Resultants of the dynatomic polynomials of a Chebyshev map

res(Phi_(d,n), Phi_(d,m)) = Phi_(m/n)(dⁿ)^(E) with E = deg Phi_(d,n) (n >= 2) and E = d-1 (n = 1), on all 50 cells of gridSyl (2 <= d <= 7, deg sum <= 320) plus the four larger cells (2,3,9), (5,2,4), (3,2,6), (3,3,6); 11 of the 54 lie in the regime 1 < n | m where the source proves only an inequality; both polynomials are built from Definition 1.1 and the resultant is the determinant of the Sylvester matrix by fraction-free Bareiss elimination, so no step uses the source's Lemma 3.1 or the closed form

Six named exact values in the undetermined regime – res(Phi_(2,2),Phi_(2,4)) = 5², res(Phi_(2,2),Phi_(2,6)) = 21² = 441, res(Phi_(2,3),Phi_(2,6)) = 9⁶ = 531441, res(Phi_(2,2),Phi_(2,8)) = 17², res(Phi_(2,4),Phi_(2,8)) = 17¹2 = 582622237229761, res(Phi_(3,2),Phi_(3,4)) = 10⁶ – and four more at Sylvester sizes 510 to 720: 73⁶, 26²0, 91⁶, 28²4; every one is a perfect deg Phi_(d,n)-th power

Two ways the source's Theorem 4.4(2) falls short of the truth, each pinned by a determinant: at (d,n,m) = (2,3,6) it gives v₃(res) >= 6 while the exact value is 3¹2, a factor 729 short; and at (2,2,6) the resultant is 441 = 3² * 7² while Phi₆(2) = 3, so the prime 7 is outside the set of primes the theorem quantifies over. The same happens at (3,2,6) (prime 13) and (3,3,6) (prime 2)

The source's Theorem 4.4(2), read in product form as Phiₘ(d)^(deg Phi_(d,n)) | res, is an equality exactly when rad(n) divides m/n; the divisibility of the BASES always holds; on the 826 cells with 2 <= d <= 8, n | m, n < m <= 40 the bound is sharp on 448 and strict on 378

The base of the power in two forms: Phi_(m/n)(dⁿ) = (dᵐ - 1) / lcm d^(m/q) - 1: q prime, q | m/n on all 826 cells, whose n = 1 case is Phiₘ(d) = (dᵐ - 1)/lcm d^(m/q) - 1: q | m – the identity the source proves as its Lemma 3.9 (vₚ(Phiₘ(d)) = |S(p,d,m)|)

Engine control: on all 435 pairs 1 <= a < b <= 30 the Sylvester/Bareiss determinant of Phiₐ and Phi_b equals p^(phi(a)) when b/a is a power of a prime p and 1 otherwise, plus textbook values res(x²-1,x²+1) = 4, res(x-1,x-2) = -1, res(Phi₃,Phi₃) = 0 and res(x, Phi_(2,6)) = Phi_(2,6)(0) = 1

The source's Lemma 3.1 (Phi_(d,n) = prod_(ordₖ(d) = n) Phiₖ, with the extra factor x at n = 1) reproduced on 31 cells (2 <= d <= 7, dⁿ <= 2000) from Definition 1.1 against a cyclotomic construction sharing no code, together with the source's one printed factorisation Phi_(2,6) = Phi₆3 * Phi₂1 * Phi₉

The source's Proposition 3.3 (deg Phi_(d,n) = sum_(k|n) mu(n/k) dᵏ; palindromic of even degree for n >= 2) and Corollary 3.10 (Phi_(d,n)(0) = 1 and Phi_(d,n)(1) = Phiₙ(d) for d,n >= 2), reproduced on the same 31 cells

The source's Proposition 3.5 (for n >= 2, Phi_(d,n) irreducible over Q iff dⁿ - 1 is prime) reproduced through the cyclotomic factor count on the 31-cell grid; the only irreducible cells there are (d,n) = (2,2), (2,3), (2,5), (2,7)

The source's Theorem 4.3 (res(Phi_(d,1),Phi_(d,m)) = Phiₘ(d)ᵈ⁻¹) on the 19 cells of gridSyl with n = 1, and its Theorem 4.4(1) (res = 1 iff n does not divide m) in both directions on gridSyl: all 24 cells with n not dividing m give 1, all 7 with 1 < n | m give more than 1

Four refutations: reading Theorem 4.4(2) as an equality (res = Phiₘ(d)^(deg Phi_(d,n))) fails at (2,2,6) (9 against 441) and (2,3,6) (729 against 531441); using deg Phi_(d,1) = d instead of d-1 as the exponent fails at (3,1,2) (64 against 16); dropping the inner power (Phi_(m/n)(d) instead of Phi_(m/n)(dⁿ)) fails at (2,2,6) (49 against 441); and using n instead of deg Phi_(d,n) as the exponent fails at (2,4,8) (17⁴ against 17¹2)

Non-vacuity of the hypotheses on gridSyl: 26 cells have n | m and all 26 have resultant > 1; 24 have n not dividing m and all 24 have resultant exactly 1; 19 of the 26 are the n = 1 case the source settles and 7 are the regime it leaves open

OEIS A059499 (number of m with multiplicative order of 2 mod m equal to n) reproduced for n = 1..15 as the cyclotomic factor count of Phi_(2,n), and OEIS A027375 reproduced for n = 1..15 as deg Phi_(2,n)

res(Φ_(E_d,n),Φ_(E_d,m)) = Φ_q(dⁿ)ᵃ·Φ_q(−dⁿ)ᵇ·Φ_q(d²ⁿ)ᶜ for n | m (q = m/n) and = 1 for n ∤ m, against 56 exact Sylvester determinants of sizes 4–1020 built from Definition 1.1 by iterating E_d; plus a third, independent route (product of pairwise Ψ resultants) agreeing on 35 cells

fifteen named exact values (9 + 4 + 2 across the three files), res(Φ_(E₂,3),Φ_(E₂,6)) = −250047 = −(9³·7³), res(Φ_(E₂,4),Φ_(E₂,8)) = 17⁴·15⁸, res(Φ_(E₃,2),Φ_(E₃,6)) = 91²·73⁴, res(Φ_(E₂,5),Φ_(E₂,10)) = −(33¹⁵·31¹⁵) (46 digits), … each an exact N × N Bareiss determinant, N up to 1020

the counting rule in closed form: with n = 2ᵛ u, u odd, b(d,n) = (D(d^(2ᵛ),u) − δ)/2, δ = [u=1][d odd], c(d,n) = n=1, a = D(d,n) − b − c — verified against the direct root count on all 54 cells with 2 ≤ d ≤ 12, dⁿ ≤ 5000 (and on 267 cells to dⁿ ≤ 3·10⁹ in the prototype)

Φ_(E_d,n) = ∏ Ψ_N: ord±_N(d) = n on 43 cells (2 ≤ d ≤ 16, dⁿ ≤ 260), with the un-Möbius'd form ∏ Ψ_N: N | dᵏ−1 or N | dᵏ+1 = E_(dᵏ)(x) − x on 34

the unit is +1: res(Φ_(E_d,n),Φ_(E_d,m)) is negative exactly when m = 2n and b(d,n) is odd (5 of 26 divisor cells of the grid), and Φ_q(−dⁿ) rewrites to a cyclotomic value at +dⁿ in four cases (4 | q, q odd, q ≡ 2 mod 4, q = 2)

the multiplier assignment as a polynomial divisibility: Ψ_N | (E_(dⁿ))' − λ_N with λ_N = +dⁿ, −dⁿ, d²ⁿ by the residue of dⁿ mod N, on all 34 cells with dⁿ ≤ 130; both signs occur on 31 of them

engine control: the generic dynOf engine pointed at xᵈ reproduces dynatomic d n and the dynatomic-res closed form Φ_(m/n)(dⁿ)^(E_(d,n)) on 35 cells (row DR1 re-derived through the Chebyshev pipeline); Definition 1.1 divides exactly for both maps on 43 cells; and Φ_(E_d,n) ≠ Φ_(xᵈ,n) on every one of those 43 while the degrees agree

E₁... E₇ written out and E_d o Eₑ = E_(de) on 1 <= d,e <= 8, hence E_dᵒᵏ = E_(dᵏ) for dᵏ <= 130 with 2 <= d <= 16

Loper–Werner's two printed tables (V₀ … V₄ and Ψ₁ … Ψ₁0) reproduced exactly from this family's own constructions; OEIS A187360 (coefficients of the minimal polynomial of 2cos(π/n)), all 17 published rows, against psiPoly(2n); deg Ψ_N = φ(N)/2 for 3 ≤ N ≤ 60

four refutations, each by one Sylvester determinant — exchanging a and b at (3,2,4) (640000 vs 409600), the xᵈ closed form at (2,2,4) (25 vs 9), the symmetric split a = b at (2,2,4) (−15 vs 9), dropping the critical fixed points at (2,1,2) (−1 vs −5) — with the census that they differ from the truth on 20, 25, 24 and 19 of the 26 divisor cells

b(2,n) = OEIS A011946 = n·A000048(n) for n = 1..14, computed both by the closed form and by the direct root count; the companion a(2,n) = 0,0,3,4,15,24,63,112,252,480,… is not in OEIS

Loper–Werner's resultant theorem reproduced on all 780 pairs 1 ≤ N < M ≤ 40: res(Ψ_N,Ψ_M) = ±1 unless N | M and M/N is a prime power (84 of 780), where |res| = p^(deg Ψ_N) with the single exception res(Ψ₁,Ψ₂) = 4; and N | M ⟹ ord±_N(d) | ord±_M(d)