Claims · 38 dated claims on Fill polynomials, each with its tag and its witness; newest 2026-09-21.
Conjecture The discriminant staircase is gapless and counts 2(dim-1) new discriminants per dimension: the negative fundamental discriminants (d = 0 or 1 mod 4) carried by the irreducible quadratic factors of fill polynomials at dimension dim form a gapless initial segment of length dim(dim-1), dim = 2 giving -3, -4, dim = 3 adding -7, -8, -11, -12 for 6, dim = 4 adding -15, -16, -19, -20, -23, -24 for 12, with 20 at dim = 5, exhaustive over 17424 signatures, gapless to -40; at dim = 6, exhaustive over 1053696 signatures, the peeled-remainder reading is gapless from -3 to -63 with length 31 and the reading over every irreducible quadratic factor gives length 80 to -160, deepest -899, so the dim(dim-1) count law predicting 30 is Refuted at dim = 6 under both readings while the run stays gapless; gapless and unbounded forces every imaginary quadratic order to appear at some finite dim, the geometric content of the imaginary completeness conjecture; exhaustive at dim = 2..6 by exact factorization. Witness: lab/py/fill-polynomials.
Refuted Seven dim = 4 fill-polynomial remainders labelled irreducible over Q of degree 4 - palindromic signatures (1,0,1,0,1), (1,0,2,0,1), (1,1,2,1,1), (1,2,3,2,1), (1,3,2,3,1), (1,4,2,4,1), (1,4,5,4,1) - split over Q into two centered-polygonal quadratics k n^2 + k n + 1; rational-root peeling proves irreducibility only through degree 3, and no qualify/fail verdict moves since no factor is linear. Witness: lab/py/fill-polynomials.
Proved The odd-side fill is a product of norm forms, one per irreducible factor of the weight enumerator W(t) = sum_j s_j t^j: with m = deg W and W = cont(W) prod_i g_i^(e_i) over Z, primitive irreducible g_i, the fill P(n) = (n+1)^dim W(n/(n+1)) is (n+1)^(dim-m) cont(W) prod_i g_i*(n)^(e_i) with g*(n) = (n+1)^(deg g) g(n/(n+1)) = lc(g) prod_theta ((1-theta) n - theta), the norm form of Q(theta); the constant is the content, never the leading coefficient, and is 1 on the origin-filled box. Exact on all 16, 256 and 65536 designs at dim = 2, 3, 4, in the factored and the resultant form. Witness: lab/py/field-ladder.
Proved The norm form of a factor has the discriminant of the factor: the map t = n/(n+1) is the Mobius map of [[1,0],[1,1]] in SL_2(Z), so Q(theta/(1-theta)) = Q(theta) and disc(g*) = disc(g) for every factor with g(1) != 0, which is deg g* = deg g and is automatic for an irreducible factor of degree at least 2; checked over the 6, 32 and 350 origin-filled signatures at dim = 2, 3, 4, 0 mismatches on the 2, 25 and 343 factor slots of degree at least 2. Witness: lab/py/field-ladder.
Proved The bare product form P = s_dim prod_theta ((1-theta) n - theta) needs the lift (n+1)^(dim - deg W) on exactly half the designs: deg W < dim iff s_dim = 0 iff the all-odd corner is empty, and the complement in that corner is a fixed-point-free involution, 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. Witness: lab/py/field-ladder.
Proved With the origin filled every rational root of W is -1/k and every linear factor of the fill is (a n + 1): W has nonnegative coefficients and W(0) = 1 so no factor has a positive real root, W is primitive with constant term 1 so every irreducible factor has g(0) = 1, and (1 + k t)* = (k+1) n + 1; this is why the divisor tribe is the all-rational floor and why its factors are never (a n + b) with b > 1, and with the origin empty the law fails, signature (0,2,1) at dim = 2 having W = t^2 + 2t and fill n(3n + 2). Witness: lab/py/field-ladder.
Proved The pure quadratic layer realizes exactly the imaginary quadratic fields of discriminant at least -2 dim (dim-1): the pure signature (1, b, c) with 0 <= b <= dim and 0 <= c <= C(dim, 2) has W = 1 + b t + c t^2, fill (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 in [-4C(dim,2), -1] occurs at b = 0 or b = 1 and nothing deeper occurs, and dividing out the conductor leaves exactly the fundamental discriminants of absolute value at most 4C(dim,2) = 2 dim (dim-1), which is 2, 5, 10, 14, 21 fields at dim = 2..6. Witness: lab/py/field-ladder.
Verified The whole origin-filled box adds no further imaginary quadratic field at dim <= 6: the imaginary quadratic field discriminants carried by every irreducible factor of every signature are exactly the fundamental discriminants of absolute value at most 2 dim (dim-1), so the run is gapless and its first gap is the next fundamental discriminant, 7, 15, 31, 43, 67 at dim = 2..6, exhaustive over 6, 32, 350, 8712 and 526848 signatures. Witness: lab/py/field-ladder.
Proved Amendment to the discriminant staircase row: its dim(dim-1) count law is exact under the order reading on the origin-filled box, and what needs amending is the gloss "fundamental", since -12 = -3 * 2^2, -16, -20 and -24 are discriminants of orders and not fundamental. The pure layer gives exactly the dim(dim-1) values 0 or 1 mod 4 in [-4C(dim,2), -1], and exactly 2, 5, 10, 14, 21 fields at dim = 2..6. The two rows sweep different boxes, 1053696 signatures with s_0 free against 526848 with s_0 = 1, so -63, -160, -899 and -60 do not meet: -899 sits at the origin-empty (0,0,15,1,15,0,0), W = t^2 (15t^2 + t + 15). Witness: lab/py/field-ladder.
Proved Imaginary completeness holds for the origin-filled box: every imaginary quadratic order of discriminant d and every imaginary quadratic field of discriminant d occurs as a quadratic factor of a fill polynomial at every dim with 2D(dim-1) >= abs(d), so at every dim at least (1 + sqrt(1 + 2 abs(d)))/2. Witness: lab/py/field-ladder.
Verified Positive-disc completeness does not follow the imaginary law: the pure layer reaches only d = b^2 - 4c <= dim^2 - 4 on the real side, the census at dim = 6 runs 5, 8, 12, 13, 17, 21, 24, 28, 29, 33 and stops at 37 with the last two values coming from signatures of degree above 2, and the real quadratic run is the binding constraint on the degree-2 Hunter bound at every dimension. Witness: lab/py/field-ladder.
Verified The ladder by degree over the origin-filled box: signatures by the top degree of the irreducible factors of W are 4 rational and 2 quadratic at dim = 2, 7, 13, 12 at dim = 3, 12, 62, 130, 146 at dim = 4, 19, 266, 955, 3522, 3950 at dim = 5, and 30, 1173, 7305, 45292, 222437, 250611 at dim = 6, exhaustive over the 6, 32, 350, 8712 and 526848 signatures of the box, the last carrying 2^63 oriented designs; counted by oriented design the dim = 4 row is 504 rational, 6884 quadratic, 13241 cubic and 12139 quartic of 32768. Witness: lab/py/field-ladder.
Verified Cubic ownership: every cubic field of absolute discriminant at most 307 is a norm form of a fill polynomial at dim = 6, the signature (1,1) run being the whole 100-entry table 23 to 815 with no gap and the totally real run 49, 81, 148, 169, 229, 257 stopping at 316, against (1,1) runs stopping at 44, 244 and 652 at dim = 3, 4, 5. Witness: lab/py/field-ladder.
Verified The field-discriminant run of the box stops at a first gap per degree and field signature, reading dim = 3, 4, 5, 6 where the class is nonempty: degree 2 at 15, 31, 43, 67 for (0,1) and 8, 13, 24, 37 for (2,0); degree 3 at 44, 244, 652 and past 815 for (1,1) and 81, 316 for (3,0); degree 4 at 225, 981, past 2156 for (0,2), 400, 1423, 3275 for (2,1) and 1125 for (4,0); degree 5 at 7684 and past 12752 for (1,2), 5783 and past 13883 for (3,1), and nothing at all for (5,0). Witness: lab/py/field-ladder.
Verified The Hunter-type bound of the box: with B(d, dim) the largest bound such that every field of degree d and absolute discriminant at most B is reached, B(2, dim) = 7, 12, 23, 35, B(3, dim) = 31, 44, 76, 307, B(4, dim) = none, 189, 697, 1107 and B(5, dim) = none, none, 5753 with B(5, 6) at least 12752 at dim = 3, 4, 5, 6, the first miss at dim = 6 being 37 at signature (2,0), 316 at (3,0) and 1125 at (4,0), while the box height max_j C(dim, j) is only 3, 6, 10, 20. The merge is by field: 8 is two fields and only -8 is reached at dim = 3, and a discriminant the table lists twice is credited only on two non-isomorphic factors. Witness: lab/py/field-ladder.
Proved No field signature is excluded by the sign condition: no irreducible factor of W has a positive real root, and for any field K with generator gamma the element theta = -1/(gamma + N) with N above every real conjugate generates K, has all real conjugates negative and has 1/theta an algebraic integer, so its primitive minimal polynomial meets both conditions a factor of W meets. Witness: lab/py/field-ladder.
Verified The totally real classes are the sparse side of the ladder: 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 <= 6, the smallest totally real quintic field being 14641. Witness: lab/py/field-ladder.
Verified Every field discriminant of the census is computed by PARI nfdisc on the reversed monic model and guarded against the polynomial discriminant, which must be a square multiple of it, and against 0 or 1 mod 4: all 256179 distinct irreducible factors of degree 2 to 5 over dim = 2..6 pass, so no run rests on an unchecked value and no factor is unresolved. A run counts discriminants, since two fields can share one, the first repeat inside a printed run being 576 twice in degree 4 signature (0,2); the Hunter bounds are lifted to fields by nfisisom. Witness: lab/py/field-ladder.
Conjecture Every number field appears at some finite dim and its discriminants arrive in order: the box s_0 = 1, 0 <= s_j <= C(dim, j) reaches every field of degree 2, 3, 4 of absolute discriminant at most 35, 307, 1107 at dim = 6 and every quintic field of the tables read, to 12752, each a gapless initial run, and the two necessary conditions on a factor, constant term 1 and no positive real root, are met by a generator of every field. Witness: lab/py/field-ladder.
Refuted The swap clause, that when one side factors completely over Q the other carries an irreducible factor, forbids the P+ V+ cell alone, and that cell is not empty: over origin-filled oriented designs the table by fill split and void-core split reads 17, 4, 67, 40 at dim = 3 and 413, 91, 4994, 27270 at dim = 4, so 0.13 of the designs at dim = 3 sit in the forbidden cell, the smallest on corners 000 and 001 with fill (n+1)^2 (2n+1) and void core (2n+1)(3n+2). The exception class was already named with the clause; the count is the news, and the P- V- cell is not forbidden. Witness: lab/py/field-ladder.
Proved The sponge rule, keep a cell with at most one odd coordinate, is the signature (1, dim, 0, ..., 0) and fills (n+1)^dim + dim n (n+1)^(dim-1) = (n+1)^(dim-1)((dim+1) n + 1) at odd side 2n+1, a product of dim linear factors of exponent pattern (dim+1, 1, ..., 1), hence d(x^n) for x = 2^(dim+1) 3 * 5 * ... * p_dim, the tower 4, 24, 240, 3360 at dim 1..4, with n = 1 column (dim+2) 2^(dim-1) = A001792. Witness: lab/py/field-ladder, divisor-avatars. Claims heading: Divisor avatars.
Verified The Q column of the ladder over the origin-filled box, 4, 7, 12 signatures at dim 2, 3, 4, is in bijection with the exponent patterns of prod_i (a_i n + 1) under the avatar map, so the rational floor is named integer by integer: 30, 60, 120, 180, 240, 360, 900 at dim 3 and 210, 420, 840, 1260, 1680, 2520, 3360, 5040, 6300, 7560, 12600, 44100 at dim 4, and it is closed under products. Witness: lab/py/field-ladder. Claims heading: Divisor avatars.
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. Witness: lab/py/field-ladder. Claims heading: Fill polynomials.
Verified Over the 350 origin-filled signatures of the box at dim 4 the 6884 quadratic oriented designs split into 6518 imaginary, 105 real and 261 mixed, carried by 55, 4 and 3 signatures. Witness: lab/py/field-ladder verb ladder.
Verified Over the 8712 origin-filled signatures of the box at dim 5 the 2147483648 oriented designs read 1209703 rational, 102969641 quadratic (92090824 imaginary, 85372 real, 10793445 mixed), 213022933 cubic, 956166567 quartic and 874114804 quintic by top irreducible factor degree. Witness: lab/py/field-ladder verb ladder.
Verified Counted by oriented design the quadratic class of the origin-filled box splits by field sign at every dim 2..6: imaginary, real, mixed read 3, 0, 0 at dim 2, 60, 3, 0 at dim 3, 6518, 105, 261 at dim 4, 92090824, 85372, 10793445 at dim 5 and 37730659810317711, 282117813525, 4034425906484660 at dim 6, the three cells partitioning the quadratic column, 6884 at dim 4; by signature 2, 0, 0 then 12, 1, 0 then 55, 4, 3 then 223, 12, 31 then 939, 30, 204. Witness: lab/py/field-ladder verb ladder.
Verified Field by field over the origin-filled box, a signature carrying a quadratic field when an irreducible quadratic factor 1 + b t + c t^2 of W has that field discriminant, the fundamental part of b^2 - 4c by exact integer arithmetic agreeing with PARI quaddisc on all 43 distinct values: the fields per dim number 2, 5, 10, 14, 21 imaginary and 0, 1, 3, 6, 10 real at dim 2..6, and 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. Witness: lab/py/field-ladder verb sign.
Verified By oriented design the count per imaginary quadratic field falls strictly with abs(d_K) at dim 4, 5, 6 and weakly at dim 3, with -3 and -4 at the top at every dim; by signature it does not, -20 outranking -19 at dim 6 (74 against 69), -40 outranking -39 at dim 6 (13 against 12) and -24 outranking -23 at dim 5 (5 against 4); on the real side the designs column falls at dim 5 and 6 and not at dim 4, where 12 carries 21 designs against 15 for 8. Witness: lab/py/field-ladder verb sign.
Refuted The pure law of the first dim of an imaginary quadratic field, the least dim with 2 dim (dim - 1) >= abs(d_K), fails at dim 7: (1 - t + 22 t^2)(1 + t) = 1 + 21 t^2 + 22 t^3 is the origin-filled signature (1,0,21,22,0,0,0,0) at dim 7, its factor has fundamental discriminant -87 = -3 * 29, and no field below -59 occurs at dim 6, so -87 first appears at dim 7 where the pure law says dim 8. Witness: lab/py/field-ladder verb beyond.
Proved The family (1 - t + c t^2)(1 + t) = 1 + (c - 1) t^2 + c t^3 with c = C(dim, 2) + 1 lies in the origin-filled 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 4 C(dim, 2); 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. Witness: lab/py/field-ladder verb beyond.
Proved Every root theta of an origin-filled W at dim has abs(theta) >= 2^(1/dim) - 1 = R, since 1 = abs(sum_(j >= 1) s_j theta^j) <= (1 + abs(theta))^dim - 1; so every irreducible quadratic factor 1 + b t + c t^2 has c <= 1/R^2, every imaginary field of the census has abs(d_K) <= 4c <= 4/R^2, and every real one has 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. Witness: lab/py/field-ladder verb beyond.
Verified The walk over 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 and reproduced to the last W by an unpruned control at dim 6 and dim 7 H = 3 (6179 and 27809 W), is exhaustive at H = dim - 2: it returns exactly the 6, 13, 20, 31 quadratic fields of the full census at dim 3..6 and the full quadratic census at dim 7, 28 imaginary and 14 real fields over 211529 signatures, the real ones every real fundamental discriminant to 44 with 53 the first miss; at H = 4, 2, 2 for dim 8, 9, 10 it is a cut reading 37 + 21, 49 + 26, 59 + 34. Outside the pure layer it finds -87 at dim 7, then -115, -116, then -148, -151, -152, then -183, -184, -187 on the imaginary side and 40, 44, then 53, 57, 61, 65, 69, then 76, 88, 92, then 85, 89, 93, 97, 101, 105, 109, 113 on the real side, every real one (dim + 1)^2 - 4c carried by (1 + (dim + 1) t + c t^2)(1 - t + c' t^2) with c + c' = dim + 1, the mechanism behind 29 = (1 + 7t + 5t^2)(1 - t + 2t^2) and 33 = (1 + 7t + 4t^2)(1 - t + 3t^2) at dim 6; all 29 fields found outside the pure layer at dim 6..10 have pure dim one more than the dim they are found at. Witness: lab/py/field-ladder verb beyond.
Conjecture A quadratic field first appears in the origin-filled box at its pure dim, the least dim at which a pure signature (1, b, c) carries it, or one dim before it, never earlier: exhaustive at dim 2..7 (21 imaginary fields at their pure dim, 29 and 33 one before at dim 6, -87, 40, 44 one before at dim 7) and every one of the 24 further fields of the cut at dim 8..10 one before; 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. Witness: lab/py/field-ladder verbs sign and beyond.
Proved Every number field K of degree d appears in the origin-filled box at some finite dim, in the pure layer of its own degree: for an algebraic integer generator gamma and an integer N above every real conjugate, the sign-lemma generator theta = -1/(gamma + N) has minimal polynomial g(t) = prod_i (1 + (gamma_i + N) t) with coefficients e_k(gamma + N) = sum_j C(d - j, k - j) N^(k - j) e_j(gamma), polynomials in N with leading term C(d, k) N^k, positive for N large, so W = g is the pure signature (1, e_1, ..., e_d, 0, ...) at the least dim with e_k <= C(dim, k); Polya's theorem gives the sharper route for one generator, (1 + t)^m g nonnegative for m past the Powers-Reznick exponent (d^2 - d) L(g) / (2 lambda(g)) - d and in the box at dim = m + d + e for any e with g_k <= C(d + e, k), by Vandermonde. Witness: the note, and lab/py/field-ladder verb polya for the exponents.
Verified From the LMFDB polynomial of each field with nfdisc recomputed, the sign-lemma generator of the totally real fields 37, 316, 1125, 14641 the box misses at dim 6 has Polya exponent 0 at its least translate, a product of 1 + alpha_i t with every alpha_i > 0, and fits the box by dim 9, 10, 9, 11, while -87 has exponent 1 and fits at dim 7 by (1 + t)(1 - t + 22 t^2); each is an upper bound from one generator, and the pure layer reaches 37 at dim 7. Witness: lab/py/field-ladder verb polya.
Conjecture The discriminants of each degree and field signature arrive in order: the Hunter bounds B(2, 6) = 35, B(3, 6) = 307, B(4, 6) = 1107, B(5, 6) at least 12752 are gapless initial runs, and the quadratic runs of the cut at dim 7..10 are gapless to 87, 116, 152, 187 on the imaginary side and 44, 69, 77, 101 on the real side. Witness: lab/py/field-ladder verbs hunter and beyond.
Verified Amendment to the box comparison: the origin-filled box at dim 6 carries the order discriminant -63 by (1,0,15,16,0,0,0) with W = (1 + t)(1 - t + 16 t^2), an order of the field -7, so it is the field discriminants of the box that stop at -59 while its order discriminants reach -63; the origin-free box of the staircase row reaches -63, -160 and -899 as orders. Witness: lab/py/field-ladder verbs fields and sign.
Proved Cross-reference for the line of this heading that conjectures every number field appears at some finite dim and its discriminants arrive in order: the existence half is Proved on the integers note, THE FIELD LADDER, by the translate generator g(t) = prod_i (1 + (gamma_i + N) t) with positive coefficients e_k(gamma + N) for N large, a pure signature of the box at the least dim with e_k <= C(dim, k), and the order half stays a Conjecture with the Hunter bounds and the quadratic runs to dim 10 as evidence. Witness: the note, and lab/py/field-ladder verb polya.