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

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=-φ (register L121). This note explains the mechanism and reduces the "why -φ" 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 Hvv=α𝟙+β A1, Hvf=t F, Hff=h 𝟙 (F the vertex–face incidence; face centres never share a simplex, so Hff is exactly scalar). The sharing-count identity FF⊤=deg·𝟙+m A1+A2 — m the number of pentagons per edge, A2 the pentagon-diagonal shell — makes the Schur complement exactly two-shell for every stiffness, with g=-(s)/(β-ms), s=(t2)/(h), hence g=-φ⇔ (s)/(β)=(φ)/(1+mφ). In three dimensions (m=2, dodecahedron with lifted face centres) the condition is the golden rung 1/φ2 itself and is met at exactly one lift, b2/a2=(5+√5)/5, equivalently ρ*/ρplanar=4-√5, where ceff=-1 and the spectrum is M(-φ)+𝟙. In four dimensions (m=3) the condition reads s/β=1/(√5 φ)=(5-√5)/10, and the 120-cell satisfies it by itself: translation zero modes evaluated row-wise give h=-5t RF/Rv and α+4βcosθ1+6tcosγ=0; pentagon planarity gives cosγ=RF/Rv exactly with (RF/Rv)2=(5+2√5)/10; and the single remaining input is the identity α/β=4, which we prove: the triangles (C,F,v) and (C,Z,v) are right-angled at F and Z (facet planarity v· F=RF2, cell hyperplanarity v· Z=RZ2), so their areas are stationary in rv and the vertex row sees only the four edge hinges; the v↔ w mirror of each simplex fixes the edge hinge and forces ∂δ/∂ rv=∂δ/∂ rw; counting (a vertex lies in four edge hinges, an edge in one) gives α/β=4. An independent reconstruction reproduces the register coefficients of L121 to six digits. Since the Möbius map x↦ -x/(1-mx) has rational coefficients it commutes with the Galois involution σ; if the star partner of the 120-cell carries the conjugate stiffness σ(s/β)=φ/√5, its coupling is forced to g=σ(-φ)=+1/φ — 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.