IndisputableMonolith.RecogGeom.EffectiveManifold
The EffectiveManifold module defines refinement relations on recognizers and the conditions under which sequences of refinements converge to stable manifold structures. It is cited by researchers formalizing the geometric limits of composite measurements in recognition geometry. The module proceeds by introducing the finer-than ordering, proving its basic lattice properties, and then layering directed refinement, convergence, and non-collapse conditions on top of the quotient and composition foundations.
claimLet $R_1$ and $R_2$ be recognizers on a configuration space $C$. Write $R_1$ is at least as fine as $R_2$ when $c_1$ is identified with $c_2$ under $R_1$ implies the same identification holds under $R_2$. The module further defines directed refinements, convergent refinement sequences, dimension invariants, and non-collapse conditions that guarantee the limiting object behaves as an effective manifold.
background
Recognition geometry begins with the quotient construction $C_R = C / {~}$ from the Quotient module, where ${~}$ is the indistinguishability relation induced by a recognizer. The Composition module supplies the Refinement Theorem for composite recognizers acting on the same space. EffectiveManifold sits between these two, equipping the quotient with a partial order on recognizers so that successive refinements can be compared and shown to stabilize.
proof idea
This is a definition module. It first declares the IsFinerThan' relation together with its reflexivity and transitivity lemmas. It then introduces DirectedRefinement and RefinementConverges, followed by the DimensionInvariant and NonCollapseCondition predicates. The remaining lemmas establish that monotonicity and non-collapse hold automatically once the ordering is respected.
why it matters in Recognition Science
The module supplies the ordering and convergence apparatus required by the Refinement Theorem in Composition and the quotient constructions in Quotient. It thereby closes the geometric scaffolding needed to treat physical measurement as the stable limit of refining recognizers, consistent with the eight-tick octave and dimension-forcing steps of the overall framework.
scope and limits
- Does not contain the Refinement Theorem itself.
- Does not compute explicit quotients or numerical invariants.
- Does not address mass ladders or coupling constants.
- Does not treat non-monotone or collapsing refinement sequences.
depends on (2)
declarations in this module (12)
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def
IsFinerThan' -
theorem
isFinerThan'_refl -
theorem
isFinerThan'_trans -
structure
DirectedRefinement -
structure
RefinementConverges -
structure
DimensionInvariant -
structure
NonCollapseCondition -
theorem
monotone_separation_of_refinement -
structure
EffectiveManifoldHypotheses -
theorem
nonCollapse_monotone_automatic -
theorem
convergence_implies_identity -
def
status_summary