partial_removal_preserves_info
plain-language theorem explainer
Recognition Science forces constant field values across any connected J-minimum cache, so a single remaining vertex suffices to recover the global field. Neuroscientists modeling holographic memory or functional preservation after partial removal cite this result. The argument reduces directly to the ratio_rigidity lemma applied to the cache structure.
Claim. Let $C$ be a local cache on vertex set $V$ with field $f:V→ℝ_{>0}$, where the underlying graph is connected and $J(f(v)/f(w))=0$ for every adjacent pair. For any subset $S⊆V$ containing a vertex $v_0$ and any $w∈V$, $f(w)=f(v_0)$.
background
A LocalCache is a structure on vertices $V$ consisting of an adjacency relation whose reflexive-transitive closure connects every pair, a positive real-valued field, and the at-J-minimum condition that $Jcost(f(v)/f(w))=0$ whenever vertices are adjacent. This models a brain region as a J-cost-optimal local cache derived from GCIC. The module derives brain holography from RS first principles via the chain T5 (J uniqueness) → GCIC Graph Rigidity (zero cost implies global constancy) → Local-Global Information Theorem (every connected subgraph vertex determines global state) → Holographic Cache Property (boundary encodes bulk) → Surface Area Scaling in D=3 → Partial Removal Resilience.
proof idea
The proof is a one-line term-mode wrapper that applies the ratio_rigidity lemma to the cache's graph_connected property, its field_positive condition, and the at_J_minimum relation between the arbitrary vertex w and the distinguished remaining vertex v_remain.
why it matters
This theorem occupies the Partial Removal Resilience position in the module's derivation chain, showing that information access depends only on connectivity, not on the volume of the remaining set. It directly supports the master certificate brain_holography_inevitable and aligns with the eight-tick octave and D=3 surface-area scaling. No downstream uses appear in the module.
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