How matter curves the network: local defect energies, a sector-dependent sign, and the horizon route to the Newton constant in SQG

The second geometry companion of Spectral Quantum Gravity (SQG) found that the spatial base is an ordered honeycomb times an activated scale axis ω. This third companion asks how matter curves that network. On the honeycomb, curvature is carried by disclinations (a pentagon is a cone of deficit π/3, a heptagon of excess π/3), and in general relativity a mass produces a conical deficit through a local constraint.Results:The network form of the constraint is proved: for a static metric the Einstein action reduces exactly to the Regge form, which gives δ/(8πG) = E_m at every vertex together with its sign convention. Summed over a closed base it gives a sum rule: the total energy equals χ/(4G), i.e. 1/(2G) for a spherical base.Matter has no energetic preference for positive curvature: on the two-dimensional base this is excluded by Gauss–Bonnet, and explicit computations show only Friedel-type sign alternation.The vacuum energy is concave in a local lapse (proved); its stiffness decays as r^−5 (scaling argument; computed exponent 5.06 on the infinite lattice), so the lapse must be integrated along a rotated contour.The deficit-linear defect vacuum energy has a sector-dependent sign: scalars and the honeycomb hopping fermions give the sign of G > 0, Kähler–Dirac fields and spinors with spin connection the opposite sign. An exact Hodge identity shows that the Kähler–Dirac field is spectrally four bosonic scalars with fermionic sign. The hopping fermions add a dominant ring energy even in the deficit. Local defect energies therefore do not fix the sign of G.Across the SQG horizon, perpendicular to the scale axis, the honeycomb fermions obey an area law. At fixed base momentum the problem decouples exactly into two ionic chains; the area-law coefficient is a Brillouin-zone integral of a one-chain entropy given by elliptic integrals, equals 0.4956 per base cell, and grows as (1/3)[ln(8 t_ω) − m(1+x+y)], with m the Mahler measure. With four Weyl nodes per honeycomb species (proved) this gives G > 0 and places the ω spacing at 2.3 Planck lengths, against 0.55–0.77 in the main text; the logarithm assigns the factor to the regulator.All computations (P360.1–P360.23) were registered before they were run; the registrations, result logs and a table of the predictions met and not met are in the code archive. Every statement carries a grade (proved, computed, argued).