The entropy counts the defect, and the defect fixes the curvature
Boltzmann and Planck: the entropy is the logarithm of the number of configurations.
On a cut with M crossing links, k quanta give S = k log M.
The 2g links of a surface of genus g all cross the cut after site 0, so
the surface is one of those configurations, and Gauss–Bonnet turns its defect into curvature:
S(Σ_g) = (log M / 2π) · (4π − ∫K dA)
Σ_g on the cut after site 0
Entropy against curvature
The entropy of a cut counts its quanta
Top: handleLinks_mem (the 2g links of Σ_g are a configuration of
2g quanta on the cut at 0), entropy_eq_curvature and entropy_handle_step
(one handle: +2 log M of entropy, −4π of curvature). δ∞ cancels in the
ratio and stays positive in each link: the defect is not swept away. Bottom: k quanta on the
M crossing links of a cut of a path with 21 sites have entropy k log M, proportional to
the defect (entropy_counts_defect); k distinct links are the C(M, k) graphs
with k new cycles, never above M log 2. A far site cannot tell them apart while the
cone has not reached the cut (horizon_hides_entropy). Gauss–Bonnet is the declared hypothesis
HGaussBonnet; the Bekenstein–Hawking factor 1/4 is a declared bridge, not derived.
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