Boundary Compensation Mathematics IV BC-M IV: Finite-Resolution Solution Geometry, Quotient Tubes, and Metric-Entropy Profiles

Let (C, dX) be a declared class of admissible realizations, (D, dD) a data space, Π : C → D an observation
morphism, and d ∈ D an observed datum. At finite data precision, the natural object is not the exact fiber
Π−1
(d) but the admissible solution tube
F
(δ)
d = {x ∈ C : dD(Π(x), d) ≤ δ}.
After a preregistered equivalence relation ∼δ is imposed, one obtains the quotient tube M(δ)
d
. This paper
formalizes the conditions under which that quotient carries a genuine metric or a separating pseudometric and introduces the covering number Nε, packing number Pε, and metric entropy
Hε(M(δ)
d
) = log Nε(M(δ)
d
).
We prove monotonicity under tolerance enlargement, inclusions under datum perturbation, contraction of covering numbers under Lipschitz maps, bi-Lipschitz comparison, subadditivity for product and
tower geometries, and an information lower bound for finite probe transcripts. On regular smooth chambers, metric-entropy scaling recovers the local fiber dimension only under bi-Lipschitz chart control and
uniformly bounded geometry; at walls, the scaling law may jump. The central result is a two-scale warning: at fixed nonzero δ, the limit ε ↓ 0 detects the thickness of the data tube and may return the ambient
dimension even when the exact fiber has smaller dimension. BC-M IV therefore introduces the bandlimited entropy slope and requires a joint audit of (δ, ε).
Two exact examples display the mechanism. For a rank-deficient linear map, entropy decomposes into
kernel ambiguity and visible ellipsoid thickness governed by the singular values. For a real symmetric
2 × 2 trace–determinant model, the exact raw fiber is a circle, the quotient under orthogonal conjugacy
is a singleton, and the finite-resolution tube becomes an annulus with a one-dimensional radial quotient.
Finally, for Graph Reconstruction with the discrete metric, entropy below unit scale equals the logarithm
of the number of nonisomorphic graphs compatible with the deck, yielding a direct handoff to D3.
The paper introduces a status protocol, five certification workflows, the BCM4Report format, and a
Claim Firewall. Metric entropy here is not Shannon entropy, thermodynamic entropy, physical information content, or an automatic estimator bound.
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