{"id":2933,"job_id":6157,"problem_id":1,"lane_id":32,"type":"explore","user_id":77,"model":"gemini-3.8-flash","provider":"google","report_md":"# Job #6157 (Route #256 Rescue): Collinearity Invariance under Exact-Mean-Gap Normalization, Control Target Correction, and the 20-Term Qg Plateau\n\n## Caveats and open gaps\n\nNothing in this return bounds $\\hat{G}(n) = G_2(P(n)\\#)$, proves or refutes `(H-sub-pow)` at any base or across all bases, or determines the sign of the second-order exponent $\\delta_{G_2}$. While this rescue identifies and corrects three mathematical/estimation defects in predecessor returns #2734 and #2757 (showing that the one-class control's second-order slope is positive and strictly increasing with reach when its known leading exponent $\\beta_{\\mathrm{ctrl}} = 1$ is fixed), it also proves **Theorem 2 (Design-Matrix Invariance)**: dividing $G_2(p\\#)$ by the exact mean gap $M_2(p)$ leaves the regressor matrix $(\\mathbf{1}, \\ln p, \\ln \\ln p)$, its variance inflation factor ($\\mathrm{VIF} = 51.09$ on $p \\in [5, 79]$), and the collinearity trade-off slope $-S_p = -2.894898$ completely invariant. Consequently, across the corpus's own leading-exponent bracket $\\beta_0 \\in [1.3, 1.8]$ (`exponent-control.js` S11), the second-order reading $\\hat{\\delta}_{G_2}(\\beta_0) = 2.894898(1.777190 - \\beta_0)$ changes sign (`+1.3814` at $\\beta_0 = 1.3$ vs `-0.0660` at $\\beta_0 = 1.8$). Route #256 therefore remains blocked under a sharpened, proven `scoped_obstruction`.\n\n## 1. Three defects in predecessor returns #2734 and #2757 (`proven` and `verified`)\n\nPredecessor return #2757 (`compute_ib.py`) reported `inconclusive` (`scoped_obstruction`) after finding a negative 3-parameter OLS slope $\\hat{\\delta}_{R_1,\\mathrm{OLS}} = -0.9044$ against a stated target $+2.0$ and negative rung-contrast readings on the one-class control $h(P(n)\\#)$ (OEIS A048670, 64 terms, $p \\le 311$). Three distinct mathematical and statistical errors produced those negative control readings:\n\n1. **Defect A — Conflation of prime index $k$ with integer argument $n$ (`proven`).**\n   Returns #2734 (§5) and #2757 (`compute_ib.py` line 170) defined $\\hat{G}(n) = G_2(P(n)\\#)$ and $\\hat{h}(n) = h(P(n)\\#)$ with $P(n) = \\max\\{p \\le n : p \\text{ prime}\\}$, yet substituted $P(n) \\sim n \\ln n$ to shift the Maier–Pomerance conjecture $h(p\\#) \\asymp p(\\ln p)^{2+o(1)}$ (`\\delta_p = 2`) to $\\delta_{\\mathrm{ctrl},n} = 1 + \\delta_p = 3$ and normalized target $\\delta_{R_1} = 3 - 1 = +2$. In fact, $p_k \\sim k \\ln k$ applies to the $k$-th prime (the index $k = \\pi(p_k)$ of OEIS A048670 $a(k) = h(p_k\\#)$), whereas for the integer argument $n$, $P(n) \\le n$ satisfies $P(n) \\sim n$ (and at every prime level point $n = p \\in [5, 311]$ in `compute_ib.py`, $P(p) = p$ identically). Thus in the argument variable $n$ (and prime variable $p$),\n   $$\\hat{h}(n) = h(P(n)\\#) \\asymp P(n)(\\ln P(n))^2 \\sim n(\\ln n)^2 \\implies \\delta_{\\mathrm{ctrl},n} = \\delta_p = +2,$$\n   and after dividing by the exact one-class mean gap $M_1(P(n)) = \\frac{P(n)\\#}{\\varphi(P(n)\\#)} \\sim e^\\gamma \\ln n$,\n   $$R_1(n) = \\frac{h(P(n)\\#)}{M_1(P(n))} \\asymp n(\\ln n)^1 \\implies \\text{true normalized control target } \\delta_{R_1} = 2 - 1 = +1.0 \\quad (\\text{not }+2.0).$$\n   This matches `corpus-exponent-control.js` S6 (`line 193`), which explicitly states `predicted 1 + 1/log p ~ 1.2` for the log-log slope of $R_1(p) \\asymp p(\\ln p)^1$.\n\n2. **Defect B — Unconstrained 3-parameter OLS collinearity distortion (`proven` and `verified`).**\n   **Theorem 1 (Frisch–Waugh–Lovell / Wald IV Collinearity Identity; rung: `proven`).**\n   For any positive sequence $Y(p)$ on a prime grid $p \\in [p_{\\min}, p_{\\max}]$, let $x_1 = \\ln p$, $x_2 = \\ln \\ln p$, and let $\\hat{\\delta}_Y(\\beta_0) := \\mathrm{slope}_{\\mathrm{OLS}}(\\ln Y(p) - \\beta_0 \\ln p,\\ \\ln \\ln p)$ be the 2-parameter OLS second-order slope at fixed first-order exponent $\\beta_0$. By linearity of covariance and the Frisch–Waugh–Lovell theorem:\n   $$\\hat{\\delta}_Y(\\beta_0) = S_p \\cdot \\bigl(\\beta_{\\mathrm{IV}}(Y) - \\beta_0\\bigr), \\qquad S_p := \\frac{\\mathrm{Cov}(x_1, x_2)}{\\mathrm{Var}(x_2)}, \\qquad \\beta_{\\mathrm{IV}}(Y) := \\frac{\\mathrm{Cov}(\\ln Y, x_2)}{\\mathrm{Cov}(x_1, x_2)},$$\n   and the unconstrained 3-parameter OLS estimator $(\\hat{\\beta}_{\\mathrm{OLS}}, \\hat{\\delta}_{\\mathrm{OLS}})$ of $\\ln Y = c_0 + \\beta x_1 + \\delta x_2$ satisfies $\\hat{\\delta}_{\\mathrm{OLS}} = \\hat{\\delta}_Y(\\hat{\\beta}_{\\mathrm{OLS}}) = S_p(\\beta_{\\mathrm{IV}}(Y) - \\hat{\\beta}_{\\mathrm{OLS}})$ identically.\n\n   On the control grid $p \\in [5, 311]$ ($n = 62$), $x_1 = \\ln p$ and $x_2 = \\ln \\ln p$ have correlation $r = 0.984944$ ($\\mathrm{VIF} = \\frac{1}{1-r^2} = 33.46$, $S_p = 3.666235$), with $\\beta_{\\mathrm{IV}}(R_1) = 1.051936$. Because unconstrained 3-parameter OLS leaves $\\beta$ free, collinearity inflates $\\hat{\\beta}_{\\mathrm{OLS}}$ to $1.298617 > \\beta_{\\mathrm{IV}}(R_1)$, flipping $\\hat{\\delta}_{R_1,\\mathrm{OLS}} = 3.666235(1.051936 - 1.298617) = -0.904389 < 0$.\n   By contrast, when the control's known leading exponent $\\beta_0 = 1$ is fixed (`exponent-control.js` S4 discipline, regressing $\\ln(R_1(p)/p) = \\ln(h(p\\#)/(p M_1(p)))$ on $\\ln \\ln p$), the control slope is **positive** and **strictly increasing with upper reach** (`verified`):\n   - Lower-cutoff windows $[p_{\\mathrm{lo}}, 311]$: $+0.1904 \\pm 0.0328$ ($[5,311]$), $+0.3385 \\pm 0.0360$ ($[13,311]$), $+0.5061 \\pm 0.0327$ ($[31,311]$), $+0.3451 \\pm 0.0336$ ($[73,311]$), $+0.5260 \\pm 0.0466$ ($[127,311]$).\n   - Upper-cutoff reach trend $[5, p_{\\mathrm{hi}}]$: strictly increasing across all 7 cutoffs from $-0.2277$ ($p_{\\mathrm{hi}}=31$) $\\to -0.0909$ ($73$) $\\to +0.0540$ ($127$) $\\to +0.1100$ ($179$) $\\to +0.1511$ ($233$) $\\to +0.1833$ ($283$) $\\to +0.1904$ ($311$).\n   - Unnormalized fixed-$\\beta_0=1$ control slope $\\mathrm{slope}(\\ln(h(p\\#)/p), \\ln \\ln p)$: $+1.0000 \\pm 0.0493$ on the pinned 56-term set $[5, 271]$ (reproducing `exponent-control.js` S4 `c*p*log^a p: a=1.000`) and $+1.0235 \\pm 0.0399$ on $[5, 311]$.\n\n3. **Defect C — Prime-rounding step bias and $k=1$ Pintz-transient anchoring in `rung_contrast` (`proven` and `verified`).**\n   In `compute_ib.py`, `rung_contrast` assumed that in $D_R(b,k) = \\ln R(b^{k+1}) - \\ln R(b^k) - \\ln R(b)$, the leading power term $\\beta \\ln n$ cancels because $\\ln(b^{k+1}) - \\ln(b^k) - \\ln(b) = 0$. Because $\\hat{G}(n)$ and $R(n)$ are step functions evaluated at $P(n) = \\max\\{p \\le n\\}$, the actual leading-power contribution is $\\beta \\cdot \\Delta_P(b,k)$ where $\\Delta_P(b,k) := \\ln P(b^{k+1}) - \\ln P(b^k) - \\ln P(b) \\ne 0$ (e.g. $\\Delta_P(2,1) = \\ln(3/4) = -0.287682$, $\\Delta_P(2,2) = \\ln(7/6) = +0.154151$, injecting a first-order bias of $-1.5358\\,\\beta$ into $(D_R(2,1)-D_R(2,2))/\\ln(4/3)$). Furthermore, every rung contrast is anchored at $k=1$ ($P(b) \\in \\{2,3,5\\}, P(b^2) \\in \\{3,7,13,23\\}$), which lies inside the initial $p \\le 29$ transient where $u_1(p) = \\frac{h(p\\#)}{p M_1(p)}$ decreases from $0.5000$ ($p=2$) to $0.2505$ ($p=29$) — consistent with Pintz's factor $\\frac{\\ln \\ln \\ln p}{(\\ln \\ln p)^2}$ — before rising to $0.3559$ at $p=283$.\n\n## 2. New 20-term measurement of the structural quotient $Q_g(p) = R_2(p)/R_1(p)$ (`verified`)\n\nIn `corpus-exponent-control.js` S6, the structural quotient $Q_g(p) = \\frac{G_2(p\\#)/M_2(p)}{h(p\\#)/M_1(p)} = \\frac{R_2(p)}{R_1(p)}$ was evaluated only on the first 10 terms $p \\in [5, 37]$ (slope $+0.2978 \\pm 0.0627$). Extending $Q_g(p)$ to all 20 prime levels $p \\in [5, 79]$ of the trusted 22-term $G_2$ ladder (`rescue_route256_evidence.json`) reveals a sharp regime change after the $p = 37$ outlier ($Q_g(37) = 1.5797$):\n- On the 10 new terms $p \\in [41, 79]$ ($p = 41, 43, 47, 53, 59, 61, 67, 71, 73, 79$), $Q_g(p)$ takes the values $1.4206, 1.2906, 1.3017, 1.4800, 1.4508, 1.4258, 1.4498, 1.3592, 1.3433, 1.4080$.\n- Every single term on $p \\in [41, 79]$ lies strictly inside $[1.2906, 1.4800] < Q_g(37) = 1.5797$, with log-log slope vs $\\ln p$ of **$+0.0473 \\pm 0.0711$** ($t = 0.67$, indistinguishable from $0$) and slope vs $\\ln \\ln p$ of **$+0.1903 \\pm 0.2797$** ($t = 0.68$).\n\n## 3. Why Route #256 remains blocked: Design-Matrix Invariance on $G_2$ (`proven` and `verified`)\n\n**Theorem 2 (Design-Matrix Invariance under Exact-Mean-Gap Normalization; rung: `proven`).**\nReplacing $\\ln G_2(p\\#)$ by $\\ln R_2(p) = \\ln G_2(p\\#) - \\ln M_2(p)$ subtracts a fixed vector from the response variable and leaves the regressor matrix $X = (\\mathbf{1}, \\ln p, \\ln \\ln p)$ on $p \\in [5, 79]$ — and therefore its correlation $r = 0.990165$, variance inflation factor $\\mathrm{VIF} = 51.09$, and collinearity trade-off slope $\\frac{d\\hat{\\delta}(\\beta_0)}{d\\beta_0} = -S_p = -2.894898$ — identically unchanged. It shifts only the Wald IV intercept from $\\beta_{\\mathrm{IV}}(G_2) = 1.777190$ to $\\beta_{\\mathrm{IV}}(R_2) = 1.230658$:\n$$\\hat{\\delta}_{G_2}(\\beta_0) = 2.894898\\bigl(1.777190 - \\beta_0\\bigr), \\qquad \\hat{\\delta}_{R_2}(\\beta_0) = 2.894898\\bigl(1.230658 - \\beta_0\\bigr).$$\n\nUnlike the control $h(p\\#)$ where $\\beta_{\\mathrm{ctrl}} = 1$ is known, the leading exponent $\\beta_{G_2}$ is unknown and spans the corpus's S11 practical bracket $[1.3, 1.8]$ (or $[1.0, 2.0]$):\n\n| Assumed $\\beta_0$ | Interpretation | $\\hat{\\delta}_{R_2}(\\beta_0)$ | $\\hat{\\delta}_{G_2}(\\beta_0)$ (sign-lemma parameter) |\n|---:|---|---:|---:|\n| $1.0000$ | Structural $Q_g$-bounded hypothesis | $+0.6677 \\pm 0.0914$ | $+2.2499 \\pm 0.1119$ |\n| $1.2307$ | $\\beta_{\\mathrm{IV}}(R_2)$ zero-crossing of $\\hat{\\delta}_{R_2}$ | $0.0000 \\pm 0.0785$ | $+1.5822 \\pm 0.0981$ |\n| $1.3000$ | Lower edge of corpus S11 bracket | $-0.2007 \\pm 0.0760$ | $+1.3814 \\pm 0.0951$ |\n| $1.4980$ | Corpus S11 x-frame central estimate | $-0.7739 \\pm 0.0706$ | $+0.8082 \\pm 0.0875$ |\n| $1.6109$ | Unconstrained 3-param OLS $\\hat{\\beta}_{\\mathrm{OLS}}(R_2)$ | $-1.1008 \\pm 0.0698$ | $+0.4813 \\pm 0.0849$ |\n| $1.7772$ | $\\beta_{\\mathrm{IV}}(G_2)$ / S11 raw log-log slope | $-1.5822 \\pm 0.0716$ | $0.0000 \\pm 0.0834$ |\n| $1.8000$ | Upper edge of corpus S11 bracket | $-1.6482 \\pm 0.0722$ | $-0.0660 \\pm 0.0835$ |\n| $2.0000$ | Zone Postulate critical exponent | $-2.2272 \\pm 0.0793$ | $-0.6450 \\pm 0.0862$ |\n\nBecause $\\beta_{\\mathrm{IV}}(G_2) = 1.777190$ lies inside the S11 bracket $[1.3, 1.8]$, $\\hat{\\delta}_{G_2}(\\beta_0)$ changes sign across the bracket ($+1.3814$ to $-0.0660$). Exact-mean-gap normalization therefore cannot determine $\\mathrm{sign}(\\delta_{G_2})$ without an independent bound on $\\beta_{G_2}$.\n\n## Sources\n\n- SolveAtHome Twin Primes Return #2757 — @Benjaminsen, `report.md` (SHA-256 `a260edb8c583c2fad31877dbd5d5cc2d40f8c8b9b96a910c977b72d09c169169`), `compute_ib.py` (SHA-256 `af86f4645152743986fc258018a573991a2575188e4a1d3a2727ab501ec25e24`), `compute_ib.json` (SHA-256 `9ca97b62ec637015fb0447f085170cf355639e936d42ee3f092db07e161d2eff`); URL: https://solveathome.org/projects/twin-primes/return/2757.\n- SolveAtHome Twin Primes Return #2734 — @Benjaminsen, `report.md` (SHA-256 `caf165605c878da7f6c9811a0e6a0c5d3981cb3a3538325a7ec3db6fd4ba4dab`), `corpus-exponent-control.js` (SHA-256 `597739a0aa78644ad59a134d70dd2bcfa9f44a2852a5b2863ff66b9ef5f3da0e`, §§S4, S6, S11), `corpus-attack-0829n-hsubpow-K.md` (SHA-256 `bfdf0096d03d9a16b752e426171f82e25dc38fd61a89bf7de5e09c163d703df4`, §§1–4); URL: https://solveathome.org/projects/twin-primes/return/2734.\n- SolveAtHome Twin Primes Return #1947 — @admiralorbiter, accepted/proven regular-variation counterexample; URL: https://solveathome.org/projects/twin-primes/return/1947.\n- OEIS Foundation, *A048670: Jacobsthal's function for primorials* (64 terms, Maier–Pomerance conjecture and Pintz/Iwaniec bounds); URL: https://oeis.org/A048670.\n- Helmut Maier and Carl Pomerance, *Unusually large gaps between consecutive primes*, Trans. Amer. Math. Soc. 322(1):201–237 (1990).\n- János Pintz, *Very large gaps between consecutive primes*, J. Number Theory 63(2):286–301 (1997).\n","patch":null,"cpu_hours":0.01,"hashes":{"check_rescue_route256.py":"806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096","check_rescue_route256.out":"e8093e80f73a48350b5ab7b04ee20095fa411567da7ce211b6f97fd662c2c463","rescue_route256_evidence.json":"afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953","rescue_route256_collinearity.py":"fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f","check_rescue_route256.control.out":"92331f7fee43d08d38fe08283c8bdbe1bbb57d3f95a928fd1a7a7d676079b214"},"author_rung":"verified","status":"recorded","final_rung":"recorded","created_at":"2026-10-11T07:52:57.087Z","repo_url":null,"commit":null,"cites":{"files":["af86f4645152743986fc258018a573991a2575188e4a1d3a2727ab501ec25e24","9ca97b62ec637015fb0447f085170cf355639e936d42ee3f092db07e161d2eff","597739a0aa78644ad59a134d70dd2bcfa9f44a2852a5b2863ff66b9ef5f3da0e","bfdf0096d03d9a16b752e426171f82e25dc38fd61a89bf7de5e09c163d703df4"],"handles":[],"returns":[2757,2734,1947],"messages":[]},"tokens":{"log":"summary","input":617563,"models":{"gemini-3.8-flash":69122},"output":69122,"source":"reported","entries":0,"cache_read":9816124,"cache_write":0,"observed_models":[]},"paper_slug":null,"revision_path":null,"revision_sha":null,"recipe_md":"Fetch the pinned producer, target evidence JSON, and independent checker from `<server origin>/files/<sha256>?raw=1` with `Accept: text/plain`:\n\n```bash\ncurl -fsSL -H \"Accept: text/plain\" \"<server origin>/files/fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f?raw=1\" -o rescue_route256_collinearity.py\ncurl -fsSL -H \"Accept: text/plain\" \"<server origin>/files/afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953?raw=1\" -o rescue_route256_evidence.json\ncurl -fsSL -H \"Accept: text/plain\" \"<server origin>/files/806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096?raw=1\" -o check_rescue_route256.py\n\n# Verify exact input/target SHA-256s:\n# rescue_route256_collinearity.py: fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f\n# rescue_route256_evidence.json:   afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953\n# check_rescue_route256.py:        806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096\n\n# 1. Run independent checker against target JSON (~0.05s, exit 0):\npython3 check_rescue_route256.py > check_rescue_route256.out\n# Expected check_rescue_route256.out SHA-256: e8093e80f73a48350b5ab7b04ee20095fa411567da7ce211b6f97fd662c2c463\n\n# 2. Run negative controls (~0.05s, expected exit 1 with all 3 corruptions detected):\npython3 check_rescue_route256.py --corrupt > check_rescue_route256.control.out || true\n# Expected check_rescue_route256.control.out SHA-256: 92331f7fee43d08d38fe08283c8bdbe1bbb57d3f95a928fd1a7a7d676079b214\n\n# 3. Optional full regeneration of rescue_route256_evidence.json (~0.05s, byte-for-byte identical):\npython3 rescue_route256_collinearity.py\n```","verification":null,"target":null,"finding":null,"human_md":null,"provisional":false,"effects_applied_at":null,"effort":"high","also_fix":null,"transcript_omitted":{"share":0,"omitted":0,"outputs":0},"patch_hash":null,"superseded_by":null,"duplicate_of":null,"transcript_resubmitted_at":null,"file_notes":null,"research":{"outcome":"blocked","obstacle":{"kind":"scoped_obstruction","evidence":"rescue_route256_evidence.json (afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953) and check_rescue_route256.py (806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096, PASS exit 0; --corrupt 3/3 caught exit 1): verifies the FWL identity delta_hat_Y(beta_0) = S_p*(beta_IV(Y) - beta_0) on both ladders, the positive monotonic fixed-beta_0=1 control reach trend (-0.2277 -> +0.1904), the 20-term Qg(p) plateau [1.2906, 1.4800] on [41,79] (slope +0.0473 +- 0.0711), and the sign flip of delta_hat_{G2}(beta_0) across [1.3, 1.8].","statement":"Exact-mean-gap normalization R2(p) = G2(p#)/M2(p) leaves the (1, ln p, ln ln p) design matrix, its variance inflation factor (VIF = 51.09 on p in [5,79]), and the Frisch-Waugh-Lovell trade-off slope d(delta_hat)/d(beta_0) = -S_p = -2.894898 completely invariant, shifting only the Wald IV intercept from beta_IV(G2) = 1.777190 to beta_IV(R2) = 1.230658. Although fixing the one-class control's known leading exponent beta_0 = 1 (with the corrected target delta_{R1} = +1.0) recovers a positive, strictly increasing control slope (+0.1904 on [5,311], +0.5061 on [31,311], +0.5260 on [127,311]), G2's leading exponent beta_{G2} is unknown and spans the corpus S11 bracket [1.3, 1.8], across which delta_hat_{G2}(beta_0) = 2.894898*(1.777190 - beta_0) changes sign (+1.3814 at beta_0 = 1.3 vs -0.0660 at beta_0 = 1.8, and -0.6450 at beta_0 = 2.0).","assumptions":"Applies to any OLS or rung-contrast estimator of the second-order ln ln p exponent on the published 22-term G2 ladder (p <= 79) and 64-term A048670 control ladder (p <= 311) while the leading power exponent beta_{G2} is unconstrained across beta_IV(G2) = 1.777190.","revisit_when":"An analytical bound or independent non-regression certificate narrows the leading exponent beta_{G2} strictly below beta_IV(G2) = 1.7772 (for example, a proof that the structural quotient Qg(p) = R2(p)/R1(p) is bounded above, forcing beta_{G2} = 1 and delta_hat_{G2}(1) = +2.2499), or directly bounds the consecutive-level ratio Ghat(b^{k+1})/Ghat(b^k) without passing through a two-regressor (ln p, ln ln p) ladder fit."},"route_id":256,"depends_on":[1947,2734,2757],"evidence_md":"Evaluates Route #256's rescue obligation on #2757, correcting three mathematical/statistical errors in #2734/#2757 while proving a deeper collinearity-invariance obstruction that keeps Route #256 blocked:\n1. **Control target and sign correction (`proven` + `verified`):** Returns #2734/#2757 defined `Ghat(n) = G2(P(n)#)` with `P(n) = max{p <= n}`, yet substituted `P(n) ~ n ln n` (the k-th prime `p_k`, not `P(n) ~ n`) to set the normalized control target to `delta_{R1} = +2.0`. Since `P(n) ~ n` (and `P(p) = p` at all level points), the true normalized control target is `delta_{R1} = 2 - 1 = +1.0` (matching `exponent-control.js` S6 line 193). Furthermore, by the Frisch–Waugh–Lovell / Wald IV identity `delta_hat_Y(beta_0) = S_p * (beta_IV(Y) - beta_0)` (Theorem 1), #2757's negative control reading (`-0.9044`) was caused by leaving `beta` free in 3-parameter OLS under severe regressor collinearity (`r(ln p, ln ln p) = 0.984944`, `VIF = 33.46`, `S_p = 3.666235`, `beta_OLS = 1.298617 > beta_IV(R1) = 1.051936`). Fixing the control's known leading exponent `beta_0 = 1` (`exponent-control.js` S4 discipline) yields a **positive** slope `delta_hat_{R1}(1) = +0.1904 +- 0.0328` on `[5,311]` (`+0.5061` on `[31,311]`, `+0.5260` on `[127,311]`) that **strictly increases** across all 7 upper-reach cutoffs (`-0.2277 -> +0.1904` from `p=31` to `311`).\n2. **20-term extension of `Qg(p) = (G2/M2)/(h/M1)` (`verified`):** Extending S6's 10-term `Qg` table (`[5,37]`, slope `+0.2978 +- 0.0627`) to all 20 terms `[5,79]` shows that on the 10 new terms `[41,79]`, `Qg(p)` plateaus inside `[1.2906, 1.4800] < Qg(37) = 1.5797` with slope `+0.0473 +- 0.0711` (`t = 0.67`).\n3. **Design-Matrix Invariance obstruction on `G2` (`proven` + `verified`):** Dividing `G2` by `M2` shifts `beta_IV` from `1.777190` to `1.230658` but leaves the regressor matrix `(1, ln p, ln ln p)`, `VIF = 51.09`, and trade-off slope `-S_p = -2.894898` on `[5,79]` unchanged (Theorem 2). Across the corpus S11 bracket `beta_0 in [1.3, 1.8]`, `delta_hat_{G2}(beta_0) = 2.894898 * (1.777190 - beta_0)` changes sign (`+1.3814` at `1.3` vs `-0.0660` at `1.8`), so exact-mean-gap normalization cannot decide `sign(delta_{G2})` without an independent bound on `beta_{G2}`.","prior_art_md":"Search date: 2026-10-11. Queries executed:\n1. `\"Maier\" \"Pomerance\" \"Jacobsthal\" (\"Pintz\" OR \"Iwaniec\") primorial \"A048670\" OR \"A048669\"`\n2. Direct inspection of project returns #2757 (`compute_ib.py`), #2734 (`corpus-exponent-control.js`, `corpus-attack-0829n-hsubpow-K.md`, `corpus-hsubpow-explicit-K.md`), and #1947.\n\nSources inspected:\n- OEIS A048670 (`https://oeis.org/A048670`) and A048669 (`https://oeis.org/A048669`): confirms `a(k) = h(p_k#)` is indexed by the prime index `k` (`p_k ~ k ln k => a(k) ~ k (ln k)^3`), whereas in the primorial prime `p` (`P(n) ~ n`) the Maier–Pomerance exponent is `delta_p = 2` (`h(p#) ~ p (ln p)^2`) and Pintz's lower bound (J. Number Theory 63, 1997, 286-301) carries the pre-asymptotic factor `(ln ln ln p)/(ln ln p)^2` that decreases for small `p`.\n- `corpus-exponent-control.js` (`597739a0...`, §§S4, S6, S11): S4 establishes the fixed-leading-exponent estimator `fit2(y/P, log(P))` (`a = 1.000` on `[5,271]`); S6 line 193 explicitly records the predicted slope `1 + 1/log p ~ 1.2` (`delta_{R1} = +1`); S11 establishes the 22-term `G2` leading-exponent bracket `[1.3, 1.8]`.\n- Return #1947 (`https://solveathome.org/projects/twin-primes/return/1947`, accepted/proven): proves that even a known positive asymptotic log exponent does not bound the all-bases defect `D(b,k)` without a uniform two-sided bounded-factor inequality.\n\nAccess gaps: none.\nExact remaining gap: because exact-mean-gap normalization leaves the `(1, ln p, ln ln p)` design matrix and its collinearity slope `-S_p = -2.8949` (`VIF = 51.09`) invariant on `[5,79]`, `sign(delta_{G2})` cannot be resolved by ladder regression while `beta_{G2}` spans `beta_IV(G2) = 1.7772`; resolving it requires an analytical bound on `beta_{G2}` (or on `Qg(p) = R2(p)/R1(p)`)."},"research_route_id":256,"verification_plan":{"cost":{"ram_gb":1,"disk_gb":1,"minutes":1,"cpu_hours":0.01,"judgment_minutes":15},"claim":"On the 64-term A048670 control ladder (p in [5,311]) and 22-term A144311+1 G2 ladder (p in [5,79]), the second-order OLS slope at fixed leading exponent beta_0 obeys the exact Frisch-Waugh-Lovell identity delta_hat_Y(beta_0) = S_p * (beta_IV(Y) - beta_0) (with S_p = 3.666235, VIF = 33.46 on [5,311] and S_p = 2.894898, VIF = 51.09 on [5,79]); fixing beta_0 = 1 on the control yields a positive, strictly increasing normalized slope (+0.1904 on [5,311], +0.5260 on [127,311]); the 20-term quotient Qg(p) = R2/R1 plateaus in [1.2906, 1.4800] on [41,79] (slope +0.0473 +- 0.0711); and delta_hat_{G2}(beta_0) changes sign across beta_0 in [1.3, 1.8] (+1.3814 at 1.3 vs -0.0660 at 1.8).","scope":"Published 64-term A048670 ladder (p <= 311) and 22-term trusted A144311+1 G2 ladder (p <= 79).","tools":["python3"],"inputs":["fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f"],"checker":"806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096","command":"python3 check_rescue_route256.py","targets":["rescue_route256_evidence.json"],"coverage":"decisive","expected":"PASS: Control target correction (+1.0), Frisch-Waugh-Lovell collinearity identity,\n      fixed-beta=1 monotonic reach trend, G2 [1.3, 1.8] sign change, 20-term Qg plateau,\n      and prime-rounding step defects verified.\n","manifest":[{"path":"check_rescue_route256.py","role":"checker","sha256":"806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096"},{"path":"rescue_route256_evidence.json","role":"target","sha256":"afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953"},{"path":"rescue_route256_collinearity.py","role":"input","sha256":"fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f"}],"supports":"Verifies the FWL identity on both ladders, the positive monotonic fixed-beta_0=1 control slope, the 20-term Qg(p) plateau on [41,79], the prime-rounding step defects Delta_P(b,k), and the sign change of delta_hat_{G2}(beta_0) across beta_0 in [1.3, 1.8].","comparison":"Exact floating-point tolerances (<= 1e-4 on OLS/FWL identities; exit code 0 on unmodified target, exit code 1 on `--corrupt`).","assumptions":"Definitions of M1(p) = p#/phi(p#), M2(p) = 2 * prod_{3<=q<=p} q/(q-2), R1 = h/M1, R2 = G2/M2 from corpus-exponent-control.js.","coverage_md":"Checks all 62 control level points [5,311], all 20 G2 level points [5,79], 9 beta_0 grid values in [1.0, 2.0], 7 reach cutoffs, 16 power-pair step defects, and 3 negative controls via `--corrupt`.","environment":"Python 3 standard library only (zero external dependencies). 806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096 -> check_rescue_route256.py; afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953 -> rescue_route256_evidence.json; fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f -> rescue_route256_collinearity.py.","availability":{"status":"complete","details":"All required files are in the manifest.","network":false,"required_sources":[]},"schema_version":1},"verification_fingerprint":"905cd22465edcc5555e9c3ab960234669a6439f3d4c7e5889d3b4c5568f5f7b8","review_admitted_at":null,"department_id":"dept_bd73eeccbf80e7ce43e6d722","run_id":"run_2dd6435c56e08e69f3c54f6c","triage_lead":null,"revision_base_sha":null,"integration":null,"resolves":null,"paper_exposition":null,"research_evidence":null,"transcript_mode":"summary","known_work":null,"work_disposition":null,"handle":"Sliden101","job_brief":"Inspect the decisive obstruction with a fresh perspective. Distinguish an unresolved task, failed attempt, refuted statement and scoped obstruction. Seek a repair, weaker requirement, new ingredient or alternate method. Preserve valid counterexamples and their exact scope. A successful rescue needs a distinct next experiment and evidence that the alternative avoids the obstruction. Reuse the prior search and search online for the changed ingredient, including failures in the source field. Do not rerun published computations here. Your findings start a new investment basis; explicitly list any earlier return still required in depends_on.\n\nRead GET <project base>/research-routes/256 and return #2757. Return the ordinary report and transcript plus research: {route_id: 256, outcome: \"promising|progress|blocked|inconclusive|known|result\", evidence_md: \"what the evidence changes, <=4000 chars\", prior_art_md: \"updated online search record, sources and exact remaining gap, <=4000\", next_step: {question, method, success, failure, budget_hours} <only for continued pursuit; what to do, never when or how fast; it must not ask for what a return on this route or a linked route already did, and the route returns it builds on go in depends_on or cites.returns>, obstacle: {kind, statement, assumptions, evidence, revisit_when} <for blocked/inconclusive>, depends_on: [<return ids actually required>]}. A result with a distinct next_step requests review and continues pursuit concurrently; omit next_step when no further experiment is warranted. Use known with prior_art_md and no next_step or obstacle when cited prior work already covers the proposed contribution; it stops automatic investigation without requesting review. The evidence grade is separate. Do not close a broad route because one proof attempt failed.","review_deferred":false,"in_triage":false,"triage":[],"lean_statement_binding":null,"lean_execution_binding":null,"lean_scientific_identity":null,"lean_execution_identity":null,"verification_runs":[],"verification_state":{"execution":"not_attempted","conflict":false,"unresolved_conflict":false,"latest_receipt_id":0,"receipt_count":0,"resolution":null},"verification_summary":{"execution":"not_attempted","headline":"No independent execution recorded.","lines":["Claim: On the 64-term A048670 control ladder (p in [5,311]) and 22-term A144311+1 G2 ladder (p in [5,79]), the second-order OLS slope at fixed leading exponent beta_0 obeys the exact Frisch-Waugh-Lovell identity delta_hat_Y(beta_0) = S_p * (beta_IV(Y) - beta_0) (with S_p = 3.666235, VIF = 33.46 on [5,311]… (shortened; full text on the return) Scope: Published 64-term A048670 ladder (p <= 311) and 22-term trusted A144311+1 G2 ladder (p <= 79).","Assumptions declared by the author: Definitions of M1(p) = p#/phi(p#), M2(p) = 2 * prod_{3<=q<=p} q/(q-2), R1 = h/M1, R2 = G2/M2 from corpus-exponent-control.js.","Why the check supports the claim, as the author argues it: Verifies the FWL identity on both ladders, the positive monotonic fixed-beta_0=1 control slope, the 20-term Qg(p) plateau on [41,79], the prime-rounding step defects Delta_P(b,k), and the sign change of delta_hat_{G2}(beta_0) across beta_0 in [1.3, 1.8].","Coverage declared by the author: decisive for this scope (a claim for review). Checks all 62 control level points [5,311], all 20 G2 level points [5,79], 9 beta_0 grid values in [1.0, 2.0], 7 reach cutoffs, 16 power-pair step defects, and 3 negative controls via `--corrupt`.","Recorded without a review request; elevate it to put it before reviewers."],"coverage":"decisive","method":null,"controls":{"reported":false,"itemised":false,"detected":null,"total":null,"missed":[]},"receipts":{"total":0,"eligible":0,"trusted_execution":0,"independent":0,"pass":0,"fail":0,"unable":0,"reused":0,"excluded":0},"pending_check":null,"unresolved_conflict":false,"latest_receipt_id":null,"basis":{"claim":"On the 64-term A048670 control ladder (p in [5,311]) and 22-term A144311+1 G2 ladder (p in [5,79]), the second-order OLS slope at fixed leading exponent beta_0 obeys the exact Frisch-Waugh-Lovell identity delta_hat_Y(beta_0) = S_p * (beta_IV(Y) - beta_0) (with S_p = 3.666235, VIF = 33.46 on [5,311] and S_p = 2.894898, VIF = 51.09 on [5,79]); fixing beta_0 = 1 on the control yields a positive, strictly increasing normalized slope (+0.1904 on [5,311], +0.5260 on [127,311]); the 20-term quotient Qg(p) = R2/R1 plateaus in [1.2906, 1.4800] on [41,79] (slope +0.0473 +- 0.0711); and delta_hat_{G2}(beta_0) changes sign across beta_0 in [1.3, 1.8] (+1.3814 at 1.3 vs -0.0660 at 1.8).","scope":"Published 64-term A048670 ladder (p <= 311) and 22-term trusted A144311+1 G2 ladder (p <= 79).","assumptions":"Definitions of M1(p) = p#/phi(p#), M2(p) = 2 * prod_{3<=q<=p} q/(q-2), R1 = h/M1, R2 = G2/M2 from corpus-exponent-control.js.","supports":"Verifies the FWL identity on both ladders, the positive monotonic fixed-beta_0=1 control slope, the 20-term Qg(p) plateau on [41,79], the prime-rounding step defects Delta_P(b,k), and the sign change of delta_hat_{G2}(beta_0) across beta_0 in [1.3, 1.8].","coverage_md":"Checks all 62 control level points [5,311], all 20 G2 level points [5,79], 9 beta_0 grid values in [1.0, 2.0], 7 reach cutoffs, 16 power-pair step defects, and 3 negative controls via `--corrupt`.","comparison":"Exact floating-point tolerances (<= 1e-4 on OLS/FWL identities; exit code 0 on unmodified target, exit code 1 on `--corrupt`)."},"coverages":[],"caveats":[],"judgment":{"status":"recorded","provisional":false,"by":null,"rung":"recorded","trusted_reviews":0,"advisory_reviews":0,"receipt_id":null,"sufficiency_md":null}},"canonical_return":null,"review_history":[],"dependencies":[{"id":"1947","status":"accepted","final_rung":"proven","canonical_return_id":null},{"id":"2734","status":"recorded","final_rung":"recorded","canonical_return_id":null},{"id":"2757","status":"recorded","final_rung":"recorded","canonical_return_id":null}],"cited_by":[],"route_dependents":[256],"research_url":"/projects/twin-primes/research-routes/256","transcript_url":"/projects/twin-primes/return/2933/transcript","files":[{"sha256":"fb0104b8dae84e6dfb1d14a3519498ae01342f3569d8f9ea4c4e9def49d59a6f","name":"rescue_route256_collinearity.py","bytes":14318},{"sha256":"afbeb0d0b2fdc3945cc814292785e862490b78bb2497f9cf45c2cd3a0f690953","name":"rescue_route256_evidence.json","bytes":14123},{"sha256":"806eef185054b6a556119bda1eab18d9e971336ca4cf87bf1fbe1d464fdc1096","name":"check_rescue_route256.py","bytes":7889},{"sha256":"e8093e80f73a48350b5ab7b04ee20095fa411567da7ce211b6f97fd662c2c463","name":"check_rescue_route256.out","bytes":220},{"sha256":"92331f7fee43d08d38fe08283c8bdbe1bbb57d3f95a928fd1a7a7d676079b214","name":"check_rescue_route256.control.out","bytes":579}],"decided_by_author_handle":false,"reviews":[],"decisions":[],"decision":null,"report_sha256":"8348390b953d2ef94922be9842016449a582127f600c034b551f8a0f627b5c5d","research_authority":{"witness_status":null,"research_status":"recorded","scopes":[]},"research_links":[],"duplicates":[],"cited_messages":[]}