The Stiffness Condition: Sharing Counts, Schur Complements, and Why the 120-Cell Couples at −φ (INDYNA Research Note 23)

Abstract

Note 20 derived the two-shell gravitational operator of the 120-cell cone by Schur elimination of its stiff pentagonal face centres, with coupling ratio $g=-\varphi$ (register L121). This note explains the mechanism and reduces the "why $-\varphi$" question to a single rational identity. For a cone over a pentagon-faced body whose face centres are stiff, locality and transitivity force the block form $H_{vv}=\alpha\mathbb{1}+\beta A_1$, $H_{vf}=t\,F$, $H_{ff}=h\,\mathbb{1}$ ($F$ the vertex–face incidence; face centres never share a simplex, so $H_{ff}$ is exactly scalar). The sharing-count identity $FF^{\top}=\deg\cdot\mathbb{1}+m\,A_1+A_2$ — $m$ the number of pentagons per edge, $A_2$ the pentagon-diagonal shell — makes the Schur complement exactly two-shell for every stiffness, with \[ g=-\frac{s}{\beta-ms},\qquad s=\frac{t^{2}}{h},\qquad\text{hence}\qquad g=-\varphi\iff \frac{s}{\beta}=\frac{\varphi}{1+m\varphi}. \] In three dimensions ($m=2$, dodecahedron with lifted face centres) the condition is the golden rung $1/\varphi^{2}$ itself and is met at exactly one lift, $b^{2}/a^{2}=(5+\sqrt5)/5$, equivalently $\rho^{*}/\rho_{\mathrm{planar}}=4-\sqrt5$, where $c_{\mathrm{eff}}=-1$ and the spectrum is $M(-\varphi)+\mathbb{1}$. In four dimensions ($m=3$) the condition reads $s/\beta=1/(\sqrt5\,\varphi)=(5-\sqrt5)/10$, and the 120-cell satisfies it by itself: translation zero modes evaluated row-wise give $h=-5t\,R_F/R_v$ and $\alpha+4\beta\cos\theta_1+6t\cos\gamma=0$; pentagon planarity gives $\cos\gamma=R_F/R_v$ exactly with $(R_F/R_v)^2=(5+2\sqrt5)/10$; and the single remaining input is the identity $\alpha/\beta=4$, which we prove: the triangles $(C,F,v)$ and $(C,Z,v)$ are right-angled at $F$ and $Z$ (facet planarity $v\cdot F=R_F^2$, cell hyperplanarity $v\cdot Z=R_Z^2$), so their areas are stationary in $r_v$ and the vertex row sees only the four edge hinges; the $v\leftrightarrow w$ mirror of each simplex fixes the edge hinge and forces $\partial\delta/\partial r_v=\partial\delta/\partial r_w$; counting (a vertex lies in four edge hinges, an edge in one) gives $\alpha/\beta=4$. An independent reconstruction reproduces the register coefficients of L121 to six digits. Since the Möbius map $x\mapsto -x/(1-mx)$ has rational coefficients it commutes with the Galois involution $\sigma$; if the star partner of the 120-cell carries the conjugate stiffness $\sigma(s/\beta)=\varphi/\sqrt5$, its coupling is forced to $g=\sigma(-\varphi)=+1/\varphi$ — the four-dimensional Kepler–Poinsot pairing of the coupling ratio itself. Stated as a prediction. 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.