# Attack G (rerun): the hybrid bound — exact head, sieved tail **Producer:** `research/attack-hybrid-bound.js` (no arguments, ~25 s, exit 0, run 2026-08-18). Every number below is in that file's `OUTPUT` block. --- ## Headline The glue closes, it is a real theorem, and it buys **one factor of 1.93 in x and nothing asymptotic**. > **Theorem HY [VERIFIED — proof in the script banner, evaluated in §B].** > Work in slot coordinates (`j ↔ 6j+5`; prime `p` kills `j ≡ r_p` and > `j ≡ r_p − s_p (mod p)`, `s_p = 3⁻¹ mod p`, `r_p` free — the covering > formulation of `two-class-lower-bounds.md` §1). Fix `5 ≤ x₀ ≤ x`, put > `H = {p : 5 ≤ p ≤ x₀}`, `T = {p : x₀ < p ≤ x}`, `Q = ∏_{p∈H} p`, `n_T = |T|`, > and for odd `m` > `A_m = Σ_{j≤m} (−1)^j e_j({2/p}_{p∈T})`, `B_m = Σ_{j≤m} C(n_T, j) 2^j`. > If `A_m > 0`, then no admissible choice of two-class pairs over the primes > `5..x` covers `[1, L]` for any `L > Q·(B_m/A_m + 1)`. Writing `β_hy(x) = min_{x₀, m} ln(6·Q·(B_m/A_m + 1)) / ln x`, in the same convention `sift-limit-attack.md` §7 uses for `β_pure`: | | pure Brun `β_pure` | hybrid `β_hy` | |---|---|---| | beats 4.2665 up to | **x = 227** | **x = 439** | | first loss | x = 229 | x = 443 | | returns? | never | never (5972 further primes swept to x = 60000) | | x = 101 / 1009 / 10⁵ / 10⁶ | 3.983 / 5.383 / 8.418 / 8.878 | 3.408 / 4.655 / 6.501 / 7.303 | **Is it worth having? Read §5 before answering — the short version is: as a remark, yes; as a theorem, no.** At the crossover the whole improvement is a factor of **1.27** in a bound already **6.8 orders of magnitude** above the measured truth. --- ## 1. What the glue is, and why it is not a case split The brief distinguishes the trivial glue ("below `X₀` use the certificate, above it use the sieve") from the non-trivial one that feeds the finite result *into* the asymptotic argument. HY is the second kind, and the mechanism is one line: **the head-survivor set is a union of `K = ∏_{p∈H}(p−2)` residue classes mod `Q`, and on each such class `a + Qt` the tail primes act as an ordinary two-class sift in `t`**, because `Q` is invertible mod every `p ∈ T`. So Brun's truncation applies verbatim to a line whose head is already handled exactly, with no head terms in `A_m` or `B_m` at all. The head therefore enters through **one number**, `Q = e^{θ(x₀)}`, and through nothing else. HY is a single inequality with `x₀` a free parameter, valid at every `x`: **uniform, not a case split.** What *is* a case split is the comparison against 4.2665 — `min(β_hy(x), 4.2665)` improves the exponent only on `x ≤ 439` — but that is a statement about which of two bounds is smaller, not about HY's form. `β_hy ≤ β_pure` at every `x` by construction (`x₀ = 0`, the empty head, is in the sweep), so the glue never loses. --- ## 2. The measured result Section B of the script. `x₀*` is the optimal head, `m` the Bonferroni depth. | x | `β_pure` | `x₀*` | m | `θ(x₀*)/ln x` | `β_hy` | gain | |---|---|---|---|---|---|---| | 13 | 2.806 | 5 | 1 | 0.627 | 2.484 | 0.322 | | 101 | 3.983 | 5 | 3 | 0.349 | 3.408 | 0.575 | | 227 | 4.253 | 7 | 3 | 0.655 | 3.709 | 0.544 | | **439** | 5.202 | 11 | 3 | 0.978 | **4.227** | 0.975 | | 443 | 5.210 | 13 | 3 | 1.398 | 4.278 | 0.932 | | 1009 | 5.383 | 7 | 5 | 0.514 | 4.655 | 0.728 | | 10⁵ | 8.418 | 13 | 7 | 0.740 | 6.501 | 1.918 | | 10⁶ | 8.878 | 23 | 7 | 1.262 | 7.303 | 1.575 | Custody: section A recomputes `β_pure` from scratch and returns 2.80, 3.98, 4.25, 5.38, 8.88 at x = 13, 101, 199, 1009, 10⁶ and the crossover 227 / 229 — the published values of `sift-limit-attack.md` §7, digit for digit, so the hybrid column is comparable to a number already in the corpus. --- ## 3. Why there is no asymptotic gain, measured rather than asserted **The cap is the primorial.** The crude "one error per residue class" is the only bound available for a `Q`-periodic set inside a window shorter than `Q`, so exactness over `H` costs `Q = e^{θ(x₀)}` — exponential in the head's reach. Requiring the head price alone to stay under a bounded exponent forces `θ(x₀) ≤ β·ln x`, hence `x₀ = O(ln x)`. Measured (§D2): `x₀*/ln x` = 1.08, 1.01, 1.85, 1.13, 1.66 across x = 101 … 10⁶, with no drift. **The head can only reach a vanishing share of the Mertens mass.** The depth is set by `W_T = Σ_{p∈T} 2/p`; primes below `O(ln x)` carry `2 lnlnln x + O(1)` against the `2 lnln x + O(1)` of all primes to `x`. The removable fraction is `lnlnln x / lnln x → 0` **[INFERRED, from `x₀ = O(ln x)`]**. Over the accessible range it has not turned over: measured 0.204 → 0.324 across x = 101 … 10⁶, which is what that ratio looks like below 10⁶ and is flagged rather than read as growth. **The floor that makes it a divergence.** `B_m ≥ C(n_T,m)2^m ≥ (2n_T/m)^m`, and `A_m ≤ δ_T ≤ 1` because an odd-order Bonferroni truncation under-counts (checked at every row, §D3). So `β_sieve ≥ m₀·ln(2n_T/m₀)/ln x`, where `m₀` is the first admissible depth. Measured floor against achieved: 1.775/3.059, 3.029/4.141, 4.813/5.761, 5.076/6.041. With `m₀/W_T` climbing (1.92 → 2.52 over the range, toward the 3.594 that `attack-beta2-05` (E2) verifies exactly to `W = 1024`), the floor grows like `W_T` and `W_T` loses only `lnlnln x` to the head. **`β_hy` diverges at the same leading rate as `β_pure`**; the glue moves a `lnlnln x` term, not the `lnln x` one **[INFERRED]**. **The gain is a staircase, not a curve.** §D1, at x = 10⁶: removing 11 and removing 23 each cut the depth by 2 and pay net +1.186 and +0.946 of exponent; every other head prime pays between −0.075 and −0.281. Cost `ln p/ln x` is smooth; gain only arrives at a depth drop. That is why `x₀*` jumps around (5, 7, 11, 13, 17, 19, 23) instead of tracking a smooth optimum. --- ## 4. The literal glue the brief named — A144311 as base case — is dead, on a type mismatch The brief's DP3 slot wants the *certified finite bound* used as the base case. Done in §C, and it fails for a reason worth recording. A head certificate is **order** information: A144311 says the head cannot cover `m₀` consecutive slots, so every window of `m₀+1` slots holds a head survivor and `[1,L]` holds at least `⌊L/(m₀+1)⌋` of them. Bonferroni cannot consume that — it needs counts in arithmetic progressions, i.e. **congruence** information — so a gap bound can only be spent against the Theorem P union count of `attack-beta2-05`, which needs `1/(m₀+1) > W_T`. Measured: at every `x₀` from 11 to 79, the **first** tail prime alone already exceeds the budget. At `x₀ = 79`: `1/(m₀+1) = 3.509e−3` against `2/83 = 2.410e−2`. | x₀ | 11 | 23 | 43 | 79 | |---|---|---|---|---| | largest x reached | 11 | 23 | 43 | 79 | The route admits **not one further prime** at any level. It buys strictly nothing beyond the exact term it starts from. This is `sift-limit-attack.md` §7c's wall seen from the other side. There the dilation was an isomorphism of *congruence* structure and not of *order*, and the sieve was asking an order question. Here the certificate carries *order* and the sieve is asking a congruence question. Same mismatch, opposite direction. --- ## 5. What a finite-range improvement buys, and what it does not Attack E measured the object: `G₂(x#) ≈ 0.762 x ln²x lnln x` over x = 11..79 **[MEASURED]**, pure power law excluded by 10.6 AICc units (`history/staging/attack-growth-law.md`). That changes what counts as a win here, so, head on: §E of the script, with the sieve side idealised at constant 1 — the convention §7 already uses for `β_pure`, and the reason the last column is a floor on the sieve's real weakness rather than the sieve's actual bound: | x | exact `G₂` | law `0.762 x ln²x lnlnx` | hybrid certificate | log₁₀(cert/truth) | log₁₀(x^4.2665/truth) | |---|---|---|---|---|---| | 13 | 66 | 6.14e1 | 5.85e2 | 0.9 | 2.9 | | 79 | 1710 | 1.70e3 | 2.24e6 | 3.1 | 4.9 | | 101 | — | 2.51e3 | 6.77e6 | 3.4 | 5.2 | | 227 | — | 8.61e3 | 5.47e8 | 4.8 | 6.1 | | 439 | — | 2.24e4 | 1.48e11 | 6.8 | 6.9 | **Is it a theorem worth writing down?** As a remark in the covering section, yes. As a theorem of its own, no. It is elementary — the head-survivor line is a two-class sift and Brun applies to it unchanged — and no novelty is claimed for the move. **What would it let the corpus claim that it cannot claim today?** Only this: *the covering economy's unconditional window against 4.2665 runs to x = 439, not x = 227.* Nothing else. It does not touch the asymptotic exponent, it does not touch `β₂`, it does not touch any statement about `G₂`'s growth law, and it does not touch the twin wall. **Is the glued statement uniform or a case split?** HY itself is uniform (§1). The *improvement over 4.2665* is a case split on `x ≤ 439`. **And the honest arithmetic on the window.** Exact terms run to x = 79, so the only range where HY is simultaneously the best available bound and the truth is unknown is `(79, 439]`. In that range the bound is wrong by 3.4 to 6.8 orders of magnitude, and attack E's law already describes the object to within a factor of about two. At the crossover itself the improvement is `1.880e11 / 1.479e11 = 1.27` — a 21% cut in a bound seven orders of magnitude too large. **Verdict: the glue buys little.** It is worth one paragraph in `sift-limit-attack.md` §7 updating "227" to "439 with an exact head", and it is worth the §4 negative, which is the more durable half. --- ## 6. Regress, and whether the range grows with effort **No regress.** §7c killed exact strata because exactness at level `j` needs the upper sieve function of a dilated tile at level `j−1` — it terminates at every finite `z` and at no uniform `z`. HY has no recursion and no re-entry: it is one closed-form inequality. It escapes the regress precisely *because* it buys exactness at one scale outright and pays cash (`θ(x₀)`) rather than borrowing. Bounded price, bounded gain — the two are the same fact. **The range does not grow with effort.** §F: the certificate at x = 439 evaluates in under a millisecond. There is no search to deepen and no ladder to extend. `x = 439` is where `θ(x₀)` overtakes the depth saving, not where the compute ran out. The brief's target shape — "certified below `x^{4.2665}` for all `x ≤ N`, with `N` reachable by `C` hours of compute" — **is not the shape of this result**: `N` is closed form and `C` is zero. --- ## 7. One methodological note on the head price §G exists because the head price *is* the result. Pricing the head at `3^{π(x₀)}/∏(1−2/p)` instead of at `Q` under-prices it by 1.83e1 at `x₀ = 13`, 3.27e4 at `x₀ = 29`, and **1.89e19** at `x₀ = 79`, because `θ(x₀) ~ x₀` while `1.0986 π(x₀) ~ 1.0986 x₀/ln x₀`. Any hybrid whose head cost grows slower than the primorial is not bounding the CRT error of a `Q`-periodic set inside a window shorter than `Q`, and no such bound exists. The distinction decides whether the crossover is 439 or a number in the tens of thousands. --- ## Draft CHANGELOG entry (not applied — this agent may not edit CHANGELOG.md) > **2026-08-18 — the hybrid bound: exact head, sieved tail (attack G rerun).** > `research/attack-hybrid-bound.js`; write-up > `history/staging/attack-hybrid-bound.md`. Theorem HY glues an exactly-handled > head of primes onto Brun's truncated tail: the head-survivor set is > `∏(p−2)` classes mod `Q = x₀#`, and on each class the tail acts as an ordinary > two-class sift, so no head term appears in `A_m` or `B_m`. Uncoverable for > `L > Q(B_m/A_m + 1)`. It widens the covering economy's window against 4.2665 > from **x ≤ 227 to x ≤ 439** (first loss 443, never returns to x = 60000) and > gains 0.29 to 2.01 of exponent over `β_pure` across x = 13..10⁶, but does not > change the divergence rate: exactness costs `e^{θ(x₀)}`, so the head reaches > only `x₀ = O(ln x)` (measured `x₀*/ln x` = 1.01 to 1.85) and removes only > `lnlnln x` of the `lnln x` Mertens mass that sets the Bonferroni depth. The > literal "A144311 as base case" glue is **dead on a type mismatch**: a > certificate is order information, Bonferroni needs congruence information, and > spent against Theorem P the certificate admits not one further prime at any > `x₀` from 11 to 79. Verdict: worth updating §7's "227" to "439"; nothing else. > At the crossover the improvement is a factor 1.27 in a bound 6.8 orders of > magnitude above attack E's measured law. --- ## Suggested edit to `research/sift-limit-attack.md` §7 (not applied) The sentence "It **beats 4.2665 for x ≤ 227**, loses from x = 229, never returns" should gain: "An exactly-handled head of primes below `x₀` replaces the head's Bonferroni terms at a cost of `x₀#`, which widens that window to `x ≤ 439` (`attack-hybrid-bound.js`) and changes nothing asymptotic: the head can only reach `x₀ = O(ln x)`."