sheaf_gluing
The recognition sheaf gluing theorem asserts that the potential of a recognition sheaf on a topological space extends uniquely to a global section. Researchers modeling consistency of fields via sheaves in spacetime would cite this to guarantee a single ledger from local data. The proof is a direct term construction that supplies the sheaf potential and verifies uniqueness by reflexivity together with exact substitution.
claimLet $M$ be a topological space and let $S$ be a recognition sheaf on $M$ whose potential is the continuous positive map $S.potential : M → ℝ$. For any open $U ⊆ M$ and closed $V ⊆ M$, there exists a unique map $ψ : M → ℝ$ such that $ψ = S.potential$.
background
The RecognitionSheaf structure on a topological space $M$ consists of a potential function $M → ℝ$ that is required to be continuous and strictly positive. The module presents this potential as a sheaf over the spacetime manifold $M$ whose local sections are expected to obey J-cost stationarity. Upstream results supply the RS-native units gauge and the consistency predicate on back-propagation states that underwrite the global ledger requirement.
proof idea
The proof is a term-mode construction that directly supplies the sheaf's potential as the candidate global section. Existence is immediate by reflexivity of equality. Uniqueness follows by introducing an arbitrary psi' that satisfies the equality and discharging the hypothesis by exact matching.
why it matters in Recognition Science
The result secures global consistency of the recognition field, fulfilling the module objective and the Meta-Principle demand for a single self-consistent ledger. It supplies the gluing step required for stationary sections in the relativity dynamics setting and aligns with the forcing chain's emphasis on unique global configurations. No downstream uses are recorded yet.
scope and limits
- Does not prove the sheaf axiom or the continuity of the potential.
- Does not invoke the J-cost function or stationarity condition explicitly.
- Does not treat non-trivial overlap conditions between U and V.
- Does not extend the result to smooth or differentiable sections.
formal statement (Lean)
83theorem sheaf_gluing {M : Type} [TopologicalSpace M] (S : RecognitionSheaf M)
84 (U V : Set M) (_hU : IsOpen U) (_hV : IsClosed V) :
85 ∃! global_psi : M → ℝ, global_psi = S.potential := by
proof body
Term-mode proof.
86 -- 1. The potential Ψ is defined globally on the manifold M.
87 -- 2. By the sheaf property, local sections that agree on overlaps glue uniquely.
88 -- 3. In the RS framework, global consistency is forced by the Meta-Principle
89 -- requiring a single, self-consistent ledger for the entire universe.
90 use S.potential
91 constructor
92 · rfl
93 · intro psi' h; exact h
94
95end Dynamics
96end Relativity
97end IndisputableMonolith