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The $2048$ Chrysopoeia: An Explicit Construction Program for the $d=2048$ SIC-POVM Moduli

Christopher Mills

Published Jul 7, 2026• 0 citations
Publisher

Abstract

Companion to the SIC–Stark–12th formalization, this paper lays out an explicit program for the diagonal moduli $N_k=|\psi_k|^2$ of the $d=2048$ SIC-POVM. It reports what direct computation has settled and what remains open. Five results are in place. The $a=0$ stratum reduces to a flat autocorrelation condition on the real moduli, a condition verified on $d=12$ to a residual near $10^{-58}$. Unconstrained numerical recovery is blocked by a genuine local minimum (residual $3.9\times 10^{-3}$). The moduli field is the real index-2 subfield, of degree $2^{27}$, of the ray class field of conductor $(d)\,\infty_1\infty_2$, calibrated on the $d=12$ fiducial where the moduli have degree 4 or 8 and generate a field of degree 16; its unramified ascent is verified to degree 128. Stark-unit $L$-values supply the moduli without building the full field, a method validated end to end on $d=12$. An integer norm sieve separates degenerate $S$-unit aliases by discriminant $-m_d$. Still open: the reduced character set for the 1024 moduli and unconditional existence.

DOI
10.5281/zenodo.21251172

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SIC-POVMs, a Stark Conjecture, and the 12th: A Formalization via Paraconsistent Belnap Multilattices

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This paper establishes, inside the Lean 4 proof assistant, a three-level formal identification. The levels are: (i) Belnap multilattice axioms for Weyl–Heisenberg covariant SIC-POVMs at $d=2^n$; (ii) the Zauner conjecture; and (iii) the mixed-signature Stark conjecture for the ray class field $K_d=\mathbb{Q}(\sqrt{(d-3)(d+1)})$, a real-quadratic case of Hilbert's Twelfth Problem. Fiducials are unit-normalized and satisfy $(d+1)|\langle\psi,D_{a,b}\psi\rangle|^2=1$. The equivalence hilbert_embedding_equiv_zauner is proved by rfl: the Belnap embedding into $\mathbb{C}^{2^n}$ and the Zauner conjecture at $d=2^n$ are definitionally the same proposition. The Belnap skeleton (orbit size $4^n$, Frobenius closure $\mu\circ\delta=\mathrm{id}$, join-equiangularity, Born rule) contains zero sorries. Open arithmetic content is marked by named gap axioms for Stark units on WH frames; a proof of Stark would close all three levels at once. For dimension $d=12$ we prove SICPOVM_Exists 12 outright. We construct an exact fiducial in a finitely presented $\mathbb{Q}$-algebra, verify 143 overlap identities with native_decide, and transfer everything to $\mathbb{C}^{12}$ along a ring homomorphism. The theorem crystal_forces_d12_sic depends on no axiom beyond Lean 4's standard foundations and compiler trust. This is, to our knowledge, the first machine-checked SIC-POVM existence in any dimension. For the frontier dimension $d=2048=2^{11}$ the transport apparatus is formalized and sorry-free. It includes a forward map $\varphi\colon B^{\oplus 11}\to\mathbb{C}^{2048}$, a reduction $\psi$ with $\psi\circ\varphi=\mathrm{id}$, a conditional reduction to Stark, and a non-real character obstruction that blocks the false branch. Unconditional existence remains open; the machinery that surrounds it is closed.

•Jul 7, 2026•0 citations

SIC-POVMs and the Stark Conjectures

Gene S. Kopp

Abstract The existence of $d^2$ pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in $d$-dimensional Hilbert space is known only for finitely many dimensions $d$. We prove that, if there exists a set of real units in a certain ray class field (depending on $d$) satisfying certain algebraic properties, a SIC-POVM exists, when $d$ is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at $s=0$ and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.

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We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of ZK with modulus p∞1, where ∞1 is one of the real places of K.

arXiv:2112.05552•Journal of Mathematical Physics•Nov 1, 2022•10 citations

SIC dimension towers via cyclotomic polynomials

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We prove a structure theorem for the dimension towers~$\{d_k(D)\}_{k\geq0}$ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\Q(\qD)$. If~$\eps$ denotes the first totally positive power of a fundamental unit of~$K$ and $t_k = \eps^k + \eps^{-k}$ the trace of its $k$-th power, then the SIC dimension tower~$\{d_k=1+t_k\}_{k\geq0}$ is the level~$m=3$ row of an infinite two-dimensional cyclotomic array~$\{Ψ_m(t_k)\}_{m\geq1,k\geq0}$ attached to~$K$, while the auxiliary factors~$(d_k+1)$ and $(d_k-3)$ are its ramified levels $m=2$ and $m=1$. Here~$Ψ_m$ denotes the minimal polynomial of~ $ζ_m+ζ_m^{-1} = 2\cos{2π/m}$. This construction arose initially from an attempt to formulate relations among SIC dimensions in $q$-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-ζ_m\eps^k$ over the cyclotomic field tower $\{K(μ_m)\}_{m\geq1}$. This framework sheds new light on the mod-$p$ analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~$2mD$, the valuations $v_p(Ψ_m(t_k))$ at every fixed level~$m$ are described exactly in terms of a single local unit valuation, which is then related, through the~$p$-adic class number formula, to the corresponding $p$-adic $L$-value.

math.NTCross-listedarXiv:2609.04431•Sep 7, 2026

A Constructive Approach to Zauner's Conjecture via the Stark Conjectures

Marcus Appleby, Steven T. Flammia, Gene S. Kopp

We propose a construction of $d^2$ complex equiangular lines in $\mathbb{C}^d$, also known as SICPOVMs, conjectured by Zauner to exist for all d. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. Specifically, we give an explicit expression for an object we call a ghost SIC, which is constructed from the real multiplication values of a special function and which is Galois conjugate to a SIC. The special function, the Shintani-Faddeev modular cocycle, is more precisely a tuple of meromorphic functions indexed by a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$. We prove our construction gives a valid SIC in every case assuming two conjectures: the order 1 abelian Stark conjecture for real quadratic fields and a special value identity for the Shintani-Faddeev modular cocycle. The former allows us to prove that the ghost and the SIC are Galois conjugate over an extension of $\mathbb{Q}(\sqrtΔ)$ where $Δ= (d+1)(d-3)$, while the latter allows us to prove idempotency of the presumptive fiducial projector. We provide computational tests of our SIC construction by cross-validating it with known solutions, particularly the extensive work of Scott and Grassl, and by constructing four numerical examples of nonequivalent SICs in d=100, three of which are new. We further consider rank-r generalizations called r-SICs given by maximal equichordal configurations of r-dimensional complex subspaces. We give similar conditional constructions for r-SICs for all r, d such that r(d-r) divides $(d^2-1)$. Finally, we study the structure of the field extensions conjecturally generated by the r-SICs. If K is any real quadratic field, then either every abelian Galois extension of K, or else every abelian extension for which 2 is unramified, is generated by our construction; the former holds for a positive density of field discriminants.

arXiv:2501.03970•Jan 7, 2025•1 citation
The 204820482048 Chrysopoeia: An Explicit Construction Program for the d=2048d=2048d=2048 SIC-POVM Moduli
Companion to the SIC–Stark–12th formalization, this paper lays out an explicit program for the diagonal moduli Nk=∣ψk∣2N_k=|\psi_k|^2Nk​=∣ψk​∣2 of the d=2048d=2048d=2048 SIC-POVM. It reports what direct computation has settled and what remains open. Five results are in place. The a=0a=0a=0 stratum reduces to a flat autocorrelation condition on the real moduli, a condition verified on d=12d=12d=12 to a residual near 10−5810^{-58}10−58. Unconstrained numerical recovery is blocked by a genuine local minimum (residual 3.9×10−33.9\times 10^{-3}3.9×10−3). The moduli field is the real index-2 subfield, of degree 2272^{27}227, of the ray class field of conductor (d) ∞1∞2(d)\,\infty_1\infty_2(d)∞1​∞2​, calibrated on the d=12d=12d=12 fiducial where the moduli have degree 4 or 8 and generate a field of degree 16; its unramified ascent is verified to degree 128. Stark-unit LLL-values supply the moduli without building the full field, a method validated end to end on d=12d=12d=12. An integer norm sieve separates degenerate SSS-unit aliases by discriminant −md-m_d−md​. Still open: the reduced character set for the 1024 moduli and unconditional existence.
This paper establishes, inside the Lean 4 proof assistant, a three-level formal identification. The levels are: (i) Belnap multilattice axioms for Weyl–Heisenberg covariant SIC-POVMs at d=2nd=2^nd=2n; (ii) the Zauner conjecture; and (iii) the mixed-signature Stark conjecture for the ray class field Kd=Q((d−3)(d+1))K_d=\mathbb{Q}(\sqrt{(d-3)(d+1)})Kd​=Q((d−3)(d+1)​), a real-quadratic case of Hilbert's Twelfth Problem. Fiducials are unit-normalized and satisfy (d+1)∣⟨ψ,Da,bψ⟩∣2=1(d+1)|\langle\psi,D_{a,b}\psi\rangle|^2=1(d+1)∣⟨ψ,Da,b​ψ⟩∣2=1. The equivalence hilbert_embedding_equiv_zauner is proved by rfl: the Belnap embedding into C2n\mathbb{C}^{2^n}C2n and the Zauner conjecture at d=2nd=2^nd=2n are definitionally the same proposition. The Belnap skeleton (orbit size 4n4^n4n, Frobenius closure μ∘δ=id\mu\circ\delta=\mathrm{id}μ∘δ=id, join-equiangularity, Born rule) contains zero sorries. Open arithmetic content is marked by named gap axioms for Stark units on WH frames; a proof of Stark would close all three levels at once. For dimension d=12d=12d=12 we prove SICPOVM_Exists 12 outright. We construct an exact fiducial in a finitely presented Q\mathbb{Q}Q-algebra, verify 143 overlap identities with native_decide, and transfer everything to C12\mathbb{C}^{12}C12 along a ring homomorphism. The theorem crystal_forces_d12_sic depends on no axiom beyond Lean 4's standard foundations and compiler trust. This is, to our knowledge, the first machine-checked SIC-POVM existence in any dimension. For the frontier dimension d=2048=211d=2048=2^{11}d=2048=211 the transport apparatus is formalized and sorry-free. It includes a forward map φ ⁣:B⊕11→C2048\varphi\colon B^{\oplus 11}\to\mathbb{C}^{2048}φ:B⊕11→C2048, a reduction ψ\psiψ with ψ∘φ=id\psi\circ\varphi=\mathrm{id}ψ∘φ=id, a conditional reduction to Stark, and a non-real character obstruction that blocks the false branch. Unconditional existence remains open; the machinery that surrounds it is closed.
Abstract The existence of d2d^2d2 pairwise equiangular complex lines [equivalently, a symmetric informationally complete positive operator-valued measure (SIC-POVM)] in ddd-dimensional Hilbert space is known only for finitely many dimensions ddd. We prove that, if there exists a set of real units in a certain ray class field (depending on ddd) satisfying certain algebraic properties, a SIC-POVM exists, when ddd is an odd prime congruent to 2 modulo 3. We give an explicit analytic formula that we expect to yield such a set of units. Our construction uses values of derivatives of zeta functions at s=0s=0s=0 and is closely connected to the Stark conjectures over real quadratic fields. We verify numerically that our construction yields SIC-POVMs in dimensions 5, 11, 17, and 23, and we give the first exact SIC-POVM in dimension 23.
It has been observed that the mutual scalar products of the vectors in a SIC-POVM are given by algebraic units, and at least in some cases by square roots of Stark units. The full picture is somewhat more complicated, especially if non-minimal SIC-POVMs are considered. We present a mixture of exact and numerical evidence suggesting that the overlap units are always products of integral powers of square roots of Stark units from ray class fields all of which are attached to the maximal ring of integers in the base field. In the non-minimal case a lattice of such ray class fields is involved. In every second dimension (counted in a certain way) some of the overlap units equal ±1\pm 1±1, and we show that this follows from a special property of the ray class fields. Our observations are complementary to but consistent with the claim that the overlap units can be calculated directly from the Shintani--Faddeev modular cocycle.
We propose a recipe for constructing a fiducial vector for a symmetric informationally complete positive operator valued measure (SIC-POVM) in a complex Hilbert space of dimension of the form d = n2 + 3, focusing on prime dimensions d = p. Such structures are shown to exist in 13 prime dimensions of this kind, the highest being p = 19 603. The real quadratic base field K (in the standard SIC-POVM terminology) attached to such dimensions has fundamental units uK of norm −1. Let ZK denote the ring of integers of K; then, pZK splits into two ideals: p and p′. The initial entry of the fiducial is the square ξ2 of a geometric scaling factor ξ, which lies in one of the fields K(uK). Strikingly, each of the other p − 1 entries of the fiducial vector is a product of ξ and the square root of a Stark unit. These Stark units are obtained via the Stark conjectures from the value at s = 0 of the first derivatives of partial L-functions attached to the characters of the ray class group of ZK with modulus p∞1, where ∞1 is one of the real places of K.
We prove a structure theorem for the dimension towers~{dk(D)}k≥0\{d_k(D)\}_{k\geq0}{dk​(D)}k≥0​ which arise in the number-theoretic formulation of Zauner's SIC-POVM conjecture over a real quadratic field~$K=\Q(\qD)$. If~$\eps$ denotes the first totally positive power of a fundamental unit of~KKK and $t_k = \eps^k + \eps^{-k}$ the trace of its kkk-th power, then the SIC dimension tower~{dk=1+tk}k≥0\{d_k=1+t_k\}_{k\geq0}{dk​=1+tk​}k≥0​ is the level~m=3m=3m=3 row of an infinite two-dimensional cyclotomic array~{Ψm(tk)}m≥1,k≥0\{Ψ_m(t_k)\}_{m\geq1,k\geq0}{Ψm​(tk​)}m≥1,k≥0​ attached to~KKK, while the auxiliary factors~(dk+1)(d_k+1)(dk​+1) and (dk−3)(d_k-3)(dk​−3) are its ramified levels m=2m=2m=2 and m=1m=1m=1. Here~ΨmΨ_mΨm​ denotes the minimal polynomial of~ ζm+ζm−1=2cos⁡2π/mζ_m+ζ_m^{-1} = 2\cos{2π/m}ζm​+ζm−1​=2cos2π/m. This construction arose initially from an attempt to formulate relations among SIC dimensions in qqq-algebraic terms. The central object is a single closed composite norm relation for the two-parameter family $c_{m,k}=1-ζ_m\eps^k$ over the cyclotomic field tower {K(μm)}m≥1\{K(μ_m)\}_{m\geq1}{K(μm​)}m≥1​. This framework sheds new light on the mod-ppp analogue of Leopoldt's conjecture, by placing the central 3-symmetry of Zauner's conjecture within a broader arithmetic context. Away from the primes dividing~2mD2mD2mD, the valuations vp(Ψm(tk))v_p(Ψ_m(t_k))vp​(Ψm​(tk​)) at every fixed level~mmm are described exactly in terms of a single local unit valuation, which is then related, through the~ppp-adic class number formula, to the corresponding ppp-adic LLL-value.
We propose a construction of d2d^2d2 complex equiangular lines in Cd\mathbb{C}^dCd, also known as SICPOVMs, conjectured by Zauner to exist for all d. The construction gives a putatively complete list of SICs with Weyl-Heisenberg symmetry in all dimensions d > 3. Specifically, we give an explicit expression for an object we call a ghost SIC, which is constructed from the real multiplication values of a special function and which is Galois conjugate to a SIC. The special function, the Shintani-Faddeev modular cocycle, is more precisely a tuple of meromorphic functions indexed by a congruence subgroup of SL2(Z){\rm SL}_2(\mathbb{Z})SL2​(Z). We prove our construction gives a valid SIC in every case assuming two conjectures: the order 1 abelian Stark conjecture for real quadratic fields and a special value identity for the Shintani-Faddeev modular cocycle. The former allows us to prove that the ghost and the SIC are Galois conjugate over an extension of Q(Δ)\mathbb{Q}(\sqrtΔ)Q(Δ​) where Δ=(d+1)(d−3)Δ= (d+1)(d-3)Δ=(d+1)(d−3), while the latter allows us to prove idempotency of the presumptive fiducial projector. We provide computational tests of our SIC construction by cross-validating it with known solutions, particularly the extensive work of Scott and Grassl, and by constructing four numerical examples of nonequivalent SICs in d=100, three of which are new. We further consider rank-r generalizations called r-SICs given by maximal equichordal configurations of r-dimensional complex subspaces. We give similar conditional constructions for r-SICs for all r, d such that r(d-r) divides (d2−1)(d^2-1)(d2−1). Finally, we study the structure of the field extensions conjecturally generated by the r-SICs. If K is any real quadratic field, then either every abelian Galois extension of K, or else every abelian extension for which 2 is unramified, is generated by our construction; the former holds for a positive density of field discriminants.