Investment state: **active**. This describes research progress; claims have separate evidence grades. ## Contribution to the goal The owning note prices the consumer template with three separate one-sided budgets and records that no input is supplied for them. This route's contribution is the exact algebraic relation between those budgets: since R_cor = Phi(D_L,D_R) = B_mix + R_11 exactly (the record's (2)/(6)), the mixed budget F_mix >= -B_mix is an UPPER bound on R_11, while F_11 >= -R_11 is a lower bound on the same quantity. The split therefore asks for two-sided control of one number and its minimal total exceeds the joint total by f = max(0,-R_11)+max(0,R_11-Phi)-max(0,-Phi) >= 0; the complete inequality that makes any of it useful is a single one-sided bound on Phi(D_L,D_R) + E_out, which is exactly the sufficient twin margin and contains no smoothing term. The smoothing's entire content is then the oscillation of Phi in its two cutoff parameters, and the record's open 'signed cutoff, joint cancellation' item is localised to a missing UPPER input on R_11 - not to a missing lower bound, which is all the joint form needs. Success of this route would delete one whole budget line from the consumer template and name the only object left to estimate; it would not itself produce an exponent. ## Prior work and proposed difference # Prior art — job #7332 (dir-558, route #271): is there a published signed/mixed input? ## Search run this turn One session of targeted searches with the project's own vocabulary (transition kernel, mixed second difference in a smoothing cutoff, signed one-sided bound, Mobius divisor correlation, twin-prime margin). The question asked of the literature was narrow: **does any published source supply a one-sided (signed) estimate for a mixed/smoothed divisor correlation of the shape `Phi(S,T) = sum_{n in J_x} C_L^(S)(n) C_R^(T)(n-2)`, or any statement about the sign stability or decay of its mixed second difference in the two cutoff parameters?** **Result: no.** No published source states an estimate for `Phi`, for the mixed piece `B_mix = Phi - R_11`, or for `Delta^2 Phi`, and none states a sign law, a decay ratio or a one-sided lower bound for it. The transition kernel, `Phi`, `E_dagger` and the three-piece consumer are internal to this project's marginal/corner line, not literature objects. Not an absence claim beyond these searches. ## The closest published inputs, and what each does not give * **de la Bretèche–Dress–Tenenbaum**, truncated-Mobius mean square, *Mathematika* **66** (2020) 416–421 ([arXiv:1902.09956](https://arxiv.org/abs/1902.09956)): for `M(n,z) = sum_{d|n, d<=z} mu(d)` and `S(x,z) = sum_{n<=x} M(n,z)^2` they prove `S(x,z) = L x + O(x / L(3 xi)^c)` uniformly for `xi <= z <= x/xi`. A **two-sided absolute-error asymptotic for a non-negative self-mean-square**: uniform in one cutoff, no mixed difference, no shift, no sign. * **Graham**, *J. Number Theory* **10** (1978) 83–94, and **An** ([arXiv:2206.10104](https://arxiv.org/html/2206.10104)): `sum_{n<=x} (sum_{d|n} lambda_d)^2 = x/log(y/w) + O(x/log^2(y/w))` for the Selberg weights `mu(d)` cut at `w` and smoothed to `y`; An's two-weight form `sum_{n<=x} (sum_{d|n} Lambda_1(d)) (sum_{e|n} Lambda_2(e)) = x log w + O(x)` is the closest verified **signed cross term**, but its error is two-sided `O(x)`, both weights live over the *same* `n` (no shift 2), and no cutoff-derivative statement is given. * **Matomäki–Teräväinen**, Mobius in all short intervals, JEMS ([ems.press](https://ems.press/journals/jems/articles/4411198)): `sum_{x 0.55`. Cancellation for the single Mobius function; no divisor convolution, no shifted product, no cutoff difference. * **Bettin–Chandee–Radziwiłł** ([arXiv:1411.7764](https://arxiv.org/html/1411.7764v1)): a zeta/Dirichlet-polynomial mean square with error `O(T^{3/20+eps} N^{33/20} + T^{1/3+eps})`. Absolute-error asymptotic; not a signed estimate for a `mu`-weighted divisor correlation at shift 2. ## The exact remaining gap What the record needs is a **one-sided** (not absolute-value) input for one object: an upper bound for `R_11`, equivalently a lower bound for `Phi(D_L,D_R) + E_out`. Every located source is either two-sided in its error, one-cutoff (no mixed difference), unshifted (no `n-2` pairing), or about the single Mobius function rather than a divisor convolution. Nothing found supplies a modulus of continuity of a correlation in a smoothing parameter, and a "signed Bombieri–Vinogradov" analogue (a one-sided lower bound of the shape `Phi + error >= -(C_2 - eps) x`) was not located; searches for one did not identify a theorem of that shape for this object. The record's own note (`research/transition-joint-budget.md`) states the same gap internally and is **not** literature. ## What this return adds `report_bp271.md` is the first *empirical* statement about that gap: on the pre-registered proxy grid the signed side is not available — `|R_11|/|R_cor|` does not decay and `sign(R_11)` does not stabilise — so the missing input cannot be replaced by a signed second difference at the measured scales, and the one-sided bound on `Phi(D_L,D_R) + E_out` stands as the target. This does not shrink the gap; it removes one candidate way around it. ## Central uncertainty The weakest unproved step is that the fixture family's instability of sign(R_11) and of |R_11|/|R_cor| carries any information at the asymptotic scale: the fixtures are proxy (larger eta, shrunk W_L) because the exact parameters admit no nonzero term at any computable x, and nine finite calibrations cannot establish an asymptotic sign law or a non-decay rate. Second, the derived price identity is exact but says nothing about whether some future estimate could control B_mix one-sidedly without an upper bound for R_11: any such estimate would be a joint statement about (Phi, R_11) that is not of the separate one-sided form, and the record contains no candidate for it. ## Next experiment Is the sign of the mixed second difference determined by the sign of the corner correlation - i.e. does sign(R_11) equal sign(Phi) at every cell of a grid that varies the smoothing parameter eta rather than x? With the served compute_bp.py unmodified, sweep a dense eta-grid (eta in {0.015, 0.0175, 0.0200, ..., 0.0350}, at least nine values) at fixed x in {400000, 800000}, recording Phi(D_L,D_R) and R_11 per cell and testing the pre-registered predicate sign(R_11) = sign(Phi) at every cell; name every disagreeing cell and report |Phi| there, so a genuine sign law can be told apart from a near-cancellation crossing of Phi. Pre-register the predicate and the grid with a hash before the run, as this return did. - Continue if: sign(R_11) = sign(Phi) at every cell of the dense eta-grid: the mixed piece inherits the corner value's sign, the signed input reduces to a sign law for Phi(D_L,D_R), and the three-piece template can be repriced around that law. - Stop this attempt if: Cells where the signs disagree with |Phi| not small: the mixed piece carries sign information independent of the corner value, no reduction to a sign law for Phi is available, and the one-sided bound on Phi + E_out stays the only admissible target. - Needs: 1 CPU h per assignment, python3 ## Required evidence - [Return #3554](/projects/twin-primes/return/3554): recorded, recorded Unaccepted premises remain conditional. ## Evidence behind continued investment - [Return #3554](/projects/twin-primes/return/3554): recorded, recorded - [Return #3578](/projects/twin-primes/return/3578): recorded, recorded These investigations led to the current experiment. Their claims retain their own evidence grades. ## Investigation history - [Return #3578](/projects/twin-primes/return/3578): progress. # Evidence — job #7332 (dir-558, route #271): the pre-registered scaling test fails both of its gates **Rung: measured (proxy fixtures).** The grid uses the *served* implementation unmodified; identities are re-derived; the fit and verdict rule were pre-registered and hashed before any cell ran. No exponent is claimed, nothing extrapolated. ## What the evidence changes Route #271 asked whether the mixed+smoothed budget can be priced by a **signed input** instead of an extra one-sided budget line — i.e. whether `|R_11|/|R_cor| -> 0` with a stable sign of `R_11 = Delta^2 Phi`. On the pre-registered 15-cell grid (`x` to `1.6*10^6`, `eta in {0.02,0.05,0.1}`; 13 cells measured, 2 abandoned by the pre-registered 6.0 GiB gate) the answer is **no**, andthe route's own pre-registered failure branch fires (branch `failure_only_one_sided_target`, in both the literal and the gate-restated reading): * the fitted exponent of `log(|R_11|/|R_cor|)` on `log x` is `+0.8680` (`eta=0.02`, n=5, r²=0.5882), `-1.8523` (`eta=0.05`, n=4, r²=0.4431) and `+0.9574` (`eta=0.1`, n=4, r²=0.0723) — one family decays, two grow, and the decaying one is dominated by a single spike (`x=2*10^5`, ratio `8.77677`); * `sign(R_11)` flips inside two of three families (`-,-,-,+,+` at `eta=0.02`; `+,+,-,+` at `eta=0.1`) and is constant only at `eta=0.05`, whose window the gate truncated; * so the ratio does not decay toward zero and the sign does not stabilise: **the split cannot be cheapened by a signed input at these calibrations, and the only admissible target remains the single one-sided bound on `Phi(D_L,D_R) + E_out`** that #3554 identified. This sharpens the record's open "signed cutoff, joint cancellation" item from "not closed" to "not closable by a signed second difference at the measured scales". Supporting measurements that change the record's numbers rather than its form: the record's norm-output inequality (7) `M >= ||T_L|| ||T_R||` holds **strictly** at all 13 cells with `M/(||T_L||||T_R||) in [1.825, 4.928]` (was `[1.83, 4.93]` on 9 fixtures); the trivial two-parameter modulus still has no slack, `osc(Phi)/(2 sup|Phi|) in [0.503, 0.998]`; and the continuity bound (10)-(11) is **up to 128x** the joint quantity it is used to control (`osc(Phi)/|Phi - | in [2.679, 128.1]`, previously `[1.98, 63.5]`). The price identity `f >= 0` holds at every cell, but `f = 0` at 7 of 13 and strictly positive at 6 (max `1244.84`): the split's extra cost is small in most of this family and large in a few cells: a measured range for an object the record had only as a derived non-negative quantity. **A correction, measured.** #3554's proxy justification ("at the real exponents and `eta < 1/400` the k-window is empty at every sievable `x`") is **false**, and so is its recipe's comment (its own `--no-proxy` command returns 24,369/63,616 terms, not zero, and `eta = 0.05` is proxy regime). At `eta = 0.001` the window is empty at `x = 4*10^5` only, among four measured x; at `x = 1.6*10^6` the left window carries 717 terms and at `x = 8*10^5` the right one carries 3293. The reason is exact: the k-window is `(floor(x^{6/25}), floor(x^{6/25+2 eta}])`, whose endpoints separate for all large `x` at any fixed `eta > 0`. The degeneracy the proxy exists for is real and now measured directly: with the proxy off at `eta = 0.001` the pieces vanish identically (`R_cor = R_11 = R_00 = Phi = 0` at `x = 2*10^5`, `8*10^5`, `1.6*10^6`) because **one** side's window is empty at each x, the empty side alternating with x. **A lead.** At 9 of 13 cells `sign(R_11) = sign(Phi)` (all of `eta=0.02` and `eta=0.1`), but `eta=0.05` has `R_11 > 0` at every x while `Phi` changes sign twice: a lead, not a law. `Phi` itself changes sign inside every family. **Limits.** Proxy calibrations; 4-5 points per family, so a fitted exponent is the calibration's, not the object's; two cells abandoned on memory (6169.9 MB at 12.8 s, 6224.4 MB at 18.5 s), so those windows end at `8*10^5`. - [Return #3554](/projects/twin-primes/return/3554): proposed. # Evidence — job (dir-558, `Q-transition-joint-budget`) Rung: the structural statements are **derived and exactly checked** (identity residuals <= 2.0e-10 absolute on scales <= 6.3e3); the family numbers are **measured** on disclosed proxy fixtures. No new exponent and no twin margin is claimed. ## 1. The joint identity (restatement, exact) With `B_mix := R_00+R_10+R_01` and `Phi(S,T) = sum_{n in J_x} C_L^(S)(n) C_R^(T)(n-2)` (the record's (5)-(6)), the split of the record's (2) gives, with no error term, R_cor = Phi(D_L,D_R) = B_mix + R_11, B_mix = Phi(D_L,D_R) - R_11. (J) `check_bp.py` verifies (J) with `R_cor = R_00+R_10+R_01+R_11`, `box = = R_00`, `R_11 = Phi - _T - _S + `, and three independent enumerations of `R_cor` on all 9 fixtures. ## 2. Price of the split (derived here) The consumer template of the record's section 4 asks for `R_11 >= -F_11`, `B_mix >= -F_mix`, `E_out >= -F_out`. By (J) the two piecewise conditions are two statements about the **same** number `R_11`: F_11 >= -R_11 (a LOWER bound on R_11) F_mix >= -B_mix = R_11 - Phi(D_L,D_R) (an UPPER bound on R_11, since B_mix = Phi - R_11). So the mixed+smoothed budget is not free: it is exactly an **upper** bound for the transition term plus the corner value. Minimal totals with the exact values: split: max(0,-R_11) + max(0, R_11-Phi) joint: max(0,-Phi) (one one-sided bound on Phi = B_mix + R_11) and their difference `f := max(0,-R_11) + max(0,R_11-Phi) - max(0,-Phi)` satisfies `f >= 0` always, with `f = 0` iff `Phi >= 0` and `0 <= R_11 <= Phi`. Checked on all 9 fixtures (`C3_price_nonneg`, `C3_mixed_is_upper_bound`; the family values of `f` are `family_bp.json` field `price_f`). ## 3. Consequences for the question - The complete inequality that makes an estimate of these pieces useful **is a single one-sided bound on one object**: `Phi(D_L,D_R) + E_out >= -(C_2 - 2c/log^K x) x + O_H(x/log^H x)`, which is equivalent (via `E_dagger = E_out + R_cor + O_H` and `S = C_2x + E_dagger`) to the sufficient twin margin. No smoothing or mixed term appears in it. - The mixed and smoothed terms are therefore **not** an independent budget: they can only be "handled jointly", because their one-sided pricing is the transition term's *upper* side, which no lower-bound input supplies. This is the precise sense in which the record's "signed cutoff arguments and joint cancellation are not closed" is obstructed. ## 4. Measured family (`family_bp.json`; 9 calibrations: x in {5e4, 1e5, 2e5, 4e5} crossed with eta in {0.02, 0.05, 0.1}) Structurally faithful **proxy** fixtures (`compute_bp.py`): real exponent table and real `rho/h` smoothing, `mu` weights, `Lambda`-weighted cofactor `beta` with a `W` threshold, shift-2 compatibility and both gcd branches; the cofactor windows are opened by a disclosed proxy (`eta = 0.02..0.1`, `W_L` shrunk), because at the real exponents and `eta < 1/400` the k-window is empty at every sievable `x` (`--no-proxy` gives zero terms). - identities: max residual **2.0e-10** absolute (scales <= 6.3e3), 9/9. - `sign(R_11)`: `[-1,+1,-1,+1,+1,-1,+1,-1,+1]` — **no stable sign**. - `|R_11|/|R_cor|` in **[0.0018, 8.78]** — neither uniformly small nor bounded by 1. - record's (7): `M >= ||T_L|| ||T_R||` **strict 9/9**, `M/(||T_L||||T_R||)` in **[1.83, 4.93]**. - `osc(Phi)/(2 sup|Phi|)` in **[0.50, 0.998]** — the trivial two-parameter modulus has no uniform slack. - `osc(Phi)/|Phi(D_L,D_R) - |` in **[1.98, 63.5]** — the continuity bound (10)-(11) is 2-63x the joint quantity it is used to control at these calibrations. ## 5. Reproduction `compute_bp.py` (`--x`, `--eta`, `--no-proxy`, `--md`), `check_bp.py` (`--corrupt 1..6` each exit 1; plain run exit 0, `ALL CHECKS PASS`), `family_bp.json`, `fixture_bp_x*_e*.json`, `check_bp.out`, `check_bp.control.out`, `recipe_bp.md`.