# The registry's integers

- 2026-08-31 [Proved] The registry's written set is finite at every ceiling: a row renders at most 48 terms, so the 18066 rows write at most `48 * 18066 = 867168` integers however far the ceiling is pushed, and the miss density tends to 1. The adversarial read kills the way the bound was first used - `11133` was read as `0.01284` of that cap, a comparison that is vacuous inside the census window because `867168` exceeds the ceiling `100000`; the honest saturation is `11133/100000 = 0.11133`, and the finiteness is the only claim of the lane untouched by a change of cap. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The census of `1..=100000` over the whole registry: 18066 rows, `7692` closed, `5044` convolved, `2665` side grid and `2665` level grid, each tier matched against an independent derivation from `SPACES`, `ledger::designs` and `Measure::applies`, none unread; stops `5529` ceiling, `6802` cap, `5735` budget, `390` silent; never/once/multiple `41/31/928` at `1000`, `3589/765/5646` at `10000`, `88867/2897/8236` at `100000`, shares written `0.9590`, `0.6411`, `0.1113`; miss density by decade `0`, `0`, `0.045556`, `0.394222`, `0.947533`; `347308` `(row, integer)` incidences against `360703` `(row, index, integer)`, so `13395` double counts are refused, and `29144` terms at or below zero are excluded and reported. Two independent refolds of `rows.csv` by readers sharing no code with the sweep reproduce every one of these numbers with zero mismatches, and add four checks the study did not run - no duplicate key, `max |written| = 48` never exceeded, every in-range head term present in `written`, every ceiling row's head strictly increasing. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The miss set's arithmetic: `269`, a prime, is the first missed integer and `1..268` the longest written run; the longest missed run is 447 wide on `95265..95711` with both neighbours written; `100000` is written by 103 rows; the written share on `10000..100000` by greatest prime factor falls `0.5798`, `0.1406`, `0.0506`, `0.0313`, `0.0117` over the bands `1..10`, `10..100`, `100..1000`, `1000..10000`, `10000..100000`; the tail written count by residue mod 12 runs `1175, 440, 145, 194, 715, 176, 420, 224, 531, 358, 229, 116` for a ratio `10.13`, and mod 6 `1595, 664, 676, 552, 944, 292` for `5.46`; primes `750/9592` with only `158` of the `8363` above `10000`; cubes `46/46`, fourth powers `17/17`, fifth `10/10`, sixth `6/6`, squares `176/316` with the largest written `97969 = 313^2`, carried by the single row `sequence_dim=4_code=28662_measure=voids_axis=side` on `4k^4 - 8k^3 + 8k^2 - 4k + 1`. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The first missed square is a cap artifact and not arithmetic: row multiplicity at `96^2, 97^2, 98^2, 99^2, 100^2` is `321, 19, 480, 0, 123`, and deepening the rendered window to 96 terms writes at least 228 of the 316 squares and moves the first missed square from `9801` to `38809 = 197^2`. The adversarial read kills the mechanism first offered for it - that `9801` is odd and so outside the family `(2k+2)^2` - because `97969 = 313^2` is odd and written, and because that family was selected by grepping a head prefix out of the study's own rows; what survives is the frontier, printed by the generator over `96..100` rather than read off a chosen family. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The rendered window is measured rather than assumed harmless: the census at 8, 32 and 48 rendered terms writes `5263`, `8749` and `11133` integers, so `5870` of the written set arrive only past term 8 and `2384` only past term 32, and the 8-term miss set opens `269, 281, 302, 311` where the 48-term one opens `269, 362, 422, 443`. Rebuilding a row's `written` column from its 8-term head and the pinned stop rule alone, by Newton forward extension, passes `1306` of `1306` ceiling-stopped rows and `1325` of `1333` cap-stopped rows, the 8 failures being exactly the degree-6 detections an 8-term head cannot certify; the `969` budget-stopped rows carry no rendered length in their head and are declared untestable. Extending only the 1325 rebuilt cap rows to 96 terms gives a strict lower bound on a deeper census: at least `11898` written, first miss moved from `269` to `362`, longest written run at least `361`. The adversarial read kills the first statement of the rebuild check, which reported a pass-set size as a population size and omitted the budget stop kind entirely; the generator now prints the population by stop kind and the failing degrees. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The champions are small perfect powers: `16` at 2858 rows, `9` at 2811, `4` at 2559, `12` at 2303, `36` at 2270, `64` at 2176, `3` at 1951, `6` at 1883, `8` at 1790, `33` at 1777, all twenty of the top twenty below 65 and carrying `39007` of the `347308` incidences, a share `0.1123`; the 366 perfect powers of the window carry `58906` incidences, a share `0.1696` against a density `0.003660`, `46.34` times their weight. The adversarial read kills the normalisation the enrichment was first printed under - a straddle-coverage ratio whose value at the integer 1 is definitional, since a written span containing 1 must start at 1, and which supplies `27.6%` of the squares mean - so the study prints unconditional means on `1..1000` instead: `193.42` over all integers, `995.26` over the squares, `920.58` over the perfect powers. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] A champion is a property of a measure column, not of a design: `euler.side` writes 1 in 695 of its 859 rows, `peak.side` writes 12 in 809 of its 1261, `heights.side` writes both 9 and 33 in 765 of its 1261, and the eight integers below 100 that `heights.side` writes most often are `9, 17, 25, 33, 41, 49, 57, 65`, every one `1 mod 8`, which is what puts `33 = 3 * 11` tenth in a census otherwise made of powers. The adversarial read kills the mechanism first offered - that `heights.side` rows are the progressions `1 + s(k-1)` - since only 840 of the 1261 rows have arithmetic heads and only 5 start at 1, witness `sequence_dim=2_code=6_measure=heights_axis=side` reading `2, 4, 6, 8`; the leader set and its `1 mod 8` law stand as printed. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] The tiers split the census cleanly: of the 11133 written integers the closed tier covers 7628 with 3983 exclusive, the side grid 6203 with 2603, the level grid 1826 with 541, the convolved tier 792 with 130; above `30000` there are 2174 written integers and the closed tier covers 1853 of them. Witness: lab/rs/integer-census.
- 2026-08-31 [Verified] Two OEIS collisions, both explained and neither an identification: the ascending champion set opens `2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 18`, a window of A100290 and of A336231 and of no other record in a dump of 398817, all three parting at the thirteenth term with 21, 19 and the census's 20; and the written-per-decade run `9, 90, 859` sits inside A209631 alone, which continues 6689 where the census gives 5452. Both searches index every window of the census sequence and walk every record, so they are exhaustive on both sides rather than sampled at offsets. Witness: lab/rs/integer-census, A100290, A336231, A209631.
- 2026-08-31 [Verified] No recognizable family is systematically missed: 173 records of the dump hold at least ten distinct integers of `1..=100000` and lie wholly inside the miss set, the longest being A361796 at 41 terms, which at a miss density of `0.88867` has probability about `10^-2.1` and is ordinary across 398817 records. Witness: lab/rs/integer-census, A361796.
- 2026-08-31 [Conjecture] That a missed integer is written by no row at any depth. The miss set is a statement about the rendered window: 6802 rows are cut by the 48-term cap, the 96-term lower bound already moves at least 765 misses across, `269` among them, and the cost of a true deeper census is cubic in the cap on the dimension-2 side grid. Witness: lab/rs/integer-census.
- 2026-08-31 [Conjecture] That the miss set is new to the OEIS. It is clean only in its dense head: no record of the dump carries a 4-term window of the miss set at offsets `0..416`, the first hit being offset 417 in A049537, and above that the miss set hits near-interval records - 852 hits at `k = 4`, 130 at `k = 10`, 37 at `k = 15` and 15 at `k = 20`, the 20-term witnesses A112820 and A118471. The absence rests on a dump, which is a snapshot, so it stays a conjecture under the standing caveat. Witness: lab/rs/integer-census, A049537, A112820, A118471.
- 2026-08-31 [Conjecture] That the write-once set is absent from the OEIS. The 2897 integers written by exactly one row have no hit at any offset of any record at `k = 4, 10, 15, 20`, a cleaner absence than the miss set's because the once set is thin where the miss set is an interval complement; the same dump caveat applies. Witness: lab/rs/integer-census.
- 2026-08-31 [Conjecture] That the miss set has no arithmetic characterisation. No modulus to 64 separates written from missed, the written share has no common growth order, and the set is closed under nothing; the finiteness bound is the only theorem the lane offers. Witness: lab/rs/integer-census.
- 2026-08-31 [Conjecture] That no bounded union of named families reaches the written tail. Above `30000` the rows' written sets collapse to 953 distinct families of which 875 own a tail integer no other family writes, covering 2005 of the 2174 tail integers, so every cover needs at least 875 families; the families are de-duplicated by written set and not by generating rule, so 953 is itself a lower bound on the number of rules. Witness: lab/rs/integer-census.
- 2026-08-31 [Conjecture] That no reparametrisation of the multiplicity function `a(n) =` rows writing `n` is submittable. It is absent from the dump at every offset tested, but it is a reading of the registry's own shape - the tier mix, the cap and the ceiling - rather than a function of `n`, so its terms move with the instrument. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] The miss set is a union of residue classes - all 2079 classes mod `2..64` hold a written integer on `10000..100000`, exhaustively, so no modulus in that range separates written from missed. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] The champions are the highly composite integers - on `1..1000` the mean row count is `193.42` over all integers, `995.26` over the squares and `920.58` over the perfect powers, but only `170.60` over the 413 integers with at least eight divisors, below the overall mean; being a small perfect power is what a champion is, and being divisor-rich reads slightly against it. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] Multiplicity is driven by each row's first rendered term - dropping every row's first term removes `17036` of the `347308` incidences, `4.9%`, and changes nothing: the written set stays `11133`, the never counts stay `41`, `3589`, `88867` in all three windows, and the leaders stay `36` at 2212, `64` at 2112, `16` at 2000 and `9` at 1999, the same integers in a different order. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] The multiplicity spectrum is geometric or a power law - with `S(m)` the count of integers written by at least `m` rows, `S(1) = 11133` and `S(2) = 8236` give a ratio `0.7398` predicting `S(64) = 6.312e-5` against the observed `977`, wrong by seven orders; the spectrum takes 410 distinct values to a maximum of 2858 and its local log-log slope is convex, so no single exponent fits. The adversarial read also kills the fit the power law was rejected by: the exponent first printed was a two-point secant with its amplitude pinned at `S(1)`, not a fit, so the misfit figure it carried is not the minimax one - the rejection stands, the number behind it does not. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] The written tail is the union of a few dominant families - above `30000` the rows' written sets collapse to 953 distinct families of which 875 own a tail integer no other family writes, so every cover of the tail needs at least 875 of them. Witness: lab/rs/integer-census.
- 2026-08-31 [Refuted] No OEIS record contains any contiguous window of the miss set - the search behind that universal sampled 40 offsets spread over 88867 terms, and an exhaustive walk of every window against every record finds hits at every length tested: 852 at `k = 4`, 130 at `k = 10`, 37 at `k = 15` and 15 at `k = 20`, with A112820 and A118471 each carrying 20 consecutive misses and A043635 lying wholly inside the miss set. What survives is the restricted statement, absence at `k >= 4` for offsets `0..416`. Witness: lab/rs/integer-census, A112820, A118471, A043635.
- 2026-08-31 [Refuted] Every square below the ceiling is written - 140 of the 316 are missed, the first being `9801 = 99^2`. Witness: lab/rs/integer-census.