This note places on record, before any comparison with data, the magnetic specific heat implied by the icosahedral cluster magnon spectrum, so that a later comparison with the published low-temperature specific heat of the ferromagnetic quasicrystal i-Au65Ga20Gd15 (TC=23 K) and the antiferromagnetic i-Au56In28.5Eu15.5 (TN=6.5 K) is a genuine test. Model. Twelve harmonic magnon modes per icosahedral cluster (twelve rare-earth spins), each contributing an Einstein term: Cmag/(NRkB)=(1/12)Σm gm xm2exm/(exm-1)2, xm=Em/kBT. Ferromagnet: E=0(×1), [(5-√5)+g(5+√5)]J1S(×3), 6(1+g)J1S(×5), [(5+√5)+g(5-√5)]J1S(×3); antiferromagnet: three Goldstone modes, E5=2√(5-√5)/5 J1S(×5), E4=4sin(π/5)J1S(×4), both scaling as E(g)=E(0)√(1-g)(1-φ2g) towards the boundary g=φ-2 (verified against linear spin-wave theory to four digits at g=0.1,0.2,0.3,0.38). Conventions declared in advance. The single scale is fixed by mean field with the icosahedral coordination z=5: kBTC,N=(S(S+1))/(3) 5J1(1+g), S=7/2; no parameter is fitted. A second, shape-only comparison with J1S as the one declared free parameter is reported alongside. Numbers (g=0). Gd: levels 8.48/18.40/22.19 K (degeneracies 3/5/3, ratios 1:2.171:2.618); Cmag/R=0.071, 0.295, 0.502, 0.637, 0.721 at T=2,4,6,8,10 K, gapped as e-8.5 K/T at low T, saturating towards 11/12. Eu: E5=1.29 K, E4=2.04 K, plateau 0.75R above ≈4 K. The v1.1 convention (z=6) gives 7.06/15.33/18.49 K and 0.110, 0.386, 0.591, 0.707, 0.773; the g-family is tabulated in the deposited file. Network check with the published geometry (added before deposition). Using the single-crystal refinement of the isostructural Au–Ga–Ce 1/1 approximant by Tamura's group (Suzuki et al., arXiv:2308.10070: a=14.8872 Å, rare-earth site 24g at (0,0.18780,0.30521)), the rare-earth icosahedron has circumradius 5.335 Å and edges 5.59/5.61 Å, while each rare-earth atom has five neighbours in adjacent clusters at 5.48 Å (×4) and 5.80 Å (×1) — as close as, or closer than, the intra-cluster edge. The rare-earth sublattice of the approximant is therefore a genuine three-dimensional network with ten near-equal neighbours per spin, not a lattice of isolated clusters, and an inter-cluster exchange J'≈ J1 is expected. Bloch spin-wave theory on this network, with the mean-field scale kBTC=(S(S+1))/(3)(5J1+5J') fixed by TC=23 K, removes the cluster gap (acoustic bands, C∝ T3/2) and gives an almost universal curve, Cmag/R≈0.10, 0.36, 0.59, 0.73, 0.82 at 2,4,6,8,10 K, for J'/J1 anywhere between 0.5 and 1.5; the density of states shows two broad maxima near 15–16 and 20–21 K instead of the three cluster levels. The same holds one step closer to the quasicrystal: in the 2/1 approximant refined by the same group (Labib, Takakura, Ishikawa, Tamura, arXiv:2305.09334, Ga50Pd35.5Tb14.5, Pa-3, a=23.1449 Å; 96 icosahedral Tb reconstructed from the published 24d sites, cluster centres at (0.155,0.155,0.155), icosahedron radius 5.04 Å), each icosahedral rare-earth atom has five intra-cluster neighbours at 5.22–5.43 Å and 4.5 inter-cluster neighbours at 5.30–5.77 Å, plus the acute-rhombohedron rare-earth atoms; Bloch spin-wave theory on this 96-spin cell gives Cmag/R=0.101, 0.365, 0.594, 0.732, 0.814 at 2–10 K for J'=J1 — identical to the 1/1 network curve. The network prediction is therefore the same for the 1/1 and the 2/1 approximant and, since the quasicrystal is built from these two units, for the quasicrystal. Using instead the exchange law employed by Tamura's group itself (RKKY, J(r)∝(-xcos x+sin x)/x4, x=2kFr, free-electron kF=1.38 Å-1; Labib et al., arXiv:2310.14292) for every rare-earth pair within 10 Å of the refined 1/1 geometry gives a ferromagnetic J1 at 5.59 Å, J'/J1=0.94 and 0.86 for the two inter-cluster distances, and J2/J1=-0.24 on the second icosahedral shell; the resulting network is ferromagnetically stable and yields Cmag/R=0.106, 0.364, 0.592, 0.731, 0.814 at 2–10 K, unchanged for a phase offset of ±0.3 rad, and 0.121, 0.385, 0.597, 0.730, 0.811 when the phase is instead anchored to the observed AFM–FM boundary of the e/a series (δ≈-1.6 rad) — the same universal curve, independent of the electronic phase. This network curve is the primary prediction for the 1/1 and 2/1 approximants and for the quasicrystal; the isolated-cluster curve (0.071 at 2 K, gap 8.5 K) is retained as the limiting case. A consequence stated openly: the resolved φ2 branch pair of the v1.1 note requires J'≲0.25J1, i.e. clusters far better isolated than the published geometry indicates; in the network regime the Galois structure survives in the centroid, in the sum rules, and — exactly — at the zone boundary: for J'=J1 the Bloch spectrum of the bcc network is rational at Γ and H (0,6,8,10,12,14 in units of J1S), while at the N point (π/a)(1,1,0) it contains the Galois pair 11∓√5 (fourfold each): the network shifts the rational centre of the cluster pair and leaves the ±√5 splitting invariant; for general inter-cluster exchange the N-point pair is exactly (5+6J'/J1)∓√5 in units of J1S (verified to four digits for J'/J1=0,0.25,0.5,0.75,1,1.5), so a single spectrum at N measures both J1S (from the splitting 2√5 J1S) and J'/J1 (from the centre, (c-5)/6) without any fitted parameter. This gives a neutron-scattering signature in the network regime: integer level ratios 6:8:10:12:14 at the zone centre and two lines 2√5 J1S apart about 11J1S at N. Limits stated in advance. The cluster model omits the spin-wave bands of the inter-cluster network (C∝ T3/2 FM, T3 AFM); data above the curve at 2–5 K diagnose dispersive magnons and do not rescue the model. The critical anomaly at TC,N lies outside the model; the test window is T≤ TC/2. The lattice contribution is not negligible in the window (Debye estimate 1.6–3.9 J molGd-1K-1 at 10 K) and must be removed with a non-magnetic reference or a declared Debye fit above TC. Falsification. Network (primary): Cmag/R at 2 K well below 0.10 (a gap) would indicate isolated clusters (J'≲0.25J1) and revive the cluster branches; values well above 0.12 at 2 K, or a curve far from 0.36/0.59/0.73/0.82 at 4–10 K after phonon subtraction, falsify the network model at the mean-field scale (the shape-only fit then reports the required J1S). Cluster limit: absence of a gapped rise with ≈8.5 K (or 7.1 K under z=6), absence of the 3:5:3 shape between 4 and 10 K, or Cmag/R 11/12 well below TC/2, falsifies the cluster picture; AFM: a plateau far above 0.75R above 2–4 K falsifies it. The entropy check is not a test: the harmonic model saturates at 11R/12 while the full spin entropy is Rln8, reached only across TC. Structural companion. The antiferromagnetic ground state is the tangential ("whirling") 3' order with bond angle 116.57°; the ferromagnetic branch ratio is the J2-meter r(g) (addendum v1.2). The outcome, whatever it is, will be entered in the public register. Deposited data: predicted curves 0.5–30 K for all declared conventions (Cmag_vorhersage_v2_z5.csv, Cmag_vorhersage_cluster.csv, Cmag_vorhersage_netz_bcc.csv) and high-resolution network spin-wave data computed on the published 1/1 and 2/1 geometries (Cmag_netz_hires.csv: C_mag/R in 0.25 K steps for 15 model variants; dos_netz.csv: magnon density of states; spektren_GHNP.csv: Bloch spectra at Γ, H, N, P). Register entries L347–L386 (DOI 10.5281/zenodo.21891088). Licence CC-BY-4.0.
Arber Gishto· Zenodo (CERN European Organi...· 0 citations
This note places on record, before any comparison with data, the magnetic specific heat implied by the icosahedral cluster magnon spectrum, so that a later comparison with the published low-temperature specific heat of the ferromagnetic quasicrystal i-Au65Ga20Gd15 (TC=23 K) and the antiferromagnetic i-Au56In28.5Eu15.5 (TN=6.5 K) is a genuine test. Model. Twelve harmonic magnon modes per icosahedral cluster (twelve rare-earth spins), each contributing an Einstein term: Cmag/(NRkB)=(1/12)Σm gm xm2exm/(exm-1)2, xm=Em/kBT. Ferromagnet: E=0(×1), [(5-√5)+g(5+√5)]J1S(×3), 6(1+g)J1S(×5), [(5+√5)+g(5-√5)]J1S(×3); antiferromagnet: three Goldstone modes, E5=2√(5-√5)/5 J1S(×5), E4=4sin(π/5)J1S(×4), both scaling as E(g)=E(0)√(1-g)(1-φ2g) towards the boundary g=φ-2 (verified against linear spin-wave theory to four digits at g=0.1,0.2,0.3,0.38). Conventions declared in advance. The single scale is fixed by mean field with the icosahedral coordination z=5: kBTC,N=(S(S+1))/(3) 5J1(1+g), S=7/2; no parameter is fitted. A second, shape-only comparison with J1S as the one declared free parameter is reported alongside. Numbers (g=0). Gd: levels 8.48/18.40/22.19 K (degeneracies 3/5/3, ratios 1:2.171:2.618); Cmag/R=0.071, 0.295, 0.502, 0.637, 0.721 at T=2,4,6,8,10 K, gapped as e-8.5 K/T at low T, saturating towards 11/12. Eu: E5=1.29 K, E4=2.04 K, plateau 0.75R above ≈4 K. The v1.1 convention (z=6) gives 7.06/15.33/18.49 K and 0.110, 0.386, 0.591, 0.707, 0.773; the g-family is tabulated in the deposited file. Network check with the published geometry (added before deposition). Using the single-crystal refinement of the isostructural Au–Ga–Ce 1/1 approximant by Tamura's group (Suzuki et al., arXiv:2308.10070: a=14.8872 Å, rare-earth site 24g at (0,0.18780,0.30521)), the rare-earth icosahedron has circumradius 5.335 Å and edges 5.59/5.61 Å, while each rare-earth atom has five neighbours in adjacent clusters at 5.48 Å (×4) and 5.80 Å (×1) — as close as, or closer than, the intra-cluster edge. The rare-earth sublattice of the approximant is therefore a genuine three-dimensional network with ten near-equal neighbours per spin, not a lattice of isolated clusters, and an inter-cluster exchange J'≈ J1 is expected. Bloch spin-wave theory on this network, with the mean-field scale kBTC=(S(S+1))/(3)(5J1+5J') fixed by TC=23 K, removes the cluster gap (acoustic bands, C∝ T3/2) and gives an almost universal curve, Cmag/R≈0.10, 0.36, 0.59, 0.73, 0.82 at 2,4,6,8,10 K, for J'/J1 anywhere between 0.5 and 1.5; the density of states shows two broad maxima near 15–16 and 20–21 K instead of the three cluster levels. The same holds one step closer to the quasicrystal: in the 2/1 approximant refined by the same group (Labib, Takakura, Ishikawa, Tamura, arXiv:2305.09334, Ga50Pd35.5Tb14.5, Pa-3, a=23.1449 Å; 96 icosahedral Tb reconstructed from the published 24d sites, cluster centres at (0.155,0.155,0.155), icosahedron radius 5.04 Å), each icosahedral rare-earth atom has five intra-cluster neighbours at 5.22–5.43 Å and 4.5 inter-cluster neighbours at 5.30–5.77 Å, plus the acute-rhombohedron rare-earth atoms; Bloch spin-wave theory on this 96-spin cell gives Cmag/R=0.101, 0.365, 0.594, 0.732, 0.814 at 2–10 K for J'=J1 — identical to the 1/1 network curve. The network prediction is therefore the same for the 1/1 and the 2/1 approximant and, since the quasicrystal is built from these two units, for the quasicrystal. Using instead the exchange law employed by Tamura's group itself (RKKY, J(r)∝(-xcos x+sin x)/x4, x=2kFr, free-electron kF=1.38 Å-1; Labib et al., arXiv:2310.14292) for every rare-earth pair within 10 Å of the refined 1/1 geometry gives a ferromagnetic J1 at 5.59 Å, J'/J1=0.94 and 0.86 for the two inter-cluster distances, and J2/J1=-0.24 on the second icosahedral shell; the resulting network is ferromagnetically stable and yields Cmag/R=0.106, 0.364, 0.592, 0.731, 0.814 at 2–10 K, unchanged for a phase offset of ±0.3 rad, and 0.121, 0.385, 0.597, 0.730, 0.811 when the phase is instead anchored to the observed AFM–FM boundary of the e/a series (δ≈-1.6 rad) — the same universal curve, independent of the electronic phase. This network curve is the primary prediction for the 1/1 and 2/1 approximants and for the quasicrystal; the isolated-cluster curve (0.071 at 2 K, gap 8.5 K) is retained as the limiting case. A consequence stated openly: the resolved φ2 branch pair of the v1.1 note requires J'≲0.25J1, i.e. clusters far better isolated than the published geometry indicates; in the network regime the Galois structure survives in the centroid, in the sum rules, and — exactly — at the zone boundary: for J'=J1 the Bloch spectrum of the bcc network is rational at Γ and H (0,6,8,10,12,14 in units of J1S), while at the N point (π/a)(1,1,0) it contains the Galois pair 11∓√5 (fourfold each): the network shifts the rational centre of the cluster pair and leaves the ±√5 splitting invariant; for general inter-cluster exchange the N-point pair is exactly (5+6J'/J1)∓√5 in units of J1S (verified to four digits for J'/J1=0,0.25,0.5,0.75,1,1.5), so a single spectrum at N measures both J1S (from the splitting 2√5 J1S) and J'/J1 (from the centre, (c-5)/6) without any fitted parameter. This gives a neutron-scattering signature in the network regime: integer level ratios 6:8:10:12:14 at the zone centre and two lines 2√5 J1S apart about 11J1S at N. Limits stated in advance. The cluster model omits the spin-wave bands of the inter-cluster network (C∝ T3/2 FM, T3 AFM); data above the curve at 2–5 K diagnose dispersive magnons and do not rescue the model. The critical anomaly at TC,N lies outside the model; the test window is T≤ TC/2. The lattice contribution is not negligible in the window (Debye estimate 1.6–3.9 J molGd-1K-1 at 10 K) and must be removed with a non-magnetic reference or a declared Debye fit above TC. Falsification. Network (primary): Cmag/R at 2 K well below 0.10 (a gap) would indicate isolated clusters (J'≲0.25J1) and revive the cluster branches; values well above 0.12 at 2 K, or a curve far from 0.36/0.59/0.73/0.82 at 4–10 K after phonon subtraction, falsify the network model at the mean-field scale (the shape-only fit then reports the required J1S). Cluster limit: absence of a gapped rise with ≈8.5 K (or 7.1 K under z=6), absence of the 3:5:3 shape between 4 and 10 K, or Cmag/R 11/12 well below TC/2, falsifies the cluster picture; AFM: a plateau far above 0.75R above 2–4 K falsifies it. The entropy check is not a test: the harmonic model saturates at 11R/12 while the full spin entropy is Rln8, reached only across TC. Structural companion. The antiferromagnetic ground state is the tangential ("whirling") 3' order with bond angle 116.57°; the ferromagnetic branch ratio is the J2-meter r(g) (addendum v1.2). The outcome, whatever it is, will be entered in the public register. Deposited data: predicted curves 0.5–30 K for all declared conventions (Cmag_vorhersage_v2_z5.csv, Cmag_vorhersage_cluster.csv, Cmag_vorhersage_netz_bcc.csv) and high-resolution network spin-wave data computed on the published 1/1 and 2/1 geometries (Cmag_netz_hires.csv: C_mag/R in 0.25 K steps for 15 model variants; dos_netz.csv: magnon density of states; spektren_GHNP.csv: Bloch spectra at Γ, H, N, P). Register entries L347–L386 (DOI 10.5281/zenodo.21891088). Licence CC-BY-4.0.
Arber Gishto· Zenodo (CERN European Organi...· 2 citations
The common refinement of the icosahedral edge shell and its Galois shadow (the great-icosahedron chords) on the sphere is computed exactly: each visible edge crosses exactly one shadow edge, the shadow edges cross each other twenty times, and the refined sphere is the barycentric subdivision of the icosahedron — $V=62$ ($12+30+20$: vertices, edge midpoints, face points), $E=180$, $F=120$ triangles, one triangle per group element of $H_3$. The double of two cones over this sphere is a closed Regge 3-manifold of 240 tetrahedra (the $E_8$ number) which is an exact vacuum solution at unit radii: both balls are flat and the entire curvature is concentrated on the interface, with three exact class deficits $(0.702532,\,0.067165,\,1.089679)$, all positive. Read as $3{+}1$ gravity, the deficits are cosmic strings: the double is an exact static string universe, its matter a string network on the flag complex (model A, one metric). In the bimetric reading (model B, shadow edges $\varphi$-scaled, shared radials) the matter moves into the volumes as pairs of opposite string tension $\pm\mu$ with exact nodal cancellation: empty and static from outside, two oppositely filled worlds inside — the balance principle as a matter statement. Compression transmits through the interface with a saturating counter-pull (linear response $-1.9$ at the vertices: the shadow expands where the visible is squeezed), and no bottleneck occurs down to $20\%$. Behind the construction stands the conjugation tower: the stiffness blocks of all cones are exact golden expressions ($2\sqrt5\,\varphi^{-3}/a$, $6\varphi^{-3}/a$, $-4\varphi^{-5}$, $2\sqrt5\,\varphi^{-6}$, $1/10$), the shadow block is the Galois conjugate of the visible one, and the tower exponents $\varphi^{4},\varphi^{5},\varphi^{10},\varphi^{12}$ are pure conjugation — a parameter-free hierarchy mechanism. An observer tower coupled through balanced channels is exactly self-similar: each level carries its golden ladder untouched (machine precision). Honest boundaries are recorded: the rectified 600-cell leaves the field $\mathbb{Q}(\sqrt5)$ (minimal polynomial $x^4-6x^3-31x^2+60x+100$), delimiting the $\sigma$-order conjecture. Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.
Arber Gishto· Zenodo (CERN European Organi...· 0 citations
The common refinement of the icosahedral edge shell and its Galois shadow (the great-icosahedron chords) on the sphere is computed exactly: each visible edge crosses exactly one shadow edge, the shadow edges cross each other twenty times, and the refined sphere is the barycentric subdivision of the icosahedron — $V=62$ ($12+30+20$: vertices, edge midpoints, face points), $E=180$, $F=120$ triangles, one triangle per group element of $H_3$. The double of two cones over this sphere is a closed Regge 3-manifold of 240 tetrahedra (the $E_8$ number) which is an exact vacuum solution at unit radii: both balls are flat and the entire curvature is concentrated on the interface, with three exact class deficits $(0.702532,\,0.067165,\,1.089679)$, all positive. Read as $3{+}1$ gravity, the deficits are cosmic strings: the double is an exact static string universe, its matter a string network on the flag complex (model A, one metric). In the bimetric reading (model B, shadow edges $\varphi$-scaled, shared radials) the matter moves into the volumes as pairs of opposite string tension $\pm\mu$ with exact nodal cancellation: empty and static from outside, two oppositely filled worlds inside — the balance principle as a matter statement. Compression transmits through the interface with a saturating counter-pull (linear response $-1.9$ at the vertices: the shadow expands where the visible is squeezed), and no bottleneck occurs down to $20\%$. Behind the construction stands the conjugation tower: the stiffness blocks of all cones are exact golden expressions ($2\sqrt5\,\varphi^{-3}/a$, $6\varphi^{-3}/a$, $-4\varphi^{-5}$, $2\sqrt5\,\varphi^{-6}$, $1/10$), the shadow block is the Galois conjugate of the visible one, and the tower exponents $\varphi^{4},\varphi^{5},\varphi^{10},\varphi^{12}$ are pure conjugation — a parameter-free hierarchy mechanism. An observer tower coupled through balanced channels is exactly self-similar: each level carries its golden ladder untouched (machine precision). Honest boundaries are recorded: the rectified 600-cell leaves the field $\mathbb{Q}(\sqrt5)$ (minimal polynomial $x^4-6x^3-31x^2+60x+100$), delimiting the $\sigma$-order conjecture. Part of the INDYNA Notes series (Notes 16–34). All numbers reproduce from the INDYNA law register (master data record, doi:10.5281/zenodo.21891088). Licence CC-BY-4.0.
Arber Gishto· Zenodo (CERN European Organi...· 2 citations
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