# The transfer operator and the pair count: exact at any level, with no tile in memory *Parent: `research/U-FRAME.md` §11, which carries these claims in summary. Two exact instruments, neither needing a tile in memory, and one FIT that must not be used past the range it was fitted on.* *Both instruments are implemented in `research/a3-09-histogram-operator.js` — the operator as a histogram recurrence and the pair count alongside it — which is why neither needs a tile resident in memory.* Two instruments, both exact, neither needing a tile in memory, and together they are what make the f and L questions measurable at any level in reach. The operator is PRIOR ART (Holt and Rudd 2014, U-FRAME §6a) and nothing here may be presented as new structure; what is ours is the head engine and the numbers it produces. ### The histogram transfer operator (PROVEN, VERIFIED exactly at six folds) The new gap histogram is generated from the old gap word by a transfer operator whose three clauses are: the interior of a run dies, the left end lives, the right end lives. It reproduces every measured histogram size for size, satisfies the mass identity Σcount = D(q−2), and iterated from the T₇ word alone with no sieve it walks to T₂₃ reproducing every glossary census, then past enumeration to T₂₉ and T₃₁ with D and G₂ both correct. **The structural consequence, and it is Chris's misalignment principle in its sharpest form.** The new histogram is a function of the old gap word and q, **and of nothing else**: not of W mod q, not of the tile's residues, not of the copy order. Shifting the alignment by the left endpoint's residue removes absolute position entirely. **The honest limit** is that the operator 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 means running it (`research/gate-multiplies.md` §6). ### The tail at any level, and the fit that goes with it Extending the grain-census inclusion-exclusion into a head engine gives count(d)/D at any level with no tile in memory, agreeing with enumeration to 5e−16, and yields **about 31 exact diagonal tail points over a range of 20 in x** against the seven U-FRAME §5a step 7 has from tiles. The count is not an identifier: the engine stops on a per-level time budget, so it is machine-dependent and moves between runs. FIT, flagged as such: ln(1/tail) = −0.235 + 1.2992·(2p′/m̄) with R² = 0.993, cross-checked within each level to 3%. **These points measure a different object from f, and the two fits are not comparable.** The tail here is the unconditional P(gap ≥ 2p′); f is the comb-restricted sub-tail inside it. Their ratio f/tail runs 0.032 to 0.454 and drifts, which is exactly why the slopes differ. The decomposition closes: the tail slope 1.311 over the 30 shared levels plus the comb-share drift 0.366 gives 1.677, the f slope on those same levels. So 1.2992 is not a third estimate of §5a's slope and must never be averaged against it. **QUARANTINE on the three numbers in that decomposition (2026-08-19).** The "f slope on those same levels" is CENSUS-derived, and the census producer aliased residues for primes above 32, so every f point from x = 37 up is wrong by a factor 0.62–1.05 (`research/f-decays.md` header, `history/staging/fdecay-deep.md`). The 30 shared levels run past x = 37, so 1.311, 0.366 and 1.677 are all contaminated and none of the three may be quoted. **What survives is the structural point, not the arithmetic**: the tail and f measure different objects and their slopes must differ, so 1.2992 is still not a third estimate of §5a's slope. Read the decomposition either scoped to x ≤ 31, where the census is unaffected, or re-computed against the alias-free numbers embedded in `research/fdecay-deep-01-census-defect.js`. **The structural claim is confirmed.** m̄/ln²p falls 3.70 → 2.42 and has essentially arrived: the Mertens limit is e^{2γ}/(2C₂) = **2.4026**, and exact W/D gives 2.4195 at x = 1009, 2.4086 at x = 10007 and 2.4035 at x = 200003. There is no further drift to come. Meanwhile 2p′/ln²p rises 5.8 → 69.6, so the threshold does recede from the bulk, and lnD/ln(1/tail) divided by ln²p is flat near 0.8 across p = 7 to 131, giving **L ≲ 0.8 ln²p**. **But that does not close the route, and the reason is worth carrying with the number.** Turning L ≲ 0.8 ln²p into "the route closes" goes through Σ L·m̄, which is U-FRAME §5a step 4, and step 4 is false. What the reading does buy is the branch: L ≲ 0.8 ln²p sits comfortably inside the required 0.19 to 0.31·p/ln p on the schedule U-FRAME §5a step 7 gives. The tail evidence says which branch, not that the route closes. **One caution on transferring the tail fit.** The FIT above is measured on the tile. Pushed through an extreme-value argument in the localized head it predicts R(1) = 12.0 to 12.6 across the Y = 10⁹ scan against a measured 12.6 to 18.7, so the head's gap tail is **heavier** than the tile's fitted law by 20 to 25 percent: the head's exponential rate is ≈ 1.06/m̄ against 1.2992/m̄ here. That 1.06 is the localized head's rate and has nothing to do with §5a step 7's seven-point slope of 1.062, which the digits otherwise invite a reader to connect. Do not use the fit past the range it was fitted on. ### The exact pair count (PROVEN, VERIFIED at every fold) Verified against two independent enumerations that never mention gaps: > PAIRS(T, p) = 2·Σ_{d ≡ 0 (mod p)} count(d) + Σ_{d ≡ ±2 (mod p)} count(d) | fold | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |---|---|---|---|---|---|---|---|---|---| | pairs | 2 | 0 | 6 | 72 | 1,088 | 11,870 | 243,822 | 8,025,014 | **114,874,436** | The weight 2 on the p-divisible sizes is the multiplicity factor of `research/kappa-not-L.md`, and a naive Σcount(d) misses by exactly 86 at fold 23 and 6 at fold 29. The deepest point is T₃₁ with 6,226,553,025 slots, three-way agreement across 12.45 billion kills in 2,620 s detached. **Alternation confirmed at fold 37**, which is the deepest test the Alternation Lemma of `research/kappa-not-L.md` has. The complete list of long-run words is 72+150, 150+72, 72+222, 222+72, 150+222, 222+150, 72+150+72, 150+72+150, where for p = 37 the three classes are 2p−2 = 72, 4p+2 = 150 and the neutral 6p = 222. Not one word repeats a non-neutral class, and 72 + 150 = 222 = 6p 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.*