The multipliers

The comparison behind the low-rank threshold, carried to the ranks that DO fail, does not survive. Over the census's own 370 cells at ranks 5 to 18, only 16 of the 82 failures deep enough for the champion to be a pure product at all have it among the cheapest pure cofactors of their cell, and at twelve of the fourteen ranks it is among the cheapest at no depth of its column where it is available at all. Where it is — ranks 5 and 6, at 12 and 4 depths, each of them a run reaching the top of its column from depth 19 and 27 — the rank has already failed, at 18 and 22. So that threshold says nothing about where a failing rank's band BEGINS.

What is there instead is a trade. A cyclotomic polynomial Φ_d is the minimal polynomial of a primitive d-th root of unity, and x^d − 1 is the product of Φ_e over the divisors e of d — so a pure product is a product of cyclotomics, and a pure cofactor carries a multiset of them read off its parts with no factorisation. Divide any cofactor by every cyclotomic factor and every power of x it carries, and call what is left its residual. Across this chart the residual of the ONE minimiser the census exhibits at a cell is one of five — a cell can carry others, with other residuals — and it is 1 at exactly the cells that do not fail. Three of the five are multipliers in their own right — A = 2 + 4x + 5x² + 4x³ + 2x⁴, B = 2 + 3x + 2x², and C = 3 + 5x + 3x² — the other two being 1 and the product A·B. That census is the flattened minimum on the unit circle; what it is used for here is the low edge.

The smallest specimen carries the whole shape. At rank 5 and depth 18 — width 23 — the cheapest pure product has cofactor (1+x)²(1+x+x²) and height 4420, against the cell's h of 3638, whose minimiser has cofactor (1+x)²(2 + 3x + 2x²). Both heights are the whole polynomial's, the cofactors' being 4 and 10. The cyclotomic factor 1+x+x² is gone, and B, of the same degree, stands where it was.

That makes a FAMILY and not only a description. Fix a cell; range over every pure cofactor that fits it, over every sub-multiset of that cofactor's cyclotomic factors, and over every product of A, B and C whose degree equals the degree of the factors dropped — the empty drop with the empty product included. Write hs for the least height over what comes out. Equal degrees keep each member inside the degree budget the pure cofactor already met, so every member is a nonzero vector of that cell's lattice and h ≤ hs; and the empty trade puts the whole pure family inside, so hs is at most the pure bound. Between those two, hs = h at all 83 failing cells, with no BASIS REDUCTION anywhere in the family — the lattice algorithm the census reads h with — and at all 83 the vector that algorithm returns is itself a member of the family up to sign, counted rather than assumed: 54 of the 83 carry a negative leading coefficient and none of them a factor of x.

Read against the cheapest pure product at the same cell instead of against every pure cofactor, the match is only partial: the minimiser's cyclotomic multiset divides that one's at 78 of the 83, and the degree dropped matches the residual's at 73. Both counts are upper bounds by at most six, six cells offering several cheapest pure products that classify differently, with the reading taken the one most favourable to the trade. The ten cells the partial reading misses are reached from a different pure cofactor, which is the distance between a description and a construction. And the three multipliers are not a degree accident: three polynomials matching them in degree, in height and in value at 1, and differing in not being reciprocal — a polynomial is reciprocal when its coefficient list reads the same backwards — reach h at none of the 83 and match the pure bound at all 83. They never fire.

Both inequalities are needed and each does a different job. Because hs is at most the pure bound always, those are the only two cases; and because h is at most hs always, the strict one puts h under the pure bound too. So hs strictly below the pure bound CERTIFIES a cell as failing with no reduction in it, while hs equal to it certifies nothing, the cell being free to fail below both. That certificate is cheap where the reduction is dear: along the rank-22 column at widths 30 to 52 the reduction runs from 0.03 seconds to about 7 while the family holds near 2.

What none of it survives is leaving the chart, and what breaks there is the multipliers and not the trade. Each of the ranks 19 to 26 has its column scanned upward from depth 2 to its first failing cell — depth 26, 23, 25, 30, 22, 29, 31 and 27, at widths 45, 43, 46, 52, 45, 53, 56 and 53, none of them guessable from the others — and each of those cells is asked whether its residual is a product of the multipliers already in hand. At seven of the eight it is: A alone at six and A·B at rank 25. Rank 22's is not, and it mints a fourth, D = 3 + 9x + 15x² + 17x³ + 15x⁴ + 9x⁵ + 3x⁶, whose admission drops hs from 44108 to that cell's h of 42222. So the seven is never a statement about one fixed set: at three of them the set in hand holds three multipliers and at four it holds four, rank 22 having minted D in between.

And the first cell of a rank is the wrong sample. Taking the next three failing cells of each of those eight columns, 24 more, TWO need a multiplier the four cannot make, and they sit at different ranks: rank 22 at depth 32, width 54, where h = 108376 against hs = 110932 and a pure bound of 125135; and rank 26 at depth 36, width 62, where h = 402038 against 421338 and 450551. Both carry the same residual, E = 3 + 11x + 24x² + 37x³ + 43x⁴ + 37x⁵ + 24x⁶ + 11x⁷ + 3x⁸, of degree 8 and height 43, divisible by none of A, B, C and D; admitting it drops hs to h at both. So the set of multipliers grows with the CELL and not with the RANK, and a fifth recurring at two ranks four apart is what separates a multiplier from a one-cell accident.

Walked further — ranks 27 to 29, each column read the same way at its first four decided failing cells — the list still does not close. Rank 27's first failing cell mints a sixth, F = 3 + 10x + 20x² + 29x³ + 33x⁴ + 29x⁵ + 20x⁶ + 10x⁷ + 3x⁸, of height 33 with every root back on the unit circle, which then recurs at all three ranks; and rank 28's second failing cell mints a seventh, G = 3 + 6x + 8x² + 6x³ + 3x⁴, of height 8. Each admission drops hs to h at its cell, and a new multiplier has appeared in every stretch of ranks yet walked.

E also leaves the description the chart's own five residuals fit inside. Substituting y = x + 1/x turns it into 1 + 4y + 12y² + 11y³ + 3y⁴, whose exact count of real roots in (−2, 2) — the interval y = 2cos θ carries onto the unit circle — is two out of four, so four of E's eight roots lie off that circle, where every residual the chart's CENSUS exhibits — the one minimiser per cell above, not everything a cell can carry — has all of them on it.

And the certificate's silence tracks the set in hand and not the cell. Over the eight first failing cells of ranks 19 to 26 it fires at the seven the set in hand reaches and is silent at rank 22's, the one that mints D; at the two minting cells of ranks 27 to 29, carrying the larger set, it fires — hs strictly between h and the pure bound, the set improving on pure without reaching h. Read forward it is a cheap sufficient condition for a cell to fail; read as a scanner for new multipliers it is the wrong instrument in both states, quiet certifying nothing and firing not clearing a cell.

Scope. Rule, exhaustive over the census's rectangle, for hs = h at the 83 failing cells and for the champion counts at ranks 5 to 18. Observation for the residual named at any one cell: the reduction returns ONE minimiser and a cell can carry several, with different residuals. Off the chart, the eleven first failing cells and the cells above them are a rule over the depths scanned — each column decided from depth 2 upward, a first failure claimed only where every cell below it was decided, one cell of rank 28 undecided at its search budget with decided cells on both sides — and at each minting cell the two statements separate. That the multipliers in hand cannot reach h there is hs > h, a statement about the family's whole minimum with no minimiser in it, and so is the admission closing the gap; that the residual IS the polynomial named is read off one vector. None of this is a closed form and none of it is an algorithm: no finite set of multipliers is known to make every residual the lattice can exhibit — the walk mints them one cell at a time and nothing so far predicts the next — and that is a question about the lattice and never about the ambient class, which is infinitely generated. Every root on the unit circle describes the five this chart exhibits and does not describe E, so what survives outside the chart is the weaker statement that a minimising cofactor splits as cyclotomics times a residual.

verifiers: explore_flatten_near.py, explore_flatten_swap.py, explore_flatten_class.py, explore_flatten_gen.py (the walks past the chart's corner)

Six of the seven multipliers fit one description: non-monic, primitive — the coefficients sharing no common factor — carrying no cyclotomic factor, and with every root on the unit circle. The non-monic condition costs nothing to impose: at positive degree a monic integer polynomial with all its roots on the circle is a product of cyclotomics (Kronecker), so cyclotomic-free already forces it. Reading the seven as a short list makes that class look small, and it is not: it holds an infinite one-parameter family at the lowest degree it occupies at all. Set W_c = c + (2c−1)x + cx² for an integer c ≥ 2. Each W_c is reciprocal; primitive, since c and 2c−1 share no factor; irreducible over the rationals, its discriminant 1 − 4c being negative; non-monic and therefore not cyclotomic; and both its roots lie on the unit circle, since under y = x + 1/x it reduces to cy + 2c−1, whose one root −(2c−1)/c lies in (−2, 2). The first two members of the family ARE multipliers — W₂ is B and W₃ is C — so the family is the members' own, continued past them. And no finite set generates it: the W_c are pairwise distinct primitive irreducibles with positive leading coefficients, hence pairwise non-associate.

Degree 2 is the lowest the class reaches: degree 1 admits no member at all, ax + b with its root on the circle forcing b = ±a and the polynomial a(x ± 1), whose factor x ± 1 is cyclotomic. So the short list was never evidence that the class is small. Seven multipliers minted across every cell examined, against an infinite family at degree 2, puts the scarcity in the LATTICE: what stays open is whether the class the lattice EXHIBITS is finitely generated — whether some finite set of multipliers makes every residual a minimiser can carry. The family also says the walk is not enumerating its class in order: W₄ = 4 + 7x + 4x², of height 7 — between C's 5 and D's 17 — has never been exhibited.

Scope. Property: every membership fact and the non-generation are proved, for every c ≥ 2. E is the one multiplier the description does not cover, four of its eight roots lying off the circle; what covers all seven is only the split of a minimising cofactor as cyclotomics times a residual.

verifiers: explore_flatten_deep.py, explore_flatten_gen.py

The first six multipliers' values at 1 — 17, 7, 11, 71, 193 and 157 — are all prime, and the seventh's is not: G(1) = 26 = 2·13. A primality law was never available anyway, the value at 1 being the single point this subject has no constraint at. Every bound it owns comes from evaluating a vector at a root of unity — P(z) = (z−1)^JQ(z) with |P(z)| ≤ Mh forces |Q(z)| ≤ Mh/|z−1|^J — and at z = 1 the flattening IS the constraint, P(1) = 0 saying nothing about the cofactor. What the first six share instead sits at −1: every one takes the value 1 there, exactly — and with it every residual a product of them makes, the value being multiplicative. G takes the value 2 there, and it is the multiplier minted furthest from the chart.

Read over every minimiser rather than the one the reduction returns — holding the enumeration radius fixed at the winning height collects a cell's whole minimiser set — the shared value is a rule with a threshold. At every one of the 462 cells of this chart with height 4 or more, every vector of least height has a residual with the value 1 at −1: exhaustive, no cell unreached, the 20 cells at height exactly 4 decided under a twentyfold raise of the per-cell search budget. An earlier reading of the threshold as height 5 was that budget's edge — two cells the original budget had left unexamined — and never a property.

One step lower the statement is false, so the threshold is exact. Sixteen cells carry a residual with another value at −1, every one of them of height 1, 2 or 3, and across them the value takes eleven values — every integer from −5 to 5 except 0, which cannot occur, the residual having every factor of x + 1 divided out, and 7. And the class count is the wrong measure of that failure: one cell of height 1 carries 232 distinct residual classes by itself, 121 of the 163 that lose the value, so what the low cells report is degeneracy and not a spread.

The rule deflates the primality, and G ends it. A residual's values at 1 and −1 differ by twice its odd-index coefficient sum, so the value 1 at −1 forces the value at 1 ODD — and six odd numbers between 7 and 193 all being prime sits near one in twenty-four at the prime density among odds. No law about the class could supply the rest — W_c(1) = 4c − 1 is composite at eleven of c = 2 to 24, the first being W₄'s 15 — and what briefly survived, the possibility that the lattice never exhibits a multiplier of composite value, is refuted: it exhibits G.

And the value is independent of the circle. Minimisers whose residuals leave the unit circle stand at 24 cells and reach height 6, above the failures' top at 3 — so cells leave the circle and keep the value, and E, off the circle with E(−1) = 1, is the first instance of a separation and not an exception. Whether the value is FORCED on the chart is open, and the hypothesis of any derivation now binds on both sides: the statement is false at height 3 and below on the chart, and false off the chart at G's cell, of height 67323 — so no hypothesis in the height alone can state it, and both handles in hand are height- and position-free, the minimiser's primitivity and the bound |Q(−1)| ≤ Mh/2^J, the latter discharged by the cyclotomic part by itself wherever x + 1 divides the cofactor.

Scope. Rule for height 4 and above, at every minimiser of every one of the census rectangle's 462 cells at that height, proved at none — the rectangle's and not the plane's, one off-chart cell exhibiting the value 2. Observation for the sixteen failing cells and for the 24 that leave the circle, both read off a narrower pass — width to 40, depth to 20 — that closes 405 of its 550 cells, the widened chart having closed only 105 of the 233 cells below height 4. The multiplicativity and the oddness forcing are properties.

verifiers: explore_flatten_endvalue.py, explore_flatten_deep.py, explore_flatten_recur.py (the widened chart whose two cells at height 4 decided the threshold), explore_flatten_gen.py (the off-chart cell)