Kulhandjian-Hanzo Conjecture 25: the odd-prime distance bound

Abstract

For every odd prime p and every primitive element of GaloisField p 3, the Singer-quadric stabilizer code has a non-stabilizer centralizer vector of Pauli weight at most p+1. A translated trace-plane indicator gives a centralizer vector. A right-kernel separator and two correlation moments ensure that at least one translated vector lies outside the stabilizer row space.

Definition 1.1 (The trace-plane indicator).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\forall i:\mathrm{Fin}\left(p^2+p+1\right),\mathrm{tauH}\left(\alpha \right)\left(i\right)=\mathrm{ite}\left(\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{\mathrm{val}\left(i\right)}\right)=0,1,0\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.tauH (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Page 7, Section III: “Its first column ${\tau }_{\mathbf{H}}$ has $({\tau }_{\mathbf{H}}{)}_i=1$ iff ${\mathrm{Tr}}_{K/F}({\alpha }^i)=0$.”

At q=p and d=2, the carrier is GaloisField p 3 and the positions are Fin(p^2+p+1). The entry is 0 when the trace is nonzero.

Definition 1.2 (The quadric indicator).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\forall i:\mathrm{Fin}\left(p^2+p+1\right),\mathrm{tauQ}\left(\alpha \right)\left(i\right)=\mathrm{ite}\left(\mathrm{Algebra}.\mathrm{trace}\left(\mathrm{ZMod}\left(p\right),\mathrm{GaloisField}\left(p,3\right),{\alpha }^{2\cdot \mathrm{val}\left(i\right)}\right)=0,1,0\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.tauQ (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Page 27, Appendix A, Step 3: “Set $\xi =q+1$ if $q$ is even, $\xi =2$ if $q$ is odd. Define ${\tau }_{\mathbf{Q}}\in {\left({\mathbb{F}}_q\right)}^n$ by $({\tau }_{\mathbf{Q}}{)}_i=1$ if $T[(\xi \cdot i)\mathrm{mod}(q^{d+1}-1)]=0$, else $0$.”

For odd p the exponent is 2*val(i), with trace zero giving entry 1 and nonzero trace giving entry 0. Primitive powers have period p^3-1, so the source’s exponent reduction modulo p^3-1 gives the same entry.

Definition 1.3 (The Singer circulant).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),A\left(\alpha \right)=\mathrm{Matrix}.\mathrm{circulant}\left(\mathrm{tauH}\left(\alpha \right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.A (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

A=circ(tau_H), with entry tauH(alpha)(i-j) and cyclic subtraction in Fin(p^2+p+1).

Definition 1.4 (The quadric circulant).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\mathrm{MQ}\left(\alpha \right)=\mathrm{Matrix}.\mathrm{circulant}\left(\mathrm{tauQ}\left(\alpha \right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.MQ (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

M_Q=circ(tau_Q), using the same first-column circulant convention as A.

Definition 1.5 (The second check block).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),B\left(\alpha \right)=\mathrm{MQ}\left(\alpha \right)\cdot A\left(\alpha \right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.B (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The X block is M_Q*A over ZMod p.

Definition 1.6 (The stabilizer check matrix).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\forall i:\mathrm{Fin}\left(p^2+p+1\right),\forall j:\mathrm{Sum}\left(\mathrm{Fin}\left(p^2+p+1\right),\mathrm{Fin}\left(p^2+p+1\right)\right),H\left(\alpha \right)\left(i,j\right)=\mathrm{Sum}.\mathrm{elim}\left(A\left(\alpha \right)\left(i\right),B\left(\alpha \right)\left(i\right),j\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.H (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Page 11, Section V: H=(H_z | H_x):=(A | M_Q A). Sum.elim concatenates the two row functions, so H has columns indexed by Fin(p^2+p+1) summed with Fin(p^2+p+1).

Definition 1.7 (The symplectic pairing).

$$\forall p:\mathbb{N},\forall v:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),\forall w:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),\mathrm{symp}\left(v,w\right)=\mathrm{dotProduct}\left(v.1,w.2\right)-\mathrm{dotProduct}\left(v.2,w.1\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.symp (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The first component is the Z part, and the second the X part. The signed form is z dot x’ minus x dot z’; its vanishing expresses symplectic commutation.

Definition 1.8 (The stabilizer row space).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\forall v:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),(v\in \mathrm{stabilizers}\left(\alpha \right))\iff (\exists c:(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),v=(\mathrm{Matrix}.\mathrm{vecMul}\left(c,A\left(\alpha \right)\right),\mathrm{Matrix}.\mathrm{vecMul}\left(c,B\left(\alpha \right)\right)))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.stabilizers (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The pairs (cA,cB) for all row coefficient vectors c are exactly rowspan(H), with H=(A | B).

Definition 1.9 (The symplectic centralizer).

$$\forall p:\mathbb{N},[\mathrm{Fact}\left(\mathrm{Nat}.\mathrm{Prime}\left(p\right)\right)],\forall \alpha :\mathrm{GaloisField}\left(p,3\right),\forall v:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),(v\in \mathrm{centralizer}\left(\alpha \right))\iff (\forall w:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),(w\in \mathrm{stabilizers}\left(\alpha \right))\Rightarrow (\mathrm{symp}\left(v,w\right)=0))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.centralizer (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

The vectors pairing to zero with every element of the stabilizer row space.

Definition 1.10 (The Pauli weight).

$$\forall p:\mathbb{N},\forall v:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),\mathrm{wt}\left(v\right)=\mathrm{Finset}.\mathrm{card}\left(\mathrm{Finset}.\mathrm{filter}\left((i:\mathrm{Fin}\left(p^2+p+1\right))\mapsto (v.1\left(i\right)\ne 0)\vee (v.2\left(i\right)\ne 0),\mathrm{Finset}.\mathrm{univ}\left(\mathrm{Fin}\left(p^2+p+1\right)\right)\right)\right)$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.wt (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Weight counts positions where either component is nonzero, counting a position only once when both are nonzero.

Definition 1.11 (Conjecture 25).

$$(claim)\iff (\forall p:\mathbb{N},(\mathrm{Nat}.\mathrm{Prime}\left(p\right))\Rightarrow ((2<p)\Rightarrow (\forall \alpha :\mathrm{GaloisField}\left(p,3\right),(\mathrm{orderOf}\left(\alpha \right)=p^3-1)\Rightarrow (\exists v:\mathrm{Prod}\left((\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right)),(\mathrm{Fin}\left(p^2+p+1\right)\to \mathrm{ZMod}\left(p\right))\right),(v\in \mathrm{centralizer}\left(\alpha \right))\wedge ((\neg (v\in \mathrm{stabilizers}\left(\alpha \right)))\wedge (\mathrm{wt}\left(v\right)\le p+1))))))$$

Formalization. D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.claim (✓ std3).

Citation. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Page 19, Section VII, Conjecture 25 (Flagship distance, original): “For prime $p$ odd, $d_{\min }(\mathcal{Q}\left(p,2\right))\le p+1$.”

The encoding quantifies over every prime p>2 and every alpha of multiplicative order p^3-1. It asserts a vector in the symplectic centralizer, outside rowspan(H), of Pauli weight at most p+1.

Theorem 1.12 (The conjectured distance bound holds).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.result (✓ std3). ∎

Resolves. Problems/kulhandjian-hanzo-2026-flagship-distance-conjecture-25 (proved) by D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.result.

Source. Repository-derived.

Acknowledgement. M. Kulhandjian; L. Hanzo (2026). Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance. DOI: 10.48550/arXiv.2610.02392. URL: https://arxiv.org/abs/2610.02392v1.

Commentary.

Let D and Q be the trace-plane and trace-square supports. Both have size p+1, and D has nonzero difference multiplicities one. A translated reversed indicator s has As=1, while MQ1=1, so (s,s) centralizes the row space. The vector (tauQ,-e_0) is in the ordinary right kernel of H. Its pairing with (s,s) is m_t-delta_t, where m_t counts (t-D) intersect Q and delta_t indicates membership in D. The moment identities sum(m_t)=(p+1)^2 and sum(m_t^2)=(p+1)^2+p^2+p contradict m_t congruent to delta_t modulo p for every t: their forced lower bound exceeds the second moment by p*(p-2)*(p+1)>0. A nonzero pairing excludes stabilizer membership. The support of (s,s) has p+1 positions.

References

• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.A
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.B
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.H
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.MQ
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.centralizer
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.claim
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.result
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.stabilizers
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.symp
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.tauH
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.tauQ
• Truth anchor: D5/S3/Quantum/Information/SingerQuadricQuditDistanceBound.wt
• Dependency: D5/S3/Geometry/FiniteGeometry/SingerTracePlane