boundary_scales_as_area
plain-language theorem explainer
In three spatial dimensions the boundary of a region of radius R scales with area rather than volume. Researchers modeling holographic information storage in brains or deriving scaling laws from graph rigidity would cite this to connect the forced dimension to surface encoding. The proof is a one-line term that introduces the hypotheses and directly substitutes the spatial dimension value of three to exhibit a positive constant.
Claim. In three spatial dimensions, for any positive integer radius $R$, there exists a positive real constant $C$ such that the boundary scaling exponent equals two.
background
The module derives brain holography from GCIC by chaining T5 J-uniqueness to graph rigidity, local-global information theorems, holographic cache properties, and surface-area scaling under D=3. The J-cost function, induced by the multiplicative recognizer on positive ratios, enforces constancy on connected graphs at minimum cost. The spatial dimension is fixed at three by the eight-tick octave in the unified forcing chain, yielding the surface-to-volume relation stated in the doc-comment.
proof idea
The term proof introduces the radius and positivity hypotheses, then exhibits the constant 1 together with a norm_num check for positivity and a simp reduction on the spatial dimension definition to confirm the exponent equals two.
why it matters
This declaration supplies the D=3 surface-area step in the brain holography chain, feeding the master certificate that holography follows from RCL through J-uniqueness, GCIC rigidity, and the local cache theorem. It aligns with the eight-tick octave forcing three dimensions and the holographic principle that boundary encodes bulk. No open scaffolding questions are closed here.
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