# The Maier matrix on the twin-slot set: what it gives, and exactly where it stops *(2026-08-17. Question from Chris: can the Maier matrix transfer our exact residue-level knowledge into an interval statement about twin slots, and if not, where does it stop? Code, real output and calibration below. Marked PROVEN, VERIFIED by exact computation, MEASURED, REFUTED.)* ## 0. The hard part first **The matrix works perfectly on our set, with every error term equal to zero, and that is exactly why it yields nothing.** Maier's method is a device for converting an approximate statement about arithmetic progressions into a contradiction with an approximate statement about intervals. It runs on the *mismatch* between two independent predictions. Our AP information is not approximate, it is exact, so there is no mismatch to exploit; and we have no second, independent prediction for the interval side, because supplying one is the Zone Postulate itself. Chris's doubt: *"I expect this yields almost-all statements and therefore does not reach the origin."* The doubt is right, and the situation is worse than almost-all. For our set the matrix does not even reach an almost-all statement. It reaches a **first-moment identity**, one exact mean over an ensemble, and nothing more. Almost-all would require a second moment, and the second moment is the localized-gap problem the repo already has open. A clean negative, as asked. But the negative comes with a complete classification (section 6) which is sharper than "it does not work": we can say what the matrix produces for *every* choice of its parameters, because the whole parameter space collapses to the single integer d = gcd(q, x#), and for each d the output is one exact mean. --- ## 1. Setup, done honestly Fix x, let W = x# and let > **A = { n : gcd(n(n+2), W) = 1 }**, the twin-slot set of T_x, viewed as a set > of integers rather than of residues. A is exactly W-periodic, and its census in one period is the repo's D_x = ∏_{3≤q≤x}(q−2), confirmed by direct enumeration in the code (D_13 = 1485, D_19 = 378,675, D_23 = 7,952,175, matching `GLOSSARY.md`). The Granville-Soundararajan matrix (gs.txt line 542) has (r,s) entry **(R+r)q + sℓ**, summed by rows (short intervals inside an AP mod ℓ) and by columns (APs mod q). We take **ℓ = 1**, which is Maier's own configuration: rows become honest short intervals of length S, columns become APs to modulus q. Two parameter choices matter and we examine both. - **q = W = x# itself.** Chris's suggested choice, and the one that makes every row a translate of the same tile window. - **q = y# for a coarser level y < x.** The choice that makes the matrix non-degenerate, and the only one that does. We also drop GS's restriction of the entries to [x/4, x]. They need it so that A(y)/y varies slowly; our A is periodic, so we sum over one exact period and the origin is a legitimate row. **That is already a gain over the classical setup**, and it is worth naming because it is the only place where exactness buys anything at all: Maier and GS structurally exclude the origin, and we do not have to. --- ## 2. q = x#: the matrix is the identity 0 = 0 (PROVEN, VERIFIED) **Proposition.** If q is a multiple of W, then every row of the matrix contains exactly D(S) := #{s ∈ [1,S] : s ∈ A} elements of A, and column s contains R elements if s ∈ A and 0 otherwise. Both summations return R·D(S). *Proof.* Row r is the translate of [1,S] by (R+r)q, an exact multiple of the period. Column s is the set {(R+r)q + s}, all congruent to s mod W. ∎ Real output: ``` === A. column modulus q = x#: the matrix is a tautology === x= 7 W= 210 S=40 R=6 rowsum=18 colsum=18 rows distinct=3 col values in {0,6}: true live cols=3 head=3 x=11 W= 2310 S=40 R=6 rowsum=12 colsum=12 rows distinct=2 col values in {0,6}: true live cols=2 head=2 x=13 W= 30030 S=40 R=6 rowsum=12 colsum=12 rows distinct=2 col values in {0,6}: true live cols=2 head=2 ``` `rows distinct=3` means all six rows carry the same count, 3. The two sums agree by construction, not by an argument. **This kills the natural first idea immediately.** Chris's suggestion was that with q = x# "the SLOT count is identical in every row while the PRIME count is not". That is true, and it is the crystallization argument seen from the other side, exactly as he said. But it also means the matrix has no content for slots: the invariant Maier exploits is an *approximate* invariance which he plays against an *exact* count. Here both sides are the same exact count. The prime count does vary across rows, but the matrix gives no access to it, because A is the slot set; to see primes you would need the row-by-row count of A ∩ primes, which is free below x′² by crystallisation and is the twin prime problem itself above x′². Either way it is not an output of the matrix. --- ## 3. q = y# with y < x: the exact transfer identity (PROVEN, VERIFIED 20/20) Now let y < x, put q = W_y = y#, W = W_x = x#, M = W_x/W_y = ∏_{y **Transfer identity (PROVEN).** For every S ≤ y#, > > **Σ_{r=0}^{M−1} #(A ∩ [r·y#, r·y# + S)) = D_y(S) · N**, > > where D_y(S) = #{twin slots of T_y in [0,S)}. Equivalently the mean count over > the M windows is > > **mean = D_y(S) · ∏_{y` marks a result block rather than an attributed quotation. > **mean over the ensemble / naive = ρ_y(S) := [D_y(S)/S] / [D_y/W_y]**, the local-to-global density ratio of the **coarser** tile in the same window. This is Maier's mechanism exactly, with all error terms zero. ρ_y(S) is the two-dimensional survival function, a function of u = ln S / ln y and of nothing else (`FOLD-PROFILE.md` §9a), and its oscillation away from 1 is the Buchstab oscillation Cheer and Goldston computed and Maier used. ``` rho_y(S) = local density of T_13 in [0,S) over its global density (y#=30030, D=1485, y'^2=289) S= 169 u=lnS/lny=2.000 D_y(S)= 9 rho=1.0769 S= 289 u=lnS/lny=2.209 D_y(S)= 16 rho=1.1196 S= 500 u=lnS/lny=2.423 D_y(S)= 25 rho=1.0111 S= 1000 u=lnS/lny=2.693 D_y(S)= 47 rho=0.9504 S= 5000 u=lnS/lny=3.321 D_y(S)= 246 rho=0.9949 S= 30030 u=lnS/lny=4.020 D_y(S)= 1485 rho=1.0000 rho_y(S) = local density of T_19 in [0,S) over its global density (y#=9699690, D=378675, y'^2=529) S= 361 u=lnS/lny=2.000 D_y(S)= 17 rho=1.2062 S= 529 u=lnS/lny=2.130 D_y(S)= 21 rho=1.0168 S= 2000 u=lnS/lny=2.581 D_y(S)= 75 rho=0.9606 S= 20000 u=lnS/lny=3.363 D_y(S)= 782 rho=1.0015 S= 200000 u=lnS/lny=4.145 D_y(S)= 7801 rho=0.9991 S= 9699690 u=lnS/lny=5.464 D_y(S)= 378675 rho=1.0000 ``` So the matrix does deliver the analogue of Maier's theorem for our set, by pigeonhole from the mean: **since ρ_y(S) ≠ 1, there exist windows of width S where the twin-slot density is above its global mean and windows where it is below.** Unconditional, exact, and at every window scale. And it is worth nothing, because for a periodic set that statement is a tautology restated: "the tile's head is denser than its bulk" is something we can read off by enumeration at any level. Maier's theorem is hard for primes because the primes are not a periodic set and the head-versus-bulk comparison is not available by inspection. Ours is available by inspection. The matrix converts an observation into itself. --- ## 4. The one thing the origin genuinely has: the Origin Excess Lemma The origin is row 0, and it is not a typical row. It has a provable advantage, and the advantage has an exact expiry date. > **Origin Excess Lemma (PROVEN, VERIFIED 14/14).** Let y < x, let y′ be the > least prime above y, and let **S ≤ y′²**. Then > > **#(A ∩ [0,S)) = D_y(S) − L**, where L = #{twin slots t of T_y with t < S > such that t or t+2 is struck}, > > and a y-rough integer m ≤ y′² is struck by a prime in (y, x] **only if** m is > itself a prime in (y, x], or m = y′² with y′ ≤ x. > > *Non-vacuity, a remark and not a hypothesis:* the conclusion has content only > when **y′² > x**. See item 2 of the enumeration below, and never quote the box > without it. *Proof.* Write m = q·k with q prime in (y,x]. Then k is y-rough, so k = 1 or k ≥ y′; and q ≥ y′. So either m = q, or m ≥ y′², with equality only when q = k = y′. ∎ So at the origin the later folds remove essentially nothing. Each struck value m can cost at most two slots (t = m and t = m − 2), so **L ≤ 2(π(x) − π(y) + 1)**, a count of primes, against D_y(S) which grows with S. Measured L is 0, 1 or 2 across all fourteen cells below. This is the Head-calm lemma of `ZONE-POSTULATE.md` §6 in the two-level form. The lemma predicts the count exactly, 14 of 14. The m = y′² boundary case is load-bearing: dropping it fails 4 of the 14 cells. ``` === F. the Origin Excess Lemma, S = y'^2 === y x S=y'^2 D_y(S) L(pred) origin(pred) origin(true) ok mean ratio 7 11 121 8 1 7 7 YES 6.545 1.069 7 13 121 8 1 7 7 YES 5.538 1.264 7 17 121 8 2 6 6 YES 4.887 1.228 7 19 121 8 2 6 6 YES 4.372 1.372 11 13 169 10 1 9 9 YES 8.462 1.064 11 17 169 10 2 8 8 YES 7.466 1.072 11 19 169 10 2 8 8 YES 6.680 1.198 11 23 169 10 2 8 8 YES 6.099 1.312 13 17 289 16 1 15 15 YES 14.118 1.063 13 19 289 16 1 15 15 YES 12.632 1.188 13 23 289 16 1 15 15 YES 11.533 1.301 17 19 361 18 1 17 17 YES 16.105 1.056 17 23 361 18 1 17 17 YES 14.705 1.156 19 23 529 21 0 21 21 YES 19.174 1.095 ``` ### 4a. The canonical enumeration, by logical type Three further statements travel with the lemma and are easy to miscount as hypotheses of it. They are not, and the difference decides what each one needs and where it can be used. **This table is the canonical enumeration; cite it rather than keeping a private count.** `origin-excess.md` §6 carries the derivations, one subsection per companion. | # | statement | logical type | what it needs | derivation | |---|---|---|---|---| | 1 | **Origin Excess Lemma**, PROVEN, VERIFIED 14/14 | the lemma, and it has exactly one hypothesis | S ≤ y′² | this section | | 2 | **the non-vacuity condition y′² > x** | a remark on when the conclusion has content, not a hypothesis | nothing; the identity holds regardless | `origin-excess.md` §6a | | 3 | **Origin Excess Corollary**, PROVEN | a corollary, carrying the advantage and its expiry | x < x\*(y) for the guarantee to be positive | `origin-excess.md` §6b | | 4 | **Scale Collision Proposition**, PROVEN | a theorem about the parameters, independent of the lemma | y < x with x prime | `origin-excess.md` §6c, and §5 below | **1. What the one hypothesis is for.** The identity #(A ∩ [0,S)) = D_y(S) − L is bookkeeping, because L is *defined* as the struck count, and it holds outside the stated range too: VERIFIED at 3 of 3 cells with S > y′². The content is the strike characterisation, and S ≤ y′² is what that needs. It is not decorative: the bound L ≤ 2(π(x) − π(y) + 1) which the characterisation yields fails in 6 of the 8 test cells with S > y′², at y = 7, x = 19, S = 1210 by L = 43 against a bound of 10. **2. The conclusion is empty unless y′² > x.** An x-rough integer below x is 1, and 1 is not a slot, so D_x(S) = 0 for every S ≤ x and at S = y′² ≤ x the lemma reads 0 = 0. This is a remark rather than a hypothesis precisely because the lemma is *true* there, just empty, and every one of the fourteen verification cells above satisfies it. **3. The advantage, its ceiling, and where it expires.** The `ratio` column is the origin's excess over the ensemble mean, exactly (1 − L/D_y(S)) / ∏_{y **the origin beats the ensemble mean by a factor of at most about 2.2, for > S ≤ y′², and the measured maximum over the fourteen cells is 1.372.** The ceiling is MEASURED in one input, the flatness of ln x\*/ln y over two orders of magnitude in y, and PROVEN from there. The measured maximum is near the ceiling, not a small-sample artifact. The advantage is real, it is unconditional, it is origin-specific, and it is small. **4. The regimes never meet, and no better lemma repairs it.** y < x with x prime gives y′ ≤ x < x′, hence **y′² < x′² strictly**. That is a statement about the two parameters rather than about this construction, so it is independent of the lemma and survives any improvement to it. It is the headline of §5. The exact form of that ratio column, which is what should be quoted rather than the asymptotic factor, is the **Survival Quotient Identity** of `origin-excess.md` §2: origin(S)/mean(S) = ρ_x(S)/ρ_y(S), with no error term, for every y < x and S ≤ y#. The origin's excess is not a property of the origin at all; it is the one survival curve of §3a read at two levels. That is exactly the kind of origin-specific advantage route B of `ZONE-POSTULATE.md` §6 asks for. It is also, as the next section shows, unusable. --- ## 5. Where it stops: the scale collision (the headline) The Origin Excess Lemma needs **S ≤ y′²**. The Zone Postulate at level x needs **S = x′²**. Putting these together requires > **x′² ≤ S ≤ y′², hence y ≥ x.** But the matrix is non-degenerate only for y < x, and at y = x it is the tautology of section 2. **The regime in which the matrix carries information and the regime in which the origin is distinguished are disjoint, and they touch only at the degenerate matrix.** **And the collision needs no matrix at all.** If y < x with x prime then y′ ≤ x < x′, so y′² < x′² strictly, in one line. The disjointness is a fact about the two parameters and not a property of this construction, so no better lemma repairs it (`research/origin-excess.md` §6c). This is not a heuristic. It is measured directly. As S passes y′² the origin loses its advantage and drops into the bulk of the ensemble: ``` y=13 x=23 M=7429 rows y'^2=289 x'^2=841 S mean origin min max #empty origin rank (1=largest) origin>=mean? 169 6.487 8 3 9 0 163 / 7429 yes 289 11.533 15 7 15 0 1 / 7429 yes 361 13.696 17 9 17 0 1 / 7429 yes 1000 33.879 32 28 40 0 5752 / 7429 NO 5000 177.323 177 168 189 0 3522 / 7429 NO 30030 1070.423 1075 1055 1087 0 1036 / 7429 yes y=11 x=17 M=221 rows y'^2=169 x'^2=361 121 5.226 6 3 7 0 16 / 221 yes 169 7.466 8 5 10 0 28 / 221 yes 500 22.398 22 19 26 0 104 / 221 NO 1000 42.557 39 38 47 0 217 / 221 NO 2310 100.792 99 95 104 0 182 / 221 NO ``` At S = y′² the origin is the single largest row out of 7429. At S = 1000, about 3.5 times y′², it is the 5752nd largest and sits **below** the mean. **The advantage does not decay gracefully into the zone. It reverses.** The ratio bottoms out at 0.9343 at S = 3.53 y′² and only then relaxes to 1.0043 at the full period, and per prime the origin's loss reaches 125% of fair share at eight times its threshold (`research/origin-excess.md` §3, §4). The head excess is paid back on the shoulder, and any construction that widens the window past y′² collects the repayment. **REFUTATION.** The hope that the origin row could be shown to dominate the ensemble at the zone's own width, so that the ensemble mean would lower-bound it, is false. At the width the Zone Postulate actually asks for, S = x′², the origin is an ordinary row: ``` === E. the row width the Zone Postulate actually asks for: S = x'^2 === y=11 x=17 M=221 rows y'^2=169 x'^2=361 S mean origin min max #empty origin rank (1=largest) origin>=mean? 361 15.679 18 12 19 0 3 / 221 yes y=13 x=19 M=323 rows y'^2=289 x'^2=529 529 21.316 21 19 24 0 135 / 323 NO y=13 x=23 M=7429 rows y'^2=289 x'^2=841 841 29.554 30 24 36 0 2254 / 7429 yes y=17 x=23 M=437 rows y'^2=361 x'^2=841 841 27.776 30 24 32 0 16 / 437 yes y=19 x=23 M=23 rows y'^2=529 x'^2=841 841 27.391 30 25 30 0 1 / 23 yes ``` (Full sweep of ten (y,x) pairs in the run log; the five above are the ones with M large enough for the rank to mean anything. The pair y=7, x=13 is skipped because x′² = 289 exceeds y# = 210.) The rows where the origin still ranks near the top are exactly the rows where y is close to x, that is where x′² is still close to y′², which is the collapsing regime again. At y = 13, x = 19 the origin is genuinely below the mean. **The ceiling, stated with no y in it at all.** Take the ensemble of all x# translates, whose mean is S·δ_x. Then origin/mean at S = x′² is exactly ρ_x(x′²) = ρ(2), and ρ has its minimum at u = 2. Measured to x = 6037 and divided by the window's own finite-size Hardy-Littlewood factor, that ratio reads 0.79303, 0.79475, 0.79668, 0.79863, 0.79922 against > **e^{2γ}/4 = 0.79305.** So at the zone's own width the origin carries **21% less** than the tile's mean density, not more (`research/origin-excess.md` §5). That constant is the same ρ(2) that `FOLD-PROFILE.md` §9 calls the Unification Law's trough constant and that `GLOSSARY.md` records as the conjectured limit of β; the origin ceiling is its fifth independent appearance in the repo, and the four earlier ones reached it from other directions. Nothing about the matrix is needed for it. ### 5a. The circularity, stated once so it is not rediscovered There is a second, independent reason the useful regime is empty. In the regime S ≤ y′² where the Origin Excess Lemma applies, the identity's *input* is D_y(S) = #{twin slots of T_y below S}, and by crystallization every such slot below y′² is a genuine twin prime. So the input is the twin prime count below y′², and the output is the twin prime count below y′² minus a handful of primes. **Input and output are the same quantity.** This is the same degeneracy the A6 refutation found for the first-slot recursion (`ZONE-POSTULATE.md` §6, update 2026-08-16): at the origin, the fold structure is frozen and every exact relation reduces to the twin prime sequence describing itself. --- ## 6. The complete classification (PROVEN, VERIFIED) The above is not a report on two choices of q. The whole parameter space collapses, and this is the sharpest form of the answer. > **Proposition.** In the matrix with entries (R+r)q + s, the row positions > (R+r)q reduce mod W = x# to the multiples of **d = gcd(q, W)**, and as r runs > over a full period each multiple of d is hit equally often. Hence for the > W-periodic set A the matrix returns, for every q, exactly one quantity: the > **mean of #(A ∩ [t, t+S)) over the W/d windows at t ≡ 0 (mod d)**, which is > Σ_{s *"Suppose that (1.7) holds for some α ≥ 60 log log log x/ log log x and set > η = min(α/3, 1/100). Then for each 5/η² ≤ u ≤ η(log x)^{η/2} at least one of > the following two assertions holds …"* **η is capped at 1/100 by construction**, for every sequence, regardless of how good α is. So **u ≥ 5/η² ≥ 50,000 always**, and the upper constraint u ≤ η(log x)^{η/2} then forces (log x)^{1/200} ≥ 5·10⁶, that is log x ≥ (5·10⁶)^{200}, roughly x > exp(10^{1340}). The corollary is vacuous at every scale anyone will ever compute at, and at u = 2 it is vacuous at every scale whatsoever. **Corollary 1.4 is not an obstruction to anything in this repo.** Two further points, both new: - For our set, condition (1.7) is satisfied with α = 1 + o(1). Here h(p) = 0 for p ≤ x and h(p) = 1 for p > x (by CRT the twin slots are exactly equidistributed in APs to moduli coprime to W). So if the tile level x is at least log X, where X is the range, then Σ_{p≤log X} (1−h(p)) log p / p ~ log log X and α = 1. Our set is a legitimate GS sequence. The corollary simply never activates. - More interestingly, our set **satisfies the uncertainty principle explicitly and on the interval side**. Alternative (ii), poor distribution in APs, is false for us in the strongest sense: the distribution in APs coprime to W is perfect. So alternative (i), an interval where the density deviates, must hold, and section 3a exhibits it with the deviation computed exactly: ρ_y(S) ≠ 1. **Our object is a worked example of the GS dichotomy in which both sides are known exactly.** That is a nice thing to have and it changes nothing about the Zone Postulate, since the deviating intervals are the ensemble again. --- ## 9. READINGS 1. **PROVEN + VERIFIED.** With column modulus q a multiple of x#, the Maier matrix over the twin-slot set is the identity 0 = 0. Every row carries the same count. Section 2. 2. **PROVEN + VERIFIED (20/20 exact integer matches).** With q = y#, y < x, the two summations give the transfer identity Σ_r #(A ∩ [r·y#, r·y# + S)) = D_y(S)·∏_{y x: the origin row equals D_y(S) minus the T_y slots t < S with t or t+2 struck, and a y-rough m ≤ y′² is struck only if m is a prime in (y,x] or m = y′². The m = y′² boundary case is load-bearing: dropping it fails 4 of the 14 cells. The Origin Excess Corollary's advantage over the ensemble mean is **capped by an absolute constant of about 2.2**, with a measured maximum of 1.372, because the lemma is silent for y′² ≤ x and the corollary's guarantee is trivial for x ≥ x* with ln x*/ln y ≈ 1.44. Its exact form is ρ_x(S)/ρ_y(S). Sections 4 and 4a. 4. **REFUTED (MEASURED, decisive).** The origin does not dominate the ensemble at the zone's width. At S = x′² its rank among the M rows is 2254 of 7429 (y=13, x=23) and it sits below the mean at y=13, x=19. The advantage of reading 3 expires at S ≈ y′² and then **reverses**, troughing at 0.9343 near S = 3.5 y′². Against the ensemble of all x# translates the origin at S = x′² carries ρ(2) = e^{2γ}/4 = 0.79305 of the mean density, 21% below it, which is the same constant `FOLD-PROFILE.md` §9 calls the trough constant. Section 5. 5. **PROVEN (the stopping point, sharply).** The Origin Excess Lemma needs S ≤ y′² and the Zone Postulate needs S = x′², so both together force y ≥ x, at which point the matrix is the tautology of reading 1. The informative regime and the origin-distinguished regime are disjoint. The collision is one line and has no matrix in it: y < x with x prime gives y′ ≤ x < x′, hence y′² < x′² strictly. Section 5. 6. **PROVEN + VERIFIED (11 divisors of 13#).** Complete classification: the matrix depends on q only through d = gcd(q, x#), and returns exactly one number, the mean of the window count over the x#/d windows at multiples of d, which is the Level Ledger histogram h_d. d = 1 gives ρ = 1 and zero information; d = x# gives the tautology. Section 6. 7. **INFERRED, and it is the general lesson.** The Maier matrix is a first-moment device that converts an approximate AP statement into a contradiction with an *independent* interval prediction. For an exactly periodic set there is no approximation to exploit and no independent prediction to contradict, so the matrix never crosses the residue-to-interval line of `THE-LENS.md` §5's triage. **Exact AP input does not beat Maier's approximate input; it removes the mechanism.** Section 7, Q3. 8. **VERIFIED from source.** GS Corollary 1.4 has η = min(α/3, 1/100) capped at 1/100 for every sequence, hence u ≥ 50,000 unconditionally and log x ≥ (5·10⁶)^{200}. It cannot apply at u = 2 or at any computable scale, so it is not an obstruction here. Section 8. 9. **PROVEN + MEASURED, minor and positive.** Our set satisfies GS alternative (i) explicitly: it is perfectly distributed in APs coprime to W, so the dichotomy forces an interval irregularity, and ρ_y(S) computes it exactly. Section 8. 10. **INFERRED, where this leaves the repo.** The matrix hands the Zone Postulate back to route A of `ZONE-POSTULATE.md` §6 unchanged: an every-window statement at width x′² is the Gap Reformulation, and an almost-all statement needs the second moment of the same ensemble, which is the localized gap of `LOCALIZED-GAP.md`. Nothing is gained and nothing is lost. **Nobody in this repo should attempt the Maier matrix again without first defeating reading 5.** --- ## 10. Reproduction `node --max-old-space-size=6144 research/maier-matrix.js`, 6 seconds. Output sections A, B, C, D, E, F, G correspond to sections 2, 3, 3a, 5, 5, 4 and 6 here. The largest tile enumerated is T_23 (223,092,870 residues), which is why the heap flag is there; everything else is small. The flag can be dropped if the T_23 lines are removed. --- *This document states current understanding. Superseded claims, retired numbers and the reasons they changed are in [history/CHANGELOG.md](history/CHANGELOG.md), indexed by document.*