Water at the Boundary: The 4D-Simplex Angle, the Icosahedron in Water, Salt as the Full Cube, Bismuth as the Half-Locked Cube, and the Nautilus Between the Rungs (INDYNA Research Note 33)

Abstract

We read matter through the flag universe (Notes 24–32), separating measured facts, register statements and candidates. Bismuth (fact): the A7 structure is a simple-cubic lattice Peierls-distorted along the body diagonal into two counter-shifted sublattices, symmetry $O_h\to D_{3d}$ — the fourfold axis lost, inversion kept: a material realisation of the half-locked cube of Note 30; hopper crystals and oxide colours are growth and thin-film effects; $u=0.2341$ vs $1/\varphi^3=0.2361$ is a curiosity, not universal (As, Sb differ). Water (fact): the H–O–H angle $104.474^\circ$ has $\cos=-1/4$ to $0.0035^\circ$, the vertex angle of the four-dimensional simplex ($\cos=-1/(n-1)$, $n=5$), while methane sits on the tetrahedron ($n=4$), BF$_3$ and SO$_2$ on the triangle ($n=3$) and H$_2$S, H$_2$Se, PH$_3$ near $90^\circ$; among fourteen molecules only water occupies the 4D angle. Water is planar with both sides occupied by lone pairs, and in ice every hydrogen bond carries its proton on one of two sides (Pauling's residual entropy) — the register's boundary object (Note 29: time zero on the boundary, the shell "thinks both exist"). Tetrahedron theorem (exact): octahedron $=T\cap-T$, cube $=\mathrm{hull}(T\cup-T)$, the hexagon is the threefold projection of the stella, the icosahedron is $\mathrm{snub}(T,-T)$ (Note 30), the dodecahedron is the hull of five disjoint tetrahedra (its twenty vertices admit exactly ten regular tetrahedra, two chiral compounds of five). The icosahedron in water (fact): five tetrahedra about an edge leave the Frank gap $360^\circ-5\arccos(1/3)=7.356^\circ$ and the pentagon angle $108^\circ$ lies $1.47^\circ$ below the tetrahedral angle, so hydrogen-bond five-rings are nearly strain-free: clathrate $5^{12}$ cages (twenty molecules, the hull of five tetrahedra, with a guest inside) and icosahedral clusters; liquid water prefers five-rings (denser, maximum at $4^\circ$C), ice Ih six-rings (hexagonal, less dense) — the density anomaly read as a change of roof, gold to twelvefold. Salt (fact): Na$^+$ is octahedrally hydrated; evaporation at the surface crystallises rock salt, each ion in an octahedron, the crystal a full cube with fourfold symmetry — the ionic bond has no direction, hence no offset: the sea produces the perfect cube the golden universe cannot lock. Depth (fact): clocks slow gravitationally by $g\Delta h/c^2$ ($1.1\cdot10^{-15}$ per 10 m); cold water becomes more mobile under pressure (Bridgman anomaly), so the network inversion speeds up with depth; a surface wave's motion decays as $e^{-2\pi z/\lambda}$ while its frequency is depth-independent — the register's shadow "ticks but does not move". Time scales: under the identification breathing $=m_1=0.50$ meV the observer's inversion lies at $1.82$ meV $=2.27$ ps and the tesseract clock at $\ge2.66$ meV $\le1.56$ ps, inside the hydrogen-bond window; the inversion is not an $E_8$ line (ledger 18, falsifiable). Nautilus (candidate): the shell is logarithmic but not golden (growth $2.8$–$3.3$ per turn, $\varphi^4=6.85$); two register constructions bracket it — five steps of $72^\circ$ with the space-shortage factor $1/\cos^2(\arcsin 1/\sqrt5)=5/4$, giving $(5/4)^5=3.052$, and the visibility exponent $\varphi^{2+0.297}=3.021$ (the present fraction at the time–shape crossing) — to be separated by data at the 1% level. Golden growth is established where fivefoldness grows: phyllotaxis, $360^\circ/\varphi^2=137.5^\circ$. Data from eighty Smithsonian shells (mean $1.310$ per quarter turn, i.e. $2.94$ per turn, species range $2.53$–$3.30$) leave both constructions inside the biological spread without confirming either: the Nautilus does not single out a register number. For the water angle itself, a four-domain repulsion model (two bonds, two lone pairs of weight $w$) reproduces $\cos=-1/4$ under Coulomb repulsion at $w=1.2455$ ($5/4$ to $0.36\%$, but not exactly, and potential-dependent); the exact origin of the 4D-simplex angle remains open. 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.