8 Komorebi Space, realization fibers, and informative questions
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Komorebi Space supplies a natural setting for studying the proof field's coherent realizations. Here the connection is relative to an explicitly chosen system of views. Nothing in a finite proof hypergraph canonically supplies one universal category of all proofs, and a formal proof term is not automatically equivalent to an intended semantic object. The view system must be stated, together with its restrictions and the observations that are required to survive.
Definition 8.1: Coherent views of a proof field
Choose a finite collection of views \(V\), fibers \(F_v\) of local proof states or annotations, and for each declared directed edge \(e:v\to w\) a restriction map \(e^*:F_w\to F_v\). The strict realization set is
The extension fiber of prescribed information \(s\) on \(U\subseteq V\) consists of those global coherent families whose restrictions equal \(s\). If the edges form a category, the usual functorial composition equations are additional explicit obligations.
The finite implementation ambient.py realizes this edge-defined model exactly. The gluing condition should not be weakened to covering the objects alone: a compatibility equation is attached to an edge and must appear in at least one patch.
Theorem 8.2: Finite arrow-cover gluing
Suppose patches \(U_i\subseteq V\) cover every view and every declared edge has both endpoints in some \(U_i\). Let \(s_i\) be coherent assignments on \(U_i\) that agree on all overlaps. There is a unique global coherent assignment \(s\in\Gamma(F)\) restricting to every \(s_i\).
Proof
For each view \(v\), choose any patch containing it and define \(s(v)\) to be the value supplied by that patch. Agreement on overlaps makes this independent of the choice. For a declared edge \(e:v\to w\), choose a patch containing both endpoints. The local assignment on that patch obeys \(e^*s(w)=s(v)\), so the global assignment is coherent. It is the only possible glue because every coordinate was prescribed by at least one patch.
Example 8.3: Object coverage cannot replace arrow coverage
Let \(V=\{v,w\}\) with an edge \(v\to w\), both fibers \(\{0,1\}\), and identity restriction. Singleton patches prescribe \(s(v)=0\) and \(s(w)=1\). These assignments cover all views and agree on their empty overlap. There is no global coherent assignment with those coordinates. The missing ingredient is the edge condition, not the availability of the two individual values.
The theorem is a finite instance of the strict gluing construction in Komorebi Space [1]. It does not claim a general stack theorem for nonabelian proof objects. Gluing up to isomorphism requires the identifications and their coherence witnesses, which are retained in the corresponding groupoid setting.
8.1 Exact finite realization and obstruction certificates
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A second connection is quantitative. Work over \(\mathbb F_2\) with \(E=\mathbb F_2^n\). Let \(A:E\to\mathbb F_2^m\) impose coherence, \(P:E\to\mathbb F_2^p\) retain prescribed information, and \(x\in E\) be a raw candidate. Define
This is a finite linear realization fiber. The equation \(My=b\) does not by itself produce a solution. Compatibility of the prescribed information must first be established.
Theorem 8.4: Finite realization alternative and ambiguity
Exactly one of the following holds:
\(My=b\) has a solution \(y_0\). Its complete solution set is \(y_0+H\), where \(H=\ker A\cap\ker P\), and
\[ |\mathfrak K_P(x)|=2^{n-\operatorname{rank}(M)}. \]A row vector \(\lambda\) exists with \(\lambda M=0\) and \(\lambda b=1\). It is a directly checkable certificate that \(\mathfrak K_P(x)=\varnothing\).
Proof
If \(My_0=b\), any other solution differs from \(y_0\) by an element of \(\ker M=H\), and every such difference gives another solution. Rank nullity gives \(\dim H=n-\operatorname{rank}(M)\). If \(b\) is not in the column space of \(M\), linear algebra supplies a linear functional that vanishes on that column space and takes value \(1\) on \(b\). Expressing it as a row vector gives \(\lambda\). Conversely such a witness contradicts any proposed equation \(My=b\), since it would imply \(0=\lambda My=\lambda b=1\). These cases are mutually exclusive and exhaustive over a field.
This negative certificate is a concrete instance of the protected obstruction principle from Komorebi Space. Its scope is exactly the finite linear system \(My=b\). It is not a certificate that some broad mathematical conjecture is unprovable.
8.2 The information gained by a new observation
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Suppose the realization fiber is nonempty and a new linear observation \(Q:E\to\mathbb F_2^q\) becomes available. The right question is not how many bits appear in the output of \(Q\) but how many remaining degrees of freedom in the fiber it distinguishes.
Theorem 8.5: Exact observation gain
Give the nonempty finite affine fiber \(y_0+H\) the uniform distribution. Then the Shannon entropy of the observed value \(Qy\) is
For every observation value that actually occurs, the corresponding conditional fiber contains
realizations. Thus the uncertainty reduction is exactly \(\operatorname{rank}(Q|_H)\) bits.
Proof
The image \(Q(H)\) is a vector subspace of dimension \(r=\operatorname{rank}(Q|_H)\), so the observed values form the affine set \(Qy_0+Q(H)\) of size \(2^r\). Every such value has a preimage that is a coset of \(H\cap\ker Q\), whose dimension is \(\dim H-r\) by rank nullity. Equal fiber cardinalities make the induced output distribution uniform on \(2^r\) values. Its entropy is \(\log_2(2^r)=r\), and each conditional fiber has cardinality \(2^{\dim H-r}\).
Example 8.6: One informative question
Let \(y=(a,b,c)\in\mathbb F_2^3\) with \(a+b+c=0\) and protected value \(a=0\). The realization fiber is \(\{(0,0,0),(0,1,1)\}\). It has one bit of ambiguity. Observing \(b\) resolves that bit exactly. Observing \(a\) supplies no new information, despite being a perfectly valid measurement. This difference is measured by the rank of the observation restricted to \(H\), rather than by the output width of the observation.
The module realization.py computes these quantities through exact binary row reduction, and returns an explicit linear witness when the protected realization is impossible. The program does not enumerate all solutions, although the tests compare its results with exhaustive enumeration on small cases. The entropy statement uses Shannon entropy [3] and assumes uniform sampling of the compatible realizations. It does not assert a universal prior over mathematical discoveries, or that the most informative observation is automatically the most valuable research question once costs and task priorities are included.