field-ladder

• Reads the odd-side fill of a base-2 design as a product of norm forms and censuses the number fields those forms carry.
• Objects: a design is a corner set C of {0,1}^dim, its signature is s_j = #{c in C : weight(c) = j}, its weight enumerator is W(t) = sum_j s_j t^j, its fill at an odd side is P(side) = sum_(c in C) (side+1)^(dim-weight(c)) side^weight(c) = (side+1)^dim W(side/(side+1)).
• The box is the origin-filled box s_0 = 1, 0 <= s_j <= C(dim, j): 6, 32, 350, 8712, 526848 signatures at dim 2..6, carrying 2^(2^dim - 1) oriented designs.
• norm checks P(n) = (n+1)^(dim - deg W) cont(W) prod_i g_i*(n)^(e_i) over W = cont(W) prod_i g_i^(e_i), with cont the content and g*(n) = (n+1)^(deg g) g(n/(n+1)), on every design at dim 2..4; it checks the resultant form P(n) = (n+1)^(dim - deg W) Res_t(W(t), n - t(n+1)) on the same designs, disc(g) = disc(g*) over the origin-filled signatures on every irreducible factor of degree at least 2, where g(1) != 0 holds automatically, and the linear-factor law that every linear factor of an origin-filled fill is (a n + 1). The constant is the content and never the leading coefficient: W = 1 + 2t at dim 2 has lc(W) = 2 and fill (n+1)(3n+1).
• ladder factors W over Q for every signature of the box and counts signatures and oriented designs by the top degree of the irreducible factors, exhaustively at dim 2..6.
• fields takes every distinct irreducible factor of degree 2 to 5, batches them into one PARI script, and reads the field discriminant from nfdisc of the reversed monic model x^d g(1/x), the field signature from polsturm and the polynomial discriminant from poldisc. Every value is guarded: it is accepted only if it is 0 or 1 mod 4 and divides the polynomial discriminant with a square quotient, the square root of that quotient being the index [O_K : Z[1/theta]]. It prints per dim, degree and field signature the run against the table of smallest field discriminants and the first gap. A value failing the guard would be carried as an unresolved factor and could only widen a gap, never close one; 0 of 256179 fail.
• swap counts oriented origin-filled designs at dim 3, 4 by whether the fill splits into linear factors over Q and whether the void core does, V(n) = (2n+1)^dim - P(n) = n Q(n).
• hunter prints, per degree and dim, the largest B with every field of that degree and absolute discriminant at most B reached by the box, and the first miss with its field signature. The merge over signatures is signature aware, so one absolute value carried by two signatures is two fields; a value the table lists twice in one signature is two fields and is credited only when the box carries two non-isomorphic factors at it, decided by PARI nfisisom over at most 24 carriers. The bound over discriminants rather than fields is printed beside it, and the merged table is valid only below the smallest per-signature table maximum, which is printed.
• Tables: the quadratic tables are generated here from the definition of a fundamental discriminant; the cubic, quartic and quintic tables are the smallest field discriminants by signature read from the LMFDB, 100 entries each for degree 3 and 4 signatures (1,1), (3,0), (0,2), (2,1) and degree 5 signature (1,2), 20 entries for the others.
• PARI is required for fields and hunter: gp must be on PATH. The run is one gp batch per 20000 factors, and the guard above is what stands between a machine answer and a printed one.
• The census counts discriminants, not fields: two fields can share a discriminant, the first repeat inside a printed run being 576 twice in degree 4 signature (0,2). hunter lifts its bound to fields with nfisisom; a run in fields is a run of discriminants.
• Domain: all 2^(2^dim) designs at dim 2..4 for norm; all 526848 signatures at dim 6 for ladder; all 256179 distinct irreducible factors of degree 2 to 5 over dim 2..6 for fields and hunter.

RUN

uv run python research/lab/py/field-ladder/ladder.py norm ladder fields swap hunter

• Verbs may be given in any combination and run in order in one process, which is how fields and hunter share one field census.
• Measured on 8 workers: norm 1 s, swap 1 s, ladder 25 s, fields and hunter together 35 s. The whole study is inside a wall-clock cap of 10 minutes, and no verb may cross it.

WITNESSES

• integers, THE FIELD LADDER - the norm-form law, the lift on the half of designs with s_dim = 0, and the linear-factor law: norm.
• integers, THE FIELD LADDER - the ladder by degree at dim 2..6: ladder.
• integers, THE FIELD LADDER - the oriented-design counts of the ladder, the quadratic split by field sign at dim 4 (6518 imaginary, 105 real, 261 mixed of 6884) and the dim 5 row (1209703, 102969641, 213022933, 956166567, 874114804 of 2^31): ladder.
• integers, THE FIELD LADDER - the quadratic layer, the imaginary run and its first gaps 7, 15, 31, 43, 67, the real run and the two extras 29, 33 at dim 6: fields.
• integers, THE FIELD LADDER - the field-discriminant runs by degree and signature, the Hunter bounds B and the field-level merge behind B(2,3) = 7: fields and hunter.
• integers, THE FIELD LADDER - the fill and void table 17, 4, 67, 40 at dim 3 and 413, 91, 4994, 27270 at dim 4: swap.
• sequences, THE ODD-SIDE FILLS - the divisor tribe is the all-rational floor of the ladder: norm and ladder.