diff b34550f..f6bf760 hadamard_order_six_fourier_v2.txt
diff --git a/hadamard_order_six_fourier_v2.txt b/hadamard_order_six_fourier_v2.txt
index cc96437..4b1474a 100644
--- a/hadamard_order_six_fourier_v2.txt
+++ b/hadamard_order_six_fourier_v2.txt
@@ -27,22 +27,6 @@ The central object of both theorems is the character sum
defined for balanced integer vectors a with sum a_i = 0.
-CONTENTS
---------
-
- 1. Hadamard matrices
- 2. Mutually unbiased bases
- 3. Notation and character sums
- 4. The dichotomy for a single Hadamard matrix
- 5. Classifying the cubic alternative
- 6. Character identities of a complete family
- 7. Cubic matrices cannot sit inside seven MUBs
- 8. Six positive forms that rule out seven MUBs
- 9. Exact verification
- 10. A small worked example
- 11. Glossary
-
-
1. HADAMARD MATRICES
====================
@@ -1327,142 +1311,4 @@ finite set gives enough reduction, and then checks the resulting
identity exactly.
-10. A SMALL WORKED EXAMPLE
-==========================
-
-To see the pattern in a simpler setting, work with d = 2.
-
-Take
-
- H = [[1, 1],
- [1, -1]].
-
-A balanced charge is a = (a, -a). Compute:
-
- g_H((1, -1)) = (1/2) ( H(0,0) H(1,0)^{-1}
- + H(0,1) H(1,1)^{-1} )
- = (1/2) ( 1 * 1 + 1 * (-1) )
- = 0.
-
- g_H((2, -2)) = (1/2) ( H(0,0)^2 / H(1,0)^2
- + H(0,1)^2 / H(1,1)^2 )
- = (1/2) ( 1 + 1 )
- = 1.
-
-So the (1, -1) character sum vanishes, but the (2, -2) character sum
-is 1. The order-one vanishing and the order-two collapse are the 2D
-analog of the order-one / order-two cases that drive Section 4.
-
-The 2D analog of alpha = (1,1,1,-1,-1,-1) is a' = (1, -1). And
-g_H(a') = 0 for EVERY 2x2 Hadamard matrix H. So the "Theorem 1"
-analog in d = 2 is: order-2 Hadamard matrices have unconditionally
-vanishing order-one character sums. (There is no exceptional matrix
-in d = 2 because H^{circ 2} is automatically a multiple of an all
-ones matrix, not a Hadamard, so case (C) cannot occur.)
-
-The contraction identity also has a 2D analog. In d = 2, the p = 3
-homogeneous relation (analog of p = 7) reads
-
- sum_{l != i} m_H(A_{i -> l}) = 0 if p = 3,
-
-which has no analog of the "7 - p" coefficient that appears for
-p <= 6. In dimension 2, the analog of the constant 7 is 3 (two labels
-plus one potential "extra" assignment); the p = 3 boundary case has
-no room for averaging, so it is homogeneous in the same way.
-
-The mixed-moment construction of Section 8 does NOT have a clean 2D
-analog: the specific integer data (the lists ws_lambda and matrices
-Z_lambda) is chosen to violate the bound S >= -1/5 that is specific
-to the order-six structure. So Theorem 2 is genuinely a dimension-six
-statement, whereas Theorem 1's setup principle (contrast a single-
-matrix character with an orthogonal configuration) is dimension-
-independent.
-
-
-11. GLOSSARY
-============
-
-Hadamard matrix. A d x d matrix H with |H(i,j)| = 1 and
-H^* H = dI (i.e. columns orthogonal and length sqrt(d)).
-
-Character sum. g_H(a) = (1/d) sum_k prod_i H(i,k)^{a_i}, for
-balanced integer vector a.
-
-Balanced charge. Integer vector a with sum a_i = 0. Balanceness makes
-g_H insensitive to column phases.
-
-Order of a charge. ord(a) = sum of the positive entries.
-
-alpha. The distinguished charge (1,1,1,-1,-1,-1). Vanishing of
-g_H(pi alpha) at every permutation pi is the Fourier-vanishing
-conjecture (Matolcsi-Ruzsa-Weiner 2013).
-
-Cubic matrix T. The 6x6 Hadamard defined in Section 1, whose
-entrywise square is also Hadamard. Equivalent to the classic example
-studied by Tao (2004), Moorhouse (2001), Tadej-Zyczkowski (2006).
-
-MUB family. Pairwise-mutually-unbiased orthonormal bases of C^d.
-Maximum size is d + 1; attained for prime-power d (Wootters-Fields
-1989).
-
-N(d). Maximum size of a MUB family in C^d. Theorem 2 establishes
-N(6) <= 5.
-
-Injective moment m_H(A). The average over K in {H, H^T} and over
-uniform injections rho, kappa of the product of matrix entries with
-integer exponents A_{uv}. Converts between "character-sum" and
-"sampled-column" descriptions.
-
-Contraction. The operation of summing a low-order row against
-column orthogonality of H, giving a linear relation among moments.
-Reduces total number of rows by one.
-
-Equality pattern. The set partition of sampled column positions
-induced by which positions coincide. Gives the coefficient
-(6)_{|S|} / 6^n in the expansion of a product of character sums.
-
-S-Hadamard matrix. A Hadamard matrix H for which H^{circ 2} is also
-Hadamard. T is essentially the unique such matrix of order six.
-
-Dephasing. Multiplying rows and columns by phases so that the first
-row and column of H are all ones. Preserves the Hadamard properties.
-
-Complete family. A family of d + 1 MUBs in C^d. Theorem 2 says none
-exists for d = 6.
-
-Weiner completion theorem. Any m < d + 1 MUBs in C^d extend to a
-complete family. So N(d) = d + 1 iff a complete family exists.
-
-Zauner 2-design identity. The projector relation
-Tr(X Y) = sum_{s,k} Tr(X P_{s,k}) Tr(Y P_{s,k}) for trace-zero
-X, Y. Source of order-two vanishing in a complete family.
-
-Mixed moment M(s; a). Average of Re(prod_j g_{s_j}(a_j)) over
-sampled configurations (reference basis, permutation, bijection
-between labels and other bases). Decomposed by (I)-(V) of Section
-8.3.
-
-Shared-column functions F_0, F_1. Character sums tied to a
-common column z of H_0. Force Gram entries to be three-factor
-mixed moments, which is the entry point to the mixed certificate.
-
-Young symmetrizer. The signed sum over row-group and column-group
-permutations of a Young diagram shape. Organizes the six positive
-forms of Section 8.4.
-
-Positive-definite Gram matrix. A symmetric real matrix Z with
-x^T Z x > 0 for every nonzero real x. All six Z_lambda are verified
-by exact Schur-complement elimination.
-
-
-FINAL REMARK
-============
-The two theorems share one method. Every relation arose from
-orthogonality of a Hadamard matrix; every bound arose from
-|g_H(a)| <= 1. The Fourier-vanishing theorem is a statement about
-one matrix; the MUB bound is a statement about a family. Both come
-from the same "sum of nonnegative terms equals zero" argument,
-applied once with two probe charges (q_cross and q_same) to force a
-dichotomy on H, and once with six positive-definite quadratic
-forms to force a contradiction with nonnegativity.