Pith. sign in
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IndisputableMonolith.Cosmology.PolarizedBirthInterfaceCost

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Recognition cost of an ordered adjacency whose cells differ by d phi-rungs equals J(φ^d), with J the forced cost and φ the golden ratio. Carried monochromatic edges cost zero; bichromatic interface edges cost J(φ) each. Cosmologists auditing the polarized birth ledger cite the aggregates. Arguments are algebraic evaluations of J at powers of φ plus zero-cost identities for equal-level edges.

claimFor an ordered adjacency of cells whose charges differ by $d$ rungs on the $\varphi$-ladder, the recognition cost is $J(\varphi^d)$, where $J(x)=(x+x^{-1})/2-1$ and $\varphi$ is the golden ratio. Monochromatic (carried) edges cost $0$; unit interface edges cost $J(\varphi)$. Aggregate interface and carried costs are those unit costs times the corresponding edge counts.

background

Recognition Science forces a unique nonnegative cost $J$ on positive reals by the Recognition Composition Law and fixed-point normalization (T5): $J(x)=(x+x^{-1})/2-1$. The self-similar scale is the golden ratio $\varphi$ (T6). Costs of ladder steps are therefore values $J(\varphi^d)$.

The polarized birth field assigns each lattice cell a charge in ${+1,0,-1}$. Upstream LatticeBallEdges counts ordered adjacencies of the L1 ball and splits them into monochromatic (same-sign, carried-internal) versus bichromatic (forced-distinction) edges. Interface counts are closed forms: $8t-4$ in 2D and $8t^2-8t+4$ in 3D; the remainder is carried for free.

This module attaches the cost functional to that split. It introduces $J$-powers of $\varphi$, the level difference of an edge, the per-edge cost, and the two aggregate ledgers (interface versus carried).

proof idea

Definition layer first: Jpow $d := J(\varphi^d)$, level_diff from the two cell charges, edgeCost as Jpow of that difference, then interfaceCost and carriedCost as sums over the two edge classes.

Algebraic lemmas evaluate the base cases: Jpow at $0$ vanishes, at $\pm 1$ recovers $J(\varphi)$, and absolute value one is enough by evenness of $J$ in the log coordinate. Positivity $J(\varphi)>0$ is recorded once.

The structural facts are then immediate: edgeCost is zero on carried (equal-level) edges and equals $J(\varphi)$ on unit interface edges. Aggregates multiply those unit costs by the counts imported from LatticeBallEdges. No analytic estimates; pure rewriting on the forced $J$ and $\varphi$.

why it matters in Recognition Science

Closes the binary (sign-valued) half of the polarized-birth cost ledger: carried bulk is free, the interface pays $(\mathrm{count})\cdot J(\varphi)$. Downstream GradedRungCost imports the module and lifts the same law from a single sign flip to a multi-valued phi-rung profile, the live engine's "coarsest allowed rung" field. That parent doc names this file as Phase 55 of the ledger.

In the broader forcing chain the cost is the T5 $J$-unique functional and the rung step is the T6 fixed point $\varphi$. The eight-tick and $D=3$ landmarks sit one layer up, in the adjacency counts this module consumes rather than re-proves. Without the zero-carried / $J(\varphi)$-interface split, the graded generalization has no binary base case.

scope and limits

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