Integers

Which integers this work writes, which it never writes, and which it writes many times - taken over the whole registry rather than over one favourite sequence. The sequences ledger asks whether a given sequence is known; this page asks the opposite question, what the union of every sequence the registry holds covers, and answers it by census.

The generator is lab/rs/integer-census, one pass over ledger::keys, which prints its definition before any table and writes rows.csv, multiset.csv and a manifest into a directory given on the command line. Every number below is a line of that run. The registry it walks is the one sequences is rendered from and the sequences demo searches live; the closed forms it replays are the fill law and the exposure recurrence of the sequence-census paper. The integers demo reads that union integer by integer: which of the first thousand the designs write, how many rows write each, and which the pinned window misses. The plot demo draws a row of the same ledger rather than listing it, with the smallest linear recurrence its terms satisfy, its characteristic polynomial and its growth beside it.

The definition

The census is only as good as its window, so the window is pinned and printed, never assumed.

• A registry row is one (design, measure, axis) key of ledger::keys over the four cost tiers.
• 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; the count of rows truncated this way is printed, never assumed to lose nothing.
• Row R writes n iff n is a term of R inside R's rendered window and 1 <= n <= 100000.
• Multiplicity counts rows, not (row, index) pairs: a row writing n at several indices counts once.
• An integer n appears iff some row writes it, and is missed iff no row writes it.

The census

• The registry is 18066 rows and every tier count is derived twice. Verified. 7692 closed, 5044 convolved, 2665 side grid, 2665 level grid, each matched against an independent count over SPACES, ledger::designs and Measure::applies; no row goes unread. Truncation is declared and counted: 5529 rows stop at the ceiling, 6802 at the 48-term cap, 5735 at a cell budget, and 390 write no integer in the window at all.
• The two readings of multiplicity differ, and the honest one is smaller. Verified. There are 347308 (row, integer) incidences against 360703 (row, index, integer) incidences, so 13395 times a row writes the same integer twice and the row reading refuses to count it twice. 29144 rendered terms are at or below zero - Euler characteristics, the voids of a solid design - and are excluded and reported, never folded in.

• The written set is finite, so the miss density tends to 1. Proved. A row renders at most 48 terms, so whatever the ceiling, the registry writes at most 48 * 18066 = 867168 integers. The registry is a fixed finite object and the integers are not: past 867168 the census is almost all miss, and no growth of the ceiling changes that. This is the one statement on the page that survives any change of window.

The miss set

• Every integer to 268 is written and the first miss is a prime. Verified. 269 is missed, and 1..268 is the longest written run in the window; the longest missed run is 447 wide, on 95265..95711, with 95264 and 95712 both written. The ceiling itself, 100000, is written by 103 rows. The first thirty misses are 269, 362, 422, 443, 446, 487, 502, 538, 607, 611, 618, 626, 643, 653, 659, 668, 677, 691, 698, 701, 709, 723, 758, 773, 787, 797, 803, 835, 857, 878.
• The miss set is not a union of residue classes. Refuted. All 2079 classes mod 2..64 hold a written integer on 10000..100000, exhaustively; no modulus in that range separates written from missed.
• The bias is divisibility and smoothness, not congruence. Verified. On the tail 10000..100000 the written count by residue mod 12 runs 1175, 440, 145, 194, 715, 176, 420, 224, 531, 358, 229, 116, a ratio of 10.13 between residue 0 and residue 11, and mod 6 it runs 1595, 664, 676, 552, 944, 292, a ratio of 5.46. Sorted by greatest prime factor the written share on the same band falls 0.5798, 0.1406, 0.0506, 0.0313, 0.0117 across the bands 1..10, 10..100, 100..1000, 1000..10000, 10000..100000. Of the 9592 primes, 750 are written and only 158 of the 8363 above 10000; the first missed prime is 269, the first missed integer.
• Every cube, fourth, fifth and sixth power is written; the squares are not. Verified. Cubes 46/46, fourth powers 17/17, fifth 10/10, sixth 6/6, all of 1..100000. Squares run 176/316: every square to 98^2 = 9604 is written and 99^2 = 9801 is not, and the largest written square is 97969 = 313^2, written by exactly one row, sequence dim 4, code 28662, voids, side, whose closed form 4k^4 - 8k^3 + 8k^2 - 4k + 1 is the square of the centered square numbers.
• The square frontier is the cap, not arithmetic. Verified. Row multiplicity at 96^2, 97^2, 98^2, 99^2, 100^2 is 321, 19, 480, 0, 123: the dense square families are exhausted, not excluded. Deepening the window to 96 terms, in the section below, writes at least 228 of the 316 squares and moves the first missed square from 9801 to 38809 = 197^2. Oddness excludes nothing: 97969 is odd and written, and 9801 is missed for want of depth.

The depth of the window

The whole miss set is a statement about the rendered window, and the study measures how much of one rather than asserting it is harmless.

• More than half the written set arrives past the head. Verified. 5870 of the 11133 written integers appear only past term 8 and 2384 only past term 32, so a census read off the ledger's own 8-term heads sees less than half of what 48 terms see, and its miss set starts 269, 281, 302, 311 rather than 269, 362, 422, 443.
• A row's written column is rebuilt from its head and the stop rule alone. Verified. 3608 rows have a head whose finite differences terminate at order 6 or less, by degree 207, 1104, 569, 518, 1202, 0, 8. Extending the head by Newton forward differences and applying the pinned stop rule reproduces the row's written column exactly for 1306 of 1306 ceiling-stopped rows and 1325 of 1333 cap-stopped rows; the 8 failures are exactly the rows whose head degree reads 6, which eight terms cannot certify. The 969 budget-stopped rows carry no rendered length in their head and are not testable this way, which is said rather than hidden.
• Deepening the cap moves every window-relative number except the longest missed run. Verified. Extending only the 1325 cap-stopped rows the rebuild reproduces, out to 96 terms, gives a strict lower bound on the 96-term census: at least 11898 integers written, the first miss moved from 269 to 362, the longest written run at least 361, at least 228 of 316 squares. 269 becomes written; the run 95265..95711 does not move.
• That a missed integer is written by no row at any depth is Conjecture. 6802 rows are cut by the cap and their deeper terms are not rendered here; the 96-term reading is a lower bound, not a census, and the true frontier of the written set is not known at any depth.

The champions

• The whole top of the census is small and mostly a power. Verified. All twenty champions lie below 65 - ascending, 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 18, 20, 21, 24, 25, 33, 36, 49, 64 - and they carry 39007 of the 347308 incidences, a share of 0.1123. The 366 perfect powers of 1..100000 carry 58906 incidences, a share of 0.1696 against a density of 0.003660: 46.34 times their weight.
• The champions are not the divisor-rich integers. Refuted. 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; being highly divisible is not, and reads slightly against it.
• Multiplicity is not driven by each row's first term. Refuted. Dropping every row's first rendered term removes 17036 of the 347308 incidences, 4.9%, and changes nothing that matters: 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, 9 at 1999 - the same integers in a different order.
• The multiplicity spectrum is neither geometric nor a power law. Refuted. 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, which predicts S(64) = 6.312e-5 against the observed 977 - wrong by seven orders. The spectrum takes 410 distinct values with a maximum of 2858.
• The effect is a property of a measure column, not of a design. Verified. 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. A champion is an integer that one reading of the geometry returns for most designs at once.
• The one champion that is neither small-smooth nor a power is an offset. Verified. 33 ranks tenth and is 3 * 11. The eight integers below 100 that heights.side writes most often are 9, 17, 25, 33, 41, 49, 57, 65, every one of them 1 mod 8: the column runs arithmetic progressions whose common difference is a power of two, and 33 - 1 = 2^5. The arithmetic of the champion is the arithmetic of the step, not of the integer.
• The closed tier carries the census and the grid tiers carry its tail. Verified. 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.
• The tail is not a few dominant families. Refuted. Restricted to 30000..100000 the rows' written sets collapse to 953 distinct families, and 875 of those families own a tail integer no other family writes - between them 2005 of the 2174 tail integers. So every cover of the written tail needs at least 875 families, and the tail is a wide superposition rather than a handful of dominant sequences.

Against the OEIS

Every search below is exhaustive on both sides: every window of the census sequence is indexed and every record of a local copy of the OEIS stripped dump is walked against that index, so no sampling of offsets is involved. The dump read holds 398817 records. Under the standing caveat of sequences a dump is a snapshot, so every absence here is Conjecture and needs a live re-read before it is repeated.

• The miss set is new to the OEIS only in its dense head. Conjecture. No record carries any 4-term window of the miss set at offsets 0..416; the first hit is at offset 417, in A049537. Above that the miss set does hit, and the hits are near-interval records rather than identifications: 852 hits at window length k = 4, 130 at k = 10, 37 at k = 15, 15 at k = 20, the 20-term witnesses being A112820 and A118471, each a sequence that runs a block of consecutive integers through a region the census misses wholesale. The head is the informative part and is clean: the string ,269,362,422,443, appears nowhere in the dump.
• The write-once set is absent at every offset. Conjecture. 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 champion set meets two records for exactly twelve terms. Verified. Ascending, the twenty champions open 2, 3, 4, 6, 7, 8, 9, 12, 14, 15, 16, 18, which is a window of A100290 and of A336231 and of no other record. All three part at the thirteenth: A100290 gives 21, A336231 gives 19, the census gives 20. Both records are binary-digit conditions, which is the right neighbourhood - the registry's designs are corner subsets of a parity cube - and neither is the champion set.
• The written-per-decade run meets one record and parts at the next term. Verified. The written counts by decade are 9, 90, 859, 5452, 4722 with 100000 itself, summing to 11133. The prefix 9, 90, 859 sits inside A209631 alone, an exponential-transform array, which continues 6689 where the census gives 5452.
• No recognizable family is systematically missed. Verified. 173 records hold at least ten distinct integers of 1..100000 and lie wholly inside the miss set, the longest being A361796 at 41 terms. At a miss density of 0.88867 a 41-term run of misses has probability about 10^-2.1, which 398817 records make ordinary: the census excludes nothing a catalogue would recognise, it just runs out of depth. The tour demo runs the other way in a dozen cards, each drawing a design beside the sequence it counts and the OEIS record that holds the terms.

What is left

• Whether any integer of 1..100000 is written by no row at any depth. The 96-term reading is a lower bound and already moves at least 765 of the misses across, 269 among them; the honest frontier needs a cap the dimension-2 side grid can pay for, and that tier costs cap^3. Conjecture.
• Whether the miss set has any arithmetic characterisation at all. No modulus to 64 separates it, no growth order does, and it is closed under nothing; the only theorem on offer is the finiteness bound above. Conjecture.
• Whether the 953 tail families are 953 rules. The families are de-duplicated by written set and not by generating rule, so two rules with equal truncated value sets merge and 953 is a lower bound on the number of rules, never an upper one. Conjecture that no bounded union of named families reaches the written tail.
• Whether the multiplicity function, a(n) the number of rows writing n, is worth an entry. It is absent from the dump, but it is a function of the registry's own shape - the tier mix, the cap, the ceiling - and not of n alone, so it is a reading of this instrument and not a sequence of the integers. Conjecture that no reparametrisation of it is submittable.

THE FIELD LADDER

A design is a corner set C of the parity cube {0,1}^dim, its signature is s_j = #{c in C : weight(c) = j} and its weight enumerator is W(t) = sum_j s_j t^j. The fill at odd side 2n + 1 is P(n) = sum_(c in C) (n+1)^(dim - weight(c)) n^weight(c) = (n+1)^dim W(n/(n+1)), the polynomial sequences counts with. This section reads that polynomial as a product of norm forms, one per irreducible factor of W over Q, and censuses the number fields those forms carry. The generator is lab/py/field-ladder, eight verbs norm, ladder, sign, fields, swap, hunter, beyond, polya; the box is the origin-filled box s_0 = 1, 0 <= s_j <= C(dim, j), which is 6, 32, 350, 8712, 526848 signatures at dim 2..6 carrying 2^(2^dim - 1) oriented designs.

• The fill is a product of norm forms, one per irreducible factor of the weight enumerator. Proved. Write W = cont(W) prod_i g_i^(e_i) over Z with each g_i irreducible and primitive, cont(W) the content, and m = deg W. Substituting t = n/(n+1) and clearing (n+1)^dim gives P(n) = (n+1)^(dim-m) cont(W) prod_i g_i*(n)^(e_i) with g*(n) = (n+1)^(deg g) g(n/(n+1)), and g*(n) = lc(g) prod_theta ((1 - theta) n - theta) over the roots theta of g, the norm form of Q(theta) evaluated at n(1 - theta) - theta. The constant is the content and never the leading coefficient: at dim 2 the signature (1,2,0) has W = 1 + 2t, cont(W) = 1, lc(W) = 2 and fill (n+1)(3n+1). On the origin-filled box cont(W) = 1 always, since W(0) = s_0 = 1. The substitution is the Mobius map of matrix [[1,0],[1,1]] in SL_2(Z), so Q(theta/(1 - theta)) = Q(theta), and disc(g*) = disc(g) whenever g(1) != 0, which is deg g* = deg g, automatic for irreducible g of degree at least 2. The identity and its resultant form P(n) = (n+1)^(dim-m) Res_t(W(t), n - t(n+1)) are exact on all 16, 256 and 65536 designs at dim 2, 3, 4 over 12, 64 and 700 signatures; the discriminant equality is checked over the 6, 32 and 350 origin-filled signatures of those dimensions, 0 mismatches on the 2, 25 and 343 factor slots of degree at least 2 (norm).
• The bare form P = s_dim prod_theta ((1 - theta) n - theta) needs the lift (n+1)^(dim - deg W) on exactly half the designs. Proved. deg W < dim iff s_dim = 0 iff the all-odd corner is empty, and C -> C xor {all-odd} is a fixed-point-free involution of the designs, so the count is 2^(2^dim - 1), which is 8 of 16, 128 of 256 and 32768 of 65536 at dim 2, 3, 4 (norm).
• With the origin filled every rational root of W is -1/k and every linear factor of P is (a n + 1). Proved. W has nonnegative coefficients and W(0) = s_0 = 1, so W(x) >= 1 for x >= 0 and no factor of W has a positive real root; W(0) = 1 makes W primitive, so cont(W) = 1 and prod_i g_i(0)^(e_i) = 1, and each g_i(0) is 1 or -1, and g_i(0) = -1 with positive leading coefficient forces a positive real root, so g_i(0) = 1. A linear factor is then 1 + k t with k >= 1 and (1 + k t)* = (k+1) n + 1. This is why the divisor tribe of sequences is the all-rational floor of the ladder and why its factors are (a n + 1) and never (a n + b) with b > 1. With the origin empty the law fails: signature (0,2,1) at dim 2 has W = t^2 + 2t and P(n) = n(3n + 2) (norm).
• The sponge rule fills a divisor form in every dimension, with minimal avatar 4, 24, 240, 3360 at dim 1 to 4. Proved. Keeping the cells with at most one odd coordinate is the signature (1, dim, 0, ..., 0), so W = 1 + dim t and the fill is (n+1)^dim + dim n (n+1)^(dim-1) = (n+1)^(dim-1)((dim+1) n + 1), a product of dim linear factors (a n + 1) of exponent pattern (dim+1, 1, ..., 1); the avatar map of the divisor avatars paper then reads it as the divisor function d(x^n) for x = 2^(dim+1) 3 * 5 * ... * p_dim, the tower 4, 24, 240, 3360, whose first three terms are A005408, A000567 and A103532. Its n = 1 column is (dim+2) 2^(dim-1), A001792, the level-1 cell count of the Menger sponge in every dimension (ladder).
• The rational floor is named integer by integer, one minimal avatar per signature. Verified. The Q column of the ladder below counts the signatures whose W splits into linear factors over Q, 4, 7, 12 at dim 2, 3, 4, and the avatar map is a bijection from them to the exponent patterns of prod_i (a_i n + 1), so each names one smallest integer: the seven at dim 3 are 30, 60, 120, 180, 240, 360, 900 and the twelve at dim 4 are 210, 420, 840, 1260, 1680, 2520, 3360, 5040, 6300, 7560, 12600, 44100. The floor is closed under products, since a Kronecker product of designs multiplies fills and concatenates exponent patterns (ladder).
• Adding one corner to the sponge leaves the rational floor for Q(sqrt 5). Verified. At dim 3 the signature (1,3,1,0), the sponge with one weight-2 corner added, has W = 1 + 3 t + t^2 of discriminant 5 and fill (n+1)(5 n^2 + 5 n + 1), the norm form of the real quadratic field of discriminant 5, which is the first value of the real side of the pure layer (norm, fields).
• The pure quadratic layer realizes exactly the imaginary quadratic fields of discriminant at least -2 dim (dim - 1), and no others. Proved. The pure signature (1, b, c) has W = 1 + b t + c t^2 with 0 <= b <= C(dim,1) = dim and 0 <= c <= C(dim,2), fill P(n) = (n+1)^(dim-2)((1 + b + c) n^2 + (b + 2) n + 1) and discriminant b^2 - 4c on both sides. Every d = 0 or 1 mod 4 with -4C(dim,2) <= d < 0 occurs, by b = 0, c = -d/4 and by b = 1, c = (1-d)/4, and no smaller value occurs since b^2 - 4c >= -4C(dim,2); those are dim(dim-1) values, the count the discriminant staircase row of CLAIMS states. Passing to fields divides out the conductor, and the fields realized are exactly those of fundamental discriminant d_K with abs(d_K) <= 4C(dim,2) = 2 dim (dim - 1), which is 2, 5, 10, 14, 21 fields at dim 2..6 and not dim(dim-1) = 2, 6, 12, 20, 30 (fields).
• The whole census adds no further imaginary quadratic field at dim at most 6. Verified. Over every signature of the box the imaginary quadratic field discriminants are exactly the fundamental discriminants of the window above, so the run is gapless and the first gap is the first fundamental discriminant past 2 dim (dim - 1), namely 7, 15, 31, 43, 67 at dim 2..6 (fields).
• The dim(dim-1) count law counts orders, and under the field reading the same layer counts 2, 5, 10, 14, 21. Proved. The two counts are the same statement read twice: dim(dim-1) counts the values b^2 - 4c that are 0 or 1 mod 4 in [-4C(dim,2), -1], which are discriminants of quadratic orders and not all of them fundamental, -12 = -3 * 2^2 being the first that is not; dividing out the conductor leaves 2, 5, 10, 14, 21 fields at dim 2..6. The two boxes must not be read against each other either: this section sweeps the 526848 origin-filled signatures at dim 6 and its field discriminants stop at -59, while the discriminant staircase row sweeps the 1053696 signatures with s_0 free (lab/py/fill-polynomials) and its order discriminants reach -63, -160 and -899; the deepest, -899, is carried by the origin-empty signature (0,0,15,1,15,0,0), whose W = t^2 (15t^2 + t + 15) has no constant term and so falls outside the box of this section, and -63 is inside it, carried by (1,0,15,16,0,0,0) with W = (1 + t)(1 - t + 16 t^2), an order of the field -7 (fields, sign).
• The real quadratic side is where the census beats the pure layer. Verified. At dim 6 the pure layer gives 5, 8, 12, 13, 17, 21, 24, 28 and the census adds 29 and 33, both from signatures whose weight enumerator has degree above 2, before its first gap at 37 (fields).

• The ladder is a census by the top degree of the irreducible factors of W. Verified. The table counts signatures, exhaustively at dim 2..6, the dim 6 row over 526848 signatures carrying 2^63 oriented designs; the next table counts the same box by oriented design, a signature carrying prod_j C(C(dim, j), s_j) of them (ladder).

• Counted by oriented design the quadratic class splits by field sign, and the real side is the thin one at every dim. Verified. The table counts the 2^(2^dim - 1) origin-filled oriented designs by the top degree of the irreducible factors of W, the quadratic column split three ways by the sign of the discriminant of its quadratic factors: imaginary when every quadratic factor has negative discriminant, real when every one has positive discriminant, mixed when both signs occur, so the three cells partition the quadratic column and 6518 + 105 + 261 = 6884 at dim 4. By signature the split reads 2, 0, 0 at dim 2, 12, 1, 0 at dim 3, 55, 4, 3 at dim 4, 223, 12, 31 at dim 5 and 939, 30, 204 at dim 6. At dim 5 the top two degrees carry a share of 0.85 of the designs while the rational floor carries less than one in a thousand (ladder).

• Field by field, an imaginary quadratic field first appears exactly where the pure layer puts it, at every dim at most 6. Verified. A signature carries the field K when some irreducible quadratic factor 1 + b t + c t^2 of its W has field discriminant d_K, the fundamental part of b^2 - 4c by exact integer arithmetic, agreeing with PARI quaddisc on all 43 distinct values of b^2 - 4c over dim 2..6. The tables count the signatures carrying each field at each dim, the pure column those with deg W = 2; the first dim of each of the 21 imaginary fields is the least dim with 2 dim (dim - 1) >= abs(d_K), the pure law. The fields per dim number 2, 5, 10, 14, 21 imaginary and 0, 1, 3, 6, 10 real, and at dim 4 the three cells of the quadratic class sum to the 6884 of the ladder (sign).
• By oriented design the count per imaginary field falls with abs(d_K); by signature it does not. Verified. The designs column is strictly decreasing down the imaginary table at dim 4, 5, 6 and weakly at dim 3, with -3 and -4, the two imaginary quadratic fields with more than two units, at the top at every dim. By signature -20 outranks -19 at dim 6 (74 against 69), -40 outranks -39 at dim 6 (13 against 12) and -24 outranks -23 at dim 5 (5 against 4): the signature count follows the number of representations d_K f^2 = b^2 - 4c inside the box, which favours d_K = 0 mod 4, so it is not a function of abs(d_K) alone. On the real side the designs column falls at dim 5 and dim 6 and not at dim 4, where 12 carries 21 designs against 15 for 8 (sign).
• The pure law of the first dim fails at dim 7, and the failure is a family. Refuted. (1 - t + 22 t^2)(1 + t) = 1 + 21 t^2 + 22 t^3 is the signature (1,0,21,22,0,0,0,0) of the box at dim 7, its factor has discriminant -87 = -3 * 29, fundamental, and no field below -59 occurs at dim 6, so the field -87 first appears at dim 7 where the pure law says dim 8. The family (1 - t + c t^2)(1 + t) = 1 + (c - 1) t^2 + c t^3 with c = C(dim, 2) + 1 is in the box iff C(dim, 2) + 1 <= C(dim, 3), which is dim >= 6, and carries the order discriminant -(2 dim (dim - 1) + 3), three past the pure window; it is a field discriminant whenever 2 dim (dim - 1) + 3 is squarefree, which over dim 6..15 is -87, -115, -183, -223, -267, -367 at dim 7, 8, 10, 11, 12, 14 and an order at dim 6, 9, 13, 15, where 63 = 7 * 3^2 and 147 = 3 * 7^2. The family is Proved; the whole census at dim 7 is below (beyond).
• Every quadratic factor of a box signature at dim has c <= 1/(2^(1/dim) - 1)^2, so the imaginary side is boxed between the pure window and about four times it. Proved. A root theta of W has 1 = abs(sum_(j >= 1) s_j theta^j) <= (1 + abs(theta))^dim - 1, so abs(theta) >= R = 2^(1/dim) - 1; the two roots of 1 + b t + c t^2 have product 1/c, so c <= 1/R^2 and abs(d_K) <= 4c - b^2 <= 4c; on the real side the root of smaller modulus is 2/(b + sqrt(d)), so b + sqrt(d) <= 2/R. The bound reads c <= 14, 27, 45, 66, 92, 122, 156, 194 at dim 3..10 against the pure C(dim, 2) = 3, 6, 10, 15, 21, 28, 36, 45, and 1/R^2 grows like (dim / log 2)^2; the first dim of an imaginary field lies between the least dim with 4 / R^2 >= abs(d_K) and the pure dim, and the census sits at the pure end (beyond).
• The census at dim 7 and the cut at dim 8..10 find one mechanism on each side, and every field found outside the pure layer sits one dim before its pure dim. Verified. beyond walks every W = (1 + b t + c t^2) h with (b, c) inside the root bound and h in Z[t] of degree at most H, pruned by the partial sums at the roots of the factor, abs(sum_(i <= j) W_i theta^i) <= sum_(i > j) C(dim, i) abs(theta)^i, a pruning an unpruned control at dim 6 and at dim 7, H = 3 reproduces to the last W (6179 and 27809). At H = dim - 2 the walk is exhaustive: at dim 3..6 it returns exactly the 6, 13, 20, 31 fields of the full census, and at dim 7 it is the full quadratic census, 28 imaginary and 14 real fields over 211529 signatures, the real ones every real fundamental discriminant to 44. At dim 8..10 it is a cut, H = 4, 2, 2, a lower bound reading 37 + 21, 49 + 26, 59 + 34. Outside the pure layer it finds -87 at dim 7, -115, -116 at dim 8, -148, -151, -152 at dim 9 and -183, -184, -187 at dim 10 on the imaginary side, and 40, 44, 53, 57, 61, 65, 69, 76, 88, 92 and 85, 89, 93, 97, 101, 105, 109, 113 on the real side; every real one is (dim + 1)^2 - 4c, carried by (1 + (dim + 1) t + c t^2)(1 - t + c' t^2) with c + c' = dim + 1, whose s_1 = dim and s_2 = 0, which is how 29 = (1 + 7t + 5t^2)(1 - t + 2t^2) and 33 = (1 + 7t + 4t^2)(1 - t + 3t^2) arrive at dim 6, and the pure dim of every field listed is one more than the dim it is found at (beyond, sign).
• A quadratic field first appears at its pure dim or one before it, never earlier. Conjecture. The pure dim of a field is the least dim at which a pure signature carries it. The evidence is exhaustive at dim at most 7: 21 imaginary fields at their pure dim and 10 real fields at dim 6, of which 29 and 33 are one before, then -87, 40 and 44 one before at dim 7; the cut at dim 8..10 adds 24 more fields outside the pure layer, all one before, 29 in all. The quadratic fields per dim number 2 + 0, 5 + 1, 10 + 3, 14 + 6, 21 + 10, 28 + 14, imaginary plus real at dim 2..7, exactly (sign, beyond).

• Each run is an initial segment of the table of smallest field discriminants, and the table above is where it stops. Verified. Each cell is the smallest field discriminant of that degree and signature the box misses; a past cell means the run covers the whole table read from the LMFDB, 100 entries for the four large classes and 20 for the rest, and the gap is beyond it; empty means no factor of that class occurs at all. The runs are printed by the generator: at dim 6 the cubic (1,1) run 23, 31, 44, 59, ..., 815 and the quintic (1,2) run 1609, 1649, 1777, ..., 12752 are 100 long and the quintic (3,1) run 4511, ..., 13883 is 20 long (fields).
• No field signature is excluded by the sign condition. Proved. No irreducible factor of W has a positive real root, so every real root of every factor is negative, and this excludes no field: for a field K with generator gamma, the element theta = -1/(gamma + N) with N above every real conjugate of gamma generates K, has all real conjugates negative, and has 1/theta an algebraic integer, so its primitive minimal polynomial has positive leading coefficient, constant term 1 and no positive real root, which are exactly the two conditions a factor of W satisfies.
• The totally real classes are the sparse side of the ladder. Verified. What the census shows is a delay, not an exclusion: signature (3,0) first occurs at dim 5 with the single field 49, (4,0) at dim 6 with the single field 725, and (5,0) does not occur at dim at most 6, where the smallest totally real quintic field is 14641 (fields).

• The box 0 <= s_j <= C(dim, j) reaches every number field of degree d up to a bound B(d, dim). Verified. B is the largest bound with every field of degree d and absolute discriminant at most B reached. The merge over signatures is by field and not by absolute value: 8 is the discriminant of two fields, -8 and +8, and at dim 3 only -8 is reached, which is why B(2,3) is 7 and not 11; a discriminant the table lists twice in one signature, 576, 1008, 1040 and 1088 below B(4,6), is credited only when the box carries two non-isomorphic factors at it, and each of those four does. none means the smallest field of that degree is already missed, and at least 12752 means the run passes the last table entry. The box height is max_j C(dim, j), which is 3, 6, 10, 20 at dim 3..6, and the bounds grow far faster than the height (hunter).

The box is not chosen for a search, it is forced by the geometry: the corner counts of the parity cube give exactly 0 <= s_j <= C(dim, j) with s_0 = 1. That a bounded-height search reaches every field of small discriminant is the classical mechanism behind the tables this section runs against: Hunter 1957 puts a generator of a quintic field of discriminant D at abs(sum rho_i) <= 2 and 5 (sum abs(rho_i)^2)^4 <= 8 abs(D), Pohst 1982 turns a bound of that kind into the computation of the minimum discriminants of sixth degree fields, and the complete lists themselves are the database of Jones and Roberts 2014, whose minimal quintic root discriminants for S_5 with two and with one complex place are 1609^(1/5) and (13 * 347)^(1/5), the 1609 and 4511 opening the runs above. What is new here is the box, the fractal reading and the fill and void involution, not the search.

• Every number field appears at some finite dim, in the pure layer of its own degree. Proved. Take an algebraic integer gamma generating K, of degree d, and an integer N above every real conjugate; the sign lemma's theta = -1/(gamma + N) has primitive minimal polynomial g(t) = prod_i (1 + (gamma_i + N) t), whose coefficient of t^k is the elementary symmetric function e_k(gamma + N), an integer, and e_k(gamma + N) = sum_j C(d - j, k - j) N^(k - j) e_j(gamma) is a polynomial in N with leading term C(d, k) N^k, positive for N large. Then W = g is the pure signature (1, e_1, ..., e_d, 0, ..., 0) of the box at the least dim with e_k <= C(dim, k) for every k, and the census question is only how early. Polya's theorem is the sharper tool for one generator with negative coefficients: a polynomial positive on [0, oo), which every g with g(0) = 1, positive leading coefficient and no nonnegative real root is, has (1 + t)^m g with nonnegative coefficients for m large. The theorem is Polya 1928, unread in the original and read in the form statement of the positive polynomial article and in the univariate restatement of Tan 2018, which carries the exponent of Powers and Reznick 2001, m > (d^2 - d) L(g) / (2 lambda(g)) - d with L = max_j abs(g_j) / C(d, j) and lambda = inf_(t >= 0) g(t) / (1 + t)^d; then W = (1 + t)^m g sits in the box at dim = m + d + e for any e with g_k <= C(d + e, k) at every k, since W_j = sum_k g_k C(m, j - k) <= sum_k C(d + e, k) C(m, j - k) = C(m + d + e, j) by Vandermonde, and the least dim over m and over eight translates N is what the verb prints. The polya verb runs both on the fields the box misses at dim 6, from the polynomial of each LMFDB label with its discriminant recomputed by nfdisc: the totally real 37, 316, 1125, 14641 have exponent m = 0 at the least translate N, since a product of 1 + alpha_i t with every alpha_i > 0 has positive coefficients, and land by dim 9, 10, 9, 11; -87 has m = 1 and lands at dim 7 by (1 + t)(1 - t + 22 t^2), one below its pure dim 8. Each is an upper bound from one generator, and the pure layer already reaches 37 at dim 7 (polya, beyond).
• The discriminants of each degree and signature arrive in order. Conjecture. The evidence is the table above, B(2, 6) = 35, B(3, 6) = 307, B(4, 6) = 1107 and B(5, 6) at least 12752 (hunter), each a gapless initial run against the LMFDB tables, and the quadratic runs at dim 7..10, exhaustive at dim 7 and a cut past it, gapless to 87, 116, 152, 187 on the imaginary side and 44, 69, 77, 101 on the real side (beyond).

• The complement on the parity cube does not swap the two tribes, and the news is the count. Refuted. With V(n) = (2n+1)^dim - P(n) = n Q(n), P+ means P splits into linear factors over Q and V+ means Q does. The swap clause forbids the P+ V+ cell alone, and that cell holds 17 of 128 origin-filled oriented designs at dim 3 and 413 of 32768 at dim 4; the smallest witness is the dim 3 design on corners 000 and 001, fill (n+1)^2 (2n+1) and void core (2n+1)(3n+2). That a witness exists was already known, the self-dual design being named as an exception where the clause is stated; what is new is that the exception is 0.13 of the designs at dim 3. The P- V- cell, 40 and 27270 designs, is not forbidden by the clause and is counted here only to show the census is dominated by it, 0.83 at dim 4 (swap).
• Each field discriminant is computed and guarded, and the runs count discriminants while the bounds count fields. Verified. The field discriminant of a factor is nfdisc of its reversed monic model and the field signature is polsturm of the factor, both from PARI; every value is then guarded against the polynomial discriminant, which must be a square multiple of it with the square root the index, and against 0 or 1 mod 4. All 256179 distinct irreducible factors of degree 2 to 5 over dim 2..6 pass, so no factor is unresolved and no run rests on an unchecked value. Two fields can share a discriminant, so a run counts discriminants; the bounds above are lifted to fields by nfisisom (fields, hunter).
• Nine of the fields the box carries are checked on their own source page. Verified. Each is an irreducible factor of a weight enumerator in the box, read through its reversed monic model: x^2 + x + 1 is LMFDB 2.0.3.1 at -3, x^2 + 3x + 1 is LMFDB 2.2.5.1 at 5, x^3 + x^2 + 2x + 1 is LMFDB 3.1.23.1 at -23, x^3 + 5x^2 + 6x + 1 is LMFDB 3.3.49.1 at 49, x^4 + 2x^2 + 3x + 1 is LMFDB 4.0.117.1 at 117, x^4 + 7x^3 + 13x^2 + 7x + 1 is LMFDB 4.4.725.1 at 725, x^5 + 2x^4 + x^3 + 4x^2 + 4x + 1 is LMFDB 5.3.4511.1 at -4511, x^4 + x^3 + 12x^2 + 19x + 11 is LMFDB 4.0.1225.1 at 1225, and x^5 + 2x^3 + 8x^2 + 4x + 1 is LMFDB 5.1.4429.1 at 4429. The label of a field is degree.r1.abs(disc).index, so 4.0.1225.1 is the totally imaginary quartic of discriminant 1225 = 5^2 7^2 and is a different field from the totally real 1125 the table above misses at (4,0). Every one agrees with nfdisc of the model (fields).
• Factor multiplicity is carried, not divided out. Verified. A repeated factor g^e contributes e copies of its field to the ladder and one discriminant to the census, and the census of distinct factors is by polynomial, so a field with several generators inside the box is counted once per polynomial in the factor counts and once per field in the runs (ladder, fields).

The rest of the tree

• README is the front door: the parity cube, the Kronecker product, and the index of every page.
• sequences is the ledger this page is the complement of: which sequences are known, against which integers are reached.
• method - how a claim here is produced and checked, worked through on the odd-side fill polynomial.
• CLAIMS - where every line above is tagged with its witness and its refutation attempt.
• REFS - every sequence id above resolved to a canonical URL.