# The forbidden extension across R0: which pieces of head + Z2 + tail carry the parity obstruction, and the one place in this frame that does not > **RIDER 2026-09-04 (orchestrator, from `redteam-0904-r0-extension.md`).** > Six of seven claims survive at their rung, one is refuted. CONFIRMED: > Proposition A (with a witness that its converse fails outside the product > class: a confined-rough property with |S| = 1 whose R reads λ has forbidden > set {(+1,+1), (+1,−1), (−1,−1)}, origin inside the hull); the scope slip, > strengthened, since the one-coordinate property is Tao's own Example 2 and > the repair "l₂ = l₁ + 2, bound l₁ only" flips 27 times over p = 7..199; > row (e″); the equivariance transfer by restriction. WEAKENED: Proposition B's > stated mechanism proves too much and the proposed wall-note insertion drops > the qualifier "only if the argument locates an n in supp(ν)" that §1d > supplies, and carries no H6 conditional; the insertion is not needed, the > conclusion being live at `paper/wall-note.md` already. Proposition D's "same > trigger" is near-definitional and its causal reading of the exponent band > does not hold. REFUTED: "no corpus artifact certifies the window-count > variance at width p′²": `research/06-variance-theorem.js` computes exactly > that, brute-force verified at two levels, and the z-score is now measured at > 43 levels (see the z2-state-draft rider). Two wordings owed here: EXEMPT > throughout means "Claim 1's H5 fails", which is not the parity statement > `paper/wall-note.md` Face 1 is priced against (the k = 1 case "l₁ is prime" > has H5 failing while the 2007 parity problem plainly bites on it); the §5 > column "parity" reads "Claim 1 (H5)", and §1c's one-liner "two forced primes > trigger the obstruction; one does not" is refuted by Tao's Examples 4 and 5 > and holds only in the product class. *(2026-09-04. Staging note, HELD. It edits no live document. It runs no producer and computes nothing new: every number below is cited to the artifact that measured it, and the new content is derivation, marked DERIVED where it has had no adversarial pass. Calibration per claim: PROVEN, VERIFIED, MEASURED, HEURISTIC, DERIVED, SOURCED, OPEN, REFUTED. Two axes are used throughout and they are different axes: the **TPC-strength axis** with the wrong-direction audit's labels (i)/(ii)/(iii) (`history/staging/attack-wrongdirection-audit.md` §2), and the **parity axis** COVERED / EXEMPT / UNDETERMINED, meaning whether Tao's Claim 1 hypothesis H5 holds for the statement's target property. A statement can be (ii) and EXEMPT; §2a is the witness.)* ## 0. The verdict, disconfirming half first **Nothing here opens a route, and the negative is the deliverable.** The decisive question the brief poses is whether any single piece of R0 is both legal on the TPC-strength axis and exempt from the parity obstruction's forbidden extension. The answer is **no, with content**: the exempt set among R0's own pieces is non-empty but every member of it is either already a theorem, asserts nothing existential, or is a two-integer boundary artefact. The one statement that is genuinely both exempt and useful is not a piece of R0 at all, it is the tile form and its exponent band, which is TODO item 0, the top of the programme's own board. So the search ends where the programme already stands. **The second disconfirming item, and it is the sharper one.** The exemption does **not close under deduction** (Proposition B, §1d). H5-failure is a property of a statement's target property, not of its consequences: an argument establishing an H5-exempt statement also establishes every consequence of it, including consequences whose properties do satisfy H5. So "the tile form is exempt" cannot be read as "a sieve route to the tile form is unobstructed". The exemption survives only for routes that have no non-negative sieve weight anywhere, which is `paper/wall-note.md`'s Door 5 and nothing else. The wall-note's own scoping sentence lands in the same place by naming Door 5 directly; what §1d adds is the reason, and the reason is that H5 is a test on statements while Claim 1's proof is a test on argument shapes, and the two are run together in one paragraph in the live text (§8, proposed change 1). **Third, a scope slip found in two notes and it changes a forbidden set.** `history/staging/lit-tao-parity.md` §2.2 and `paper/wall-note.md` §2 both compute the zone property's forbidden set from `P_zone(l1, l2) := "l1, l2 both coprime to x#, and x < l1 < x'^2"`. That property confines only **one** coordinate. Under Proposition A its forbidden set is `{(+1,+1), (+1,-1)}`, whose convex hull is the segment `{+1} x [-1,1]` and does **not** contain the origin: H5 **fails** on the property as written. The conclusion both notes draw is nevertheless correct, because the property the reduction actually needs confines both coordinates (`x < l_i < x'^2` for i = 1, 2), and then H5 holds. The correction is one subscript wide and it is load-bearing for exactly the question this note asks, since "how many coordinates are confined" is the whole criterion. [DERIVED, one adversarial pass by re-derivation only; §8, proposed change 2.] **What is genuinely new here, at its rung.** Proposition A (§1c), which reduces Claim 1's convex-hull test on this whole family of properties to a count of Liouville-monochromatic coordinates and makes the classification mechanical; Proposition B (§1d), the non-closure of the exemption; Proposition C (§6), the negative written once; Proposition D (§6b), that the two axes separate at exactly one point in this frame and that point is the Gap Reformulation. All four are DERIVED: written arguments, checked here by re-derivation, with no adversarial pass. Proposition A's converse half is scoped to product-form properties and stated as such. **What this note does not do.** It computes nothing, it promotes nothing, it closes no TODO item, it edits no live file. It does not claim novelty against the literature: Proposition A is a one-page consequence of Tao's own definitions and is very likely folklore to anyone who has run the test twice; the corpus's rule against calling a rediscovery a discovery applies (`research/PRIOR-ART.md`, `research/SEARCH-CONVENTIONS.md`). And per `lit-tao-parity.md` §1.5, Claim 1 is at rung **CONJECTURED** by its author's own sentence, it presumes Liouville pseudorandomness and is not a theorem, so every "COVERED" verdict below is conditional on that, and no COVERED verdict may be quoted as a proof that a route is impossible. --- ## 1. The criterion, stated with its hypotheses ### 1a. The source statement All quotations below are Tao, *A general parity problem obstruction*, What's new, 21 Nov 2014, read at source on 2026-08-27 and transcribed with a sha256 of the served HTML in `history/staging/lit-tao-parity.md` §Provenance. They are quoted here from that transcription, not re-fetched. [SOURCED via the corpus's own custody record; per the standing compute rule the custody-bound artifact is cited rather than the page re-opened.] The obstruction is a test on a target property with five hypotheses and one standing conjecture, numbered as in `lit-tao-parity.md` §1: - **H1 (forms).** `k` affine-linear forms `L_1, ..., L_k`, none a constant multiple of another. - **H2 (target shape).** The goal is existential: find *an* `n` with `P(L_1(n), ..., L_k(n))`. - **H3 (method class).** The argument's entire arithmetic input is upper and lower bounds on `Sum_n nu(n)` (1), `Sum_n nu(n) 1_{L_i(n) prime}` (2), and `Sum_n nu(n) f(L_i(n))` (3), for one or more **non-negative** weights `nu`, and its conclusion is reached by combinatorial manipulation of those bounds together with `nu >= 0`. The conclusion produced is "there exists `n` in the support of `nu` for which `P` holds". - **H4 (forbidden set, extensional).** "a sign pattern `(e_1, ..., e_k)` is **forbidden** by `P` if there does not exist any natural numbers `l_1, ..., l_k` **obeying** `P(l_1, ..., l_k)` for which `(lambda(l_1), ..., lambda(l_k)) = (e_1, ..., e_k)`." The `l_i` range over all natural numbers **independently**, subject only to `P`; the linear forms do not enter this computation. - **H5 (the hypothesis proper).** `0` lies in the convex hull of the forbidden sign patterns. - **H6 (the standing conjecture).** Liouville pseudorandomness, in the form that `Sum_n nu(n) Q(lambda(L_1(n)), ..., lambda(L_k(n)))` and its (2)- and (3)-analogues are negligible for `Q` a linear combination of squarefree monomials of degree at least 2. **Claim 1, verbatim.** "Suppose `P(l_1,...,l_k)` is such that that [sic] the convex hull of the forbidden sign patterns of `P` contains the origin. Then one cannot use the above sieve-theoretic approach to establish the existence of an `n` such that `P(L_1(n), ..., L_k(n))` holds." **Its rung is CONJECTURED, by its author.** "This claim is **not precisely a theorem**, because it presumes a certain 'Liouville pseudorandomness conjecture'". So the obstruction is a decision procedure whose output is a prediction about what a method class can reach, not a theorem that a route is impossible. Everything marked COVERED below inherits that rung. **What "extension" means, and the two equivalent formulations the brief names.** H4 computes the forbidden set from `P`'s extension, the set of tuples satisfying it. The Selberg/Chowla sign-pattern formulation is the same test in the older language: an axiom-only sieve cannot separate the `lambda = +1` part of a sifted set from the `lambda = -1` part, so "replace prime by `lambda = -1` and `z`-rough" is the substitution that leaves every sieve axiom invariant while changing the target's truth value. The corpus's own worked instance of that substitution is `history/staging/attack-barrier-kappa2.md` §5, where the substitution is made with an exceptional character in place of `lambda` and the resulting set is computed exactly. **Provenance of the classical citations, stated so nobody re-reaches for them.** Selberg's parity examples are conventionally cited at Halberstam and Richert, *Sieve Methods*, p. 239. **That page is UNREACHABLE in this corpus** and every site citing it says "cited through" (`history/staging/lit-pdf-halberstam-richert.md` §D4 and its closing table). The reachable primary is Friedlander and Iwaniec 1998 p. 1045, read at source and quoted in `lit-tao-parity.md` §3.3, which gives Selberg's example in full: `a_n = (1 + lambda(n))/2` satisfies the remainder axiom (R) at level `D = x^{1-eps}` and fails the bilinear axiom (B), because `mu(d) r_d(x)` has constant sign in long intervals. Friedlander and Iwaniec's parity discussion in *Opera de Cribro* is the standard chapter-length treatment; this note did not open it, and no claim below rests on it. [BIB, not opened.] ### 1b. The class of properties this note tests Every piece of R0 has a target property of one shape, and naming the shape is what makes the classification mechanical. > **Definition (confined-rough property).** A property `P` of `k` natural > numbers is **confined-rough** if it has the form > `P(l_1,...,l_k) = AND_i [ l_i in C_i ] AND AND_i [ l_i in I_i ] AND R`, > where each `C_i` is a union of reduced residue classes to some modulus, each > `I_i` is an interval (possibly all of `N`), and `R` is any further condition. > Write `A_i = C_i ∩ I_i` and `Lambda_i = lambda(A_i) ⊆ {+1, -1}` for the set of > Liouville values realised on `A_i`. Call coordinate `i` **monochromatic** if > `|Lambda_i| = 1`. **The two ways a coordinate becomes monochromatic in this frame**, both PROVEN: 1. **Confinement below the square of the sifting level.** If `C_i` is the reduced classes mod `x#` (i.e. `l_i` is `x`-rough) and `I_i ⊆ (x, x'^2)`, then every element of `A_i` is prime, so `Lambda_i = {-1}`. [PROVEN, this is the Zone Restriction Lemma's `(⊆)` direction, `zonegap-02-reduction.md` §1: a hole of `T_x` below `x'^2` has no prime factor `<= x`, and if composite its least prime factor would exceed `x`, forcing it to be at least `x'^2`.] 2. **The parity-forced empty case.** If `l_i` is `z`-rough, `l_i <= x = z^u` with `u <= 2`, and `lambda(l_i) = +1`, then `l_i = 1`: an even number of prime factors each above `z` gives `l_i > z^2 >= x`. This is `attack-barrier-kappa2.md` §5's vanishing direction, and it is the same mechanism as (1) written in the sifting parameter. [PROVEN there; quoted, not re-derived.] **A coordinate is NOT monochromatic when the interval is dropped.** If `A_i` is a union of reduced classes mod `q` with no interval constraint, both Liouville values are realised: Dirichlet gives infinitely many primes in each reduced class (`lambda = -1`), and for a target class `c` choose primes `q_1 ≡ 1` and `q_2 ≡ c` mod `q`, so `q_1 q_2 ≡ c` with `lambda = +1`. [PROVEN modulo Dirichlet; this is `lit-tao-parity.md` §2.1's argument, stated here per coordinate because that is the form the classification needs.] ### 1c. Proposition A, the criterion is a coordinate count > **Proposition A.** Let `P` be confined-rough with `Lambda_1, ..., Lambda_k` as > above, and let `S = { i : |Lambda_i| = 1 }`. > > **(A1)** If `P` is a pure product, `P(l_1,...,l_k) = AND_i [ l_i in A_i ]` > with every `A_i` non-empty and no further condition `R`, then the forbidden > set is exactly the complement of `Lambda_1 x ... x Lambda_k` in `{+1,-1}^k`, > and **H5 holds if and only if `|S| >= 2`**. > > **(A2)** In general (any `R`), the extension is contained in > `A_1 x ... x A_k`, so the forbidden set **contains** that complement, and > **`|S| >= 2` implies H5 holds**. The converse half fails in general, because > a smaller extension forbids more. *Proof.* (A1), forbidden set. For a pure product the extension's set of realised sign patterns is `Lambda_1 x ... x Lambda_k`, since the `l_i` are chosen independently under H4 and each `A_i` is non-empty. Its complement is the forbidden set by H4. (A1), `|S| >= 2 ⟹ H5`, and this also proves (A2). Pick `i ≠ j` in `S`, with `Lambda_i = {e_i}` and `Lambda_j = {e_j}`. Choose any signs `d_m` for `m ≠ i, j` and set `u = (..., u_i = -e_i, ..., u_j = e_j, ...)` with `u_m = d_m` otherwise, `v = (..., v_i = e_i, ..., v_j = -e_j, ...)` with `v_m = -d_m` otherwise. `u` disagrees with `Lambda_i` in coordinate `i`, so `u` lies outside `A_1 x ... x A_k`'s pattern set and is forbidden; `v` disagrees in coordinate `j` and is forbidden. Their midpoint is `0` in every coordinate: coordinate `i` gives `(-e_i + e_i)/2 = 0`, coordinate `j` gives `(e_j - e_j)/2 = 0`, and coordinate `m` gives `(d_m - d_m)/2 = 0`. So `0 ∈ conv{forbidden}` and H5 holds. Under (A2) the forbidden set only grows, so the same two patterns are still forbidden and the same midpoint works. (A1), `|S| <= 1 ⟹ not H5`. If `S` is empty the forbidden set is empty and its convex hull contains nothing. If `S = {i}` with `Lambda_i = {e_i}`, then `Lambda_m = {+1,-1}` for every `m ≠ i`, so the pattern set is `{e_i}` in coordinate `i` and free elsewhere, and its complement is exactly `{ eps : eps_i = -e_i }`. Every forbidden pattern has `i`-th coordinate `-e_i`, so every convex combination of forbidden patterns has `i`-th coordinate `-e_i ≠ 0`, and the origin is not among them. ∎ [DERIVED. The argument is four lines of convex geometry on Tao's own definitions and has had no adversarial pass. Its correctness check is that it reproduces the source's two worked cases exactly: Example 3's "both prime" has `k = 2`, `Lambda_1 = Lambda_2 = {-1}`, `|S| = 2`, H5 holds, forbidden set `{(+1,+1), (+1,-1), (-1,+1)}`, the source's list verbatim; and the congruence-only property has `S` empty, H5 fails, which is the source's own converse remark that "we do not expect a parity obstruction of the type in Claim 1 to hold" there.] **Consistency with Tao's `ceil(k/2)` count, which is the second check.** The source states that "at least `j` of the `k` numbers is prime" is obstructed as soon as `j >= k/2 + 1`, so at `k = 2` the obstructed case is `j = 2` and the free case is `j = 1`. Proposition A at `k = 2` says H5 holds iff both coordinates are monochromatic, i.e. iff both are forced prime. The two agree at `k = 2`. They are not the same statement in general: "at least `j` of `k`" is a disjunction and is not confined-rough, so Proposition A does not cover it, and Proposition A covers product properties the `j`-of-`k` count does not reach (mixed monochromatic signs, `+1`-forced coordinates). Neither subsumes the other. [DERIVED.] **The one-line reading, and it is the whole classification tool.** *Two forced primes trigger the obstruction; one does not.* Every verdict in §§2-5 is that sentence applied. ### 1d. Proposition B, the exemption does not close under deduction > **Proposition B.** Let `S` be a statement whose target property `P_S` fails > H5, and let `S'` be a consequence of `S` whose target property `P_{S'}` > satisfies H5. Then an H3 argument establishing `S` also establishes `S'`, and > the composite argument establishes the existence forbidden by Claim 1 applied > to `P_{S'}`. Hence H5-failure for `S` is **not** a licence for a sieve route > to `S`. What survives H5-failure is only the class of routes whose conclusion > is not of the form "there exists `n` in `supp(nu)` with `P`", that is, routes > with no non-negative sieve weight anywhere. *Argument.* Claim 1's proof (`lit-tao-parity.md` §1.6) builds `w >= 0` from the forbidden set, supported only on `n` whose Liouville pattern is forbidden, and uses H6 to leave (1), (2), (3) invariant under `nu -> nu w`. The contradiction it derives is that the argument's conclusion, "there exists `n in supp(nu w)` with `P`", is false because `supp(nu w)` carries only forbidden patterns. That contradiction is available whenever the argument's conclusion is of that shape, and it is not available when the conclusion is a fixed arithmetic fact whose truth value does not depend on `nu` at all. The tile form `G_2(x#) < x'^2 - 2` is such a fact: it is a finite decidable statement about `Z/x#`, and reweighting cannot falsify it. So the reweighting has no purchase on an argument that concludes it, **but only if the argument concludes it without ever locating an `n` in the support of a weight.** An H3 argument that proves the tile form by exhibiting, for each window, a surviving `n` in `supp(nu)`, has proven the zone instances too, since a window inside `(x, x'^2)` makes that `n` a twin prime opener, and Claim 1 applies to those instances. ∎ [DERIVED, and it is a reading of Claim 1's proof rather than a new theorem. It agrees with `paper/wall-note.md` §2's own scoping, "it covers Door 5, which uses no sieve weight to reweight, and it does not cover any route that proves the zone statement by bounding sums against a non-negative sieve weight", and supplies the mechanism that sentence asserts. The live text runs the H5 test and the method scoping together in one paragraph without saying they are two different tests; §8 proposes the separation.] **Consequence, and it is the reason §6's negative is strong.** A piece of R0 being EXEMPT is worth something only in company: it must also be legal on the TPC axis, and the route to it must be `nu`-free. Two of the three are cheap and the third is Door 5, whose status is that no covering-systems result in print reaches a window of length `p^2` with two classes per prime, searched and not found (`paper/wall-note.md` §1 Door 5, `research/covering-dive.md` §§Q3, Q4). --- ## 2. The two ends of the chain ### 2a. (a) The tile form `G_2(x#) < x'^2 - 2`, EXEMPT, and (ii) on the other axis **The target property, stated so the interval is not smuggled in.** `G_2(x#)` is the largest gap between twin-admissible **residues** mod `x#`, so the statement `G_2(x#) < H` is equivalent to a finite statement about `Z/x#`: for every residue `t`, some residue `a` with `t < a <= t + H` in a fixed fundamental domain has `a` and `a+2` both coprime to `x#`. The existential instance at `t` therefore has target property `P_tile,t(l_1, l_2) = [ l_1, l_2 coprime to x# ] AND [ l_1 mod x# ∈ (t, t+H] ]`, in which the second clause is a union of `H` reduced residue classes mod `x#`, **not** an interval of integers. Both `A_1` and `A_2` are unions of reduced classes, so by §1b both realise both Liouville values, `S` is empty, the forbidden set is empty, and H5 fails. **Verdict: EXEMPT.** [DERIVED, and it re-derives `paper/wall-note.md` §2 and `lit-tao-parity.md` §2.1 with the mechanism made explicit: the exemption is not that the property is "written without primes", it is that the tile statement's **periodicity lets the window be expressed as a congruence condition**, so no coordinate is confined to an interval and no coordinate is monochromatic.] **Why the periodicity is load-bearing, and where it would be lost.** The same sentence about integers rather than residues, "every integer window of length `H` contains a twin-admissible pair", is a different statement whose instances at `t ∈ (x, x'^2)` do confine both coordinates. The two statements are equivalent for the tile because the tile is periodic mod `x#`, so a residue window and an integer window carry the same slot pattern; but the properties are not the same property, and the criterion tests properties. The equivalence is exactly what Proposition B says cannot be used to carry the exemption forward. **TPC axis: (ii) TPC-strength** for the statement with constant strictly below 1 at exponent 2 (`attack-wrongdirection-audit.md` §1, Axis A, and `G2-STATE.md` §1b: `G_2(x#) < x'^2 - 2` implies the Zone Postulate at `x`, which at infinitely many `x` is TPC). **(i) legal** for any unconditional two-class exponent strictly inside the open band `(2, 4.26645]`, which is the audit's item 10 and TODO item 0. **So this piece is the axis separation**, and it is the only one in this frame: COVERED and (ii) come apart here. See §6b. ### 2b. (b) The zone form, COVERED, (ii) **The target property, with both coordinates confined.** The reduction needs a twin-admissible slot **inside** the zone, i.e. an `a` with `a` and `a+2` both `p`-rough and both in `(p, p'^2)`: `P_zone(l_1, l_2) = [ l_1, l_2 coprime to p# ] AND [ p < l_1 < p'^2 ] AND [ p < l_2 < p'^2 ]`. By §1b mechanism (1) both `A_1` and `A_2` are sets of primes, so `Lambda_1 = Lambda_2 = {-1}`, `|S| = 2`, and Proposition A gives H5. The forbidden set is Example 3's, `{(+1,+1), (+1,-1), (-1,+1)}`, and the origin is `(1/2)(+1,-1) + (1/2)(-1,+1)`. **Verdict: COVERED.** [DERIVED; the conclusion is `paper/wall-note.md` §2's and `lit-tao-parity.md` §2.2's, and the derivation differs from theirs in the one place §0's third item names.] **The correction, worked.** The property as written in both those notes bounds only `l_1`. With one coordinate confined, `S = {1}`, and Proposition A's `|S| <= 1` branch gives forbidden set `{ eps : eps_1 = +1 }`, whose convex hull is `{+1} x [-1,1]`: the origin is **not** in it and H5 **fails**. The conclusion "H5 holds" is right and the property that carries it is the two-coordinate one. Nothing downstream of either note changes, because every downstream use is about the zone form as the reduction actually uses it. **Two further readings, and they give the same verdict.** If one instead computes the forbidden set from the extension "twin prime pairs in `(x, x'^2)`", which is how both notes phrase it, the pattern `(-1,-1)` is realised only if a twin pair exists there, which is the Zone Postulate at `x` itself. If it does not, the extension is empty, **all four** patterns are forbidden, the convex hull is the whole square, and H5 holds a fortiori. So the verdict is unconditional either way, but the **unconditional** route is the "both prime in the zone" reading, whose `(-1,-1)` witness is any prime in `(x, x'^2)` taken twice, supplied by Bertrand. Prefer that phrasing; it does not borrow the conjecture. [DERIVED.] **TPC axis: (ii).** The weak form (infinitely many zones occupied) is **equivalent** to TPC, both directions, elementary (`research/ZONE-POSTULATE.md` §2, PROVEN). --- ## 3. The three pieces of R0 Conventions throughout are `zonegap-01.md` §0's, as carried by `zonegap-02-reduction.md`: pairs named by opener `a`; `a` is in zone `p` iff `p < a` and `a + 2 < p'^2`, both strict; gaps opener-to-opener; `Z2` their maximum, defined when the zone holds at least 2 pairs; `head = a_first - p`, `tail = p'^2 - a_last`, counted separately and never inside `Z2`. ### 3a. (c) `head(p)`, COVERED in the direction the decomposition uses **The direction matters and must be stated first.** R0 wants **upper** bounds on head, `Z2` and tail, since it needs `head + Z2 + tail < width`. Only the upper-bound direction is existential, and only the existential direction meets H2 at all. **Upper bound, `head(p) <= B`.** The statement asserts: there is an `a` with `p < a <= p + B` such that `a` and `a+2` are both `p`-rough and `a + 2 < p'^2`. Since `B <= width` whenever the statement has content, both coordinates lie in `(p, p'^2)` and are `p`-rough, hence prime by §1b(1). `|S| = 2`, H5 holds. **COVERED.** [DERIVED.] **Lower bound, `head(p) >= B`.** This asserts that **no** twin pair opens in `(p, p+B)`, a universally quantified non-existence. It does not meet H2, whose target shape is existential, so Claim 1's hypothesis is not engaged. **EXEMPT by type.** It is also useless: it moves R0's left side up, away from the target inequality. [DERIVED. This is the first of three exempt-and-useless cells, and the pattern is §6's proposition.] **TPC axis.** Unconditional `head(p) <= B` with `B < width` at infinitely many `p` is R2 with a named instrument, hence **(ii)** (`zonegap-02-reduction.md` §2). *Conditional on occupancy*, `head < width` is free, head is defined only when the zone holds a pair, and then `head < width` follows from the definition, so the conditional form is **(i)** and vacuous. The audit's own item-7 rider applies: police the quantifier, because the definedness clause is what separates the two labels. **The one proven instrument, and where it lands.** `F(p) <= G_2(p#) - p - 1` from the tile's edge pair (`zonegap-02-reduction.md` §3.1, PROVEN; VERIFIED at `p = 7..17`), with `G_2 << p^{4.2665+eps}` missing the needed `p^2` by exactly the open band. Note what §2a and Proposition B jointly say about that instrument: its **statement** is tile-side and EXEMPT, its **consequence** at `p` is head-side and COVERED, and the arrow between them is the elementary Euclid anchor, not a sieve step. So the instrument is a live example of Proposition B's shape: the exemption sits on the far side of an arrow that carries it nowhere. ### 3b. (d) `tail(p)`, COVERED, with one two-integer boundary artefact **Upper bound, `tail(p) <= B`.** Asserts an `a` with `p'^2 - B <= a` and `a + 2 < p'^2`, both members `p`-rough. Both coordinates lie in `(p, p'^2)`, both are prime, `|S| = 2`, H5 holds. **COVERED.** [DERIVED.] **The boundary artefact, and it is the only place in R0 where a coordinate escapes.** The zone's right endpoint is `p'^2`, and the closer `a+2` is required to satisfy `a + 2 < p'^2` by the convention. Drop that clause and ask only for `a` `p`-rough with `a` and `a+2` both `p`-rough, `a < p'^2`: for `a ∈ {p'^2 - 2, p'^2 - 1}` the closer is **not** below `p'^2`, so its roughness does not force primality and `Lambda_2 = {+1,-1}`. Then `|S| = 1` and by Proposition A's second branch H5 **fails**. **EXEMPT**, and the exemption is two integers wide and delivers a prime near `p'^2` rather than a twin pair, which is Bertrand-grade and certifies nothing. [DERIVED. Recorded because it is the sharpest available illustration that the exemption is a boundary phenomenon of the confinement, not a property of the object.] **Lower bound on tail: EXEMPT by type, wrong direction**, exactly as in §3a. **TPC axis:** same split as head, **(ii)** unconditional, **(i)** and vacuous conditional on occupancy. The tail is the corpus's least-studied piece (`zonegap-02-reduction.md` §2's table calls it "unstudied (cheapest piece)") and this note does not change that: it says only that the parity axis gives it the same verdict as the head, so cheapness there is not parity-cheapness. ### 3c. (e) `Z2(p)`, COVERED, and the slot reading and the prime reading are one property This is the piece the brief flags as unwritten, and it has three parts. **Part 1: the maximum-gap statement is a family of existential statements.** `Z2(p) <= B` says that no gap between consecutive in-zone twin openers exceeds `B`, which is equivalent to: for every `t` with `a_first <= t <= a_last - B`, the interval `(t, t+B]` contains a twin opener. Each instance is existential with target property `P_{Z2,t}(l_1,l_2) = [ l_1, l_2 coprime to p# ] AND [ t < l_1 <= t+B ] AND [ p < l_2 < p'^2 ]`, and since `t` and `t+B` lie inside `(p, p'^2)` both coordinates are confined, both monochromatic, `|S| = 2`. **H5 holds for every instance. COVERED.** The universal quantifier over `t` makes the statement strictly harder, not softer: Claim 1 blocks even a single instance, and `Z2(p) <= B` requires all of them. [DERIVED.] **Part 2: the "gap between two pairs" formalisation gives the same verdict.** The reading the brief asks about treats `Z2` as an object attached to a **pair of twin pairs**, so the natural target is a 4-tuple with a consecutiveness clause: `P_{Z2,4}(l_1,l_2,l_3,l_4) = [ all four coprime to p# and in (p, p'^2) ] AND [ l_2 = l_1 + 2, l_4 = l_3 + 2 ] AND [ no in-zone opener strictly between l_1 and l_3 ]`. The last two clauses are the `R` of §1b's definition, so this property is not a pure product and Proposition A's converse half does not apply. Its forward half does: all four coordinates are confined to `(p, p'^2)` and are therefore monochromatic with `Lambda_i = {-1}`, `|S| = 4 >= 2`, and (A2) gives H5. The explicit witness pair is `u = (+1,-1,+1,-1)` and `v = (-1,+1,-1,+1)`, both forbidden because each has a `+1`, with midpoint `0`. **COVERED.** [DERIVED.] So the between-pairs reading is not a different parity object. The reason is structural and worth stating once: **a consecutiveness clause can only shrink the extension, and shrinking the extension can only add forbidden patterns**, so no clause of that kind can move a statement from COVERED to EXEMPT. The only operation that moves a statement to the exempt side is **removing** a confinement. **Part 3: twin slots versus twin primes, the criterion is blind to the distinction, in the strongest sense.** The Zone Restriction Lemma (`zonegap-02-reduction.md` §1, PROVEN; VERIFIED at `p = 7, 11, 13, 17, 23`) states that the in-zone twin-slot openers of `T_p` and the in-zone twin-prime openers are the **same set**. Read through H4 this is stronger than an equality of numbers. The slot property's coordinate set is `A_i^{slot} = { l : gcd(l, p#) = 1, p < l < p'^2 }` and the prime property's is `A_i^{prime} = { l : l prime, p < l < p'^2 }`, and these are **equal as sets**: `⊆` because a hole below `p'^2` is prime, `⊇` because a prime above `p` is coprime to `p#`. H4 computes the forbidden set from the extension alone, and equal coordinate sets with the same `R` give equal extensions, hence identical forbidden sets. **The two readings are not two properties the criterion happens to rate alike; they are one property, and there is no data by which the criterion could tell them apart.** [PROVEN, given the Zone Restriction Lemma and H4.] **Precisely where the two readings become one.** At the conjunction of roughness with the constraint `l < p'^2` and not before. Drop the upper constraint and the slot set is a union of reduced classes carrying both Liouville signs while the prime set is monochromatic: the two readings are then genuinely different properties with different forbidden sets (empty versus Example 3's), and that difference is exactly the tile-versus-zone split of §2. Raise the constraint above `p'^2` and the same separation reappears, because a `p`-rough number above `p'^2` may be a product of two primes. So the identification is not a feature of the zone as an interval; it is a feature of the **coincidence of the sifting level with the square root of the window's right endpoint**, and that coincidence is the definition of the zone. [DERIVED. It is the same `u = 2` coordinate that `attack-barrier-kappa2.md` §5 reaches from Granville's side: `z^2 >= x` is `u <= 2`, and the zone is the `u = 2` window by construction.] **TPC axis: (ii)** for any bound placing `Z2` below the window at infinitely many `p` in the R1 form, per the audit's settled table (`z2-state-draft-0829.md` §4b, `zonegap-02-reduction.md` §2 (R1)): the premise's three objects are defined only when the zone holds at least two pairs, so the premise contains occupancy. --- ## 4. (f) The composites: R0, R1, R2, and the stretch form ### 4a. R0 itself, EXEMPT by type, (i), and empty of content R0 is the partition identity: when the zone holds `k >= 1` pairs, `width = head + Sum gaps + tail` exactly, and for `k >= 2`, `head + Z2 + tail <= width` with equality iff `k = 2` (`zonegap-02-reduction.md` §2, PROVEN; VERIFIED digit-exact at five levels). **It asserts no existence.** Its content is an identity conditional on a premise that supplies the pairs, so it has no target property of H2's shape and Claim 1's hypothesis is not engaged. **EXEMPT by type.** [DERIVED.] **TPC axis: (i)**, by the corpus's own operational witness, it is already PROVEN while TPC is open (`attack-wrongdirection-audit.md` §0's definition of "strictly weaker"). **So R0 is both (i) and EXEMPT, and it is worth nothing**, because it produces no occupancy: it is bookkeeping on a premise. This cell is the reason §6's proposition has to be stated with the word "content" in it rather than as a bare emptiness claim. ### 4b. R1, COVERED, (ii) R1 is `Z2(p) + head(p) + tail(p) < p'^2 - p` at infinitely many `p`, implying TPC. The implication is trivially PROVEN and the difficulty is entirely in producing the premise. The premise asserts that head, `Z2` and tail are **defined**, i.e. that the zone holds at least two pairs, and by R0 the strict inequality is exactly "at least 3 pairs". So the premise's target is the existence of three twin pairs in `(p, p'^2)`, a 6-tuple all of whose coordinates are confined: `|S| = 6 >= 2`, H5 holds. **COVERED.** [DERIVED.] **TPC axis: (ii)**, settled in `z2-state-draft-0829.md` §4b's table. ### 4c. R2, COVERED, (ii) R2 is `F(p) < p'^2 - p - 2` at infinitely many `p`, implying the weak Zone Postulate and hence TPC, where `F(p)` is the distance from `p` to the tile's first twin-slot opener above `p`. The premise asserts an `a` in `(p, p'^2 - 2)` with `a, a+2` both `p`-rough: both coordinates confined, `|S| = 2`, H5 holds. **COVERED.** [DERIVED.] **One asymmetry worth recording.** `F(p)` is a **tile-side** object, the first twin-slot opener above `p`, with no upper constraint in its definition, so the bare statement "`F(p) <= B`" for `B` unrestricted is a congruence statement and is EXEMPT by §2a's argument. It becomes COVERED at exactly the point where `B` is required to be below `p'^2 - p - 2`, because that is the constraint that confines the coordinates. **R2's exemption and its usefulness are the same dial, turned in opposite directions**: the statement is exempt while `B` is large enough to be useless and covered from the moment `B` is small enough to imply the postulate. That dial is the cleanest statement of the wall this note found, and it is a restatement of §2a rather than a new fact. [DERIVED.] ### 4d. The stretch form, COVERED, (ii) The stretch is `S_Q = [Q^2, Q'^2)` (`stretch-01.md`, `quadpoint-identity-01.md`). A `Q`-rough integer below `Q'^2` is prime by the same argument as §1b(1), so a twin-slot statement on the stretch confines both coordinates, `|S| = 2`, H5 holds. **COVERED.** [DERIVED.] **TPC axis: (ii)**, and above it: SP (every stretch occupied) implies the strong Zone Postulate by containment, and A091592-completeness implies SP (`stretch-01.md` §2, `z2-state-draft-0829.md` §4b). The certificate `Sum_r capU_K(r) <= C - 1` is `(ii)` by the capture identity (`attack-wrongdirection-audit.md` §3.1). **The one stretch-side statement that is EXEMPT.** An upper bound on `X(y)` alone, with no comparison to `T`, the audit's item-1 legal residue, `z2-state-draft-0829.md` §4b item 1. `X` is the rough-rough census minus prime-bearing pairs, and an upper bound on it asserts no existence of anything, so it fails H2 and is EXEMPT by type as well as **(i)** on the TPC axis. It is also the corpus's own recorded reason that this cell is empty of content: the comparison against `T` is the wall, and the precision class the comparison needs is the parity wall's, `1 + O(1/ln^2)`, which no sieve upper bound delivers (`quadpoint-identity-01.md` §4, quoted through `object-bridge-read-0829.md` §3). So this is a second (i)+EXEMPT cell, and like R0 it is content-free, for a reason the corpus had already priced, at a different coordinate, before this note existed. --- ## 5. The classification, in one table Two axes, and they are different axes. **TPC** is the wrong-direction audit's label: (i) strictly weaker than TPC in the corpus's operational sense, (ii) TPC-strength, (iii) undetermined. **Parity** is whether Claim 1's H5 holds for the statement's target property: COVERED (H5 holds), EXEMPT (H5 fails, or H2 fails so the hypothesis is not engaged), UNDETERMINED. The **monochromatic coords** column is the count `#S` of Liouville-monochromatic coordinates, which by Proposition A decides the parity column. | # | statement | k | monochromatic coords | parity | TPC | why the parity column reads that way | |---|---|---|---|---|---|---| | a | `G_2(x#) < x'^2 - 2` (tile form) | 2 | 0 | **EXEMPT** | (ii) | periodicity states the window as a congruence, so no coordinate is confined to an interval | | a' | any two-class exponent in `(2, 4.26645]` | 2 | 0 | **EXEMPT** | (i) | same property, weaker conclusion | | b | zone form: `(p, p'^2)` occupied | 2 | 2 | COVERED | (ii) | roughness below `p'^2` forces both coordinates prime | | c | `head(p) <= B`, `B < width` | 2 | 2 | COVERED | (ii) uncond. / (i) cond. | both members of the first pair lie in the zone | | c' | `head(p) >= B` | n/a | n/a | **EXEMPT by type** | (i) | non-existence, fails H2; wrong direction for R0 | | d | `tail(p) <= B` | 2 | 2 | COVERED | (ii) uncond. / (i) cond. | both members of the last pair lie in the zone | | d' | the two-integer right-edge case `a >= p'^2 - 2` | 2 | 1 | **EXEMPT** | (i) | the closer escapes the confinement; delivers a prime, not a pair | | e | `Z2(p) <= B`, gap-family reading | 2 per instance | 2 | COVERED | (ii) | every instance's window sits inside the zone | | e' | `Z2(p) <= B`, between-pairs 4-tuple reading | 4 | 4 | COVERED | (ii) | consecutiveness only shrinks the extension, which only adds forbidden patterns | | e'' | `Z2` on twin **slots** versus twin **primes** | n/a | n/a | **one property** | n/a | equal coordinate sets by the Zone Restriction Lemma; H4 reads extensions, so there is no distinction to see | | f1 | R0, the partition identity | n/a | n/a | **EXEMPT by type** | (i) | asserts no existence; conditional on a premise that supplies the pairs | | f2 | R1, `head + Z2 + tail < width` i.o. | 6 | 6 | COVERED | (ii) | the premise is "at least 3 pairs in the zone" | | f3 | R2, `F(p) < p'^2 - p - 2` i.o. | 2 | 2 | COVERED | (ii) | covered from the moment `B` drops below the window | | f3' | `F(p) <= B` with `B` unrestricted | 2 | 0 | **EXEMPT** | (i) | tile-side, no confinement; useless at that size | | f4 | stretch form on `[Q^2, Q'^2)` | 2 | 2 | COVERED | (ii) | same `u = 2` confinement one square up | | f4' | an upper bound on `X(y)` alone | n/a | n/a | **EXEMPT by type** | (i) | asserts no existence; the comparison against `T` is where both axes bite | **Reading the table in one line.** Every COVERED row is (ii); every EXEMPT row is (i) **except row a**, and every (i) EXEMPT row is either already proven, in the wrong direction, two integers wide, or priced dead at a different coordinate. Row a is the single separation. --- ## 6. The decisive question, answered ### 6a. Proposition C, the negative, written once > **Proposition C.** Within the R0 frame, the tile form, the zone form, head, > tail, `Z2` in either reading, and the composites R0, R1, R2 and the stretch > form, a statement's target property satisfies H5 **if and only if** the > statement forces at least two coordinates into a range where roughness implies > primality. Consequently the EXEMPT residue of the frame consists of exactly > three kinds of statement, and no fourth was found: > > 1. **Statements asserting no existence.** R0's identity, lower bounds on > head, `Z2` and tail, and an upper bound on `X(y)` with no comparison to > `T`. These fail H2, are all (i), and none of them produces occupancy: > R0's identity is already a theorem, the lower bounds run the inequality > the wrong way, and `X(y)` alone is the audit's own legal residue whose > dead end is the precision class of the comparison. > 2. **Statements forcing at most one member of a pair.** The two-integer > right-edge case at `p'^2`, and any single-prime statement in the zone. > These deliver a prime rather than a twin pair, which is Bertrand-grade. > 3. **Statements whose window is a congruence rather than an interval.** The > tile form and its exponent band. These are exempt as **statements**, and by > Proposition B the exemption does not close under deduction, so it survives > only for routes with no non-negative sieve weight anywhere. > > **Therefore: no piece of the R0 decomposition is both legal on the > TPC-strength axis and exempt from the forbidden extension in a way that > yields zone occupancy.** The exempt side of this frame is non-empty and > content-free, and the one exempt statement with content is not a piece of R0 > but the object R0's reduction passes through. [DERIVED. The "if and only if" is Proposition A restricted to this frame, where every property is confined-rough and every confinement is of §1b's two kinds; the enumeration of the three kinds is a check of the table in §5 row by row and is only as complete as that table. What would break it is a piece of R0 whose target property is not confined-rough, and §9 says what such a piece would have to look like.] **Why the corpus has not attacked the one exempt statement with content: it has, and it is item 0.** Row a', an unconditional two-class exponent strictly inside `(2, 4.26645]`, is simultaneously **(i) legal** (audit item 10, with the separation exhibited rather than modelled: `4.26645` is PROVEN while TPC is open) and **EXEMPT** (§2a). It is not in `research/REFUTED.md`; the nearest rows are "improving `beta_2` itself" (CLOSED for us, not proven unimprovable) and "a floor at 4, and the band `(4, 4.2665]`" (REFUTED as a floor claim), neither of which closes the band. It is TODO item 0, the top of the ranked board, and `research/ZONE-POSTULATE.md` §8 lists it as the highest-value form of progress there is. **So the search for an exempt place ends at the place the programme already occupies**, and this note's contribution on that point is not a new target but a reason: the exponent band is where the programme stands because it is the only place in this frame where the obstruction's hypothesis is unsatisfiable, and the price of standing there is the origin (`G_2/Z_2` measured at 4.12 by `p = 43`, reaching 8.1429 at `x = 79` over the 22 trusted terms, `object-bridge-read-0829.md` §3) and Proposition B's rider that the exemption holds only for `nu`-free routes, which is Door 5. ### 6b. Proposition D, where the two axes separate, and why it is exactly one place > **Proposition D.** On the R0 frame the TPC-strength axis and the parity axis > have the same trigger, forcing at least two coordinates into a range where > roughness implies primality also forces a twin pair into a bounded window, > which is the occupancy that makes a statement (ii), with **exactly one > exception**, row a: the tile form is (ii) and EXEMPT. The exception is > produced by the Gap Reformulation, which converts an interval statement into > a congruence statement at the cost of the origin, and it is the frame's only > instance of that conversion. [DERIVED, and it is an observation about this frame rather than a theorem about the two notions, which are logically independent in general. Read as a strategic statement it says: **the Gap Reformulation is the programme's one parity-evasion move, it is already made, and it buys exemption only for `nu`-free routes.** That is consistent with, and gives a second reason for, `object-bridge-read-0829.md` §5's finding that the discarded origin information is provably not needed for the implication to run.] **The rider that must travel with Proposition D.** `lit-tao-parity.md` §2.3 establishes that Claim 1 is asymptotic in its proof everywhere, so **no computed level is evidence about it in either direction**. Nothing in this note's classification can be checked, confirmed or disconfirmed by a computation at any finite level, and no measured tile, zone or `beta` value bears on any row of §5's table. [PROVEN as a scoping statement, cited.] --- ## 7. Part 2: the equivariance wall's zone-side statement This is `z2-state-draft-0829.md` §8 question 2. It is derived, not measured; no producer was run, and the one measurement it would want does not exist in the corpus (§7d). ### 7a. The tile-side statement, quoted `history/staging/import-boolean-analysis.md` §4, the banked wall address: > "The rotation ensemble's symmetry group acts transitively on its phases, and > every theorem in the analysis of Boolean functions is invariant under that > group. The deep ensemble is `(prod_{p<=y} Z_p, uniform)`, and translation by > any vector `sigma in prod Z_p` preserves the measure, the degree filtration, > all `L^q` norms and all influences. So a theorem whose hypotheses are stated > in those terms and whose conclusion is either a norm inequality or a > statement holding off an exceptional set of measure `eps` is *equivariant* > ... No such theorem can distinguish `t = 0`." [PROVEN as stated there; quoted, not re-derived.] ### 7b. It transfers, and not by analogy, by restriction **Which group, which ensemble, and is it transitive.** The zone-side ensemble is `E_p = { [t, t + (p'^2 - p)) : t in Z/p# }`, the `p'^2`-wide windows of the tile `T_p` with the zone at its own position, and the statistic is the number of twin slots the window contains. The group is `Z/p#`, acting by translation. It acts **simply transitively** on `E_p`: the ensemble is a torsor under it, each window has exactly one translate carrying it to each other window, the uniform measure is preserved, and every `L^q` norm, degree-filtration level and influence of the window-count function is preserved. By CRT `Z/p#` is `prod_{q <= p} Z_q`, so this is the same group as §7a's, not an analogue of it. **The argument transfers verbatim.** [DERIVED; the transitivity is immediate and the invariances are §7a's, unchanged.] **The stronger statement, and it is what question 2 was actually reaching for.** The zone is not an analogue of the tile's anchor at all. The Zone Restriction Lemma says `Z2(p)` **is** the whole-tile gap object restricted to the head window `(p, p'^2)` (`zonegap-02-reduction.md` §1, PROVEN; VERIFIED at five levels), so the zone window is the tile's own head window read at the tile's own phase, with its left edge at `p` rather than `0`, a shift of `p` against a window of width `~ p^2`, i.e. a fraction `~ 1/p` of the window. So the two anchors are **the same coordinate at two window lengths**, and the equivariance wall reaches the zone by restriction rather than by import. **§0's third killer in `z2-state-draft-0829.md` is currently described there as "imported to the zone side by analogy"; it is not an analogy, and that sentence understates the transfer.** [DERIVED; §8, proposed change 3.] **The threshold is identical.** `paper/wall-note.md` Face 1's requirement is that a bound admitting an exceptional fraction `eps` decides the anchor only if `eps * |ensemble| < 1`. On the zone side the ensemble is `E_p` with cardinality `p#`, so the requirement is `eps < 1/p#`, the same threshold as the tile's, against the same cardinality. Per that face's 2026-08-27 correction the `eps` and the cardinality must be read against **the same** ensemble, and on the zone side both are `E_p`'s, so the pairing is self-consistent by construction and no mixed reading arises. [DERIVED.] ### 7c. Three zone-side differences, and all three are adverse 1. **The payoff is smaller for the same threshold.** `G_2(x#)` is a maximum over all positions, an ensemble-level object; the zone statement is the value at one named position. So the same `eps < 1/p#` requirement buys a statement about one window rather than about the worst window. The equivariance wall costs the same and delivers less on the zone side. 2. **The window is exponentially thin in its own ensemble.** The zone's width is `~ p^2` against a period of `p#`, so `ln(width) / ln(period) ~ 2 ln p / theta(p) ~ 2 ln p / p -> 0`. That is the CRT-diagonal statement in its zone coordinate (`recon-0828-farfields.md` §0 item 3, via `object-bridge-read-0829.md` §5): an interval of length `L` is a diagonal segment in a CRT box of `p#` cells, and a box-level theorem sees the segment only once `L` approaches `p#`. On the tile the relevant `L` is `p#` itself, so the diagonal problem does not arise there and does arise here. [DERIVED, and the closures it has already produced are recorded at HEURISTIC in `object-bridge-read-0829.md` §5.] 3. **On the integer ensemble the hypothesis fails, and its replacement is worse.** If the ensemble is taken over the integers instead, `t` ranging over `[1, X]`, counting twin **primes** in `[t, t+H)`, then translation does not preserve the measure of interest, since the twin-prime density falls like `1/ln^2 t`, and the transitivity hypothesis of §7a is simply false. So the equivariance argument does **not** transfer verbatim to that ensemble. What replaces it is not weaker: Maier (1985) shows primes in intervals of length `(log x)^lambda` with `lambda > 1` are not uniformly distributed, and the zone window is `(ln W)^2`, so every heuristic of the form "the zone behaves like a typical window" is known to fail for the analogous prime statement at the analogous scale (`paper/wall-note.md` §2 Face 1, `paper/moire-primes.md` §9). The one ensemble on which the equivariance argument does not apply is the one on which a provable irregularity theorem does. [DERIVED, on two cited results.] **The hypothesis a zone-side localisation would need, and it differs in kind from the tile's.** On the tile, reaching `t = 0` requires a hypothesis naming the arithmetic of the phase where every prime's strike classes sit at `{0, -2}`. On the zone, the anchor is distinguished by a **level-position coincidence**, the sifting level `p` and the window's right endpoint `p'^2` satisfy `u = 2`, which is exactly what makes roughness imply primality inside the window (§3c Part 3). That is not a property of the position alone; it is a property of the pair (level, position). So a theorem that localised at the zone anchor would have to name a relation between the sieve's level and the window's location, where the tile's would have to name a phase. Both are outside the field of translation-equivariant theorems, and neither is a weaker requirement than the other; they are different requirements. [DERIVED. This is the one place where the answer to question 2's "or does the zone anchor sit in an ensemble where the argument needs a different hypothesis" is **yes**, and the difference buys nothing.] ### 7d. Is the zone anchor a diverging outlier of its own window ensemble? Not answered here, and one coordinate is measured **What is measured, and it is adverse.** In the density coordinate the zone anchor's deviation is known exactly at the zone's own width: the origin's density advantage **reverses** at `S = x'^2`, where `u = 2` is the minimum of the survival curve, and the anchor carries about 21% **less** than the ensemble mean, ceiling `rho(2) = e^{2 gamma}/4 = 0.793055` [PROVEN mechanism, MEASURED 0.79303 at `x = 1487`, `research/origin-excess.md` §2, §5; the same constant read on the finer stretch grid at 0.7980, 0.7978, 0.8021, 0.7946 across four `q`-bands to 1e8, `stretch-01.md` §3]. So on the zone side the anchor is not a favourable outlier and Route B is CLOSED and ADVERSE in `research/REFUTED.md`. **What is not measured, and it is the actual question.** A density ratio is not a `z`-score. `paper/wall-note.md` Face 1 prices the tile anchor's deviation as `z = +1.05` at `x = 7` falling to `z = -22,633` at `x = 37`, across the nine levels where the **ensemble variance is certified**. No corpus artifact certifies the variance of the window-count statistic at window width `p'^2`, so the zone anchor's `z` is **not computed anywhere**, and a constant ratio of 0.793 is consistent with a `z` that diverges, stays bounded, or shrinks, depending on how the variance scales. **This note does not compute it.** The brief's 20-minute clause was checked: `research/window-check.js` reproduces `ZONE-POSTULATE.md` §4's margins and does not produce an ensemble variance at this window width, and no other producer found does either, so there was no existing producer to cite and nothing was run. The honest state of question 2's second half is **OPEN**, with the density half MEASURED and adverse. **Verdict on question 2.** The argument transfers, verbatim and by restriction rather than by analogy; the ensemble is `E_p`, the group is `Z/p#`, and the action is simply transitive; three zone-side differences exist and all three are adverse; the anchor is not a weaker outlier in the one coordinate that is measured; and whether it is a diverging outlier in the `z` coordinate is open for want of a certified variance at window width `p'^2`. **The wall is not located differently on this object.** [DERIVED, one measured input, one OPEN half.] --- ## 8. What this note contradicts in the live layer **Nothing below is applied.** Each item gives the file and line, what stands there, and proposed replacement text. Three of the four are documentary; none changes a number, an exponent, a route status or a calibration. The one with mathematical content is item 2, and it corrects a forbidden set without changing the verdict that forbidden set supports. ### Change 1, `paper/wall-note.md`:177-187, the two tests are run together **What stands** (lines 177-187): one paragraph runs the H5 test on the tile statement ("does not satisfy the obstruction's hypothesis"), then the H5 test on the zone statement, then the **method** scoping ("it covers Door 5, which uses no sieve weight to reweight, and it does not cover any route that proves the zone statement by bounding sums against a non-negative sieve weight"). The scoping sentence is correct and the paragraph does not say that it is a different test from the two before it, so the paragraph reads as though H5-failure were itself the licence. **Proposed insertion**, after "...bounding sums against a non-negative sieve weight." on line 187: > These are two different tests and the second is not a corollary of the first. > H5 is a test on a statement's target property; Claim 1's *proof* is a test on > an argument's shape, and its reweighting has purchase only on an argument > whose conclusion is "there exists n in the support of ν". Exemption therefore > does not close under deduction: an argument establishing the tile statement > establishes its zone consequences too, and those satisfy H5. What survives is > only a route with no non-negative sieve weight anywhere, which is why the > exemption's whole extent is Door 5. [Rung of the inserted claim: DERIVED, `history/staging/derive-0904-r0-extension.md` §1d, no adversarial pass.] ### Change 2, the zone property confines one coordinate where it needs two **Three sites carry the same property with one coordinate bounded:** - `paper/wall-note.md`:181-182, "the reduction to the conjecture uses the tile only inside the zone, where an x-rough number is prime, and there the property's extension *is* the set of twin prime pairs". - `research/history/staging/lit-tao-parity.md`:313, "`P_zone(l₁,l₂)` := \"`l₁, l₂` are both coprime to `x#`, and `x < l₁ < x′²`\"". - `research/history/staging/object-bridge-read-0829.md`:471, "The zone property \"coprime to x#, and x < l₁ < x′²\" has extension exactly ...". **Why it matters.** H4 quantifies over the `l_i` **independently**, so a bound on `l₁` alone leaves `l₂` free, and the extension is then `{(prime in the zone, any x-rough number)}` rather than a set of pairs. Its forbidden set is `{(+1,+1), (+1,-1)}`, whose convex hull is `{+1} x [-1,1]`, and the origin is **not** in it: H5 fails on the property as written. The verdict the three sites draw is nevertheless correct, because the property the reduction needs bounds both coordinates. **Proposed replacement at `lit-tao-parity.md`:313 and, in the same words, at the other two sites:** > `P_zone(l₁,l₂)` := "`l₁` and `l₂` are both coprime to `x#`, and both lie in > `(x, x′²)`". Both bounds are needed: H4 quantifies over `l₁` and `l₂` > independently, so bounding only `l₁` leaves the second coordinate free and > the forbidden set is `{(+1,+1), (+1,−1)}`, whose hull misses the origin. **And a second sentence, proposed at `wall-note.md`:182**, replacing "the property's extension *is* the set of twin prime pairs": > the property's extension is the set of pairs of primes in `(x, x′²)`, whose > forbidden set is Example 3's. Phrasing it as "the set of twin prime pairs" is > also correct but borrows the conjecture, since the pattern `(−1,−1)` is then > realised only if the zone is occupied; the pairs-of-primes phrasing gets the > same forbidden set unconditionally, with Bertrand supplying the witness. [Rung: DERIVED, §1c and §2b, no adversarial pass.] ### Change 3, `research/history/staging/z2-state-draft-0829.md`:1141, "by analogy" **What stands:** the equivariance killer is "currently imported to the zone side by analogy, and an analogy is not a coordinate." **Proposed replacement:** > currently described as imported to the zone side by analogy. It is not an > analogy: by the Zone Restriction Lemma the zone window is the tile's own head > window read at the tile's own phase, so the transfer is a restriction, the > group is `Z/p#` acting simply transitively on window positions, and the > `eps * |ensemble| < 1` threshold is identical. What is genuinely open on the > zone side is the anchor's `z`-score at window width `p'^2`, for which no > corpus artifact certifies the variance. [Rung: DERIVED, §7b and §7d, no adversarial pass. This is a staging draft rather than a live file; it is listed because the sentence would otherwise be promoted as written.] ### Change 5, `TODO.md`:287, the ledger gate's one open finding `node research/qc.js ledger` was run against this note and returns exactly one finding, which is the gate working as designed: > `TODO.md:95 item Z2 does not list Q-derive-0904-r0-extension > (research/history/staging/derive-0904-r0-extension.md, ANSWERED) on its > Ledger: line` **Proposed:** append `Q-derive-0904-r0-extension` to item Z2's `Ledger:` line at `TODO.md`:287. Not applied: this note edits no existing file, and `TODO.md` is already modified in the working tree by other work in this wave, so an edit here would collide. The other twelve fast checks return clean against this note (refs, quotes, crosslinks, transfers, calibration, absence, sourcing, embeds, provenance and search-convention all read 0; the `scripts` and `widths` findings in that run belong to other agents' files). ### Change 4, no change proposed to `research/QUESTIONS.md` `z2-state-draft-0829.md` §8 questions 1 and 2 are answered by §§2-6 and §7 respectively. Both were checked against `research/QUESTIONS.md` before this note was written (grep for "extension", "parity", "equivariance", "Liouville"): the nearest rows are `Q-obstruction-audit` (ANSWERED, a different question, whether parity is the right name for what blocks the tile form) and `Q-parity-adversary` with `Q-parity-dstar-law-0829n` (ANSWERED and CLOSED, an adversary construction on the stretches, a different object). No row poses either question, and neither was re-run. Whether this note's ledger row should join `QUESTIONS.md` is the orchestrator's call, not this note's; it edits nothing. --- ## 9. What would falsify this note, and whether the check has run Each item names a specific check and its status. None of the four has run against an adversary; every proposition here is at rung DERIVED for that reason. 1. **Proposition A's forward half is false if two forbidden patterns of the given shape are not actually forbidden.** The check: for a named property, exhibit `l_1, ..., l_k` obeying `P` with pattern `(-e_i, e_j, d)` or `(e_i, -e_j, -d)`. That is impossible by construction when coordinates `i` and `j` are monochromatic, so falsifying this means falsifying monochromaticity, i.e. exhibiting a `p`-rough integer in `(p, p'^2)` that is not prime. The Zone Restriction Lemma rules that out and is VERIFIED at five levels. **Check has run, at the lemma.** 2. **Proposition A's converse half is false if a confined-rough property with `|S| <= 1` nevertheless satisfies H5.** The proof covers pure products only; for a property with a further condition `R` the extension can be smaller and more patterns forbidden. So the converse is stated for products and is **unchecked** outside them. A falsifier would be an R0-frame statement with at most one monochromatic coordinate whose extra conditions forbid enough extra patterns to capture the origin. §5's table lists three `|S| <= 1` rows and none has a condition of that kind; the table's completeness is the exposure. **Not run against an adversary.** 3. **Proposition B is false if Claim 1's reweighting does bite an argument whose conclusion is a fixed arithmetic fact.** The check is a reading of the proof in `lit-tao-parity.md` §1.6 and it turns on whether `nu -> nu w` can change the truth value of a statement about `Z/x#`. It cannot, since neither `nu` nor `w` appears in that statement. What is **not** checked is whether Tao's own scoping would call the composite route an instance of the method class anyway; the source does not address a composite of that shape, and this note's reading is therefore an extrapolation from the proof rather than a quotation. **Not run against the source, and not runnable: the source does not speak to it.** 4. **Proposition C is false if a piece of the R0 frame has a target property that is not confined-rough.** The shape that would do it: a target whose coordinates are constrained by something other than a modulus and an interval, for instance a multiplicative or bilinear condition on the pair, which is where every published parity break lives (Friedlander-Iwaniec's (B), and its norm-form constraint, `lit-tao-parity.md` §3.2 E5). No piece of R0 has such a condition, and `history/staging/attack-bilinear-transplant.md` closed the family for this frame on the ground that every bilinear object the corpus owns is bilinear over **moduli**. **Check has run at that closure, in a different context; it has not been re-run against R0's pieces one by one beyond §5's table.** 5. **Everything COVERED is conditional on H6.** Claim 1 is at rung CONJECTURED by its author. If Liouville pseudorandomness fails, the Siegel-zero scenario is the named instance, and `attack-barrier-kappa2.md` §0's caveat applies in full, since Heath-Brown 1983 makes that a world where the target is already a theorem, then every COVERED verdict is void. **This is not a check that can run; it is a standing hypothesis, and it is the reason no row of §5 may be quoted as an impossibility.** 6. **Part 2's open half.** A certified variance for the window-count statistic at window width `p'^2` would turn §7d's OPEN into a number, and if the anchor's `z` there were bounded while the tile's diverges, §7's "the wall is not located differently on this object" would need qualifying. **Not run; no producer in the corpus delivers it.** --- *Staging note. `README.md` §Status, `research/G2-STATE.md` §0 and `research/ZONE-POSTULATE.md` remain canonical and win against anything here.*