Raise conditional kappa to 4.611281e-4 with tightened adapter toll

[gupt1156]

Summary

This pull request sharpens the conditional Harvey-Hoeven integer multiplication upper bound from the current PR #144 record ($\kappa = 4.609169 \times 10^{-4}$) to:

$$\kappa = \frac{4611281}{10000000000} = 4.611281 \times 10^{-4}$$

Key Changes

1. Tightened Adapter Toll: Tightens the stopped-adapter toll parameter from $\text{ATOM} = 10^{-3}$ to $\text{ATOM} = 5 \times 10^{-4} = 1/2000$.
2. Exact Inequality Satisfaction: The subordination condition strictly holds:
   $$A_B = 4.615539 \times 10^{-4} < 5 \times 10^{-4} < 1 - A_B$$
   preserving all lower-order bounds for internally borrowed rows and adapter tolls without altering any upstream interface assumptions.
3. Rational Finite Assembly: With $A_B = 4.6155394715 \times 10^{-4}$ and $A_C = 4.856569 \times 10^{-4}$:
   • Minimum assembly margin expands to $4.61128262 \times 10^{-4}$.
   • Positive absorption gap over $\kappa = 4611281 / 10^{10}$: $+1.6217 \times 10^{-10} > 0$.
   • All 47 strict linear assembly inequalities and 7 margins strictly satisfied.

Verification

Passed cleanly via:

make paired-cube-verify

All moments, shared cores, finite router, and strict constraints PASS.

[Th0rgal]

Technical Review & Composition with Exact Global Bit-Subset Min-Cut (#146)

We verified the arithmetic in this PR (ATOM = Q(1, 2000) on #144's $9,543$-slot bit subset):

• Since $a_b = (1 - 1/2000) \cdot (4617656 / 10^{10}) + (1/2000) \cdot (384599 / 10^{10}) = 4.6155394715 \times 10^{-4} < 5 \times 10^{-4} = \beta_{\text{atom}}$, the subordination inequality $a_b < \beta_{\text{atom}} < 1 - a_b$ in notes/three-stage-cover-rows.tex and scripts/paired_cube_network.py (require(ATOM > AB and ATOM < 1 - AB)) continues to hold strictly, lifting $\kappa$ from $4609169 / 10^{10}$ to $4611281 / 10^{10}$.

Orthogonal Composition with #146

In #146, we kept ATOM = Q(1, 1000) unchanged and solved the $2^{11,565}$ PR #97 bit-gauge subset selection problem to provable global optimality (9,730 retained / 1,835 omitted slots) via an exact integer telescoping submodular interval-$s$-$t$-min-cut reduction, lifting coarse bit saving from $a_0 = 4617656 / 10^{10}$ to $a_0 = 461877426979 / 10^{15}$ ($\kappa = 461028508707 / 10^{15}$ at ATOM = 1/1000).

Combining your ATOM = Q(1, 2000) parameter choice here with the globally optimal $9,730$-slot bit subset from #146 yields:

• $a_0 = 461877426979 / 10^{15} \approx 4.61877426979 \times 10^{-4}$,
• $a_b = 923331436431021 / (2 \times 10^{18}) \approx 4.616657182155 \times 10^{-4} < 5 \times 10^{-4} = \text{ATOM}$,
• $\kappa = 461239827139 / 10^{15} \approx 4.61239827139 \times 10^{-4}$ (4612398 / 10^10 on the $10^{-10}$ grid).