IndisputableMonolith.RRF.Theorems
The RRF.Theorems module assembles results establishing invariance of argmin under strictly monotone rescalings of the strain function J, structural equivalence across isometric octaves, and preservation of state ordering under relevant operations. Researchers deriving scale-invariant equilibria in Recognition Science cite these to justify pattern transfer between octaves. The module organizes direct proofs from monotonicity and isometry without additional hypotheses.
claimFor strictly monotone $g : ℝ → ℝ$, argmin$(g ∘ J) =$ argmin$J$. Equivalent octaves (isometric strain functions) share identical equilibrium structures. Operations preserving ordering of states by strain also preserve meaning.
background
RRF centers on strain functions J whose physical content resides in the ordering of states rather than absolute values. The MonotoneArgmin submodule shows that any strictly monotone rescaling leaves the minimizing state unchanged, because only relative ordering matters. OctaveTransfer establishes that isometric strain functions between octaves produce identical equilibrium structures, allowing the same pattern to appear at different scales. OrderPreservation confirms that any operation respecting the strain ordering also respects the underlying meaning.
proof idea
The module imports three submodules and collects their theorems without further proof steps. MonotoneArgmin applies the definition of strict monotonicity to equate the argmin sets. OctaveTransfer uses isometry of strain functions to transfer equilibrium properties directly. OrderPreservation proceeds by showing that order-preserving maps commute with the strain comparison.
why it matters in Recognition Science
These theorems supply the invariance and transfer properties required for RRF to interface with the Recognition Science forcing chain, particularly the eight-tick octave and self-similar fixed point. They underwrite claims that structural patterns remain stable under rescaling, feeding downstream constructions of mass ladders and Berry thresholds.
scope and limits
- Does not derive numerical values for constants such as alpha or G.
- Does not treat non-monotone or non-isometric transformations.
- Does not address time-dependent or non-equilibrium dynamics.
- Does not connect directly to specific particle spectra or quantum states.