# The corner correlation at finite dyadic x: a measurement, not a rate **Twin-prime infinitude remains OPEN. The sufficient margin C_2 x + E_dagger(x) >= c_0 x/(log x)^K remains OPEN. Nothing below changes either.** This note records a finite computation. Its calibration is **MEASURED** throughout. A finite ratio at a finite x measures no asymptotic rate, proves no saving and establishes no cancellation; the most it can do is support or undercut a heuristic, and a monotone finite trend is not an asymptotic statement. Producer: [corner-measurement.js](corner-measurement.js), output bound by `node research/qc/embed.js research/corner-measurement.js` and repeated in the artifact [corner-measurement.json](corner-measurement.json). Every figure quoted here is reproduced mechanically from that artifact; the custody-bound copy is the script's OUTPUT block. --- ## 1. The object, exactly Baseline: [corner-correlation.md](corner-correlation.md) §1.4, identity (5), and [RESEARCH-HANDOFF.md](RESEARCH-HANDOFF.md) §3 for the consumer. On the s=s'=1 sub-family of the corner S_0 the residual is, up to the classes corner-correlation (5) discards, \[ K(x)=\sum_{x/2V\\ d_{\mathrm{lo}}Z\\ e_{\mathrm{lo}} Fit the slope s = d log(|K|/mass) / d log x by least squares over the j > whose r' band is non-empty. **If s >= 0 within the spread of the > random-sign control, the measurement gives no heuristic support for > cancellation in the corner correlation at these scales, and this note > says so first.** If s < 0, the reading is "consistent with" at most, > never "implies", and s is compared with the control's own decay: a > random-sign sum of N terms decays like N^(-1/2) relative to its mass, so > **a decay no faster than the control's is not evidence of arithmetic > structure.** Separately pre-registered: the sign of K at each j, and whether K ever falls below -C_2 x, which is the only thing at a finite scale that would bear on the one-sided consumer of handoff §3. Finite scales cannot settle the asymptotic either way, and a finite K > -C_2 x is not a step towards the margin; §2.2 of [corner-correlation.md](corner-correlation.md) records that the one-sided statement, given the complement, *is* the theorem. **Negative controls, all pre-registered and all computed:** 1. shift 4 in place of shift 2 must give a different signed value; 2. replacing mu(n)mu(n-2) by the constant 1 must reproduce the unrestricted mass sum_n L(n)L'(n-2) exactly, and that unrestricted mass must strictly exceed the squarefree-restricted mass somewhere; 3. the 16 seeded draws must have a mean within a few standard errors of zero; 4. a variant with the r band emptied must give K=0 with no contributing n. A fifth check is not a control but an independent reimplementation: at j in {20,22,24} the corner is enumerated exactly over (d,e,k,t) from a full Mobius sieve, and the resulting two-point value must equal the segmented pipeline's K to floating-point tolerance, with the sign identity mu(d)mu(e)=mu(n)mu(n-2) of corner-correlation §1.4 verified term by term. --- ## 3. The degeneracy at reachable x, which is the leading result This is reported before anything else because it limits everything else. The actual right cutoff is Z=floor(x^(1/20)), so the r' band of corner-correlation (5) is the set of primes in (floor(x^(1/20)), floor((x-2)/(E_1+1))]. At every x this machine can reach that interval contains **at most one prime**, and at several j it contains none, so: - the actual-parameter rows are not a measurement of the asymptotic corner. They are a measurement of a sum in which the right prime is a single fixed small prime and L'(n-2) is a fixed multiple of an indicator of one residue class; - the s>1 and s'>1 branches of the corner are unpopulated wherever x^(2eta_0) < 2, which for eta_0=1/100 means every j below 50. The s=s'=1 sub-family and the full corner therefore coincide at most of the parameters reached, and the "share of the full corner mass" question can only be answered where x^(2eta_0)>2, i.e. in the `B-eta40` rows; - `D-scaled` raises v to 1/8 so that the right band is populated. It is a model of the corner's shape and is labelled as such in every table. The exact band contents at each j are in the bound OUTPUT block; the degeneracy is visible there as the `rpband` column. --- ## 4. Verdict against the falsifier **Falsifier verdict, first: the runs give no evidence of arithmetic cancellation in the corner correlation at these scales.** |K|/mass does fall with x in all four live variants, but in none of them does it fall faster than the matched random-sign control, which is the condition the falsifier pre-registered. At the typical j, |K| is of the same size as a random-sign sum on the same support. Per the pre-registered rule, a decay no faster than the control's is not evidence of arithmetic structure, and that is what the runs show. The second disconfirming point, from §3, is prior to the first: the rows carrying the actual cutoffs have one prime in the right band, so what was measured is not the asymptotic corner. Nothing in either reading bears on whether the corner cancels asymptotically. corner-correlation §2.2 prices the absolute target at log^(2+eps) x on this sub-family; a log-power saving of that size is invisible over the factor of at most 1.8 in log x available here, so the measurement could not have detected the target even if it holds. ### 4.1 The slopes against the null Fitted slopes and the matched nulls. `randSd` is the sample standard deviation of the 16 seeded draws; `sqrt(sum w^2)` is the exact standard deviation those draws estimate, and is the sharper null. | variant | pts (j) | slope log(\|K\|/mass) | slope log(randSd/mass) | slope log(sqrt(sum w^2)/mass) | slope log\|K\| | slope log mass | signs of K | any K < -C_2 x | max \|K/(C_2 x)\| | median \|K\|/sqrt(sum w^2) | |---|---|---|---|---|---|---|---|---|---|---| | `A-eta100` | 12 (23..36) | -0.6658 | -0.5145 | -0.5003 | 0.4338 | 1.1 | `-+-+-+--+-++` | false | 0.0000889 | 0.943 | | `B-eta40` | 17 (20..36) | -0.5412 | -0.5188 | -0.5173 | 0.6174 | 1.159 | `-+--++-+---++-+++` | false | 0.000184 | 0.635 | | `C-dyadic` | 17 (20..36) | -0.4872 | -0.4461 | -0.4546 | 0.5083 | 0.9955 | `----+--+--++--+-+` | false | 0.000329 | 0.561 | | `D-scaled` | 16 (20..36) | -0.4962 | -0.545 | -0.5347 | 0.6819 | 1.178 | `+-+-++--++-+----` | false | 0.000088 | 0.499 | | `E-empty` | 0 | n/a | n/a | n/a | n/a | n/a | none | false | n/a | n/a | Band contents at the top of the range and the count of contributing n: | variant | j | V | Z | r band | r' band | available (r,r') pairs | contributing n | mass | K | |---|---|---|---|---|---|---|---|---|---| | `A-eta100` | 36 | 398 | 3 | (398,657] : 41 primes | (3,5] : 1 primes [5] | 41 | 32345791 | 333793000 | 54562.1 | | `B-eta40` | 36 | 398 | 3 | (398,1389] : 143 primes | (3,12] : 3 primes [5,7,11] | 429 | 413855192 | 5617430000 | 125446 | | `C-dyadic` | 36 | 398 | 3 | (398,796] : 60 primes | (3,6] : 1 primes [5] | 60 | 193085514 | 2068540000 | 85558 | | `D-scaled` | 36 | 398 | 22 | (398,657] : 41 primes | (22,37] : 4 primes [23,29,31,37] | 164 | 31130966 | 667635000 | -38174.3 | | `E-empty` | 36 | 398 | 3 | (398,657] : 0 primes | (3,5] : 1 primes [5] | 0 | 0 | 0 | 0 | Exact small-j corner enumeration (independent full-sieve reimplementation): | j | variant | quadruples | occurring (r,r') pairs | s=s'=1 share of full corner mass | non-squarefree share of the s=s'=1 prime mass | two-point value | matches segmented K | |---|---|---|---|---|---|---|---| | 20 | `A-eta100` | 0 | 0 | n/a | n/a | 0 | true | | 20 | `B-eta40` | 6870 | 7 | 1 | 0.2694 | -38.4913 | true | | 20 | `D-scaled` | 355 | 2 | 1 | 0.1531 | 6.16313 | true | | 22 | `A-eta100` | 0 | 0 | n/a | n/a | 0 | true | | 22 | `B-eta40` | 43689 | 11 | 0.9953 | 0.2645 | -363.154 | true | | 22 | `D-scaled` | 5032 | 3 | 1 | 0.146 | 243.658 | true | | 24 | `A-eta100` | 12170 | 5 | 1 | 0.2619 | 984.409 | true | | 24 | `B-eta40` | 228703 | 28 | 0.9858 | 0.2505 | 2033.22 | true | | 24 | `D-scaled` | 9696 | 5 | 1 | 0.09804 | -530.135 | true | Controls: `emptyBand_K_zero`=true; `muReplacedByOne_reproducesMass`=true; `massAllStrictlyExceedsMassSomewhere`=true; `shift4_differs`=true; `shift4_rows`=28; `randMeanWithin4sd`=true; `enumIdentityHolds`=true; `enumTwoPointMatchesDirect`=true; `enumSplitExact`=true; `enumMatchesSegmentedK`=true; `enumRowsChecked`=9. Range run: j=20..36, total wall clock 1393 s. Per-j wall clock at the top of the range: j=33: 77 s, j=34: 160 s, j=35: 346 s, j=36: 704 s. ### 4.2 Significance of the gap to the null Ordinary least squares standard errors from the residuals of the two fits above, computed from [corner-measurement.json](corner-measurement.json) by se(slope) = sqrt( RSS/(n-2) / sum (log x - mean log x)^2 ). **This calculation is scratchpad-grade: it is derived from the bound artifact but is not itself inside the script's output custody.** | variant | n | slope log(\|K\|/mass) | slope of the exact null | difference | difference in standard errors | |---|---|---|---|---|---| | `A-eta100` | 12 | -0.6658 +- 0.1104 | -0.5003 +- 0.0435 | -0.1655 +- 0.1187 | -1.39 | | `B-eta40` | 17 | -0.5412 +- 0.0620 | -0.5173 +- 0.0073 | -0.0239 +- 0.0624 | -0.38 | | `C-dyadic` | 17 | -0.4872 +- 0.0947 | -0.4546 +- 0.0054 | -0.0326 +- 0.0948 | -0.34 | | `D-scaled` | 16 | -0.4962 +- 0.0819 | -0.5347 +- 0.0245 | +0.0385 +- 0.0855 | +0.45 | No variant separates from its null. The largest deviation is 1.4 standard errors, in `A-eta100`, the variant with the fewest points and the most degenerate right band; and the deviations go in both directions across the four variants, which is what noise looks like rather than a signal. The null's own slope is close to -1/2 in every variant, which is the count^(-1/2) behaviour a random-sign sum has when the mass grows like the count: the measured slope of log mass against log x is between 0.9955 and 1.178. ### 4.3 Sign, and the one-sided consumer K changes sign erratically across j in every variant, with no run of one sign longer than four, so no sign bias is detectable. This is consistent with corner-correlation §2.4, which found no structural source of a sign and is not confirmed by a finite run either way. **K is never below -C_2 x.** The largest |K|/(C_2 x) reached is about 3.3e-4, in `C-dyadic`. This is not progress towards the one-sided consumer: corner-correlation §2.2 derives that the one-sided statement, given the complement, is logically equivalent to a quantitative twin lower bound, and a finite scale cannot settle an asymptotic in either direction. It is recorded because the falsifier asked for it. ### 4.4 Controls, and the independent reimplementation All four pre-registered negative controls fire, and the independent full-sieve enumeration at j in {20,22,24} reproduces the segmented pipeline's K exactly (`enumMatchesSegmentedK`), with the sign identity mu(d)mu(e)=mu(n)mu(n-2) of corner-correlation §1.4 verified term by term on every quadruple of the s=s'=1 prime class. That is a genuine cross-check: two different algorithms over different index sets agree. The non-squarefree class of corner-correlation §1.4 carries between 9.8 and 27 percent of the s=s'=1 prime-class absolute mass at these small parameters. That is smaller than the majority share corner-correlation-validation.js reports at its own parameters, and the difference is a parameter effect, not a contradiction: the two runs use different cutoff exponents and different corner cuts. The s=s'=1 share of the full corner mass is 1.000 wherever x^(2eta_0) < 2, because a cofactor k<=2V with a prime power above V dividing it is forced to equal that prime power. Only `B-eta40` at j=22 and j=24 clears that threshold, and there the share is 0.995 and 0.986. So the s>1 branch is essentially unmeasured. --- ## 5. Limitations - **The actual corner is not measured.** §3. The rows carrying the actual cutoffs carry a one-prime right band; the rows with a populated right band carry a different cutoff exponent. No argument is given here that the two have the same shape, and none should be assumed. - **The reachable range is short.** The largest j is recorded in the bound block. A least-squares slope over that range has no error bar that distinguishes a power law from a slowly varying prefactor, and log x varies by less than a factor of two across the fit. - **The mass is a term-wise absolute mass**, matching corner-correlation §1.5, so |K|/mass is a saving against the triangle inequality on the same object the note prices. It is not the ratio the consumer needs. corner-correlation §2.2 prices that at log^(2+eps) x on this sub-family and log^(4+eps) x on the full corner; a finite ratio says nothing about either. - **The random-sign control is the right null and a weak one.** Sixteen draws estimate a standard deviation to about 18 percent. The exact second moment sqrt(sum w^2) is computed alongside and is the sharper comparator; the draws are reported because they were asked for and because they check the accumulator. - **Independence of the 16 draws** comes from 16 bits of one 32-bit mixer applied to n. That is adequate for a null but is not a certified independent stream. - **Cutoffs use floating-point powers of 2.** V, Z, D_1, E_1 are `Math.floor(Math.pow(2, j*alpha))`. At an alpha where 2^(j*alpha) lands within 1e-8 of an integer the floor could move by one. No j in the run is near such a point, but the code does not check it. - **Nothing here is an input to any estimate.** No step of any argument in the corpus may cite a number from this note. --- ## 6. What would change the reading - A run reaching x with at least a few primes in the *actual* band (Z, Z x^(2eta_0)] would make the actual-parameter rows a measurement of the corner rather than of a one-prime shadow. For eta_0=1/100 and v=1/20 the band is (x^(1/20), x^(7/100)], whose prime count by the prime number theorem is about x^(7/100)/(0.07 log x) - x^(1/20)/(0.05 log x): roughly 4 primes at x=2^70 and roughly 17 at x=2^100. Neither is reachable by a sieve over (x/2,x], now or ever. The degeneracy is structural, not a compute budget problem, and it is the reason the actual corner cannot be measured directly at all. - A |K|/mass decay measurably faster than the matched control's, sustained over more than a factor of two in log x, would be the first finite evidence of arithmetic cancellation in this object. It would still be a measurement. - A single j with K < -C_2 x would matter for the one-sided reading at that scale, and would need re-deriving before being believed. - If the enumeration cross-check ever fails, the segmented pipeline is wrong and every number here is void. --- ## 7. Prior art for the numerics The search was run in the owning convention for this object recorded in [SEARCH-CONVENTIONS.md](SEARCH-CONVENTIONS.md), row "the residual with two signed factors and fixed product difference": two-point Mobius correlations, the Chowla conjecture and its averaged and logarithmic forms, and the determinant equation, with the numerics words `Chowla`+`computation`, `Liouville correlations`+numerics and the OEIS twin-count family. In that convention, and per the wave-1 literature scout §E, **no published dataset of weighted Mobius correlations along two linear forms with distinct dilations was located**. A clean negative outside the owning convention would mean nothing; this one is inside it, and it still only says that nothing reusable was found, not that nothing exists. The nearest located item is Luo and Ye, [arXiv:2401.18082](https://arxiv.org/abs/2401.18082), a numerical study of lambda(n)lambda(n+h) and of mu on squarefree n for n <= 10^8 and h <= 1000: unweighted, shift only, leading coefficients 1. It is not this object and supplies no comparison. So the run here is new, and being new is not evidence that it is informative.