IndisputableMonolith.Sociology.PolarizationFromJCost
This module applies the canonical J-cost band template to sociology, defining a polarization driver and its certificate from recognition costs on social ratios. Social physicists or complex-systems modelers would cite it when extending Recognition Science beyond physics to group dynamics. The module structure consists of four sibling definitions that instantiate the six-clause template (matched zero at unity and non-negativity) without introducing new proofs.
claimThe module supplies a polarization driver $D$ on interaction ratios $r$ together with a certificate verifying that the J-cost satisfies $J(1)=0$ and $J(r)≥0$ for all $r>0$, thereby producing polarization as an emergent imbalance under the Recognition Composition Law.
background
Recognition Science models all domains via the J-cost function $J(x)=(x+x^{-1})/2-1$, which vanishes at unity and is nonnegative elsewhere. The upstream CanonicalJBand module supplies the reusable six-clause template that every domain certificate must satisfy: matched-zero $J(1)=0$ and nonnegativity $J(x)≥0$ for $x>0$. This sociology module imports that template and specializes it to social ratios, treating polarization as the macroscopic consequence of J-cost gradients on those ratios. The local setting therefore inherits the algebraic properties of the forcing chain (T5 J-uniqueness through T8 three-dimensional space) while remaining agnostic about empirical calibration.
proof idea
This is a definition module, no proofs. It defines PolarizationDriver and PolarizationCert by direct specialization of the six-clause J-band clauses to sociological ratios, together with two auxiliary count and certification helpers.
why it matters in Recognition Science
The module supplies the sociology-domain opening in the master cert chain, feeding the B-tier whole-science structure that begins with the UnifiedForcingChain. It demonstrates that the same J-cost and Recognition Composition Law used for physics constants also generate polarization phenomena, thereby closing one of the forty-something domain certs without requiring new axioms. No open scaffolding remains inside the module itself.
scope and limits
- Does not supply numerical values for polarization strength or critical thresholds.
- Does not incorporate network topology or multi-agent dynamics.
- Does not relax the non-negativity or matched-zero clauses inherited from CanonicalJBand.
- Does not address empirical falsification against survey or election data.