IndisputableMonolith.Cosmology.PolarizedBirthInterface
On the polarized diamond field, every charge-flipping adjacency has an endpoint on the central spine x=0. The module defines that spine, proves interface confinement to it, and shows both spine and birth-field interface are sub-extensive in the L1 ball. Cosmologists bounding carried-state cost in the coarsening engine cite it. Arguments are pure sign case analysis plus closed spine cardinality against the diamond volume law.
claimOn the 2D L1 diamond $|x|+|y|\le t$ with polarized charge $\mathrm{sign}(x)$, every bichromatic unit adjacency has an endpoint on the spine $\{x=0\}\cap$ diamond. That spine has $2t+1$ cells, hence is $o(\mathrm{Vol})$ against volume $2t^2+2t+1$; the recognition-active interface is therefore sub-extensive.
background
The scale-adaptive coarsening engine grows its world as an L1 ball, one rung per recognition cycle: the 2D diamond $|x|+|y|\le t$. Upstream, LatticeBallVolume supplies the closed cell count $2t^2+2t+1$. InterfaceComponentBound already showed that locked-domain count is at most the interface size plus one for any charge field. PolarizedBirthDomains specialized to the forced conjugate-birth (polarized) field and proved carried-state sub-extensivity at the domain level.
This module isolates the geometric support of that field's active distinctions. Charge is $\mathrm{sign}(x)$, so flips can occur only between columns $x\in{-1,0,1}$. The spine is the codimension-1 slice $x=0$ inside the diamond; siblings record it as an explicit image of an interval and give its cardinality $2t+1$.
proof idea
Core lemma: any two unit-adjacent cells of opposite charge have an endpoint on the spine. Proof is pure case analysis on $\mathrm{sign}$ at each endpoint; the only unit steps that flip sign change the $x$-coordinate across $0$, forcing one endpoint onto $x=0$. The spine is identified with the image of ${-t,\ldots,t}$ under $k\mapsto(0,k)$, yielding card $2t+1$. Sub-extensivity of the interface (and of the birth field as a carried object) is then the comparison of that linear count to the quadratic diamond volume from LatticeBallVolume.
why it matters in Recognition Science
Phase-51 bridge between domain-level sub-extensivity and exact edge accounting. Downstream, PolarizedBirthInterfaceCount quotes this module as having confined the recognition-active interface to the spine and bounded spine cells, then closes the exact bichromatic ordered-edge count ($8t-4$ in 2D, $8t^2-8t+4$ in 3D) and the constant activity-per-cycle claim. LatticeBallEdges uses the same split to separate total adjacencies into monochromatic (carried free) versus interface (paid) edges. Together they finish the carried-versus-interface ledger for the polarized birth field on the coarsening world.
scope and limits
- Does not give the exact bichromatic edge count (deferred to PolarizedBirthInterfaceCount).
- Does not derive total or monochromatic adjacency formulas (deferred to LatticeBallEdges).
- Does not claim the spine argument for arbitrary non-polarized charge fields.
- Does not replace the domain-count bound of InterfaceComponentBound; it specializes geometry for the birth field.
- Does not assert extensivity statements outside the L1 diamond setting used here.