Integer Census

• Which integers MrlyMath writes, which it never writes, and which it writes many times, taken over the whole mrlylab::ledger registry rather than over one favourite sequence.
• The registry is 18066 rows, one row a (design, measure, axis) key of mrlylab::ledger::keys across the four cost tiers: 7692 closed, 5044 convolved, 2665 side grid, 2665 level grid. Every tier count is re-derived from SPACES, ledger::designs and Measure::applies and matched against the enumerator, so the registry size is checked, not trusted.
• The pinned definition, printed by the binary before any table: a row's rendered window is its first min(48, B) terms, B the leading terms whose footprint fits 100000 cells, under the ledger's own budget of 100000 cells a term; a term's footprint is 1 cell for a closed measure, number^dimension + level * span for a convolved measure, number^(dimension * level) for a grid measure; a row whose rendered terms are strictly increasing stops at the first term above 100000; row R writes n iff n is a term of R inside R's rendered window and 1 <= n <= 100000.
• Multiplicity counts rows, never (row, index) pairs. The two readings differ: 347308 (row, integer) incidences against 360703 (row, index, integer) incidences, so 13395 times a row writes the same integer twice and the honest reading refuses to count it twice.
• The window is 1..=100000. Terms at or below zero - the Euler characteristics, the voids of a solid design - are counted (29144 of them) and reported, never folded into the census.
• Truncation is declared and counted, never assumed harmless: 5529 rows stop at the ceiling, 6802 at the 48-term cap, 5735 at a cell budget, and 390 rows write no integer in the window at all.
• The 1..=100000 miss set is a statement about the rendered window. That a missed integer is written by no row at any depth is Conjecture: 6802 capped rows have terms this study does not render.
• The window dependence is measured, not assumed: the census is re-read at 8 and 32 rendered terms as well as 48, and the cap is pushed to 96 on the rows a model can extend, for a strict lower bound on a deeper census.
• The model is a Newton forward extension of a row's 8-term head, licensed only where it reproduces that row's whole written column from the head and the pinned stop rule alone. Budget-stopped rows carry no rendered length in their head and are declared untestable rather than passed.
• Two self-checks that do not read the sweep they check: every closed-form row is replayed against its own closed form - f^level, g^level - f^level, the fill polynomial at the odd-side index k, the exposure recurrence - over 34604 rendered terms with zero mismatches; and the ledger's own documented terms (8, 21, 40, 65 for the carpet side fill, 8, 64, 512 and 1, 17, 217 for the carpet level fill and void, 20, 81, 208, 425 for the sponge side fill, 72, 1056, 18048 for the sponge level surface) are found at the head of their named rows and inside the multiset.
• The histogram is folded once and read twice: the sum of the multiset against a second walk over the rows, and the leading champion's multiplicity against a linear scan for its writers.

RUN

• CARGO_BUILD_JOBS=4 cargo run --release -p integer-census -- <directory>
• Three and a half minutes on 8 cores at 4 threads, peak memory under 100 MB.
• Prints the census tables; writes rows.csv, multiset.csv and MANIFEST.md into the directory given, or into a temporary directory it names on stdout. Nothing is written beside this file.
• rows.csv carries the key, the tier, the closed form or none, the first 8 rendered terms and the full list of integers the row writes, so it rebuilds multiset.csv on its own.

WITNESSES

• The registry: 18066 rows, closed 7692, convolved 5044, side 2665, level 2665, each agreeing with its independent derivation, none unread.
• The windows: 1..=1000 never 41, once 31, multiple 928; 1..=10000 never 3589, once 765, multiple 5646; 1..=100000 never 88867, once 2897, multiple 8236. The share written falls 0.9590, 0.6411, 0.1113.
• The champions, by rows written: 16 at 2858, 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. The leader 16 is written by 2858 of the 18066 rows, and every champion of the top twenty is below 65.
• Every integer to 99 is written, and 100000 itself is written by 103 rows.
• The first missed integers: 269, 362, 422, 443, 446, 487, 502, 538, 607, 611. The least is 269, a prime.
• The miss density by decade: 0 on 1..9, 0 on 10..99, 0.045556 on 100..999, 0.394222 on 1000..9999, 0.947533 on 10000..99999. The registry writes almost every small integer and almost no large one.
• The closed-form replay: 34604 terms, zero mismatches, over every row the ledger hands a power, a difference of powers, a fill polynomial or the exposure recurrence.
• The depth: 5263 integers written at 8 rendered terms, 8749 at 32, 11133 at 48, so 5870 of the written set arrive only past term 8 and 2384 only past term 32.
• The model: 3608 rows have a head of finite-difference degree at most 6, by degree 207, 1104, 569, 518, 1202, 0, 8; the rebuild passes 1306 of 1306 ceiling-stopped rows and 1325 of 1333 cap-stopped rows, the 8 failures being exactly the degree-6 detections, and the 969 budget-stopped rows are untestable.
• The deeper window: extending the 1325 rebuilt cap rows to 96 terms writes at least 11898 integers, moves the first miss from 269 to 362 and the first missed square from 9801 to 38809.
• The arithmetic: squares 176/316, cubes 46/46, fourth powers 17/17, fifth 10/10, sixth 6/6; primes 750/9592 with 158 of the 8363 above 10000; all 2079 residue classes mod 2..64 hold a written integer on 10000..100000; the 366 perfect powers carry 58906 incidences against a density of 0.003660.
• The spectrum: 410 distinct multiplicities to 2858; the top twenty carry 39007 of the 347308 incidences; the closed tier covers 7628 written integers with 3983 exclusive; above 30000 the written sets collapse to 953 families of which 875 own a tail integer alone.
• The page these witness is integers.md.