# Does "the gate multiplies" close the u-frame branch? *(2026-08-17 late. The question: the Localized Merge Lemma died this afternoon on a no-fixed-point argument, and the line recorded was "the obstruction is not the size of the gate, it is that the gate multiplies." Does that close the ENTIRE u-frame recursion branch, or only the merge chain? Calibration marked throughout: PROVEN, VERIFIED, MEASURED, REFUTED, INFERRED. Scripts `research/gate-multiplies-01.js`, `-02.js` and `-03.js`, 22 seconds for all three, deepest object T_29 at 214,708,725 twin slots.)* ## 1. The hard part first, and the answer **No. The no-fixed-point argument closes the merge chain and one relative of it, and it does not reach TODO 0b at all.** Its hypothesis is an *accumulating index*, and 0b has none: 0b carries a single number forward and re-bases at every fold, so the running bound cancels out of the cost rather than compounding. That is the good news and it is thin, because **TODO 0b is wrong as stated, for an unrelated and more elementary reason, and the error is a factor of ln u.** > **REFUTED (PROVEN, and VERIFIED numerically below).** The inequality > `ln c <~ 2 ln^2 u / u` does **not** imply `G2(u) < u^2`. Read as a uniform > constant, no c > 1 satisfies it, since the right side tends to 0; the item is > then vacuous. Read as a per-level rate in the product telescope, the telescope > gives `12 * prod_{p<=x} c(p)`, and `sum_{p<=x} 2 ln^2 p / p ~ ln^2 x`, against > a budget of `2 ln x`. It overspends by a factor `ln x / 2`. Numerically the > bound it certifies exceeds u^2 by 4.6x at u = 37, 149x at u = 100, and > **1.6e10 at u = 1000**. > > **The sharp per-fold requirement is `ln c(p) <= 2 ln p / p`**, obtained by > differentiating the partial-sum condition `sum_{p<=x} ln c(p) <= 2 ln x - ln 12` > rather than dividing it by pi(u). That is a factor ln p tighter. And the measured multiplier is already spending that budget in full. **MEASURED, and this is the model-free statement**: `ln c(p)` against the sharp rate `2 ln p / p` on Ziller and Morack's 21 exact terms runs 1.39, 0.97, 1.36, 0.75, 1.15, 1.10, 1.18, 1.00, 0.61 at p = 41 to 73, **mean 1.06**, and on our own ladder 0.77 to 2.14. The multiplier is at the allowed rate or over it, and it is not trending down. No exponent fit enters that reading. **What the asymptotic headroom is, stated exactly.** If `G2 ~ C (ln W)^alpha` then `c_true = (theta(p')/theta(p))^alpha ~ 1 + alpha ln p / p`, so the fraction of the sharp budget the truth consumes is **alpha/2 and nothing else**: 100% at alpha = 2, 92% at 1.847, 75% at the calibrated central 1.50, 65% at the bottom of the 1.3 bracket. **So whether there is asymptotic headroom is exactly the exponent question, and that is not settled** (`U-FRAME.md` §3, `exponent-control.md` §5). What is settled is that the constant is worth a total lifetime slack of **0.88 to 1.19 nats as measured**, summed over every fold from here to infinity. The 0.598 below is a predicted asymptote and not a measurement, so it is not the figure to quote for the slack. MEASURED: the slack `ln(x^2 / G2(x#))` reads 1.06, 0.94, 0.98, 0.88, 0.95, 1.18, 1.02, 0.95 across x = 11 to 37. Flat, with a predicted asymptote of ln(1/0.55) = 0.598 if alpha = 2. Everything else follows from those two facts. What survives is named in section 9 and it is not a bound, it is an exact object. ## 2. The no-fixed-point argument in its most general form State it as a hypothesis on a proof strategy, not on the objects. The argument is about what is *provable*, not about what is true, and that distinction turns out to decide three of the verdicts below. > **NFP (no fixed point).** Let a chain seek `Q(x) <= B(x)` by telescoping N(x) > folds down to a fixed base, and suppose: > > **(NFP1) Accumulating index.** The conclusion has the form > `Q(x) <= Phi_J(base)`, where `J = sum_j kappa_j` accumulates a per-fold index > cost and the base is *not* re-established at each fold. > > **(NFP2) Deficit.** `Phi_J >= J * delta` for some delta > 0: the functional > cannot return less than delta per unit of index. For maxsum, delta = mbar, by > averaging (LOCALIZED-GAP's Deficit Lemma). > > **(NFP3) Linear cost feedback.** The best *provable* per-fold cost is > `kappa_j >= c1 * B(x) / p`. This is not an accident of any one lemma: it is > what every spacing argument gives, because a window of length B admits B/p > kills whenever the only tool limiting kills is that they sit at least ~p apart. > > **(NFP4)** `N(x) * c1 * delta / x > 1`. > > Then the chain's own hypothesis is unsatisfiable for every B. B cancels between > (NFP2) and (NFP3), leaving the linear map `B -> (N c1 delta / x) * B`, whose > multiplier is B-free and exceeds 1. The localized merge chain satisfies all four: NFP1 (`M(x_n,Y) <= maxsum_{n+1}(T_{x0},Y)`, base x0 fixed), NFP2 (Deficit Lemma), NFP3 (Fact B: `kappa <= G/(p-2) + 1`), NFP4 (`N = pi(x)`, delta = mbar ~ 2.4 ln^2 x, giving 2.4 ln x). That is localized-04-maxsum.md section 10, restated without any reference to gates. **What escapes, precisely.** Failing any one of the four is an escape, and the three interesting escapes are: 1. **No accumulating index (NFP1 fails).** A chain that collapses back to a single number at every fold pays a *relative* cost per fold, and the running bound cancels the other way: cost/B is B-free. TODO 0b is exactly this. 2. **Exactness (NFP2 fails).** An identity returns the truth, not a floor. There is no gate to stay under and no bootstrap. 3. **B-free cost (NFP3 fails).** If the per-fold cost is bounded by anything that does not mention the running bound, the map `B -> const` is constant, not expanding, and has a fixed point whenever the constant clears the target. Chris's honest doubt on the record was that an additive per-fold cost would escape entirely. **That is confirmed, and it is escape 3.** The Localized Merge Lemma tried to buy exactly that (cost 1 per fold) and failed only because its *gate* re-introduced B: the cost is 1 only while `M <= (p-2)/4`, so the honest cost is `max(1, B/p)` and the chain reverts to NFP3 the moment it leaves the gate. That is the whole content of the Traverse Bound. ## 3. The Traverse Bound is not localized, and it is the same crossover Writing the cost as `kappa(m) ~ max(1, m*mbar/p)` unifies the two regimes: | regime | condition | index behaviour | |---|---|---| | additive | `m <= p/mbar` | cost 1 per fold, J grows by 1 | | multiplicative | `m > p/mbar` | cost proportional to m, J multiplies by (1 + mbar/p) | The crossover sits at `m ~ p/mbar ~ x/(2.4 ln^2 x)`, and with the gate's alpha = 1/4 it is `x/(9.6 ln^2 x)`. **That is the Traverse Bound, and it is a general crossover, not a feature of the localized window.** The localized chain gets `pi(x)` folds' worth of demand against `x/(9.6 ln^2 x)` folds' worth of additive supply, short by 9.6 ln x. **Why the tile version is milder.** Same crossover, different target. The localized chain must stay under `x/4`; the tile chain must stay under `x^2`. The deficit budget is `target/mbar`, so the tile chain has 4x times more index to spend and the shortfall degrades from `9.6 ln x` to a constant. Section 5 prices that constant. **So the mechanism transfers and the verdict weakens.** ## 4. MEASURED: the cost is not proportional, and the counting bound is loose NFP3 is a claim about the best available bound. It is worth knowing how far that bound is from the truth, because the whole family stands or falls on closing that gap. `research/gate-multiplies-02.js` measures kappa(m), the maximum kill count inside a window of m consecutive NEW gaps, directly, by walking all `D*p` positions of the new tile in order. Four folds, the deepest being T_23 folded by 29 at 230,613,075 positions, in 17 seconds. ``` === fold T_19 by p = 23 === old: D = 378675 mbar = 25.615 G2 = 150 new: G2 = 204 true longest kill run L = 3 kappa(m) = 1:3 2:4 3:4 4:4 6:5 8:5 12:5 16:6 24:8 32:9 kappa(m)/m = 1:3.000 2:2.000 3:1.333 4:1.000 6:0.833 8:0.625 12:0.417 16:0.375 24:0.333 32:0.281 counting bound = 1:10.7 2:12.1 3:15.3 4:17.6 6:23.0 8:26.1 12:30.1 16:36.7 24:48.1 32:59.0 === fold T_23 by p = 29 === old: D = 7952175 mbar = 28.054 G2 = 204 new: G2 = 258 true longest kill run L = 2 kappa(m) = 1:2 2:3 3:3 4:3 6:4 8:4 12:5 16:5 24:8 32:8 kappa(m)/m = 1:2.000 2:1.500 3:1.000 4:0.750 6:0.667 8:0.500 12:0.417 16:0.313 24:0.333 32:0.250 counting bound = 1:10.6 2:13.2 3:15.4 4:16.6 6:21.0 8:22.6 12:27.9 16:33.7 24:43.0 32:53.9 ``` Two readings, both MEASURED at all four folds (13, 17, 19, 23 by 17, 19, 23, 29): 1. **kappa(m) is strongly sublinear in m.** `kappa(m)/m` falls monotonically from 2 or 3 at m = 1 to 0.25 to 0.28 at m = 32, at every fold. If NFP3 held as a *fact* the ratio would be flat at `mbar_new/(p-2)`, measured 1.84, 1.53, 1.51, 1.12. It is not flat and it is not at that level. 2. **The counting bound is loose by 3.5x to 6.7x, and the looseness grows with m.** At fold 29 it gives 10.6 against a true kappa(1) = 2, and 53.9 against a true kappa(32) = 8. **So NFP3 is a statement about our tools and not about the tile.** The tile's kills are far rarer than their spacing permits. That is the same fact A9's tail measurement reports from the other side, and it is why "L is polylog" is believable while unprovable. ## 5. The general obstruction that DOES bind: the Overshoot Budget The no-fixed-point argument is a special case of something simpler and harder to escape, which applies to every chain in the family including the ones where NFP3 fails. > **Overshoot Budget (PROVEN, given the measured law for G2).** Any valid chain > of upper bounds ending in `G2(x#) < x^2` passes through quantities that are all > at least `G2(x#)`. Its total multiplicative overshoot over the truth is > therefore capped at `x^2 / G2(x#)`. With `G2 ~ 0.55 (ln W)^2` and > `ln W = theta(x)`, that cap tends to `1/0.55 = 1.82`, i.e. **0.598 nats, spread > across the entire ladder to infinity.** MEASURED, and it is the single most useful number in this note. The slack `ln(x^2 / G2(x#))`: | x | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |---|---|---|---|---|---|---|---|---| | slack, nats | 1.058 | 0.940 | 0.984 | 0.878 | 0.953 | 1.182 | 1.016 | 0.953 | Flat over the whole reachable ladder, predicted 1.042 at x = 37 from `ln(1/0.55) + 2 ln(x/theta(x))` with theta(37)/37 = 0.801, measured 0.953. **There is no growing margin anywhere in this route, at any level, in any formulation.** Every per-fold statement inherits it: a chain of pi(x) folds may overshoot the true per-fold multiplier by a total of 0.88 to 1.19 nats as measured, which is `0.9 to 1.2 ln x / x` per fold on average. **For accumulating-index chains this is fatal in one line.** Such a chain ends at `G2(T_x) <= maxsum_J(T_{x0})`. Once `J >> D(T_{x0})` the word repeats and `maxsum_J = J * mbar_0 * (1 + O(D_0/J))`, so the conclusion is *equivalent* to a statement about J, and J must be pinned to within a factor 1.82 of `G2(T_x)/mbar_0`. Base-independent: choosing a larger base scales both sides by `1/mbar_0`, and cutting the ladder into blocks multiplies the per-block overshoots, so the 1.82 is a lifetime total however the chain is decomposed. > **Consequence.** An accumulating-index chain must bound the cumulative index to > within a factor under 2 of the truth, over pi(x) folds. That is not a bound on > the answer, it is the answer. The only unconditional tool available, the > spacing bound of section 4, is loose by 3.5x to 6.7x at a single fold. **This closes the accumulating-index family on the tile**, where NFP4 gives only a constant rather than a log, and it closes it for the same reason NFP does: not because the constant is wrong, but because no counting argument is exact. ## 6. Every recursion in the repo, tested one at a time | where | object | NFP1 index | NFP2 deficit | NFP3 feedback | verdict | |---|---|---|---|---|---| | 5a step 1-2 | kill law, copy reformulation | no | no (exact) | no | **SURVIVES** | | 5a step 3 | `maxsum_2 <= G2(new) <= maxsum_{L+1}` | no (one fold) | n/a | n/a | **SURVIVES**, conditional on L | | 5a step 4 | `G2(new) <= G2(old) + L*mbar` | - | - | - | already REFUTED (A2) | | 5a step 7 | `sum L*mbar ~ x ln^3 x` | - | - | - | dead with step 4 | | 5a step 3a (A4) | `maxsum_m(new) <= maxsum_{m+kappa(m)}(old)` | **yes** | **yes** | **yes** | **CLOSED** | | 5a step 3 (A4) | `maxsum_{m+1}(old) <= maxsum_m(new)` | yes | yes | n/a (lower) | SURVIVES, useless | | 5a step 2 (A4) | copy theorem for the family, exact | **no (m stays m)** | no (exact) | no | **SURVIVES** | | 10 (A5) | Theorem A, Run Cost | no | no | no | SURVIVES | | 10 (A5) | Theorem B, C | no (one-shot) | no | **supplies it** | SURVIVES as a theorem, CLOSED as a supplier | | 11 (A9) | transfer operator | no | no (exact) | no | **SURVIVES** | | 10 (A10) | effective run length m_eff | yes | yes | no | CLOSED, by step 4's refutation | | TODO 0b | `G2(u) <= 12 c^{pi(u)}` | **no (re-bases)** | no | no | **SURVIVES NFP**, but see section 7 | Item by item, with the reason rather than the label. **5a steps 1 and 2, the copy theorem for the whole family.** These read as identities and are **VERIFIED 40 of 40, over five folds and m ≤ 8** (`U-FRAME.md` §5a step 2), not proven. `maxsum_m(new) = max over the p 2-sets {a, a-2} of maxsum_m(old minus those classes)` is the one object in the family whose **index does not grow under folding**: m stays m. There is no accumulating index, no floor to stay above, and no gate. NFP cannot touch it and neither can the Overshoot Budget, because an identity overshoots by nothing. That immunity is exactly as strong as the identity, and the identity rests on a measured clause the corpus has not proved: no straddling window ever beats a single-copy one. **This is the strongest survivor in the branch**, and what would make it the safest is a proof of that clause. Its price is that the difficulty moves out of index bookkeeping and into evaluating a residue-deleted maxsum, which is a two-dimensional sieve question. **5a step 3's lower recursion.** `maxsum_{m+1}(old) <= maxsum_m(new)`, verified 40 of 40. A lower chain has no gate and nothing to bootstrap, so NFP is silent. It is also silent on TPC: it can only produce lower bounds on G2, and the Zone Postulate needs an upper one. **5a step 3a's upper recursion, A4's chain.** This is the tile analogue of the localized merge chain and it satisfies all four NFP conditions: the index accumulates (`J_{n+1} = J_n + kappa_n(J_n)`), the Deficit Lemma applies to maxsum at the base, and A5 Theorem C's structural cap supplies the feedback `kappa >= maxsum_m/(3p)`. NFP4 gives only a constant here rather than the localized 9.6 ln x, because the target is x^2 and not x/4. **It is section 5 that closes it, not section 2:** the chain must certify the cumulative index to within a factor 1.82, and the only unconditional tool is loose by 3.5x to 6.7x at a single fold. **10's A5.** Theorem A is attained with equality at every fold and is not a chain. Theorems B and C are one-shot bounds and survive as theorems. But **A5's structural cap is exactly NFP3**, discovered a day earlier in different vocabulary: "Theorem B can never prove L below G2/(3p) ~ 0.18x, which is LINEAR, the failing branch." That is `kappa >= c1 * B / p` with c1 = 1/3. So A5 is the supplier of the hypothesis that kills every chain fed by it, and redirecting L to kappa via Theorem C inherits the cap, as U-FRAME already records. **11's A9.** The transfer operator is exact and it iterates arbitrarily far: from the T_7 word alone it walked to T_23, T_29 and T_31 with D and G2 both correct. No gate, no index, no deficit. It survives everything in this note. The honest limit is that it is closed on the gap **word**, not on the histogram, so it is an exact simulator rather than a source of bounds; reading G2 off it still requires running it. **10's A10.** The negative result stands and needs no help. Its live successor, the effective run length m_eff, was priced as "constant at 3 would close 5a from the upper side, since telescoping gives sum ~ x ln^2 x, well under x^2". **That projection inherits step 4's refutation**, and worse, it contradicts the measured G2 law: `sum_{p<=x} 3*mbar(p) ~ 7.2 x (ln x - 1)` falls **below** 0.55 x^2 from x = 37 onward on the exact `mbar = W/D`, and from x = 29 on the Mertens asymptotic, so it would certify a bound smaller than the truth. A constant m_eff is therefore impossible, and the branch was already closed by its own arithmetic. ## 7. The required rate is `2 ln p / p`, and why `2 ln^2 u / u` busts the budget by `ln u` The telescoped programme is `G2(u) <= 12 * prod_{p<=u} c(p) < u^2`, and it reads two ways. The algebra from a **uniform** c is correct; the algebra from a per-level rate is a different inequality, and neither delivers `2 ln^2 u / u`. **Read as a uniform constant, it is vacuous.** `12 c^{pi(u)} < u^2` must hold at every u, and the permitted `2 ln^2 u / u` tends to 0, so the only admissible constant is c = 1. **Read as a per-level rate, it does not imply the Zone Postulate.** The telescope gives `12 * prod_{p<=x} c(p)`, not `12 * c(x)^{pi(x)}`. VERIFIED: ``` TEST: does ln c(p) <= 2 ln^2 p / p at every level imply G2 < u^2 ? running 12 * prod_{5 **The required rate (PROVEN).** Differentiating the partial-sum condition > `sum_{p<=x} ln c(p) <= 2 ln x - ln 12`, which must hold at every x, gives > > **`ln c(p) <= 2 ln p / p`**, with a **total** lifetime slack of `ln(x^2/G2(x#))` > nats, measured 0.88 to 1.19 and tending to ln(1/0.55) = 0.598. **And the measured multiplier is already spending that budget in full.** MEASURED, our own ladder against the sharp rate: | fold p | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |---|---|---|---|---|---|---|---|---|---| | ln c | 0.916 | 0.337 | 0.452 | 0.493 | 0.329 | 0.308 | 0.235 | 0.299 | 0.417 | | sharp `2 ln p/p` | 0.556 | 0.436 | 0.395 | 0.333 | 0.310 | 0.273 | 0.232 | 0.222 | 0.195 | | **ln c / sharp** | 1.65 | 0.77 | 1.15 | 1.48 | 1.06 | 1.13 | 1.01 | 1.35 | **2.14** | | against 0b's `2 ln^2 p/p` | 0.85 | 0.32 | 0.45 | 0.52 | 0.36 | 0.36 | 0.30 | 0.39 | 0.59 | The bottom row is the same data read against the softer `2 ln^2 p/p`, and it is where the impression comes from that the multiplier spends only 15 to 45 percent of its budget on a falling trend. Read against the sharp budget, which is the row above it, **the multiplier uses 77 to 214 percent of its budget and it is not trending down** — the reading `U-FRAME.md` §4 carries. Ziller and Morack's adversarial terms, recomputed the same way, run 1.39, 0.97, 1.36, 0.75, 1.15, 1.10, 1.18, 1.00, 0.61 at p = 41 to 73, mean 1.06. Their sequence is spending at exactly the allowed rate, not at a fifth of it. **The asymptotic version of the same statement, and it is an open question rather than a fact.** With `G2 ~ C (ln W)^alpha` and `ln W = theta(x)`, `c_true = (theta(p')/theta(p))^alpha ~ 1 + alpha ln p / p`, so the truth consumes exactly `alpha/2` of the sharp budget: | alpha | 2.000 | 1.847 | 1.500 | 1.300 | |---|---|---|---|---| | fraction of budget spent | 100% | 92.3% | 75.0% | 65.0% | At alpha = 2 there is no headroom at all; at the calibrated central 1.50 there is 25%, and at the bottom of the practical bracket about 35%. **So asymptotic headroom in 0b is exactly the exponent question and is not settled** (`U-FRAME.md` §3, `exponent-control.md` §5). What is measured, and what does not depend on any fit, is the table above: over every fold we can compute, the multiplier spends 77% to 214% of the budget with a mean at 1.06 and no downward trend. The constant is worth 0.88 to 1.19 nats in total as measured, and no more. ## 8. What the surviving form needs, priced The route is 5a step 3, `G2(new) <= maxsum_{1+L}(old)`. Write `rho = (maxsum_m - G2)/((m-1)*mbar)`, whose value depends on both the tile and m. MEASURED over every cell with T_11 <= T_x <= T_23 and 2 <= m <= 8: **1.02 to 1.84**, the extremes at (T_19, m=4) and (T_23, m=6). Per-level typical values, the median over m, run 1.33, 1.26, 1.15, 1.27, 1.71 at T_11 to T_23, so rho is higher at the top of the reachable ladder than at the bottom but **does not rise level by level**; see the T_29 reading below and section 10. Then the per-fold **burn** is `rho*mbar*L/G2` against a per-fold **replenishment** of `2 ln p / p`, and > **burn / replenish = rho * mbar * L * p / (2 * ln p * G2) ~ 3.27 * L * ln p / p**, > using `mbar = 2.4 ln^2 p`, `G2 = 0.55 p^2`, rho = 1.5. Two branches, and they are the same two branches the repo has been on all along: - **L at A5 Theorem B's provable ceiling, `L >= G2/(3p) = 0.18 p`:** the ratio is `0.4 * rho * ln p ~ 0.6 ln p`, which **diverges**. Dead by exactly one log. - **L at A9's measured polylog, `L <~ 0.8 ln^2 p`:** the ratio is `~2.6 ln^3 p / p`, which **tends to 0**, crossing 1 near p ~ 800 and clearing thereafter with a margin growing like `p / ln^3 p`. > **The survivor, stated as a threshold (PROVEN, given the measured G2, mbar and > rho laws).** The surviving form goes through iff > **`L <= 0.19 to 0.31 p / ln p`**, on average over the ladder. A5 Theorem B > proves `L <= 0.18 p`. The gap is a factor **`0.58 to 0.95 ln p`**, and it is the > only gap. (The favourable end is rho = 1.5, the central reading; the tight end > is T_29's rho = 2.4. rho is measured on six tiles and has no trend, §10, so > neither end may be quoted as the one the ladder is heading for.) MEASURED, the chain run with the **true** L at every fold, which is the most favourable reading the proven machinery allows: | fold p | 13 | 17 | 19 | 23 | 29 | |---|---|---|---|---|---| | true L | 2 | 2 | 2 | 3 | 2 | | bound `maxsum_{1+L}(old)` | 96 | 138 | 168 | 228 | 300 | | true `G2(new)` | 66 | 108 | 150 | 204 | 258 | | burn, nats | 0.827 | 0.738 | 0.442 | 0.419 | 0.386 | | replenish `2 ln(p'/p)` | 0.334 | 0.536 | 0.222 | 0.382 | 0.462 | Chained from `G2(T_11) = 42`, the bound reaches **698.3 at x = 29 against a target of 841**. Running slack `ln(x^2 / bound)` falls **1.058 -> 0.186 over five folds**, a net loss of 0.175 nats per fold. The next fold replenishes only `2 ln(31/29) = 0.133` nats, so the chain survives fold 31 only if `maxsum_5(T_29)/G2(T_29) < 1.376`. **It does not. VERIFIED.** `research/gate-multiplies-03.js` streams T_29, all 214,708,725 twin slots of it, out of T_23 in 4.3 seconds without storing the tile: ``` === T_29 === D = 214708725 W = 6.469693e+9 mbar = 30.1324 G2 = maxsum_1 = 258 (recorded value 258) maxsum_ 2 = 330 / G2 = 1.2791 rho = 2.389 maxsum_ 3 = 390 / G2 = 1.5116 rho = 2.190 maxsum_ 4 = 420 / G2 = 1.6279 rho = 1.792 maxsum_ 5 = 510 / G2 = 1.9767 rho = 2.091 maxsum_ 6 = 540 / G2 = 2.0930 rho = 1.872 ... step-3 chain at fold 31 needs maxsum_5(T_29)/G2 < 1.376 -> measured 1.9767 BUSTS ``` > **The step-3 chain dies at fold 31 (VERIFIED).** L(T_29, 31) = 4, so the bound > is `maxsum_5(T_29) = 510` and the chain's value at x = 31 is > `698.3 * 510/258 = 1380.5` **against a target of 31^2 = 961**. Running slack > goes 0.186 -> **-0.363**. Six folds from a base of T_11, with the true L at > every step and no slack given away anywhere. Two custody checks pass on the way: D(T_29) = 214,708,725 is exactly `prod_{3<=p<=29}(p-2)`, and G2(T_29) = 258 reproduces the published ladder from a generator that never touches a sieve. **And rho is largest at the deepest tile, which is the unhelpful direction.** `max_{m<=8} rho(m)` reads 1.58, 1.78, 1.83, 1.41, 1.84, **2.39** at T_11 to T_29: larger at T_29 than anywhere below it, but not monotone, since it dips to 1.41 at T_19. Section 8's threshold `L <= 0.31 p / ln p` used rho = 1.5; at rho = 2.4 it tightens to `L <= 0.19 p / ln p`. The asymptotics above say the burn turns around near p ~ 800 if and only if L is polylog. **That is the entire u-frame branch, and it is the L question with nothing added.** ## 9. What survives, precisely, and what it needs Three things survive, and they are not the same kind of thing. **1. The exact copy theorem for the whole maxsum family (U-FRAME section 5a, step 2).** `maxsum_m(new) = max over the p 2-sets {a, a-2} of maxsum_m(old minus those classes)`, **VERIFIED 40 of 40, over five folds and m ≤ 8**, and resting on the measured clause that no straddling window ever beats a single-copy one. **The only object in the branch whose index cost is zero**: m stays m under folding. It is immune to NFP (no accumulating index, no deficit, no gate) and immune to the Overshoot Budget (an identity overshoots by nothing), for as long as the identity holds outside the range it was checked in. What it needs is a way to bound a residue-deleted maxsum without going through a kill count, which is a genuinely different question from anything the branch has attacked. Nobody has tried it directly, meaning no branch here has; that is an internal absence, not a literature one (`SEARCH-CONVENTIONS.md`). **2. A9's transfer operator (U-FRAME section 11).** Exact, iterates arbitrarily far, and PRIOR ART (Holt and Rudd 2014 section 5). It survives everything here. What it needs to become a proof rather than a simulator is closure at the histogram level: it is currently a function of the old gap **word**, and reading G2 off it means running it. **3. The per-fold multiplier bound, which is the one surviving form of the L question.** > Bound the per-fold multiplier by `ln c(p) <= 2 ln p / p`, with a total lifetime > slack measured at 0.88 to 1.19 nats. Via 5a step 3 this is exactly > `L <= 0.19 to 0.31 * p / ln p` on average over the ladder. A5 Theorem B proves > `L <= 0.18 p`. **The gap is a factor 0.58 to 0.95 ln p (0.18/0.31 and > 0.18/0.19), and it is the only gap.** The two ends are rho = 1.5 and rho = 2.4, > and since rho is not monotone in the level neither end is the one a trend > favours. **A proof that c decays at any rate is worth nothing here.** It must decay at `2 ln p / p`, and the measured multiplier is already at that rate, mean 1.06 of budget over Ziller and Morack's 21 terms with no downward trend. Any asymptotic room is the factor `alpha/2` and nothing else, which makes it the exponent question rather than a new lever. The u-frame reformulation buys form, as U-FRAME section 6 says. It buys no measured slack at all. ## 10. Honest limits **The Overshoot Budget is conditional on the measured G2 law.** `G2 ~ 0.55(ln W)^2` is MEASURED over 12 levels, with fitted exponents running 1.48 to 2.17 depending on the window. Every reading used here is a whole-period one, which is the diagonal of `maxgap-law.md` §4, so the constant is stable in the sense that note establishes and must not be carried to a localized window, where the same object reads about twice as large. If the exponent is genuinely above 2 the Zone Postulate is false and the budget is negative; if the constant is smaller the budget is larger. What is not conditional is that the budget is a **constant number of nats spread over an infinite ladder**, whatever that constant is, and that no per-fold reformulation can create more of it. **Section 8's threshold uses three measured laws at once** (G2, mbar, rho), so its constants 0.19 to 0.31 should be read as one significant figure. What does not depend on them is the BRANCH of the answer: a factor of order `ln p` against a target of O(1). The constant in front of it is the range `0.58 to 0.95`, and the branch argument holds at either end, which is the reason it is worth making separately from the constant. **rho is measured on six tiles and it has no trend.** Its per-level median runs 1.33, 1.26, 1.15, 1.27, 1.71 at T_11 to T_23 and its per-level max 1.58, 1.78, 1.83, 1.41, 1.84, 2.39 to T_29: neither is monotone, and the high readings sit at the deepest tile rather than on a rise. We have no model for why, and the extreme-value argument that would predict rho -> 1 assumes an exponential gap tail, while the tile is measurably under-dispersed (sd/mbar = 0.51, 0.55, 0.60, 0.64, 0.67, 0.69 at T_7 to T_23, rising toward 1 but not there). Do not extrapolate rho in either direction. **Nothing here touches the parity wall or the sifting parameter.** Every argument above is bookkeeping about budgets. That is a strength as far as it goes, and it means none of it can be evidence about whether the L question is provable, only about what a proof of it would have to deliver. ## 11. Reproduction ``` node research/gate-multiplies-01.js 23 # 0.2 s, ladder to T_23, maxsum + budgets node research/gate-multiplies-02.js 29 # 17 s, kappa(m) at four folds, true L node research/gate-multiplies-03.js # 4.3 s, streams T_29, maxsum_m to m = 12 ``` Custody: `gate-multiplies-01` reproduces the G2 ladder 12, 30, 42, 66, 108, 150, 204 and `-03` reproduces 258, from a copy-theorem generator that never sieves. `maxsum_2(T_19) = 186`, `maxsum_3(T_23) = 300`, `maxsum_4(T_23) = 348`, `maxsum_8(T_23) = 528` and `maxsum_10(T_13) = 282` all match the values recorded independently in U-FRAME sections 5a and 8. D(T_29) = 214,708,725 equals `prod_{3<=p<=29}(p-2)` exactly. --- *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.*