# The dials: every variable this programme can actually push *(2026-08-17, from Chris's question: which variables can be pushed to get over the line, and which is he missing. This note is a map, not a result. Calibration marked: PROVEN, MEASURED, OPEN.)* **Scope.** This document owns the triage question — which variable a new idea proposes to move, and whether that variable is attached to anything. It does not own the state of any object (`G2-STATE.md`), the target and its logical status (`ZONE-POSTULATE.md`), or the work queue (`../TODO.md`). Where it quotes a number, the owning document is named. The 2026-09-05 realignment prioritizes `chen-fold-benchmark.md` and `chen-signed-target.md`. Restrictions below that come from certifying primality by roughness alone do not constrain every argument with additional prime-distribution or signed arithmetic input. ## 0. Chris's two, adjudicated first **Dial 1, window width. The zone's redundancy is a feature, not a defect.** `research/GLOSSARY.md` records the pane's motivation as removing the zone's redundancy: "consecutive instances overlap almost entirely, so a handful of twins satisfies vast stretches of it." Chris's point is that for a formal proof this redundancy is a feature, not a defect. A statement of the form > for every x, the interval (x, g(x)) contains a twin prime implies TPC for **any** finite g, however fast growing, because the twin found is above x and x runs to infinity. Repeats across overlapping windows cost nothing. Overlap only matters if the argument is a *count* summed over windows, and no argument here is. So the redundancy of the zone is a reason to PREFER it. This agrees with A7 from the other direction: the pane, which removed the redundancy, came out strictly harder, "4.2665 proven, 1 needed" against the zone's 2. The pane is closed twice over now. **But the width is not a free dial either, and the reason is sharp.** Widening the window while keeping the sift level at p does not give twin primes. A p-rough number below p^k can carry up to k−1 prime factors, so past the frontier the survivor is an almost-prime, not a prime. **Certification pins the right endpoint at the square of the sift level.** In sieve currency, with u = ln(window top) / ln(sift level): > **u = 2 is the square frontier for primality certified by roughness alone.** This does not forbid working at shallower depth and retaining arithmetic weights that distinguish factor counts. The long-interval Chen benchmark does exactly that for prime-or-semiprime partners. The current dimension-two DHR application gives positivity only above its achieved threshold 4.266450284…; this is not a proved universal optimum. **Dial 2, maximum-gap growth.** The achieved bound is G₂(x#) ≪_ε x^(4.266450284…+ε). A fixed exponent below 2 suffices for twins; at exponent 2 the constant and next-prime frontier matter. This remains a sufficient secondary target, not the only possible research direction. MEASURED, calibrated against the one-class control: the finite fitted slope is **1.50 central on G₂'s own 22 trusted terms (h₂'s corrected figure is 1.57), 1.3 to 1.8 practical bracket, 1 as a hard floor** (`research/exponent-control.md` §5). These are local slopes, not an estimate with asymptotic coverage. A positive logarithmic factor can bias a finite power fit upward; the controls demonstrate that bias over their measured ranges. **Quote the control line whenever a ladder exponent is reported here.** Fifty-eight terms of a quantity whose true exponent is 1 + o(1) return 1.282 ± 0.008 with no drift. A clean power-law fit with white residuals is worth nothing on data this short: on the control it beats the family containing the truth by 47 AIC units. Current margins G₂(x#) against x′²: | x | G₂(x#) | x′² | margin | |---|---|---|---| | 23 | 204 | 841 | 4.12 | | 29 | 258 | 961 | 3.72 | | 31 | 348 | 1,369 | 3.93 | | 37 | 528 | 1,681 | 3.18 | A factor of three or four in hand today. The margin does not visibly drift: on nineteen terms of Ziller-Morack's h₂ (the fit's dataset, p in [5, 73], of the 21 published — `exponent-control.md` §6), which dominates G₂, the margin against x′² is flat at 2.2 with slope +0.018 ± 0.045, its minimum of 1.880 sitting at x = 17 and never revisited. ## 1. The dials Chris did not list **Dial 3, sifting parameter u. This is the master variable and every other choice is just a way of moving it.** The ledger: | target | what it needs | what is proven | verdict | |---|---|---|---| | Pane (n², (n+2)²) | u → 1 | u > 4.26645 | strictly harder than TPC (A7) | | Zone (p, p′²) | u = 2 | u > 4.26645 | = TPC | | almost-prime pairs | u ≈ 4.27 | proven | DONE, gives ≤ 4 factors each | | Chen's theorem | weighted | proven | DONE, p and p+2 with ≤ 2 factors | Every reformulation in THE-LENS §4 lands somewhere on this line. Reading a new idea off as a value of u is the fastest triage available, and it costs minutes. **Dial 4, "every" against "infinitely often". Priced slack: all four channels closed.** (2026-08-19: the truth's oscillation is 0.3187 nats over the exact ladder and falling; alignment freedom 2–11%; power-law instruments gain nothing by PNT; the sieve depth carries no i.o. freedom because the max gap is non-decreasing under inclusion of the sieve set; and the instrument-slack channel is surveyed — across 23 instruments the loosest that bounds G₂ directly is `maxsum_{L+1}` at 0.3747 nats against a need of 6.700, no sharp-level signature survives a permutation test at any of 19 instruments (Šidák p = 0.8865), and the two signatures with a mechanism select finite sets. Records: `history/staging/attack-0c0e-level-selection.md`, `verify-monotone-depth.md`, `ioslack-survey.md`.) `research/ZONE-POSTULATE.md` §2 proves the weak form (infinitely many zones occupied) is EQUIVALENT to TPC in both directions, while the strong form (every zone) is strictly stronger. Every G₂-based route proves the strong form, because a gap bound is uniform by construction. **An argument allowed to fail on a sparse set of p would be strictly weaker and still sufficient.** Two mechanisms were looked for and both fail. An almost-all-positions bound cannot be steered to the origin, because certification pins the anchor at 0 and there is exactly one usable window per tile. Large prime gaps widen the window by 2(p′−p)/p → 0, when a factor of three is needed. The slack is real and there is currently no route that spends it. **Dial 5, how prime the members must be.** This is where all the unconditional wins are. Relaxing "both prime" to "one prime, one with at most two prime factors" is Chen, proven since 1973. Relaxing to "at most four factors each" is what u > β₂ hands over directly. The reason this works is that it moves u off 2, which Dial 1 says is the only way anything ever moves. **Dial 6, which d, and how many.** Fix d = 2 and parity blocks you. Ask instead for *some* d drawn from an admissible set of size k and the obstruction lifts: that is Zhang, Maynard, Tao, Polymath, and it is **the only dial anyone has ever successfully pushed past the wall.** Bounded gaps are proven; k = 2, which is d = 2 exactly, is still open. Worth knowing precisely where the wall was breached and what the breach cost, namely the identity of d. `research/attack-06b-difference-map.js` already measured our side of this: d = 2 is mid-pack, no bounded invariant predicts G_d, and the headroom survives for all d ≤ 210. So the repo's objects do not distinguish d = 2 either, which is consistent with the obstruction being about d being FIXED rather than about d being 2. **Dial 7, level of distribution. Inert where it counts, live where it does not.** Bombieri-Vinogradov gives θ = 1/2 and Elliott-Halberstam would give θ = 1. This is the standard dial in the literature and `research/bv-import-survey.md` returned a split verdict, not a flat one. *Inert on the target.* Assumption A in its weak form is parity-blocked at every input including GEH, and the Buchstab transfer at bounded u is the κ = 2 sifting-limit problem rather than an equidistribution problem, so neither moves. Those two legs are where "our statements contain no primes, only roughness" is true, and they are the two the survey names when it says not to spend effort. *Live on the machinery.* The prime-comb equidistribution ingredient IS a prime count, so it is a Siegel-Walfisz theorem at fixed modulus and fixed depth, and a Bombieri-Vinogradov theorem on the full wheel in the tail regime at depth q_K = T^{o(1)}. Those retire named unproven ingredients of the certificate, and one further item is provable now and elementarily. They are theorems waiting to be written up rather than research; the queue is `TODO.md` item 11. So the dial is worth nothing against the wall and worth three write-ups against the heuristics, and a triage that reads it as flatly inert kills the write-ups. ## 2. The one that looks like a dial and is not **Margin.** The zone heuristically holds about 2C₂p²/ln²p twin pairs and we need one. The expected surplus is astronomical at every level and it has been astronomical since p = 7. It buys nothing, because parity is a structural obstruction rather than a quantitative one: an argument that establishes only a large expected count, with no mechanism against parity, yields a lower bound of exactly zero, not a small one. This is the trap the repo has already walked into twice, in the pane's capacity calculation and in A5's Markov bound, and both times the diagnosis was the same. `research/a3-05-bound-L.md` states it sharply: no polynomial moment of any fixed order can give the decay, only exponential moments will, and that is quasi-independence, which is the thing being assumed. ## 3. Summary table | # | dial | status | worth pushing? | |---|---|---|---| | 1 | window width | the square frontier applies to certification by roughness alone | long intervals remain available with explicitly imported arithmetic input | | 2 | max gap growth | open, an exponent fight: 4.2665 proven, 2 needed, 1.50 measured (raw 1.78 corrected by the control's +0.28 bias over the 22 trusted terms to x = 79, OEIS A144311; bracket 1.3-1.8, floor 1); the target also implies an explicit constant-1 Jacobsthal bound (§6). **2026-08-18: The `u_sup` route was raised and CLOSED the same day — it rises at every step over z = 13..43 and its bounded asymptote is 5.46, above β₂. What moved is the loss MAP, not the gap: DP1 carries 71% of the 3.27; its one known mechanism was found on 2026-08-18 and CLOSED on 2026-08-19, provably dead for asymptotics with the required degree diverging. `sift-limit-attack.md` §7d, §7e** | secondary sufficient target; current REC route closed | | 3 | sifting parameter u | the master variable | use as triage, not as a target | | 4 | every vs infinitely often | infinitude needs only unbounded successful scales | retain this quantifier in the new signed estimate | | 5 | almost-primes | classical Chen benchmark completed on long intervals | use as a benchmark before specifying the extra input for twins | | 6 | which d / how many | the only breach ever made | not ours to redo | | 7 | level of distribution | insufficient in the old certificate; explicitly imported in the Chen benchmark | match each theorem to its sequence and interval before using it | | — | margin | not a dial | no | ## 4. The honest reading The programme can study the uniform gap target or the anchored count target, but neither has an established route to infinitude here. The current sequence is the classical long-interval Chen benchmark followed by an exact signed estimate sufficient for twins. The benchmark is complete using named theorem inputs; the signed estimate remains open. Finite measurements of slack in particular certificates do not exhaust all infinitely-often arguments. Likewise the square frontier limits roughness-only certification, not every use of longer intervals with arithmetic weights. The specific failed bounds remain closed; a new argument must identify what it estimates differently and how its error reaches the required scale. ## 5. Exceptions, and why the zone is the right window after all *(2026-08-17, Chris: the narrow windows have genuine failures, the (n², (n+1)²) family failing twelve times and last at 122, while the zone has no known failure at all. Measured in `research/window-exceptions.js`, one sieve to 1e8. Custody: 440,312 twin pairs below 1e8, matching the published count, and family A reproduces OEIS A091592 term for term without being told it.)* | family | width | exceptions | last | min twin count | min count / HL expectation | |---|---|---|---|---|---| | A. (n², (n+1)²) | 2n+1 | **12** | 122 | 0 | **0.471** in the top band | | B. (n², (n+2)²), the pane | 4n+4 | **1** | 26 | 0 | **0.575** in the top band | | C. (p, p′²), the zone | ≈ p² | **0** | – | 2, at p = 2 | **1.13, 1.18, 1.13** by band | **Chris's objection is right, and the measurement says something stronger than he claimed.** It is not only that the zone has no failures. **The zone has never once fallen below its own expectation.** Its worst case across all 1,228 windows tested sits at 1.13 times the Hardy-Littlewood count, and since that form is known to underestimate by about 12% at this height (measured independently in FOLD-PROFILE §9, S/HL = 1.1238), the zone's minimum is essentially exactly its mean. The count barely fluctuates in relative terms. The narrow families, by contrast, have minima at 0.47 and 0.58 of an already-low estimate, which are genuine tail events. The mechanism is the obvious one and worth stating anyway: the zone carries thousands to millions of twins, 8,278 at worst in the 1e6 to 1e8 band, so its relative fluctuation is O(1/√count) and vanishes. The narrow windows carry tens, so their relative fluctuation is O(1) and the minimum is drawn from the tail. The exceptions are not anomalies, they are what an O(1) count does. **Two consequences, one methodological and one cautionary.** *Methodological, and it is the real reason to prefer the zone.* A finite exception set is logically harmless: "for all n > 122" still implies TPC. But the only tool this programme has is a **uniform** gap bound, G₂(x#) < x′², and a uniform bound cannot prove a statement that has exceptions. So a window with failures is not merely inelegant, it is **incompatible with the method**. The zone is the widest window whose statement has the same shape as the tool. *Cautionary.* The zone's safety is entirely margin, and §2 of this note already recorded that margin is not a dial. Eight thousand twins where one is needed is exactly the surplus that has bought nothing since p = 7. So the right reading of the table is: the zone is the correct **target**, and its comfort is no evidence whatever that it is **tractable**. ## 6. What a sufficient gap bound actually costs (2026-08-17) *(Chris: if we can produce a formal max-gap bound smaller than the zone, are we home free? Yes. This section prices it against what is proven.)* **The reduction, restated (PROVEN, ZONE-POSTULATE §3).** The tile carries a twin slot at its edge, since p# ± 1 are both coprime to p#, so reading forward from the origin the first twin slot lies within G₂ of it, and the slot at r = 1 is never live because 3 | p#. Hence > **G₂(p#) < p′² − 2 ⇒ Zone Postulate at p ⇒ TPC.** Any of these suffice, and all are weaker than nailing a constant: G₂(p#) = O((log p#)^α) for some α < 2; or ≤ C(log p#)² with C < 1; or simply o(p²). And only infinitely many p are needed, so an asymptotic bound with an unbounded exceptional set of density zero is still enough. **The exact distance, and it is an exponent.** | object | classes per prime | sieve dimension κ | sieve limit β_κ | proven Jacobsthal exponent | |---|---|---|---|---| | ordinary Jacobsthal g(p#) | 1 | 1 | **2** | **2**, Iwaniec 1978 | | twin Jacobsthal G₂(p#) | 2 | 2 | **4.2665** | **4.2665**, `paper/beta2-note.md` | | what the Zone Postulate needs | 2 | 2 | — | **2** | *(The β_κ column is not one kind of number: 2 at κ = 1 is proven two-sided (Selberg's examples), while 4.2665 at κ = 2 is the DHR sieve's value, an upper bound on the sifting limit only; the best proven floor at κ = 2 is 2, with 1.8196 the best resting on a published inequality (Brady 2017). Added 2026-09-04, `research/history/staging/redteam-0904-sifting-limit.md`.)* **How the sieve gives a Jacobsthal bound.** A lower-bound sieve with threshold β_κ and a suitable remainder estimate supplies a survivor in every interval of length z^(β_κ+ε), for fixed ε>0. This yields a Jacobsthal upper bound at that exponent. It does not identify the optimal Jacobsthal exponent with the sieve's threshold. Iwaniec supplies exponent 2 for the one-class object; the current DHR application supplies 4.266450284… for the two-class object. A fixed exponent below 2 would prove twins, but neither that upper bound nor an impossibility theorem for other arguments follows from the threshold comparison. **And there is a second price, on the one-class object itself.** Twin slots are a subset of holes, so G₂(x#) ≥ g(x#) pointwise (PROVEN, elementary; VERIFIED at all 22 shared terms of the trusted ladders, ratios 1.00 to 8.55, `research/external-ladders-01.js`). A sufficient gap bound G₂(x#) < x′² − 2 therefore also proves g(x#) < x′² − 2, an explicit constant-1 form of the Jacobsthal bound at primorials. Iwaniec 1978 gives g(x#) ≪ x² only with an unspecified constant. The little-o target in Erdős problem #687 is different: the constant comparison gives limsup g(x#)/x² ≤ 1, whereas a fixed subquadratic power saving would give g(x#)=o(x²). These targets must be distinguished (`research/two-class-lower-bounds.md` §9). **And the floor does not collapse to an explicit elementary bound.** Checked against the sources: Kanold gives 2^{√k}, Stevens k^{Θ(log k)} and Paseman k^{O(log log k)}, all far weaker than exponent 2; the only exponent-2 statements are Vaughan 1977 for general n and Iwaniec 1971 Thm 2 / 1978 at primorials, both with **inexplicit** constants, and the constant is the whole question (`research/ZONE-POSTULATE.md` §6, `research/G2-STATE.md` §9). **What is encouraging, kept in proportion.** The band (2, 4.2665] is a proof gap and not a truth gap. Calibrated against the one-class control the two-class exponent reads 1.50 on the 22 trusted terms (1.57 is the h₂ control's own figure; `research/G2-STATE.md` §6), bracket 1.3 to 1.8, and Ziller and Morack's adversarial h₂, which dominates G₂ pointwise, measures 1.567 in the same frame over 21 terms. So the object almost certainly sits below 2. And `research/sift-limit-attack.md` records that nothing published blocks the band, since no κ = 2 extremal example is known (Halberstam 2003 p. 117); the published lower bounds on β₂, found 2026-09-04 in Selberg's reciprocal convention, all sit at or below 2 (Brady 2017: 1.8196; β(2) ≥ 2 from the one-sided dimension axiom), so they bound the band's lower endpoint and nothing inside it. That search is tabled in `SEARCH-CONVENTIONS.md` §1 and §3 and β₂'s own literature in §4. Nobody has moved β₂ in forty years, but nobody has shown it cannot move for this problem either. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*