# The identification step of lim Var/E, attacked: one open step and not two, its remainder re-split by branch type and modulus band, a mis-doubled group in the record, and the inequality that does not close relocated to friable integers in progressions *Staging note, 2026-08-30. TODO item 9, the "prove either step" move. Producer: `research/history/staging/attack-0830-varE-identification.js` (embedded by `node research/qc/embed.js`; PARTS A-G; no timing figure printed). Reads `varE-theta2-step.md` sections 4-5 for the six-level group columns and `variance-note.md` section 7 for the nine exact points; recomputes none of the nine `Var/E` readings. Every number below is from the producer's embedded block or is cited to the artifact that carries it. Nothing here is on the critical path and nothing here is TPC-strength (section 7).* ## 0. Verdict, the part that did not move first **Neither step closes.** No proof of `delta*(X - X_dec) -> 0` is given, and the one inequality this pass carries the attempt to (section 4) is stated with its quantifiers and left open. `lim Var/E = lambda_2(2) = 0.45546` stays HEURISTIC on exactly the step it rested on before. **The brief's two open steps are one.** The brief asks for "either of the two open steps": the identification half of `varE-spectral.md` and "separately" the theta = 2 mean-coefficient step. They are the same statement. The identification of the true variance with the model is Conjecture 1 of `paper/variance-note.md` section 10, `delta*(X - X_dec) -> 0`, and `varE-theta2-step.md` section 1 proves that `X - X_dec` is exactly the mean-coefficient replacement error written in real space. The record's SECOND open step, the model's own limit theorem, is PROVEN (unreviewed) in `varE-limit-theorem.md` and is not "in varE-spectral.md" as the brief has it: `varE-spectral.md`'s ledger still says "TWO open steps" and has not been updated to cite the sibling. Three more brief errors are logged in section 8. **One record defect, PROVEN by rebuild (section 2).** The `delta*X2` column of `varE-theta2-step.md` section 5, carried into `varE-theta2-proof.md` sections 3 and 7 ("measured `2X2 = 5.3`, flat near 5.3") and `paper/variance-note.md` section 10 ("loose by a factor 14 to 48 against the measured `2X2 = 5.3`"), is not the shift group it names: the producer doubles the positive-`h` half of the `h = 2 (mod p)` pattern, and that pattern is the mirror image of the `h = -2 (mod p)` pattern, not an even function of `h`. The group sum is `2X2 = -0.56, 0.53, 0.27, 0.23, 0.18` at `x = 7..19` (PART F) against the column's `|2X2| = 5.715, 5.628, 5.333, 5.352, 5.327` (`varE-theta2-proof.md` section 3, PART D). Since `X = X1 + 2X2 + Xmix` holds by definition, the corpus's `Xmix` absorbs the difference: `delta*Xmix` is `-0.034971, -0.046368, -0.025439, -0.020542, -0.016835` (this pass) against the record's `-0.241840, -0.133037, -0.076748, -0.053095, -0.038804`. The bound `|2X2| = O(ln y)` of `varE-theta2-proof.md` section 3 is unaffected, since it bounds the group and the group is smaller than the column. No live file is edited; the correction list is section 8. **Nothing here touches the conjecture.** `variance-note.md` section 4 records that `Var/E` over a uniformly random window says nothing about the one anchored window the twin problem needs. Section 7 runs the wrong-direction test anyway. With that said, what the pass produced: - **PROVEN, elementary, checked on 3000 random coprime triples (PART A).** The CRT class of a mixed lag with parts `(n0, n+, n-)` is the Kloosterman fraction `c/n = 2( inv(n0 n-, n+)/n+ - inv(n0 n+, n-)/n- ) mod 1`, and `c = 2 n0 v` with `v^2 = inv(n0)^2 (mod n+ n-)`. With `n- = 1` it is `2 inv(n0)/n+`, the phase of the additive divisor problem (section 3). - **PROVEN, checked on 2000 random `(L,n)` (PART B).** `sum_{c mod n} R_n(c) = 0`: the main term `L w/n` of every lag is the average of its count over the `n` classes, so the whole remainder `X` is a discrepancy of CRT classes against interval bumps, at every modulus, above and below `2L` alike. - **MEASURED, exact, five levels, reproducing the corpus `X` to `6.4e-8` relative or better (PART C).** `X` split by branch type (`c0`, pure shift, two-branch with the `0` branch, two-branch without it, three-branch) and by modulus band (`n <= 2L`, `2L..8L`, `8L..L ln^2 y`, above). The mixed remainder is `-1.06, -2.73, -2.51, -3.23, -3.94` at `x = 7..19`, i.e. `-0.41, -0.71, -0.49, -0.49, -0.49` times `ln y` (PART E): consistent with `O(ln y)` and settling near `-0.49 ln y` on the top three levels. Of it, `82 %` sits at `n <= 2L` at `x = 19` (`0.986, 0.916, 0.844, 0.826, 0.818` across the levels) and essentially all of it in the two-branch type carrying the `0` branch (`-3.94` of `-3.94`; the other two types read `-0.003` and `-0.0002`). - **PROVEN, checked at `x = 7, 11` to `4.2e-17` (PART D).** Above `2L` the mixed count equals the mass of the root measure inside the window bump, `sum w (L - ||c'||_{30n})^+/L`, summed over 16,763,694 (modulus, pattern) pairs at `x = 11`. - **MEASURED, exact (PART G).** The dominant cell (two-branch with the `0` branch, `n <= 2L`) split by balance: the pairs with `min(d, e) <= L^(2/5)` carry `95.7, 93.2, 94.4, 98.7, 99.7 %` of the cell's remainder at `x = 7..19`; the balanced pairs' remainder reads `-0.0087` at `x = 19`. The obstruction is unbalanced, and unbalanced is exactly where no Kloosterman- fraction bound in print reaches (section 4). - **PROVEN given a theorem read at the page (section 5).** Henriot's Corollary 2 (`arXiv:1102.1643v1`, page 7; erratum MPCPS 157 (2014) 375-377 read: "the upper bounds in Theorem 5 and Corollaries 1-2 are not sharp as claimed (although they are still valid)") applies to `F_0(h) F_1(h-2) F_1(h+2)` with its hypotheses met as stated, and gives `sum_{|h| L ln^2 y * phi(y)`, any `phi -> infinity`, contribute `o(ln^2 y)` to both the count and the main term. The open estimate's range shrinks from `n in (2L, L^3)` to `n <= L ln^(2+o(1)) y`. Measured share of the mixed remainder above `L ln^2 y`: `0.014, 0.016, 0.014` at `x = 11, 13, 17`. - **The record's named estimate is not the whole open step.** The divisor- distribution statement of `varE-theta2-proof.md` section 6 is for `n > 2L` only. The band `n <= 2L` carries `82 %` of the measured mixed remainder and the same one-logarithm gap under the kernel bound `|R_n(c)| <= n/L`. A proof of the section 6 display would leave the larger part of Conjecture 1 open. ## 1. The open step, stated once, with what stands on either side **Objects.** `L = x#`, `y` the largest prime `<= sqrt L`, `K_L` the Fejer kernel, `delta = prod_{p<=y} alpha_p/p`. `varE-theta2-step.md` section 1 (PROVEN, six levels): `X = sum_{|h| 0` there is `y_0` such that for every level `x` with `y > y_0`, | X(L) - X_dec(L) | <= eps / delta, equivalently | X - X_dec | <= eps * 3.6 ln^2 y (1 + o(1)). **What stands on each side of it.** | piece of `X - X_dec` | statement | rung | where | |---|---|---|---| | `X1 - X_dec` (`c0` type: `n+ = n- = 1`) | `delta*(X1 - X_dec) -> 0`, rate `O((ln ln y)^{-1/4})` | PROVEN, unreviewed, numerically vacuous at all levels | `varE-theta2-proof.md` sec.4 | | `2X2` (pure shift: exactly one of `n+, n-` nontrivial, `n0 = 1`) | `\|2X2\| <= 60 prod p(p-3)/(p-2)^2 = O(ln y)` | PROVEN | `varE-theta2-proof.md` sec.3; measured group sum `0.18` at `x = 19`, PART F | | `Xmix` (at least two parts nontrivial) | `Xmix = o(ln^2 y)` | OPEN | this note, sections 3-5 | | the model's limit `E[g] -> Pr[GD(2) > u]` | no correction term; bands `O(1/ln y)` | PROVEN, unreviewed | `varE-limit-theorem.md` Thms A-C | So the open step is `Xmix = o(ln^2 y)`, and by PART C it is, at the computed levels, `Xmix = X_{mix2a} + X_{mix2b} + X_{mix3}` with the first of the three carrying all of it. **The record's statement of the open step (`varE-theta2-proof.md` section 6, quoted in `lit-scourfield-2008.md` section 1 with quantifiers):** the weighted count of `y`-smooth divisors `n > 2L` of `C(C^2-4)` over `|C| < L` equals its expected count to `o(ln^2 y)`. That is the `n > 2L` band of `Xmix` only. The quantities `MAIN` and `COUNT` of PART C are exactly that display's two sides, band by band, and PART C reads the `n <= 2L` band of the mixed remainder at `-3.22` against `-0.72` above `2L` at `x = 19`. Both bands are one logarithm short under the kernel bound (`sum_{n<=2L} W_mix(n) n/L` and `sum_{n>2L} W_mix(n)` are both `O(ln^3 y)` by the `4^omega` count), so the `n <= 2L` band is not a smaller problem; it is the same problem at a different scale, `r_n/n` with `r_n = x# mod n` in place of `L/n`. ## 2. The mis-doubled group, and what it changes `varE-theta2-step.js` computes the group `X2` as `[h = 0 term] + 2 sum_{k>=1} (1 - 6k/L)(6 f_5 D_y PM[k] - 6 b_2)` with `PM[k] = prod_{p | 6k-2}(p-3)/(p-4)`: the `h = 2 (mod p)` pattern on `h > 0`, doubled. The pattern function `W_-(h) = 6[6|h] f_5(h) D_y prod_{p|h-2}(p-3)/(p-4)` satisfies `W_-(-h) = W_+(h)`, the mirror pattern, so the doubled half is `sum_{h>0} W_- + sum_{h<0} W_+`, neither group. PART F rebuilds both: | `x` | corpus column `delta*X2` | reproduced by the asymmetric formula | group sum `delta*X2` | `X2c` | `X2` | |---|---|---|---|---|---| | 7 | 0.094199 | 0.094199 | -0.009235 | 2.8574 | -0.2801 | | 11 | 0.047843 | 0.047843 | 0.004509 | 2.8140 | 0.2652 | | 13 | 0.027019 | 0.027019 | 0.001365 | 2.6666 | 0.1347 | | 17 | 0.017012 | 0.017012 | 0.000736 | 2.6760 | 0.1157 | | 19 | 0.011379 | 0.011379 | 0.000394 | 2.6633 | 0.0923 | The group sum equals PART C's pure-type remainder to `1e-6` by an independent enumeration (the per-`h` divisor expansion against the sieve), and PART C's total reproduces the corpus `X` at every level, so the split is right and the column is wrong. Consequences, each a correction owed to a live or HELD file (section 8): `2X2` is not "flat near 5.3", it is `0.18` and falling; `Xmix` is not `-9.78` at `x = 23` ("the truth is of order `ln y`", `varE-theta2-proof.md` section 5), the true mixed remainder is `-3.94` at `x = 19`, still of order `ln y` but with coefficient `-0.49`, not `-1`; and the sentence "`Xmix` the slow half" survives with the halves renamed. `X1`, `X`, `X_dec`, every `X/X_dec` ratio and every `delta*(X - X_dec)` in the corpus are untouched, since none of them passes through the `X2` column. ## 3. Attempt on the identification step: the remainder as a discrepancy of Kloosterman fractions **The reduction, PROVEN (PART A, PART B, PART D).** By PART B, `R_n(c) = phi_n(c) - (1/n) sum_{c'} phi_n(c')` with `phi_n(c) = sum_{h = c (n), |h| (M+N)^{1/4}`. With `A ~ MN/L` (the frequencies the kernel weights) and `MN ~ L` this is `M > L^{2/5}` when `M <= N` [ARITHMETIC, mine, unstamped, first term only]. Two things are NOT done and are owed before any block is called closed: the kernel `(1/n) K_L(nu/n)` couples `nu` to `de` and is not of the form `nu_a alpha_m beta_n`, so it must be separated (Mellin in `nu/(de)`, at a logarithmic cost); and the three-branch type has two coupled inverses (PART A's fraction form) and is not bilinear in any pair of its parts without a further reciprocity step. PART G measures the balanced cell's remainder at `-0.0087` at `x = 19`, so a power saving there is at least consistent with the data. 5. *The unbalanced range, which is the obstruction.* PART G: `99.7 %` of the dominant cell sits at `min(d, e) <= L^{2/5}`, and no bound in print for `e(a inv(m)/n)` with arbitrary coefficients saves anything when one variable is below the `2/5`-power of the other in this configuration. Section 4 states what is needed there. ## 4. The single inequality that does not close Take the dominant cell: two parts `(d, e)`, `d | h`, `e | h -+ 2`, `de in (L^{1-eta}, 2L]`, `min(d, e) <= L^{2/5}`, and say `d` is the small one (the mirror case swaps the roles). Then `h = 2 + e j`, `|j| <= L/e < d L^{eta}`, and `d | h` says `j = -2 inv(e) (mod d)`. For fixed `d`, the cell's remainder is E_d := sum_{e ~ L/d, e y-smooth squarefree, (e, 30d) = 1} lam1(e) [ sum_{j = -2 inv(e) (d), |2 + ej| < L} (1 - |2 + ej|/L) - L/(de) ], the discrepancy of the class of `e` modulo `d` (through its inverse, which permutes the reduced classes) against a bounded-variation window of the `j` line, weighted by `lam1(e) = (1/e) prod_{p|e} p/(p-4)`. Trivially `|E_d| = O(1)` [ARITHMETIC: `|R| <= min(1, de/L)` summed over `e <= 2L/d`], and `sum_{d <= L^{2/5}} lam0(d) = O(ln^2 y)`, so the cell is trivially `O(ln^2 y)` with an explicit constant, and PART G reads it at `-3.27`, i.e. `0.05 ln^2 y`, at `x = 19`. **What would close the cell.** For every `eps > 0` there is `y_0` such that for `y > y_0`, uniformly in `y`-smooth squarefree `d <= L^{2/5} = y^{4/5 + o(1)}` coprime to 30, | E_d | <= eps * (L/d) * sum_{e <= 2L/d} lam1(e)/e * (1/phi-share) (*) that is, the `y`-friable squarefree integers `e` in `(L^{1-eta}/d, 2L/d]`, weighted by `lam1(e)`, are equidistributed among the reduced classes modulo `d` against bounded-variation test functions, to relative precision `o(1)` UNIFORMLY in `d` up to `y^{4/5}`, in the friability range `u = ln e/ln y in (1.2, 2]`. Then `sum_d lam0(d) |E_d| <= eps * O(ln^2 y)` and the cell is `o(ln^2 y)`. The direction is legal: it is an upper bound on the absolute value of a signed discrepancy; nothing lower-bounds a count, and no window is anchored. **Why it does not close here.** It is a level-of-distribution statement for friable integers, at modulus `y^{4/5}` against sampling range `L/d >= y^{6/5}`, with a multiplicative weight and a squarefree-coprimality restriction on top. The owning convention is "entiers friables en progressions arithmetiques" / "smooth numbers in arithmetic progressions", `Psi(x, y; a, q)`, and the Bombieri-Vinogradov-type theorems for it. That convention is NOT the one the corpus searched for this step (`lit-smooth-divisors.md` searched divisors of polynomial values; `lit-scourfield-2008.md` the friable divisor sum), and it was NOT searched at the page in this pass. Candidate sources, from memory and UNVERIFIED (confirm every locator before quoting): Fouvry-Tenenbaum, Proc. LMS (3) 63 (1991); Granville, Acta Math. 170 (1993); Harper, arXiv:1208.5992; Drappeau, Compos. Math. 151 (2015). Whether any of them gives `(*)` with the uniformity in `d` up to `y^{4/5}`, the `1/e`-type weight, and the squarefree restriction is the question item 9 should ask next; the honest prior is that the range is inside what those theorems reach and the weight and restriction are bookkeeping, and that prior is worth nothing until a page is read. **What `(*)` does not cover.** The mirror case (`e` small, `d` large: friable `d` in classes modulo `e`, same shape); the balanced cell (section 3, item 4, owed the kernel separation); the three-branch and `(+,-)` types (measured `-0.0002` and `-0.003` at `x = 19`, PART C, so numerically nothing, but their `o(ln^2 y)` is not derived: the trivial bound is `O(ln^3 y)` for them too); and the `n > 2L` band, `18 %` of the measured remainder, which is the record's section 6 display with its moduli now confined to `(2L, L ln^{2+o(1)} y]` by section 5, hence with the bump width `L/n >= 1/ln^{2+o(1)} y`, a logarithmic scale and not a power scale. ## 5. The toolkit, theorem by theorem, with hypotheses checked The brief's list: Shiu, Henriot, Nair-Tenenbaum, the Erdos-Hooley `Delta` literature, Tenenbaum's book. Every entry below is an UPPER bound for a nonnegative sum; none supplies the cancellation of section 4. One of them applies as stated and earns a lemma. | source | statement as read | hypothesis against our sum | verdict | |---|---|---|---| | Shiu 1980, Crelle 313 | quoted at second hand from Nair-Tenenbaum p.119 (read at the text layer of the Acta Math. PDF, sha256 `3a2caa93...`): `sum_{x= y >= x^alpha`, `q <= y^{1-beta}` (the last range read through OCR damage; not load-bearing) | one multiplicative function of one variable; ours is a product of three functions on three shifted arguments | HYPOTHESIS UNMET as stated; superseded by the next two rows | | Nair-Tenenbaum 1998, Acta Math. 180, Theorem 1 (p.125, read) | `sum_{x= c_0 \|\|Q\|\|^delta`: `<< Delta_{D*} y prod_{g T` and `sum_{n0 | h} n0 lam0(n0) = F_0(h)`, `sum_{n+ | h-2} n+ lam1(n+) = F_1(h-2)`, COUNT(n > T) <= (15 D_y / T) sum_{|h| T) = L sum_{n>T} W_tot(n)/n << L ln^3 T / T, the second by partial summation off `sum_{n<=t} 4^{omega(n)} mu^2(n) << t ln^3 t`. With `T = L ln^2 y * phi(y)`, `phi -> infinity` arbitrarily slowly, both are `o(ln^2 y)`. Every inequality is an upper bound on a nonnegative quantity. [The exponent bookkeeping is mine; the theorem's constants depend on `g, alpha, delta, A, B` only, all fixed here, so the bound is uniform in `y`.] ## 6. Falsifiers, and whether each check has run | claim | rung | falsifier | has the check run | |---|---|---|---| | the mixed class is `2( inv(n0 n-, n+)/n+ - inv(n0 n+, n-)/n- )` | PROVEN | a coprime triple where it fails | YES, 3000 triples, residual 0 (PART A) | | `sum_c R_n(c) = 0` | PROVEN | an `(L, n)` where it fails | YES, 2000 pairs, `1.9e-12` (PART B) | | the type x band table reproduces `X`, `X1`, `X - X1` | PROVEN (exact arithmetic) | a level off the corpus | YES, five levels, worst `6.4e-8` relative (PART C) | | the corpus `X2` column doubles a positive half | PROVEN | the asymmetric formula not reproducing the column, or the group sum not matching PART C | YES, both, five levels (PART F) | | the mixed remainder is `~ -0.49 ln y` | MEASURED, 5 exact levels | growth at a new level | NO beyond `x = 19`; `x = 23` costs the descent to `4.5e8` and was not run | | `82 %` of it is below `2L`, `99.7 %` of the dominant cell is unbalanced | MEASURED, `x = 19` | the shares moving at a new level | NO beyond `x = 19` | | Henriot Cor. 2 applies and cuts the moduli at `L ln^{2+o(1)} y` | PROVEN given the theorem as read at the text layer of arXiv v1 plus the erratum | a hypothesis of Theorem 5 I misread, or the published version differing from v1 in the statement | PARTLY: v1 and the erratum read; the MPCPS published text NOT read | | `(*)` closes the dominant cell | STATED, not proven | nothing yet; it is the target | NO. Not searched in its owning convention | | any of this closes Conjecture 1 | NOT CLAIMED | | the step stands open | | this reduction is novel | NOT CLAIMED | the additive-divisor / sieve-remainder literature stating it | NO. The two-branch cell IS a truncated additive divisor problem, which is a hundred-year-old shape; do not call it new | ## 7. Wrong-direction guard Per `attack-wrongdirection-audit.md` section 1: `Var/E` is a second moment of the count in a UNIFORMLY random window; its limit constrains no single window, so it is Axis-D capacity counting and not `min_x N(x) >= 1`. Every inequality in this note is an upper bound on the absolute value of a signed remainder or on a nonnegative sum, and none lower-bounds a count anywhere. The Kloosterman- fraction shape shared with Lemma V (section 3) is an obstruction family, not a bridge: closing `(*)` gives `lim Var/E`, which `variance-note.md` section 4 already says is not a statement about the anchored window; and Lemma V closing would not close `(*)`, since Lemma V's coefficients are 1-bounded sieve weights on a different index set. Verdict (i) of the audit's scale: strictly weaker than TPC, and it stays there. ## 8. Corrections owed, and brief errors **HOLD for the orchestrator; no file is edited here.** 1. `varE-theta2-step.js` PART 3 and `varE-theta2-step.md` section 5: the `delta*X2 (each)` and `delta*Xmix` columns. Replacement values at `x = 7..19` in PART F and PART C; `x = 23` not recomputed here (the descent is priced above the session, the sieve half is cheap and the fix is one line: sum `PM[k] + PP[k]` in place of `2 PM[k]`). 2. `varE-theta2-proof.md` sections 0, 3, 5, 7: "measured `2X2 = 5.3`", "flat near 5.3", "loose by 14 to 48", "`Xmix = -9.78` against `ln y = 9.611`": the bound holds with more room, the measured group is `O(1)` and falling toward 0, and the mixed remainder is `-0.49 ln y`. 3. `paper/variance-note.md` section 10: "loose by a factor 14 to 48 against the measured `2X2 = 5.3`" and the section 6 display's `n > 2L` restriction presented as the estimate "that would close Conjecture 1": it would close `18 %` of the measured remainder. 4. `lit-scourfield-2008.md` section 1 and `research/SEARCH-CONVENTIONS.md` row 69: the moduli range `(2L, L^3)` (corrected once already to the upper endpoint by `redteam-0829-measure-c.md` E23) is now `(L^{1-eta}, L ln^{2+o(1)} y]` and includes the band below `2L`; the owning convention for the dominant part is friable integers in arithmetic progressions, not divisors of polynomial values. 5. `varE-spectral.md` ledger and section 0: "TWO open steps" should cite `varE-limit-theorem.md` for the second. **Brief errors, each verified at its record.** (a) "either of the two open steps" and "separately the theta = 2 mean-coefficient step": one statement (section 0). (b) "the MODEL half is a theorem ... (`varE-spectral.md`)": the theorem is in `varE-limit-theorem.md`; `varE-spectral.md` says two steps are open. (c) The path `research/varE-theta2-proof.md` is `research/history/staging/varE-theta2-proof.md`. (d) TODO item 9's "residuals <= 0.002 at nine exact points from x = 13" is the Monte-Carlo-model residual; against the exact model the `x = 13` residual is `+0.00291` (`varE-theta2-step.md` section 6, `redteam-0828-varE.md` section 6). Everything else in the brief checked out: the `0.45546` closed form, the two-of-three groups closed, the CRT-mixed reduction, the Scourfield verdict, the `x = 41` sentence. **Channels.** arXiv API answered HTTP 429 on two queries (a claim about the fetcher, per the campaign rules); the arXiv PDF endpoints, Project Euclid's Acta Math. download and Cambridge Core's erratum PDF answered 200 with a browser user agent, so no source is recorded unreachable. Nair-Tenenbaum was read at the text layer of a scanned PDF with OCR damage in the displays; the two statements quoted from it (Theorem 1's fixed-prime-divisor hypothesis, and Shiu's theorem) were checked against Henriot's restatement of both on his pages 1-2, which is typeset and clean.