# The head residual against the programme's own null: the null settles the direction and misses the rate by a power of ln p, and beta carries the miss STATUS: HELD, staging. Not integrated, not red-teamed. Companion script `research/history/staging/head-residual-null.js` (embedded, 1.6 s, seeded simulation, `--check` green). Its SEC 2 re-sieves [1e7,1e8) independently and reproduces the sibling's E[g], E[n], beta and CV^2 for that window to 1e-3 before it measures anything new; if that control had failed nothing below would stand. Task: the null half of TODO Z4's head question, sitting on top of `research/history/staging/head-residual-factor.md`, whose measured per-window numbers are quoted here from its embedded block and are nowhere recomputed. --- ## 0. What is still open, and what failed **The null does not reproduce the measured residual, and the miss is not subtle.** Under the programme's standard Poisson-Hardy-Littlewood null for twin gaps the residual is h - R = -4 C2 / ln p, h/R - 1 = -8 C2^2 / ln^3 p which is NEGATIVE and one power of ln p faster than the measured +8.4704/ln^2 p (8.4661 under the sibling's own error model; the two refits use the same six points and differ only in how the errors are estimated). That fitted law is local, not a limit law: under beta CV^2 -> 2 with D -> -0.95 the sibling's identity sends h - R to D, so h/R - 1 approaches 0 from BELOW and the coefficient has to change sign. On the sibling's six half-decade points the zero-parameter null sits at chi2/df = 1362 (script SEC 3). Ratio of measured to null runs -14.0, -27.7, -26.5, -36.0, -40.1 up the five decades, widening. **The head's residual is not a renewal artefact** [MEASURED, script SEC 1]. **The composed rate is not separable on this lever arm.** Both null-deviation factors carry O(lnln p / ln p) corrections, so the composed prediction is h - R ~ ((b+2c)/2) lnln p rather than a constant. Over the six half-decades lnln p moves only 15.5%, and h - R = 6.008 constant (chi2/df 1.295) against h - R = 2.126 lnln p (0.887) is not a discrimination [MEASURED, script SEC 3]. The sibling's 8.4661/ln^2 p and a lnln-p-modulated version of it are the same fit at this height. **The variance note's sub-Poisson constant does not transport, so TODO item 9's [0.46, 0.72] is not an input here.** The renewal identity Var N(L) / E N(L) -> CV^2(g) requires an L-plateau. The twin-candidate process on the wheel has none: `paper/variance-note.md` §6 shows Var/E falling monotonically in the window exponent u (0.845 at u = 0.6 down to 0.076 at u = 3), and §2's sum rule forces it to 0 at L = P exactly. So the u = 2 diagonal value 0.3958 of §7, and with it the intercept `varE-asymptotic.md` leaves unpinned across [0.46, 0.72], are a different object from CV^2(g) and neither predict nor bound it [PROVEN that the identity fails, from the note's own §2 sum rule; the numbers are the note's]. **No derivation of the excess was reached.** What follows names the object that carries it and prices it, at MEASURED and HEURISTIC. --- ## 1. Fixing the null [PROVEN, given the stated model] The null R = E[g^2]/(2E[g]) presumes is a **renewal** process on the twin openers, which is what makes R the inspection-paradox functional at all. The matching structural null in this corpus is `zonegap-03-model.md` §3's matched pure-Exp / Kourbatov-Wolf process, exponential gaps at mean ln^2 x / (2 C2). Stated so it also fixes the prime count inside a gap: - primes near p are Poisson at rate lambda = 1/ln p; - a fraction theta = lambda_2/lambda = 2 C2 / ln p of them are twin openers, the marking independent of everything else. Written as a **cluster** process so the prime density stays honest: a cluster is either a twin pair (an opener, two primes 2 apart) or a solitary prime; cluster rate lambda_c = lambda(1 - theta), a cluster is an opener with probability q = theta/(1 - theta), solitary rate lambda_s = lambda_c(1 - q) = lambda - 2 lambda_2. **CV^2_null = 1 exactly.** An independent q-thinning of a Poisson process is Poisson, so the opener gaps are Exp(lambda_2) and CV^2 = 1 with no correction term at all. **beta_null = 2 in the continuum, 1.898 discrete.** Splitting a marked Poisson process leaves the opener and solitary processes independent, so given a gap [a, a+g) the interior solitary primes are Poisson(lambda_s) on it. With n = 2 + K counting the opener a and its partner a+2, E[n | g] = 2 + lambda_s g, so alpha_null = lambda_s and beta_null = 2, where the interior is charged over the full length g. That is the continuum value; the discrete-consistent version, and the convention the measured beta lives in, is 1.898, worked out in the next paragraph. **The sibling's quoted null 2 - 2/ln p puts the two forced primes on a full-density interior** [DEFECT, see §5], which over-counts primes by 2 - 2/ln p per gap, about 14% of E[n] at [1e7,1e8). Thinning the interior to lambda_s over the g - 2 positions the pair does not occupy gives beta = 2 - 2 lambda_bulk = 1.898 in the convention the measured beta uses; 2 exactly is the continuum value, reached by letting the interior run over the full length g. The sibling's 1.887 is off by 0.011, not by 0.113, and the 0.102 between 1.898 and 2 is the same object §3 reading 3 prices as the pair's two slots. The limit is the same either way, so the sibling's §3 conclusion is unaffected. **D_null = 0 exactly.** The interior primes are uniform given their count, so S - n g/2 = g - 2 identically and A = A_forced with no remainder. **Hence the null's residual.** With E[g] = 1/lambda_2, E[g^2] = 2E[g]^2, E[n] = 1/theta, h - R = -2 / E[n] = -2 theta = -4 C2 / ln p h/R - 1 = -2/(E[n] E[g]) = -8 C2^2 / ln^3 p = -3.4865 / ln^3 p. Equivalently beta_null x CV^2_null = 2 exactly, so the two ln p terms of the sibling's identity h - R = [ln p - 4C2/ln p] + D - (beta CV^2/2) ln p cancel identically. **Under the null, h/R -> 1.** That is not what the null buys: the sibling's identity gives h/R -> 1 for any bounded beta CV^2, and what the null adds is the rate, which it gets wrong by a power of ln p, and the sign, which it gets wrong too. All six statements are checked in the continuum convention by direct simulation at 4,000,000 seeded gaps matched to [1e7,1e8) (script SEC 0): CV^2 = 0.9998, beta = 1.9984, alpha = 0.04816 against lambda_s = 0.04817, D = -0.0103, h - R = -0.1424 against -2/E[n] = -0.1497. ## 2. Where the discrepancy sits: beta, by about 5 to 2 and widening [MEASURED] Split the shortfall exactly: 2 - beta CV^2 = (2 - beta) CV^2 + 2(1 - CV^2). Script SEC 1, on the sibling's decade numbers: | window | beta | CV^2 | shortfall | beta's share | CV^2's share | |---|---|---|---|---|---| | [11,1e4) | 1.028 | 0.7436 | 1.2356 | 0.7228 (58%) | 0.5128 (42%) | | [1e4,1e5) | 1.079 | 0.7406 | 1.2009 | 0.6821 (57%) | 0.5188 (43%) | | [1e5,1e6) | 1.310 | 0.8281 | 0.9152 | 0.5714 (62%) | 0.3438 (38%) | | [1e6,1e7) | 1.314 | 0.8520 | 0.8805 | 0.5845 (66%) | 0.2960 (34%) | | [1e7,1e8) | 1.379 | 0.8941 | 0.7670 | 0.5552 (72%) | 0.2118 (28%) | beta is the larger carrier at every window and its share is rising. CV^2 is closing on 1 faster than beta is closing on 2. The 72/28 split is one of two exact decompositions of 2 - beta CV^2; taking the factors in the other order, 2 - beta CV^2 = (2 - 2 CV^2) + CV^2(2 - beta), reads 81/19 at the same window. The rise survives both orders, the split level does not. ## 3. What beta's deficit is, as an object [MEASURED] E[n] = 1/theta is an identity, not a model claim, so beta = 2 - (alpha - lambda_s)/lambda_2 exactly. A beta below 2 is therefore not an abstraction: it is a deficit of Delta = 2 - beta primes near the two openers that bound a gap, paid for by a bulk interior density above lambda_s. Measured: Delta = 0.972, 0.921, 0.690, 0.686, 0.621 up the decades, with the bulk excess +24%, +15%, +9%, +7%, +5%. Script SEC 2 measures the interior-prime density profile against distance from an opener, on the 107,722 gaps of [1e7,1e8) with g >= 300, shells aligned to 30 so the mod-30 house structure averages out inside each shell: | d | left | right | sum | |---|---|---|---| | < 30 | 0.4626 | 0.1134 | 0.5761 | | < 60 | 0.4929 | 0.1557 | 0.6486 | | < 90 | 0.4966 | 0.1421 | 0.6388 | | < 120 | 0.5295 | 0.1942 | 0.7236 | Three readings, all with the caveat that this is band-restricted (28% of the window's gaps) and its baseline lambda_bulk = 0.05101 exceeds the global OLS slope alpha = 0.05079 by 0.4%, so 0.7236 against 2 - beta = 0.621 is agreement to 16%, not an identity. 1. **The deficit is local, not log-divergent.** 0.5761 of 0.7236, 80%, sits within d < 30. The shells beyond contribute 0.0725, -0.0098, 0.0849 (+-0.0053 Poisson), non-monotone, against the steady 0.11 per nat a Montgomery-Soundararajan singular-series-average deficit would give. **The MS shape is not what the profile shows** [MEASURED, and it refutes this pass's own first guess]. The MS bracket does land on the right size, 0.446 for one-prime conditioning and 0.892 for pair conditioning against measured 0.621, but a bracket that wide and a shape that wrong is not a derivation. 2. **The left-heavy column, 0.5295 against 0.1942, is a binning artefact and not a mechanism.** The prime field around a twin pair is symmetric: measured over 381,332 pairs, the prime count per pair at centre-distance +s and -s agrees to 0.70 Poisson sd out to s = 199, 11.9752 against 11.9807 primes per pair [MEASURED, `redteam-0828-head.js`, no model]. Hardy-Littlewood says the same without measuring, the conditional density being exactly reflection-symmetric about the pair centre, and it reproduces the 4.079 ratio at 4.232 out of the binning alone (`head-residual-hl3.md` §4). The two causes are this script's skipped pair slots and the right column's 30-shell running over t = 3..32 against the left's t = 3..29. 3. **About a sixth of it is the two live positions the opener pair occupies.** The statistic above sums lambda_bulk - count_d/gaps over every d, a flat baseline, so the two excluded slots are worth 2 lambda_bulk = 0.1020 against the d < 120 profile total 0.7236, 14.1 percent, and 2 lambda_s = 0.0963 against Delta = 0.621, 15.5 percent [MEASURED, `head-residual-hl3.md` §4, confirmed on independent code]. The per-live-slot conversion 30/8 does not belong here: the continuum null charges 2 units of LENGTH at density lambda_s, not two live slots at the per-live-slot density. Everything in Delta is O(1/ln p) in this reading, so beta -> 2 follows from it, and slowly. Under Hardy-Littlewood the rate is one lnln p slower, O(lnln p/ln p) (`head-residual-hl3.md` §5), and the available lever arm does not separate the two. Neither is a proof: this is the shape of one measured profile at one height, and the residual 20% beyond d = 30 is not accounted for. ## 4. CV^2, and what the variance note can and cannot say [HEURISTIC] CV^2_null = 1 exactly (§1). The measured deficit 1 - CV^2 = 0.2564, 0.2594, 0.1719, 0.1480, 0.1059 is not derived here. The variance note's numbers are the right kind of object at the wrong scale. Gap scale is L = E[g] with sieve level y = sqrt(x), so the window exponent is u = 2 ln E[g] / ln x, which runs 0.846, 0.779, 0.704, 0.642, 0.593 up the five decades and tends to 0 like 4 lnln x / ln x. Evaluated there, §6's empirical shape ln(Var/E) ~ -(0.24 u^2 + 0.13 u) at y = 401 gives 0.7544, 0.7813, 0.8101, 0.8332, 0.8508 against measured CV^2 0.7436, 0.7406, 0.8281, 0.8520, 0.8941. Right size, right drift direction, and CV^2 -> 1 follows from u -> 0. Three reasons this is HEURISTIC and not more: the renewal identity that would license reading a count-variance ratio as CV^2(g) fails for this process (§0); the fit is at y = 401 and the note's own §6 point 2 says the ratio at fixed u drifts upward with y; and the two middle windows sit on the wrong side. ## 5. What this buys, and what it does not The head coefficient still does not derive. The sibling reduced it to CV^2 -> 1 and h/R -> 1; this pass shows both hold **under the null**, exactly, and that the null is wrong about the residual by a factor 40 and a sign at [1e7,1e8). So the reduction survives but the null cannot close it, and the object left standing is beta's endpoint deficit: local, symmetric about the pair once the binning is fixed, about a sixth of it priced by the discreteness the continuum null drops, and O(1/ln p) here against O(lnln p/ln p) under Hardy-Littlewood. Nothing in this file touches Z2, and nothing in it bears on the conjecture. ## Defects noticed in passing - `head-residual-factor.js` SEC 2b's stated null "beta = 2 - 2 lambda (the two forced left-end primes on an interior Poisson process of rate lambda)" is not density-consistent: that model has E[n] = lambda E[g] + 2 - 2 lambda, which over-counts primes by 14% at [1e7,1e8). The density-consistent value in that script's own convention is 2 - 2 lambda_bulk = 1.898, and 2 exactly is the continuum value; the script's 1.887 is off by 0.011. Its §3 conclusion is unaffected because every version tends to 2. - `head-residual-factor.md` §5 calls CV^2 -> 1 "the standard Poisson-limit statement for twin gaps" and §3 calls it "a Hardy-Littlewood consequence". Neither is cited to a source in the corpus, and this pass did not find one; it is CONJECTURED here, not PROVEN. - `paper/variance-note.md` §6's table is at fixed y = 401 while §6 point 2 records that the ratio at fixed u drifts with y. Any use of the (c_1, c_2) fit away from y = 401, including §4 above, is an extrapolation the note does not license and does not flag. - `research/GLOSSARY.md` still has no entry for **opener**, **head** or **tail**, all three load-bearing here and in Z4. The sibling flagged the last two. --- ## What would falsify this, and whether that check has run | Claim | Rung | Falsifier | Run? | |---|---|---|---| | beta_null = 2, CV^2_null = 1, D_null = 0, h - R = -2/E[n], all exact in the continuum convention | PROVEN (given the stated null) | the seeded simulation missing any of them | YES, 4e6 gaps, all five inside tolerance. The discrete-consistent beta_null is 1.898, not 2 | | the null's h/R - 1 = -8 C2^2/ln^3 p is refuted by the data | MEASURED | the null fitting the six half-decade points | YES, chi2/df = 1362, zero parameters, and the sign is wrong at every window | | beta carries 72% of the shortfall at [1e7,1e8), rising | MEASURED | the split reversing at greater height | NO. Five decades only, the split is a decomposition of one identity rather than two independent measurements, and the other exact order of the same identity reads 81/19 at that window | | Delta = 2 - beta is an endpoint prime deficit, localized within d < 30 | MEASURED | the profile flattening only past d = 120, or the deficit continuing to accrue at 0.11 per nat | PARTIAL. One window, one band (g >= 300), d <= 120 only; 20% of the total is unexplained beyond d = 30 | | the deficit is not the MS singular-series-average shape | MEASURED | shell contributions turning steady at 0.11 per nat over a longer d range | PARTIAL, d <= 120, and the non-monotone shells are not the evidence: Hardy-Littlewood's own shells are non-monotone too and correlate with the measured ones at r = 0.9518 with the two dominant shells dropped (`head-residual-hl3.md` §3). Only the accrual RATE separates them | | about a sixth of Delta is the opener pair's two live positions | MEASURED (from the statistic's own flat baseline) | the accumulator using a per-live-slot baseline after all | YES. 2 lambda_bulk = 0.1020 and 2 lambda_s = 0.0963, 14 to 16 percent; the earlier [0.289, 0.361] bracket was the per-live-slot conversion applied to a flat-baseline statistic | | the variance note's u = 2 number does not bound CV^2(g) | PROVEN (from the note's own §2 sum rule) | a renewal-plateau in L for the twin-candidate process | YES, by the sum rule: Var/E -> 0 at L = P | | note §6's shape at u_gap predicts CV^2 | HEURISTIC | the prediction missing beyond its two-window spread at 1e12 | NO. Two of five windows already sit on the wrong side | | beta -> 2 and CV^2 -> 1, hence beta CV^2 -> 2 and h - R bounded | CONJECTURED | Delta or 1 - CV^2 stalling above 1e8 | NO. Same 1e12 pass the sibling queues; this file adds beta's profile as a second thing to measure there. h/R -> 1 itself needs only that beta CV^2 stays bounded | The cheap check not run, and it is the sibling's: the same decomposition at 1e12, which would take the lever arm from ln p <= 17.9 to <= 27.6. Two additions this file wants from it: the endpoint profile of §3 at that height (does Delta fall like 1/ln p?), and the shell contributions past d = 30 over a longer range.