Which APN power functions are componentwise Walsh uniform? The classification at n=12,13,14,15, and how badly the others fail

The field model is checked, not assumed: F₂[x]/(redPoly n) is a field of order 2ⁿ for 5 <= n <= 15, the shift-and-xor product is commutative, associative and distributive on all 2³ⁿ triples for n <= 7, the parity-mask trace formula holds at every point, the tabulated generator generates F_(2ⁿ)^*, the log-indexed power table x |-> xᵈ equals square-and-multiply, and the two evaluations of T(a) = sum_(s in S)(-1)^(Tr(as)) agree at every a

Carlet's Table 2 (ePrint 2017/528, p. 28) reproduced for 5 <= n <= 11 in a stronger, exhaustive form: for every exponent d least in its cyclotomic class with |S₁(xᵈ)| = 2ⁿ⁻¹, the verdict of the full sweep over all 2ⁿ - 2 values of rho – 11 classes at n = 7, 13 at n = 9, 23 at n = 11 – together with the non-power APN function x³ + Tr(x⁹) at n = 6,7,8,9,11 and four directions a

The classification at n = 12 and n = 13, exhaustive over every APN exponent class: at n = 12 all three classes (3, 33 Gold; 159 Kasami) are CWU; at n = 13 exactly twelve of the twenty-seven classes are – the six Gold classes 3,5,9,17,33,65, the five Kasami classes 13,57,191,241,287, and 127 = 2^((13+1)/2)-1 – and the other fifteen are not, Welch (67), Niho (71), the inverse function (4095) and twelve compositional inverses among them

The classification at n = 14 and n = 15: the complete list of APN exponent classes (5 at n = 14, 21 at n = 15, 27 at n = 13) computed without any sweep, plus every verdict – all five classes CWU at n = 14, and at n = 15 exactly eight (Gold 3,5,17,129; Kasami 13,241,383; and 255 = 2^((15+1)/2)-1) against thirteen that fail, including Welch 131, Niho 2033, Dobbertin 3657 and the inverse function 16383

How badly each non-CWU class fails: the exact count of rho in F_(2ⁿ){0,1 at which the equidistribution is violated, for all three failing classes at n = 7, all seven at n = 9, Dobbertin at n = 10, all thirteen at n = 11 and all fifteen at n = 13 – 39 (count corrected 2026-08-28: 3+7+1+13+15; Deviation.lean has exactly 39 cwuOff theorems) numbers, from 84 out of 126 (Niho at n = 7) to 8190 out of 8190 (Niho at n = 13)

The same verdicts with no character sum anywhere: N_F(1,rho) counted directly over Sₐ³ in |S|² steps, giving single-rho refutations at n = 7, 9, 11, 13, 15 (e.g. the inverse function at n = 13 has N(1,2) = 8388630 against a target of 8388608) and full definition-level rho-sweeps reproducing the failure counts at n = 7, 9 and the CWU verdicts of Kasami and the Gold compositional inverse at n <= 11

Translation and scaling invariance of the count, with no finite data: N is unchanged by translating the set (so kasami-apn's Delta_F = 1 + S₁(F) and S₁(F) have the same count), by scaling the set by a nonzero t (so for a power function the direction a does not matter, since Sₐ = aᵈ S₁), and by scaling the pair (v1,v2) (so all (2ⁿ-1)(2ⁿ-2) pairs collapse to the 2ⁿ-2 values of rho)

The affine-coset theorem: if S is a coset of the kernel of a nonzero additive map f: K -> ZMod 2 in a finite field K of characteristic 2, then 8 * N_S(v1,v2) = |K|² for every pair of distinct nonzero v1,v2 – so every APN function with affine derivatives is CWU; with the computed record of which rows of the table it explains (Gold and x³+Tr(x⁹): yes; Kasami and the Gold compositional inverse: no, their S₁ is not a coset of a subspace)

Negative controls: the constant 2²ⁿ⁻³ is pinned from both sides by the definition-level sweep (twice too large, twice too small and both neighbouring integers all fail) and Z = c only for c = 0; the excluded pairs v1 = v2 and v2 = 0 give 2²ⁿ⁻², twice the value; at non-APN exponents |S₁|!= 2ⁿ⁻¹ and the verdict is false; the field test rejects reducible moduli; and the classification is two-sided (Welch at n = 13 refutes 'every APN power function is CWU', the Kasami and Gold-inverse classes at n = 13 refute 'only the functions with affine derivatives are CWU')

The term-skipping sum the sweeps evaluate is the plain sum: zAcc = zAccPlain and hence cwuOK = cwuPlainOK, for every n, every function code, every exponent and every direction, by induction

The character identity 2ⁿ N_F(1,rho) = |Sₐ|³ + Z(rho), which every n >= 12 sweep rests on, tested where it can fail: at exponents with |S₁|!= 2ⁿ⁻¹ both sides are large and rho-dependent and the identity is exact, while the constant 2²ⁿ⁻³ is wrong by a factor of nine (n = 9, d = 9: |S₁| = 64 and N(1,252) = 4096, not 32768 and not the 36352 a naive reading of Z would give)