# The origin's advantage: exactly how far it reaches, and why nothing extends it *(2026-08-17. Question from Chris: can the Origin Excess Lemma, or anything like it, be pushed to S = x′², and if not, is the scale collision a genuine theorem or an artifact of that particular argument? His doubt, recorded before the work: "I expect the answer is no, and that the collision is genuine rather than an artifact. A clean statement of WHY is what I am actually buying here." The doubt is correct. Below is the WHY, and it is sharper than the collision he stated. Code `research/origin-excess.js`, 13 seconds, all output real and pasted.)* --- ## 0. The hard part first **The collision is not an artifact, and the reason it is not an artifact has nothing to do with the Maier matrix.** Read as a clash of two inequalities, S ≤ y′² against S = x′², the obstruction looks like a bookkeeping accident of one lemma, which invites the thought that a better lemma would move the boundary. It would not, and here is why in one sentence. The origin's excess over the matrix ensemble is **exactly** a ratio of two values of the survival curve ρ, taken at the same window and two sieving levels; the Zone Postulate asks for that ratio at S = x′², which is the point u = 2 where ρ takes its **minimum**. The origin is not merely undistinguished at the zone's width. It sits at the worst point of the only curve in the problem, and against the full ensemble of all x# translates it is **below** the mean by a factor tending to e^{2γ}/4 = 0.79305. Measured, after stripping the finite-size Hardy-Littlewood factor: **0.79303 at x = 1487**. So the answer to "can it be pushed to S = x′²" is worse than no. Pushing it there lands on the one width where the origin is worst. **Stated with the right hedges.** Two of the three legs are proven outright: the Survival Quotient Identity (§2) and the parameter collision (§6c). The third, that ρ attains its minimum exactly at u = 2, is INFERRED from the squared-Buchstab form (e^γω(u))² and is not proven here; the measured trough at [0, 1.2·10⁸) sits at u = 2.0903 with the offset explained by the window's finite-size factor. But the operative fact needs no such inference: **ρ(2) < 1, measured directly at eighteen levels, and landing on e^{2γ}/4 = 0.79305 to four digits.** Whether u = 2 is the exact argmin only affects whether SOME y might still help, and §5's finite-size table answers that empirically: the band of y that helps shrinks monotonically toward y = x across nine levels. **Three things also came back that were not asked for and change the ledger.** 1. The lemma's **corollary carries a positivity threshold**, and that threshold is the binding constraint on the whole advantage. The guaranteed lower bound D_y(y′²) − 2(π(x) − π(y) + 1) is positive only for x below a threshold x\*, and ln x\*/ln y ≈ 1.44 across every level computed. So the advertised factor (ln x / ln y)² is **capped at about 2.2**, not merely small. The largest ratio in the 14-cell table is 1.372, and no choice of (y,x) can push it past about 2.2. The lemma has one hypothesis and travels with three companion statements of three different logical types; they are enumerated in `maier-matrix.md` §4a and derived in §6 here. 2. Above y′² the advantage does not decay to nothing. It **reverses**. The origin goes to a deficit with a trough at S ≈ 3.5 y′², and only then relaxes back to 1. Per prime, the origin's loss runs at 5% of fair share below the threshold, **125% of fair share** at eight times the threshold, and back to 100% at a hundred times. That overshoot is FOLD-PROFILE §5's shoulder, measured in the loss ledger and in a scale-free variable for the first time. 3. At S = x′², the residue class a_q = 0 that the origin occupies is the **worst of the q available classes** at the smallest prime q = y′ (rank 17 of 17 at y = 13, x = 23). The origin's choice is not merely non-optimal at the zone width; it is coordinate-wise the worst choice at the coordinate carrying the most weight. --- ## 1. Custody, first Everything below is computed by an engine written from scratch for this file, not by reusing `maier-matrix.js`. The slot sets are built by marking n ≡ 0 and n ≡ −2 (mod p) walking forward, where `maier-matrix.js` marks offsets 0 and W − 2; different indexing, same object. The three published censuses and the two published tables come back digit for digit. ``` 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 lemma verified 14 of 14 census custody: D_13=1485 D_19=378675 D_23=7952175 (GLOSSARY: 1485 / 378675 / 7952175) ``` The two rank measurements Chris quoted are also exact: ``` y=13 x=23 M=7429 rows y'^2=289 x'^2=841 S mean origin min max origin rank (1=largest) origin>=mean? 169 6.487 8 3 9 163 / 7429 yes 289 11.533 15 7 15 1 / 7429 yes 361 13.696 17 9 17 1 / 7429 yes 1000 33.879 32 28 40 5752 / 7429 NO 5000 177.323 177 168 189 3522 / 7429 NO 30030 1070.423 1075 1055 1087 1036 / 7429 yes ``` and the S = x′² table, including the two rows where the origin is genuinely below the mean: ``` y x S mean origin min max rank / M origin>=mean? 11 17 361 15.679 18 12 19 3 / 221 yes 13 17 361 16.765 18 15 18 1 / 17 yes 13 19 529 21.316 21 19 24 135 / 323 NO 13 23 841 29.554 30 24 36 2254 / 7429 yes 17 19 529 20.579 21 19 22 2 / 19 yes 17 23 841 27.776 30 24 32 16 / 437 yes 19 23 841 27.391 30 25 30 1 / 23 yes ``` Custody is clean: every inherited number above reproduces on an independent engine. --- ## 2. The Survival Quotient Identity, and why it settles the question This is the whole file in three lines, so it goes first. Write δ_z = D_z/z# for the twin-slot density of T_z, and > **ρ_z(S) := (D_z(S)/S) / δ_z**, the local-to-global density ratio of level z in the window [0,S). Then for every y < x and every S ≤ y#: > **Survival Quotient Identity (PROVEN, VERIFIED 39/39).** > > **origin(S) / mean(S) = ρ_x(S) / ρ_y(S).** *Proof.* mean(S) = D_y(S)·∏_{y origin/mean at width S = **ρ(ln S / ln x) / ρ(ln S / ln y)**. --- ## 3. The crossover, measured Fine scan of the intermediate range at seven pairs. The (13,23) scan, which is the pair `maier-matrix.md` measured at six widths, at forty: ``` y=13 x=23 M=7429 rows y'^2=289 x'^2=841 y#=30030 S S/y'^2 origin mean origin/mean rank/M min max note 200 0.69 11 8.65 1.2717 1/ 7429 5 11 257 0.89 13 10.09 1.2882 1/ 7429 6 13 289 1.00 15 11.53 1.3006 1/ 7429 7 15 <- y'^2 (Origin Excess Lemma stops here) 330 1.14 16 12.25 1.3057 1/ 7429 8 16 361 1.25 17 13.70 1.2413 1/ 7429 9 17 374 1.29 17 13.70 1.2413 1/ 7429 9 17 424 1.47 18 15.14 1.1891 8/ 7429 11 19 481 1.66 20 17.30 1.1561 18/ 7429 14 21 529 1.83 21 19.46 1.0790 392/ 7429 15 23 618 2.14 24 22.35 1.0740 504/ 7429 17 27 700 2.42 26 24.51 1.0609 573/ 7429 18 30 794 2.75 26 25.95 1.0019 2772/ 7429 20 32 841 2.91 30 29.55 1.0151 2254/ 7429 24 36 <- x'^2 (what the Zone Postulate asks for) 899 3.11 32 31.72 1.0089 2468/ 7429 25 38 1020 3.53 33 35.32 0.9343 6173/ 7429 29 41 1156 4.00 38 40.37 0.9414 6325/ 7429 34 46 1310 4.53 44 46.13 0.9538 6085/ 7429 39 52 1683 5.82 55 58.39 0.9420 6877/ 7429 52 65 2451 8.48 85 86.50 0.9827 4970/ 7429 78 95 4585 15.87 160 163.63 0.9778 6410/ 7429 154 175 9724 33.65 348 347.44 1.0016 2814/ 7429 335 362 30030 103.91 1075 1070.42 1.0043 1036/ 7429 1055 1087 ``` **Two readings, and the second is new.** The rank collapses far earlier than the mean does. The origin is rank 1 of 7429 up to S = 374 = 1.29 y′², is 392nd by S = 529, and 2254th at the zone width. The mean comparison is much more forgiving because the ensemble's window counts are sub-Poisson; `maier-matrix.md` §7 already noted that small measured dispersion is not a bound, and here it also means "above the mean" is a weak test. **The advantage does not decay into nothing, it reverses.** The ratio goes 1.3057 at 1.14 y′², through 1.0 near 2.9 y′², down to a trough of **0.9343 at 3.53 y′²**, and only then climbs back to 1.0043 at the full period. The origin pays back what it gained, and it overpays first. Crossover summary over the seven exactly-enumerated pairs: ``` y/x y'^2 x'^2 first crossing / y'^2 / x'^2 13/23 289 841 1020 3.53 1.213 13/19 289 529 529 1.83 1.000 13/17 289 361 529 1.83 1.465 11/23 169 841 712 4.21 0.847 11/19 169 529 529 3.13 1.000 11/17 169 361 530 3.14 1.468 17/23 361 841 1262 3.50 1.501 ``` That table on its own does not name the law, because at these levels y′² and x′² are within a factor of five of each other. The law appears at scale, in §5. --- ## 4. The per-prime anatomy: the threshold is S = q·y′, and the smallest prime binds The M rows are not a vague ensemble. They are exactly a set of residue vectors: > **Row bijection (PROVEN).** M = x#/y# = ∏_{y r·y#, and a slot s is killed by q exactly when s ≡ a_q or a_q − 2 (mod q) with > a_q = −r·y# mod q. Since y# is invertible mod q, the map r ↦ (a_q)_q is a > **bijection** onto ∏_{y vector**, and every other vector occurs exactly once. So the origin has exactly one distinguishing feature: it takes the class 0 at every prime simultaneously. Its whole advantage must live in the fact that the y-rough multiples of q are depleted at small heights. That is a statement about one prime at a time, and it has a sharp threshold: > **Per-prime threshold (PROVEN).** Put λ_q(S) = #{T_y slots t < S with q | t or > q | t+2}, the origin's loss to q; the ensemble's fair loss is 2·D_y(S)/q. If > **S ≤ q·y′** then a y-rough multiple of q below S is q·k with k y-rough and > k < y′, hence k = 1, hence λ_q(S) ≤ 2 whatever S is. The Origin Excess Lemma is > this bound applied to every q at once; since the smallest q is y′, the binding > case is **S ≤ y′²**. Measured in the scale-free variable v = S/(q·y′), pooled over six levels y ∈ {29, 61, 127, 251, 401, 641} and twenty-five primes q each, keeping only cells where the fair share is at least 4 so the ratio is not quantisation noise: ``` v = S/(q*y') cells mean lambda_q/fair_q median max (cells with fair share >= 4) 0.50 50 0.0485 0.0000 0.2089 0.75 75 0.0439 0.0000 0.2025 1.00 100 0.0754 0.0000 0.4742 1.50 125 0.6352 0.6299 1.2539 2.00 142 0.9049 0.8807 1.5000 3.00 150 1.1068 1.1259 1.6803 5.00 150 1.2240 1.2240 1.6063 8.00 150 1.2472 1.2609 1.5372 15.00 150 1.2269 1.2396 1.4253 30.00 125 1.1348 1.1521 1.3017 100.00 85 1.0041 1.0015 1.1064 ``` **This is the crossover law in its cleanest form.** Below the threshold the origin loses 4 to 8 percent of its fair share. It is at 64% by v = 1.5, at 90% by v = 2, and **overshoots to 125% at v = 8** before relaxing to 100% by v = 100. The overshoot is FOLD-PROFILE §5's shoulder, and it is why the aggregate ratio dips below 1 rather than approaching it. **No reweighting avoids the binding prime.** The loss sum is weighted by 1/q, so the smallest prime in (y,x] carries the largest weight, and its threshold is q·y′ = y′². There is no choice of y, and no choice of which primes to fold in, that removes the term with the largest weight and the earliest threshold. **And at the zone's width the origin's class is not even locally good.** Mean row count conditioned on a_q = a, over every a mod q, at S = x′²: ``` y=13 x=23 S=841 q=17: a=0 ratio 0.9398 best a=15 ratio 1.0780 rank of a=0: 17 of 17 y=13 x=23 S=841 q=19: a=0 ratio 0.9813 best a=3 ratio 1.0359 rank of a=0: 10 of 19 y=13 x=23 S=841 q=23: a=0 ratio 1.0151 best a=7 ratio 1.0418 rank of a=0: 6 of 23 y=11 x=23 S=841 q=13: a=0 ratio 0.9691 best a=10 ratio 1.0636 rank of a=0: 11 of 13 y=11 x=23 S=841 q=17: a=0 ratio 0.9520 best a=8 ratio 1.0427 rank of a=0: 16 of 17 y=11 x=23 S=841 q=19: a=0 ratio 1.0059 best a=1 ratio 1.0282 rank of a=0: 5 of 19 y=11 x=23 S=841 q=23: a=0 ratio 1.0295 best a=2 ratio 1.0514 rank of a=0: 2 of 23 ``` At the smallest prime, the one that matters, **a = 0 is the worst class of the q**. The origin at the zone width is not a lucky row that we cannot certify; it is an actively poor row. --- ## 5. The crossover law at scale, and the ceiling One incremental sieve of [0, 1.2·10⁸) with snapshots at eighteen levels gives ρ_z(S) for every level and every width at once, hence origin/mean for every (y,x) pair by the identity. Only pairs with y′² > x appear; below that the lemma is vacuous for the reason in §7 G4. u_x and u_y are the two survival arguments at the crossing. ``` y x y'^2 x'^2 crossing S /y'^2 /x'^2 u_x u_y origin/mean at x'^2 13 29 289 961 983 3.40 1.023 2.046 2.686 0.9718 (S=983) 13 127 289 17161 16414 56.80 0.956 2.004 3.784 0.9936 (S=16414) 13 257 289 69169 63408 219.40 0.917 1.993 4.311 0.9635 (S=70967) 29 367 961 139129 111352 115.87 0.800 1.968 3.451 0.9642 (S=139483) 43 1487 2209 2217121 1484745 672.13 0.670 1.945 3.778 0.9361 (S=2329678) 61 2999 4489 9006001 5124753 1141.62 0.569 1.930 3.758 0.9249 (S=8999699) 89 6037 9409 36517849 17688624 1879.97 0.484 1.917 3.718 0.9126 (S=34766432) 127 6037 17161 36517849 17688624 1030.75 0.484 1.917 3.445 0.9121 (S=34766432) 257 6037 69169 36517849 15804583 228.49 0.433 1.904 2.987 0.9045 (S=34766432) 739 2111 552049 4464769 5124753 9.28 1.148 2.018 2.339 1.0046 (S=4578908) 1049 2999 1104601 9006001 10072540 9.12 1.118 2.014 2.318 1.0074 (S=8999699) 2111 6037 4464769 36517849 38910887 8.72 1.066 2.008 2.283 1.0080 (S=34766432) 4253 6037 18139081 36517849 61054148 3.37 1.672 2.059 2.146 1.0460 (S=34766432) ``` *(the full 127-row table is in the run log; these are the rows that span the range)* **The crossover law.** In the S/y′² column the crossing wanders over three orders of magnitude, from 3.4 to 1880, so y′² is the wrong variable. In the **u_x** column it does not move. Over the 127 pairs in the table: ``` u_x at the crossing: n=127 min=1.012 p05=1.916 median=1.993 p95=2.149 max=2.224 in the band [1.90, 2.23]: 122 of 127 the 5 outside it, all with x near y'^2 where the lemma is already vacuous: y= 19 x= 367 u_x=1.072 ln x/ln y=2.01 y= 19 x= 521 u_x=1.012 ln x/ln y=2.12 y= 29 x= 739 u_x=1.043 ln x/ln y=1.96 y= 43 x= 2111 u_x=1.018 ln x/ln y=2.04 y= 61 x= 4253 u_x=1.013 ln x/ln y=2.03 ``` **122 of the 127 sit in [1.90, 2.23]**, with u_x → 1.90 when ln x/ln y is large and u_x → 2.09 when ln x/ln y → 1. Those two endpoints are exactly the two features of ρ: 1.90 is where ρ crosses 1 on the way down, and 2.09 is where it bottoms out. The law is > **S\* solves ρ(u_x) = ρ(u_y), i.e. ρ(ln S/ln x) = ρ(ln S/ln y),** which is immediate from the identity. The content is that ρ's descent and recovery pin u_x into a band of width 0.3 around 2, whatever y and x are. The five exceptions confirm rather than dent the picture: every one has ln x/ln y ≈ 2, that is x ≈ y′², which is exactly where hypothesis (a) of §6 has already failed and the origin is below the mean at S = y′² itself. **So the crossover is the zone width.** S\* = x^{1.90…2.22} against x′² = x^{2+o(1)}. The origin's advantage expires within a whisker of exactly the width the Zone Postulate asks for, and which side of the line you land on is decided by ρ(2) against ρ(2·ln x/ln y). Asymptotically ρ has its minimum at u = 2, so that comparison always goes the wrong way. The finite-size version of the same statement is the largest ln x/ln y at which the origin still beats the mean at S = x′²: ``` x 367 521 739 1049 1487 2111 2999 4253 6037 max 1.219 1.203 1.190 1.178 1.168 1.159 1.151 1.144 1.137 ln x/ln y ``` Monotone down, nine levels, no exceptions. **The band of y that helps is shrinking toward y = x**, which is the degenerate matrix. **The ceiling, stated without any y at all.** Take the d = 1 ensemble, all x# translates, whose mean is simply S·δ_x. Then origin/mean = ρ_x(x′²) = ρ(2), and that is a statement about the origin and the tile with no matrix in it: ``` x x'^2 origin S*delta_x rho_x(x'^2) HL factor rho/HL (target 0.79305) 29 961 30 32.6 0.91961 1.09750 0.83791 89 9409 187 182.7 1.02326 1.26663 0.80786 181 36481 538 536.7 1.00250 1.24283 0.80662 367 139129 1584 1638.3 0.96686 1.20691 0.80110 739 552049 4835 5082.2 0.95136 1.18365 0.80375 1049 1104601 8563 9025.6 0.94874 1.17858 0.80499 1487 2217121 16883 18044.6 0.93562 1.17982 0.79303 2111 4464769 30074 32339.3 0.92995 1.17011 0.79475 2999 9006001 53783 58169.5 0.92459 1.16056 0.79668 4253 18139081 96465 105002.3 0.91869 1.15034 0.79863 6037 36517849 173738 190487.2 0.91207 1.14120 0.79922 ``` The HL factor column is the window's own finite-size Hardy-Littlewood overshoot, measured not assumed. Dividing it out leaves **0.7930, 0.7948, 0.7967, 0.7986, 0.7992** at the top five levels against e^{2γ}/4 = **0.79305**. That is the Unification Law's trough constant of `FOLD-PROFILE.md` §9 and the conjectured limit of β in `GLOSSARY.md`, arriving at the Zone Postulate from a direction nobody had pointed it. It is the constant's **fifth** independent appearance in the repo, and the first from the origin side. > **The origin ceiling (MEASURED, HL-conditional in its exact value).** At the > zone's own width the origin carries **e^{2γ}/4 ≈ 0.79** of the tile's mean > density, not more. The raw ratio approaches it from above like 1 + c/ln x; > the measured raw values run 1.00 at x = 181 down to 0.912 at x = 6037. The curve's trough, measured directly in the same window as a check: ``` minimum of rho over 1.5 <= u <= 6: rho = 0.86392 at u = 2.0903 (y = 7331) ``` and the shape on [1,2] against e^{2γ}/u², all fifteen readings agreeing on one constant which is the window's HL factor: ``` y u y-rough pairs in (y,N) N*delta_y rho e^{2g}/u^2 1639664 1.3000 504821 243883.7 2.06993 1.87705 373121 1.4500 513766 303377.1 1.69349 1.50878 112070 1.6000 516022 369310.1 1.39726 1.23915 41369 1.7500 516757 441669.5 1.17001 1.03583 17873 1.9000 517049 520331.0 0.99369 0.87873 10954 2.0000 517139 576096.5 0.89766 0.79305 ``` ρ/(e^{2γ}/u²) reads 1.1028, 1.1224, 1.1276, 1.1295, 1.1308, 1.1319: constant to under one percent over u ∈ [1.45, 2.0]. The shape is exactly e^{2γ}/u², and the constant is the finite-size factor of the window, which tends to 1. --- ## 6. The lemma's one hypothesis, the corollary's threshold, and the collision The lemma needs exactly one hypothesis, S ≤ y′². Three further statements travel with it, and each has a different logical type: a non-vacuity remark, a corollary with its own threshold, and a proposition about the parameters that owes the lemma nothing. **The canonical enumeration of all four is `maier-matrix.md` §4a, which is the lemma's home; this section derives the three companions, one subsection each.** ### 6a. The non-vacuity condition: the conclusion is empty unless y′² > x An x-rough integer below x is 1, and 1 is not a slot because 1·3 is divisible by 3. So **D_x(S) = 0 for every S ≤ x**. At S = y′² ≤ x the lemma says 0 = 0. This is a remark and not a hypothesis, because the identity is true in that regime as well, and it is true outside S ≤ y′² too: L is *defined* as the struck count, so the bookkeeping never fails. What needs S ≤ y′² is the strike characterisation, whose consequent bound L ≤ 2(π(x) − π(y) + 1) breaks once S > y′². All fourteen verification cells of `maier-matrix.md` §4 satisfy y′² > x, so the evidence covers the non-vacuous regime only, which is the regime worth having. ### 6b. The Origin Excess Corollary, and the ceiling its threshold forces > **Origin Excess Corollary (PROVEN, positive only below a threshold).** > origin ≥ D_y(y′²) − 2(π(x) − π(y) + 1). The guarantee is positive only below a threshold x\*: ``` y y'^2 D_y(y'^2) x for which 2(pi(x)-pi(y)+1) >= D_y(y'^2) so x must satisfy 13 289 16 41 x < 41 19 529 21 61 x < 61 29 961 30 89 x < 89 43 2209 62 193 x < 193 61 4489 110 359 x < 359 89 9409 187 643 x < 643 127 17161 289 1039 x < 1039 181 36481 542 2069 x < 2069 257 69169 878 3529 x < 3529 ``` ln x\*/ln y reads 1.448, 1.396, 1.333, 1.399, 1.431, 1.441, 1.434, 1.469, 1.472 down that column: flat at about **1.44**, with no upward trend across two orders of magnitude in y. So in the regime where the lemma says anything at all, > **(ln x / ln y)² ≤ about 2.2.** **The advertised factor is not merely small, it is bounded by an absolute constant near 2**, which is a stronger statement than calling the measured factor small, and the bound does not improve with scale. The 14-cell table's maximum, 1.372, is not a small-sample artifact; it is close to the ceiling. Chris's third honest doubt, "check whether the size is useful before celebrating", is answered: it is not, and the reason is structural rather than numerical. This ceiling is on the origin's *advantage* below y′², and it is a different object from the origin's value at the zone width, which is ρ(2) = 0.79305 and a deficit rather than an advantage (§5). ### 6c. The Scale Collision Proposition: the regimes never touch > **Scale Collision Proposition (PROVEN, one line, no matrix in it).** If y < x > with x prime, then y′ ≤ x < x′, so **y′² < x′² strictly**. The two regimes are not merely disjoint, they never touch; they meet only at y = x, where the matrix is the tautology of `maier-matrix.md` §2 and the fold question A6 already closed. The strict inequality is a consequence of y′ ≤ x < x′ alone, so it is a fact about the two parameters rather than a property of the construction: **no better lemma repairs it**, and this is why §8's first reason is the hardest of the three to argue around. --- ## 7. Mechanism tests: three candidates, three failures, each for a different reason Chris asked for at least two of the mirror, the fused window and the −1/2 constant, the skeleton kernel, and head monotonicity. Three were tested. All three fail, and the three failure modes are worth keeping separate. ### G1. The mirror (`FOLD-PROFILE.md` §4) is the origin, not a second sample ``` x S head mirror equal? fused head+mirror 2*head 13 289 16 16 YES 32 32 13 1156 56 56 YES 112 112 17 361 18 18 YES 36 36 17 1444 59 59 YES 118 118 19 529 21 21 YES 42 42 19 2116 79 79 YES 158 158 23 841 30 30 YES 60 60 23 3364 116 116 YES 232 232 ``` The palindrome σ(r) = W − 2 − r is an involution on the slot set, and it maps [0,S) onto the interval immediately *below* 0 in the periodic tile. So the "fused" head-plus-mirror window is one centred window of width 2S about the point −1, and its count is exactly twice the head's. Sample size and count double together and the ratio to the mean is unchanged. **REFUTED as an independent source of origin advantage.** The mirror is not a second observation of the origin. It is the same observation written backwards. ### G2. The abutting window and the −1/2 constant (natal-cap-19, -26): the mechanism is real and it works against us Split each row into A = [0,S) and B = [S,2S). ``` y x S corr(A,B) origin A (z) origin B (z) origin A+B (z) A ratio B ratio A+B ratio 13 23 289 -0.4497 2.680 -1.254 1.327 1.301 0.809 1.090 13 23 361 -0.4439 2.395 -1.616 0.494 1.241 0.780 1.031 13 23 500 -0.0267 1.632 -2.692 -1.014 1.110 0.757 0.945 13 23 841 -0.3920 0.241 -2.056 -1.652 1.015 0.867 0.942 11 23 250 -0.3476 2.275 -0.998 1.003 1.332 0.820 1.093 11 23 400 -0.2754 1.845 -2.707 -0.981 1.212 0.642 0.927 11 19 250 -0.3082 1.906 -1.568 -0.025 1.216 0.748 0.998 11 19 400 -0.2494 1.156 -3.907 -2.525 1.106 0.586 0.846 ``` Two things, and the first is a small positive. **The −1/2 constant shows up here independently.** The abutting correlation reads −0.45, −0.44, −0.39, −0.39, −0.35, −0.31, −0.31 across most cells. natal-cap-26's exact −1/2 was proved for the flat spectral functional on the natal set; seeing −0.39 to −0.45 for abutting windows in the two-level row ensemble is a fourth independent sighting of the same negative adjacency. **And it is exactly the wrong sign for this problem.** At S = 289 the origin is +2.68 standard deviations in A and **−1.25 in B**; the fused window A+B is only +1.33 and by S = 500 it is **−1.01**. The B ratio runs 0.809, 0.780, 0.757, 0.642, 0.586: the origin is a large deficit on the shoulder. The head advantage is repaid immediately above the crystallisation frontier, and any construction that widens the window past y′² collects the repayment. **REFUTED as a route to a wider window**, and the refutation is mechanistic, not numerical: negative adjacency means a widened window sees the sum of an excess and its own compensating deficit. ### G3. The impact window and head monotonicity (`FOLD-PROFILE.md` §11): circular, exactly the A6 trap ``` x x'^2 slots in head twin primes in (x, x'^2) + [x'^2-2 rough] identical? 13 289 16 16 0 YES 17 361 18 17 1 YES 19 529 21 21 0 YES 23 841 30 29 1 YES ``` The head is the twin primes below x′² plus at most the single boundary slot x′² − 2, which is the Head Lemma's second element. So the mechanism's entire content is "the head equals the twin primes below x′²". That is the Zone Postulate written out. **CIRCULAR**, and it is exactly the failure A6 recorded for the first-slot recursion. Chris's warning about this trap was well placed: two of the three candidates that look like origin structure are crystallisation wearing a different hat, and the check that catches them is to ask whether the object being counted is π₂ of the head. ### The general obstruction, stated so it does not need re-testing The three failures are not three coincidences. > **Origin Channel Lemma (INFERRED, with the proof sketch below).** In the > two-level frame the M rows are exactly the M residue vectors (a_q)_{q∈(y,x]}, > each occurring once, and the origin is the all-zero vector (§4, PROVEN). Row > counts differ only through which T_y slots fall in the excluded classes > a_q, a_q − 2 (mod q). For a fixed a, the number of T_y slots below S in a > single class mod q is D_y(S)/q up to fluctuation, with **no systematic bias for > any a except a = 0 and a = −2**, whose bias is that a y-rough multiple of q is > q·k with k y-rough. Hence **every origin advantage in this frame is the > depletion of the y-rough numbers below S/q, and nothing else.** That depletion > is total for S/q < y′ and gone by S/q ≈ 3y′ (§4, MEASURED). Its reach is > therefore S ≈ q·y′ per prime and S ≈ y′² in aggregate, and it is > crystallisation by definition. The one loophole worth naming and closing: the origin's excluded sets overlap more than a generic vector's only in the sense that they are all multiples, but for any vector the classes a_q and a_{q'} intersect in one class mod q·q′ with the same expected count. The joint structure is identical; only the roughness of the elements differs. So there is no second channel hiding in the inclusion-exclusion either. **What that leaves of route B.** Three pieces of the origin's proven structure are aimed at route B of `ZONE-POSTULATE.md` §6 and all three miss, for the three distinct reasons above. Route B's non-fold half is narrower than it looks. --- ## 8. The verdict **Q. Can the Origin Excess Lemma, or anything like it, be pushed to S = x′²?** **No, and the target is worse than empty.** Three separate reasons, in increasing order of how hard they are to argue around. 1. **Parameters.** y < x forces y′ ≤ x < x′, hence y′² < x′² strictly. PROVEN, one line, no matrix. The regimes are disjoint and do not even touch. 2. **Size.** Even inside its own regime the lemma's advantage is capped at about 2.2, because it is silent for x ≥ y′² and trivial for x ≥ x\* with ln x\*/ln y ≈ 1.44. MEASURED at nine levels. 3. **Shape, and this is the real answer.** origin/mean = ρ(u_x)/ρ(u_y) exactly. S = x′² is u_x = 2, which is the minimum of ρ. So the zone's width is the one width at which the origin is worst, and against the full ensemble the origin carries e^{2γ}/4 = 0.79305 of the mean density. VERIFIED to four digits after stripping the window's Hardy-Littlewood factor. **Q. Is the collision genuine or an artifact?** **Genuine, and it is not really a collision.** "Collision" suggests two constraints that happen to be incompatible. What is actually going on is that the origin's advantage and the zone's width are the *same* object measured in the same variable: the advantage lives where ρ is above 1, which is u < 1.90, and the zone sits at u = 2, which is where ρ bottoms out. The crossover S\* = x^{1.90…2.22} is not near x′² by accident. Both are the point where crystallisation ends. **Q. Is there another source of origin advantage that does not route through crystallisation?** **No, in this frame, and §7's Origin Channel Lemma says why.** The origin is the all-zero residue vector; the only thing that distinguishes the zero class is that its members are q times something rough; the reach of that is q·y′. The mirror is the origin reflected, the fused window collects the shoulder's repayment, and the impact window is π₂ of the head. Anyone proposing a fourth candidate should be asked one question: *does it distinguish the class 0 mod q from a generic class by anything other than the roughness of its elements?* If not, it has the same reach. **What this leaves.** Route B of `ZONE-POSTULATE.md` §6 asked for "structure specific to the origin". The structure exists, it is exactly crystallisation, and this file measures its reach: **S ≈ y′² in the two-level frame, S = x^{1.90…2.22} in general, and the Zone Postulate needs S = x^{2+o(1)}, on the wrong side by a constant that tends to e^{2γ}/4.** Nobody in this repo should attempt an origin advantage at the zone width again without first exhibiting a mechanism that survives §7's one question. --- ## 9. READINGS 1. **VERIFIED (custody, exact).** Independent engine reproduces `maier-matrix.js` §F 14 of 14, the §D and §E rank tables digit for digit, and the censuses D₁₃ = 1485, D₁₉ = 378,675, D₂₃ = 7,952,175. §1. 2. **PROVEN + VERIFIED (39/39).** Survival Quotient Identity: origin(S)/mean(S) = ρ_x(S)/ρ_y(S) exactly, for every y < x and S ≤ y#. The origin's excess over the matrix ensemble is one survival curve read at two levels. §2. 3. **PROVEN.** Row bijection: M = ∏_{y