *needed* for a residue-only statement to reach the zone statement, the answer is PROVEN NO, and the witness is the Gap Reformulation itself: `G₂(x#) < x′² − 2` is a statement about residues alone, invariant under translation of the tile, and it implies the Zone Postulate at x (`G2-STATE.md` §1b, `ZONE-POSTULATE.md` §3, PROVEN). A rotation-invariant statement reaches the zone statement. The separation the question is reaching for therefore cannot be between "tile" and "zone", and it cannot be between "residue" and "interval" at the level of statements either. **So what is it between?** Two things, and they should not be confused. 1. **Cheap residue statements versus expensive ones.** `THE-LENS.md` §5 states the triage rule: if a statement can be phrased in residues it is free and probably classical; if it needs an interval it is hard. Read carefully, that rule sorts the residue layer by what KIND of statement it is, not by whether it is about residues. The free ones are counting and structure statements: the census ∏(q−2), the Copying Theorem, the Seam Lemma, the palindrome, the grain word, count(6) = ∏(q−4). G₂ is also a residue object and it is not free: it is the extremal statistic of the same pattern, and it is the one thing the sweep did not find in anyone's hands (`README.md` §Status, `PRIOR-ART.md`). The rule as written is a triage heuristic and the corpus labels it as such; it is not a theorem, and G₂ is the standing witness that the residue layer contains statements as hard as the interval ones. 2. **Statements versus method classes.** The only real separation instrument anyone has here is a theorem about a method class, not about a statement. Tao's Claim 1 is such a theorem, and `lit-tao-parity.md` reads it at source: H4 defines the forbidden sign patterns from the property's **extension**, never from how the property is written. That fixes the two ends of the chain in opposite positions, both PROVEN: - The bare tile property "l₁ and l₂ both coprime to x#" has an **empty** forbidden set, since each of l₁, l₂ can independently be taken prime > x (λ = −1) or a product of two such (λ = +1). H5 fails and Claim 1 is vacuous on it; the same holds for any congruence-defined property, since every reduced class carries numbers of both Liouville signs (`lit-tao-parity.md` §2.1, PROVEN modulo Dirichlet, and confirmed by Tao's own converse). - The zone property "coprime to x#, and x < l₁ < x′²" has extension exactly the twin prime pairs in (x, x′²), whose forbidden set is Example 3's (+1,+1), (+1,−1), (−1,+1), with the origin in its convex hull. **H5 holds and Claim 1 applies** (`lit-tao-parity.md` §2.2, PROVEN given H4 and zone crystallization). **Which means the exemption is real, is narrow, and is not a licence.** The finite-periodic exemption evaporates at exactly the point where the reduction bites, and the obstruction does not care that the zone property is *written* with no prime in it. What survives the exemption is Door 5 only: a proof of `G₂(x#) < x′² − 2` that is extremal or combinatorial, a covering argument, an explicit construction, a pigeonhole over ℤ/x# **with no sieve weight anywhere**, has no ν to reweight and Claim 1's proof has no purchase on it (`lit-tao-parity.md` §2.2, `wall-note.md` §2). Any route that proves the zone statement by bounding sums against a non-negative sieve weight is inside the obstruction, exemption or no exemption. And `attack-obstruction-audit.md` adds the sharper constraint from the other side: the tile-side target asks for sieve positivity at u = 2·ln x′/ln x, which exceeds 2 by 2(ln x′ − ln x)/ln x → 0, so it converges to the parity boundary from above with margin tending to zero, where the κ = 1 lower-bound function f(u) = 2e^γ log(u−1)/u vanishes (`covering-dive.md` §1.2, Granville's account transcribed). **Is there a THEOREM that no residue-only statement reaches the zone statement?** **There is not**, and nothing in this corpus or in the searched literature is a candidate. Stated plainly, so it cannot be misread: no separation theorem of that shape is on record here, the corpus's own audit says its separations are never model-theoretic, "strictly weaker" always means "the target does not imply TPC through any bridge this corpus holds", witnessed by an explicit function or by the target already being proven (`attack-wrongdirection-audit.md` §0), and the one statement of that shape that does exist runs the other way, since P4 exhibits a residue-only statement that reaches. [PROVEN for the non-existence-here; OPEN as a mathematical question.] **Is there a heuristic?** Yes, three, and none of them is close to a proof. - **The triage rule with its one paid-out prediction.** `THE-LENS.md` §5 and `ZONE-POSTULATE.md` §7 predict that the failure mode of any argument reaching an interval conclusion without an interval-versus-residue step is a silent equidistribution assumption. It was paid out once against a published programme: Holt's Legendre result (arXiv 2603.25915 Theorem 3.3) rests on his Conjecture 2.1, approximate uniformity, stated as a conjecture supported by samples, reached from inside two decades of work on these same objects (`PRIOR-ART.md`). One correct prediction against one example is evidence about the rule's usefulness, not about its truth. [HEURISTIC] - **The CRT-diagonal statement.** An interval of length L is a diagonal segment of L points in a CRT box of x# cells, and any theorem that counts, covers or bounds over the box sees the segment only once L approaches x# (`recon-0828-farfields.md` §0 item 3). This is the operational content of the equivariance wall and it has closed three import families on its own. It is a statement about which theorems apply, not a proof that none can. [HEURISTIC, with the individual closures at PROVEN or CLOSED] - **The measured price of the origin.** The separation between the anchored gap and the global gap is real and correctly measured, and it is **worth exactly one unit of exponent against a deficit of 2.2665**; further, the anchored quantity's only proven bound is β₂ = 4.26645, the identical theorem rather than a consequence, because the DHR sieve is position-uniform (`QUESTIONS.md` `Q-anchored-vs-global-gap`, CLOSED, `attack-block-09-anchored.md`). So the discarded information is priced: not negligible, and not sufficient. [MEASURED, one closure] **What would a proof of such a separation even look like?** It would have to be a theorem about a method class, quantified over arguments rather than over statements, because a statement-level separation is impossible here on the evidence above: P4 already exhibits the implication. The two instruments in this corpus of the right shape are Tao's Claim 1, which quantifies over H3 arguments whose entire arithmetic input is upper and lower bounds against a non-negative weight, and Shearer's exact criterion, which on a complete dependency graph IS the union bound and thereby closes an entire technique family (`import-shearer.md` §4, PROVEN). A separation of the kind wanted would have to name a class, "arguments using only rotation-invariant data about T_x", and give that class a closure property, and exhibit a data-preserving perturbation that changes the truth of the zone statement while fixing the class's inputs. Nothing here has such a perturbation: the tile is a single deterministic object at each level, so the natural candidate, the rotation ensemble, is exactly the thing killers 2 and 3 say cannot see the anchor. And the obstruction's own scoping cuts the other way at the finite levels this corpus can compute: Claim 1's proof is asymptotic everywhere, so **no computed level is evidence about it in either direction**, and β(x) > 0 at ten levels is not and can never be evidence that the obstruction is soft (`lit-tao-parity.md` §2.3, PROVEN). **The one sourced item that bears on the weakest form of the target.** The corpus's minimal requirement is non-annihilation infinitely often (`anchored-note.md` §8, Proposition 2, PROVEN). Tao addressed that shape directly in a 2022 comment: the obstruction does not rule out a sieve with a non-trivial non-uniform bound holding for infinitely many N but not all, but such a thing "would be a very unusual species of sieve that does not resemble any existing sieve", would "have to be sensitive to the fluctuations of the Liouville function", and is therefore "of comparable difficulty to the type of problem one is trying to attack in the first place" (`lit-tao-parity.md` §2.4, SOURCED for the quotation, HEURISTIC for the reading). So the i.o. quantifier is a genuine gap in the obstruction's coverage and it is priced as no easier than the problem. **The section's answer, at the correct rung.** The information the Gap Reformulation discards is PROVABLY NOT NEEDED for the implication to run, is MEASURED to be worth about one unit of exponent against a deficit of 2.2665, and whether it is needed for any PROOF of the tile-side statement is OPEN with no theorem on either side. The honest position is that the reduction is lossy by a factor measured at 4.12 by p = 43 with no proven ceiling, that the loss is not the reason the route is stuck, and that the reason the route is stuck is the open exponent band, which sits on the far side of the reduction where the origin was never being used. ## 6. Inconsistencies found Five, none of them a number that is wrong, all of them a place where two files say the same phrase about two different objects or state a bound for one object and use it for another. Each is given with file:line on both sides. **(I1) "window/G₂" names two different ratios, and the two bands do not match.** `research/ZONE-POSTULATE.md`:257 reads "window/G₂ sits flat at 3 to 4 with no trend", where §5's own worked example at :245 uses window = x′² − x (1644/528 = 3.11 at x = 37). `research/history/staging/zonegap-01.md`:96 reads "window/G2 is FLAT at 3.2-4.5 over the 14 exact levels", citing `research/G2-STATE.md`:345, whose column is headed **x′²/G2** and runs 3.18 to 4.50. Evaluated on the ladder, the x′² − x convention runs 3.50, 3.67, 3.67, 3.80, 3.76, 4.18, 3.19, 3.40, 4.01, 3.61, 3.84, 3.11, 3.31, 3.50 at x = 2..43 [ARITHMETIC on `G2-STATE.md` §2's terms; no producer], so its band is 3.11 to 4.18 and ZONE-POSTULATE's "3 to 4" understates the top by 0.18 while zonegap-01's "3.2 to 4.5" is the other convention's band quoted under the same name. Neither number is wrong; the word "window" is doing two jobs. **(I2) "Z₂ is not G₂ restricted" against "Z₂ IS the whole-tile gap object restricted".** `research/history/staging/zonegap-01.md`:107 says "Past level 7 the tile's extremal gap lives beyond p'^2: Z2 is NOT G2 restricted, and the zone maximum is strictly easier." `research/history/staging/zonegap-02-reduction.md`:38 says "**Z2(p) IS the whole-tile gap object restricted to the head window (p, p'^2).**" Both are true and they use "restricted" in opposite senses: the second means the gap MULTISETS agree inside the window (Lemma A), the first means the MAXIMA differ because the tile's argmax lies outside it. Read side by side without the distinction they contradict, and the second is the one a summary is likely to copy. **(I3) A bound proven for the conditionally-defined object is used for the unconditionally-defined one.** `zonegap-02-reduction.md`:89 defines F(p) as "the distance from p to the tile's first twin-slot opener above p", defined at every p, and uses it in (R2), the non-circular sufficient form. `zonegap-02-reduction.md`:122 then states and verifies "**head(p) ≤ G2(p#) − p − 1**", and "head" is the zone-side object, which §1 of the same file flags as presupposing occupancy ("'head' presupposes an in-zone first slot"). The Euclid anchor's proof works for F unconditionally, so the substitution is sound, but the file states the bound for the object it is NOT entitled to use in (R2). One sentence would fix it. **(I4) M(x, x′²) and M(x, x²) are quoted as one object on the strength of a seven-level coincidence.** `research/ZONE-POSTULATE.md`:267 says "the largest twin-slot gap inside the zone obeys M(x, x′²)/ln³x flat at 3.2 to 3.7", citing `research/maxgap-law.md` §8, whose table at :513 is headed **M(x,x^2)** and whose flatness statement at :523 is about `M(x,x^2)/ln^3 x`. The identification rests on `zonegap-01.md`:53, which reports that the two windows "happen to bind the same gap at all seven" levels. "Happen" is the right word and it is not carried forward: ZONE-POSTULATE quotes the constant for the zone window with no mention that the measurement was made on the smaller one. **(I5) A registry count that no longer matches the registry.** `research/history/staging/attack-obstruction-audit.md`:67 (2026-08-26) reads "All 83 rows were read" of `REFUTED.md`; the file today carries 67 route rows and `README.md`:82-83 says "67 closed routes". A closed-route registry does not normally shrink, so either the 83 counted something else or rows were merged under the 2026-08-19 doc-convention extension. Minor, and flagged only because a stale count in an audit is the kind of thing a later audit cites. **Where else was looked, with nothing found.** The Gap Reformulation's statement was compared across `G2-STATE.md` §1b, `ZONE-POSTULATE.md` §3, `attack-wrongdirection-audit.md` §1 and `GLOSSARY.md`: all four agree that the transition is at exponent 2 **with constant strictly below 1** and that an unspecified constant at exponent 2 implies nothing. The Zone Postulate's strong/weak split agrees across `ZONE-POSTULATE.md` §2, `stretch-01.md` §2 and `anchored-note.md` §8. The TPC-strength labels for Z2, D, A, 1d, 1e, 8, X and 0 agree between `attack-wrongdirection-audit.md` §2 and the corresponding `QUESTIONS.md` rows. The lower-bound ladder (free FGKMT, then x ln x, then K–K's y ln³y) agrees between `G2-STATE.md` §0, `ZONE-POSTULATE.md` §3 and `README.md` §Status, including the "NOT refereed" marker on both of the upper two. ## 7. Questions not in QUESTIONS.md that would sharpen the map Understanding questions only; none is a route, and each was checked against `REFUTED.md` (all 67 rows) and against `research/PRIOR-ART.md` and `SEARCH-CONVENTIONS.md`, which own the novelty position. Nothing below is claimed as new mathematics; they are measurements of objects this corpus already holds. Ranked by what they would settle. **Q1. Is G₂(x#)/Z₂(x) bounded, and if not, how does it grow?** This ratio is the exact, measurable price of the Gap Reformulation, and it is the only place the reduction's loss can be seen without an asymptotic. It is computable at every level where both objects exist: G₂ at 22 trusted terms to x = 79 (`a144311-full-ladder.js`) against Z₂ known at 27,292 zones to 10¹¹, and the corpus records it only at the fourteen exact terms (`zonegap-01.md` §5, 1.00 → 4.12). *TPC-strength label:* **(i)** as a measurement, and the label must be stated in halves, because an UPPER bound on G₂/Z₂ combined with any Z₂ bound below the width is **(ii)**. *Grep showing it is unasked:* `grep -c 'G2/Z2' research/QUESTIONS.md` returns 0; so does `grep -c 'Z2 <= G2'`. **Q2. At which x does the tile's maximal-gap interval leave (x, x²), and does it ever lie inside a zone?** `zonegap-01.md`:53 reports that the windows (x, x²) and (x, x′²) bind the same gap at all seven M levels, x = 211..9973, while `zonegap-01.md` §3 measures the zone's own binding gap sitting at u = 0.9992..1.0000 on the last fifteen envelope steps. Those two facts are about different families and they point in opposite directions on the same question, which is why the question is worth asking rather than assuming. It is also the direct test of (I2)/(I4). *TPC-strength:* **(i)**, a location statistic. *Grep:* `grep -c -i 'where the maximal gap' research/QUESTIONS.md` returns 0. **Q3. Where in the period is G₂ attained, and how does the multiplicity grow?** `QUESTIONS.md` `Q-g2-43-term` records that G₂(43#) = 618 is "attained at 8 positions of the period", at one level and as a by-product. The argmax distribution over the period is the one direct test of killer 3's operational claim on the object the programme actually holds: if the tile's extremal structure were origin-sensitive at all, this is where it would show. It costs nothing new, since both exact-ladder producers already enumerate. *TPC-strength:* **(i)**. *Grep:* `grep -c -i 'argmax' research/QUESTIONS.md` and `grep -c -i 'position of the maximum'` both return 0. **Q4. How often is the zone decomposition tight, i.e. how many zones hold exactly two pairs?** (P3) is an equality iff k = 2, and (R0)'s strict form is exactly "k ≥ 3" (`zonegap-02-reduction.md` §2). The sweep records that every zone to 10¹¹ holds at least two pairs and that zones with a single pair number zero (`zonegap-01.md` §2), but not how many hold exactly two. That count is the measure of how often head + Z₂ + tail actually saturates the width, and it is a free by-product of a sweep already run. *TPC-strength:* **(i)**. *Grep:* `grep -c 'exactly two pairs' research/QUESTIONS.md` returns 0. --- *Staging note; process record, HELD under the publication moratorium. No existing file was edited, no producer was written and no script was run: every figure above is cited to the artifact that produced it, per the standing compute rule. The only derivation carried out here is §2 (P1), whose statement the corpus already holds at `zonegap-01.md` §5 as a one-line theorem; no novelty is claimed for it or for anything else in this note. See `research/history/CHANGELOG.md` for the corpus rule.*