bibkey: pratapsi2025bargmann authors: Sagar Silva Pratapsi, Joao Gouveia, Leonardo Novo, and Ernesto F. Galvao year: 2025 title: An Elementary Characterization of Bargmann Invariants doi: 10.1103/hsnv-wpt3 url: https://arxiv.org/abs/2506.17132v3 claim: A common tuple of unit records obeys a sharp phase-dependent arithmetic-mean overlap bound, supplied by circulant averaging and the regular polygon of Bargmann roots. strata_touched: [] license: citation-only triage: anchor

Common-record polygon bound used by the five-label square

Primary text: arXiv:2506.17132v3, Theorem 1 and Theorem 5, equations (20)–(27). Published identifier: 10.1103/hsnv-wpt3. The consumer is Hidden Relations, C-29.3–C-29.5: the actual comparison graph has a square and a pendant edge.

For $m\ge 3$ common unit vectors, let

$$\Delta =\prod _{j=0}^{m-1}⟨e_j|e_{j+1}⟩,e_m=e_0,v_j=|⟨e_j|e_{j+1}⟩|.$$

Assume all adjacent overlaps are nonzero and put $\Theta =|\mathrm{Arg}\Delta |\in [0,\pi ]$. The consumed bound is

$$\underset{j}{\min }{v}_j\le \frac{1}{m}\sum _j{v}_j\le {\eta }_m(\Theta )=\frac{\cos (\pi /m)}{\cos ((\pi -\Theta )/m)},|\Delta |\le {\eta }_m(\Theta {)}^m.$$

The average bound comes from Theorem 5’s proof, not from its product-range corollary alone. Equations (20)–(21) align the edge phases on one ray; equations (23)–(24) identify the mean modulus with the magnitude of the mean overlap; equations (26)–(27) cyclically average the positive Gram matrix. Theorem 1 places this circulant first-neighbor entry $z$ in the convex hull of the $m$th roots of unity. After conjugating the orientation if necessary, $z=\overline{v}{e}^{i\Theta /m}$. The supporting line

$$\mathrm{Re}(e^{-i\pi /m}z)\le \cos (\pi /m)$$

gives the displayed arithmetic-mean inequality. AM–GM then gives the product bound. The intersection of the ray with this polygon edge is a convex combination of its two endpoint roots. Theorem 1’s circulant Gram realization has rank at most two there and equal edge moduli, attaining both bounds. This argument and construction are published results; they are not a new general polygon theorem of the project.

For $m=4$ the denominator simplifies to ${\eta }_4(\Theta )=1/(\cos (\Theta /4)+\sin (\Theta /4))$. Equivalently, the averaged square Gram has first-neighbor entry $z=\overline{v}{e}^{i\Theta /4}$ and a real opposite entry $t$; its eigenvalues are $1+t\pm 2\overline{v}\cos (\Theta /4)$ and $1-t\pm 2\overline{v}\sin (\Theta /4)$. Their nonnegativity gives the same bound. At $\Theta =\pi$, the values for $m=3,4,5,6$ are respectively $1/2,1/\sqrt{2},(1+\sqrt{5})/4,\sqrt{3}/2$; as $m\to \infty$ they tend to one. The five-label native graph does not supply a five-cycle: a polygon with another number of comparison edges requires its own common-record contract.

The paper uses the standard overlap orientation in $\Delta$. The repository channel convention is ${\Gamma }_{st}=⟨e_t|e_s⟩$, so the product $B$ along the same directed cycle is $\overline{\Delta }$. Absolute principal angles and all displayed bounds agree. Rephasing vectors is a mathematical Gram transformation; performing a calibrated system correction requires an additional physical operation. Pairwise overlaps from unrelated records cannot replace the one positive Gram matrix used here. A zero edge makes $\mathrm{Arg}\Delta$ undefined.

Refinement comparison consumed by the recovery bound

The explicit real records $e_j=(\cos (j\pi /m),\sin (j\pi /m))$, $0\le j<m$, have positive consecutive inner products $\cos (\pi /m)$ except for the closing edge, which is $-\cos (\pi /m)$. Thus

$$\Delta =-\cos (\pi /m{)}^m⟶-1.$$

Indeed $0\le 1-\cos (\pi /m{)}^m\le m(1-\cos (\pi /m))\le {\pi }^2/(2m)$. For even $m$, $e_0$ and $e_{m/2}$ are orthogonal. Consequently the consumer’s pair-state contraction bound still forces system-only recovery error at least $1/2$, although adjacent moduli approach one. On the open chain the root depth is $m-1$, and its local real-part defect is $1-\cos (\pi /m)$; the product of depth and square root of that defect does not approach zero. This comparison varies the number of labels and does not assert a fixed-five-label limit or a global impossibility theorem.