Use exact telescoping min-cut bit subset on paired-cube shared cores for kappa = 4.61028508707e-4 (4.61239827139e-4 at atom=1/2000)

[Th0rgal]

Summary

This PR extends #144 (icekylinx, commit c8b22bc5c10dba497ac25804e27d9647d818e2ff, which builds on #130 at 6a9970a530119174507904e23592fd59ede19a5d) by solving the bit-branch entrance-gauge subset selection problem on the pinned #97 (h = 23, v = 1,771, R = 28,866) readout schedule to provable global optimality over all $2^{11,565}$ subsets via an exact telescoping submodular interval-$s$-$t$-min-cut reduction, certifying:

$$ T(n) = O!\left(n (\log n)^{1 - \kappa}\right), \qquad \boxed{\kappa = \frac{461028508707}{1000000000000000} = 4.61028508707 \times 10^{-4}} \quad (\beta_{\text{atom}} = 10^{-3}), $$

and, in tightened_atom_2000 of certificates/paired-cube-network.json (composing with the tightened subordinate adapter exponent $\beta_{\text{atom}} = 1/2000$ from Rohan Gupta's #148):

$$ \boxed{\kappa_{1/2000} = \frac{461239827139}{1000000000000000} = 4.61239827139 \times 10^{-4}} \quad (\beta_{\text{atom}} = 5 \times 10^{-4}). $$

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Mathematical Contribution: Exact Telescoping Submodular Interval-$s$-$t$-Min-Cut Reduction

In #144, the complex paired-cube supplier achieves $a_c = 4856569 / 10^{10} = 4.856569 \times 10^{-4}$, so the overall semantic assembly saving $\kappa$ is 100% bound by the shared-core bit supplier ($m_b = 3h = 69$, $W_b = 2v + R = 32,408$, rank sum $2,234,128$, deficit $2,024$).

On the bit side, choosing a subset $S \subseteq \mathcal{R}$ of the $11,565$ certified entrance gauges (in their inherited readout order) trades off:

1. Individual role-transition benefits: Retaining slot $s \in \mathcal{R}$ with first-use dimension $r_s$ and entrance-gauge dimension $f_s$ replaces three stage copies of transition width $r_s$ with one exterior child of width $3f_s$ and three stage copies of transition width $r_s - f_s$, yielding a per-slot moment reduction
   $$\text{ben}(s) = 3 w(r_s) - \bigl(w(3f_s) + 3 w(r_s - f_s)\bigr),$$
   where $w(r)$ is the exact 120-bit upper-bound moment weight (including the $32 m_b^2$-child rare-class fallback on fraction $10^{-16}$ of every ideal edge). Exactly $10,888$ of the $11,565$ slots have $\text{ben}(s) > 0$.
2. Target-chain refinement costs: Retaining $s$ activates rank $f_s$ on every target $t \in \text{supp}(\text{adj}(s))$. On each target $t \in {1, \dots, 1771}$ with candidate intermediate ranks $0 = d_0 < d_1 < \dots < d_k < d_{k+1} = h - 1 = 22$, let $x_{t, p} \in {0, 1}$ indicate whether rank $d_p$ is active on target $t$. By summation by parts over consecutive active ranks, the target-chain moment on target $t$ satisfies the exact telescoping identity:
   $$\text{cost}t(x{t, 1}, \dots, x_{t, k}) = 3 w(22) + \sum_{p=1}^{k} c_{t, p, p}, x_{t, p} + \sum_{1 \le p < q \le k} c_{t, p, q} \left(1 - \prod_{m=p}^{q} (1 - x_{t, m})\right),$$
   where, by strict concavity of $r \mapsto r^{1-a_0}$, every diagonal and interval capacity is strictly positive:
   $$c_{t, p, p} = 3\bigl(w(d_p - d_{p-1}) + w(d_{p+1} - d_p) - w(d_{p+1} - d_{p-1})\bigr) &gt; 0,$$
   $$c_{t, p, q} = 3\bigl(w(d_{q+1} - d_p) + w(d_q - d_{p-1}) - w(d_{q+1} - d_{p-1}) - w(d_q - d_p)\bigr) &gt; 0 \quad (p &lt; q).$$

Each disjunctive penalty $1 - \prod_{m=p}^{q} (1 - x_{t, m})$ is represented without approximation by a single interval node $u_{t, p, q}$ with capacity $c_{t, p, q}$ to the sink $T$ and two infinite-capacity incoming DAG edges from $u_{t, p, q-1}$ and $u_{t, p+1, q}$. Scaling all 120-bit rational upper-bound weights by $W_b m_b 10^{16} 2^{120}$ to exact Python integers reduces the entire $2^{11,565}$ subset optimization problem to a single exact integer $s$-$t$ minimum cut / maximum flow on $36,760$ vertices:

Metric	PR #144 (9,543 slots)	PR #148 (9,543 slots, $\beta_{\text{atom}}=1/2000$)	This PR (9,730 slots, Global Optimum)
Retained / omitted bit gauges	9,543 / 2,022	9,543 / 2,022	9,730 / 1,835 (+165 rank 16, +22 rank 17)
Moved zero-frame dirty reads	9,594	9,594	9,179 (-415 zero-prelude reads)
Coarse bit saving $a_0$	4617656 / 10^10 (4.617656e-4)	4617656 / 10^10 (4.617656e-4)	461877426979 / 10^15 (4.61877426979e-4)
Effective bit saving $a_b$ ($\beta_{\text{atom}}=10^{-3}$)	4613422943 / 10^13 (4.613422943e-4)	—	461454009452021 / 10^18 (4.61454009452e-4)
Certified $\kappa$ ($\beta_{\text{atom}}=10^{-3}$)	4609169 / 10^10 (4.609169e-4)	—	461028508707 / 10^15 (4.61028508707e-4)
Effective bit saving $a_b$ ($\beta_{\text{atom}}=5\cdot 10^{-4}$)	—	4.6155394715e-4	923331436431021 / (2\cdot 10^{18}) (4.616657182155e-4)
Certified $\kappa$ ($\beta_{\text{atom}}=5\cdot 10^{-4}$)	—	4611281 / 10^10 (4.611281e-4)	461239827139 / 10^15 (4.61239827139e-4)

(Moreover, applying this same exact telescoping interval-min-cut on top of the $1,549$ eliminated terminal accumulators in #147 retains $8,010$ non-eliminated gauges and lifts $a_0$ to $232763446903 / (5\cdot 10^{14}) = 4.65526893806 \times 10^{-4}$, yielding $\kappa = 4.64667579244 \times 10^{-4}$ at $\beta_{\text{atom}} = 10^{-3}$ and $\kappa = 4.64880715940 \times 10^{-4}$ at $\beta_{\text{atom}} = 5\cdot 10^{-4}$; see #147#issuecomment-6074525962.)

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Verification

scripts/paired_cube_bit.py includes verify_global_mincut(), which constructs the $36,760$-vertex exact integer telescoping interval network and verifies in ~1.5s that:

1. All diagonal and interval cut capacities $c_{t, p, p}, c_{t, p, q}$ are strictly positive integers;
2. The residual $S$-reachable cut matches references/paired-cube/pr97-subset/selection.json (9,730 retained, 1,835 omitted) with exact integer cut_capacity == max_flow duality;
3. The exact integer cut value satisfies the bit-for-bit identity empty_moment - (total_pos_benefit - flow) == opt_moment < scale.

All verification targets run with assertions enabled (python3 -B):

make paired-cube-verify
make verify-joint

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Attribution & Provenance

• icekylinx: Paired-coordinate triple-cube identity, original-source involution, coordinate-star centers, cross-stage shared-core integration, and initial bit-subset pipeline (PR #144, commit c8b22bc5c10dba497ac25804e27d9647d818e2ff), extending PR #130 (6a9970a530119174507904e23592fd59ede19a5d), PR #115, and PR #104, developed with substantial OpenAI GPT-6 Astra and Codex assistance.
• Rohan Gupta (gupt1156): Tightened subordinate atom-wrapper exponent $\beta_{\text{atom}} = 1/2000$ (PR #148, commit 390e0bad6da517c5b5510c692f05a61c322e4b4a).
• an664: Completed-core cross-stage workspace-sharing principle (PR #128, 530588a019b4a74f09180680c9e3961bf649ec89), with reported OpenAI Codex assistance.
• eumemic: Positive local complex DAG (PR #117, cbb05ce504d571546d9b7794c186a613c659c3bf), with Anthropic Claude assistance.
• Zhihao Chen (jacklightChen) and Swapnil Jain: Certified physical bit word and rational frame witnesses (PR #97, e15350b66e4bf796153a68c240a6cc1b788abe3e, based on Swapnil Jain's 741e7aa078392553815df7926ee17ac5e25a8c38).
• Thomas Marchand (Th0rgal): Exact telescoping submodular interval-$s$-$t$-min-cut reduction, global-optimal 9,730-slot bit gauge subset, and integer max-flow/min-cut duality verification (Prepared by Thomas Marchand with Google Antigravity assistance).

[Mister-Meeseeks]

Thank you @Th0rgal. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Exact min-cut producer over a fixed candidate subset.

Review and limits: Fresh producer replay plus later #147/#150/#151 integration provenance. Exactness applies to the specified 2^11565 subset, not every producer or topology.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: 70355a3192028852598e96624ae29b389e664451. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.