Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4

[hpst3r]

Stacked on #144 (icekylinx). It adds research/bit-elim-144/ and one workflow; no existing file changes.

Under #144's retained interfaces, this certifies

T(n) = O(n (log n)^(1 - κ)),   κ = 4646633/10^10 = 4.646633e-4

That is +0.813% over #144 (4609169/10^10). #144 is bit-limited, so a bit-side change moves κ directly.

What changes

Only the bit word changes. #144's bit branch shares Swapnil Jain's round-seven PR97 word in dimension-69 completed cores. This package applies #122's terminal accumulator elimination (SovereignSteak) to that word: 1,549 of the 4,725 candidate terminal output slots are eliminated, and every candidate is a selected gauge slot.

Each eliminated slot is a deferred output accumulator that never sources an update. Its read, output read, inverse writes and V^-1 are deleted. Each of its forward writes is applied to its target at the same time and frame. The remaining target chains are subsequences of #144's certified chains, and every new target-chain adjacency is re-certified exactly over Q.

	roles R	W per vertex	coarse bit saving	stopped bit saving	κ
#144	28,866	32,408	4617656/10^10	4613422943/10^13	4609169/10^10
+ 1,549 eliminations	27,317	30,859	4655227/10^10	1162739093/(2.5·10^12)	4646633/10^10

The deficit per vertex (2,024) is unchanged. The complex side (4856569/10^10) still has 4.5% headroom.

#146 (Th0rgal) appeared while this was being verified and also changes #144's bit side. The two have not been composed here.

Verify

python3 research/bit-elim-144/verify.py    # stdlib only, about 5 minutes, about 0.75 GB; do not use -O

verify.py runs these checks:

1. It reproduces Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's coarse saving, stopped saving and κ exactly, using Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's own functions.
2. It rebuilds the exact frames from the pinned PR97 witness with check_lifted.py.
3. It regenerates plan.json. A subset is accepted only if every merged target chain is nested exactly over Q.
4. It regenerates the reduced row.
5. It replays the reduced word literally over Z and over F2, with arbitrary scratch and data. A missing redirected update and a sign-flipped one must both be rejected.
6. It prices the reduced row with Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's assembly (47 constraints, 7 margins), and the next grid points are rejected.

price144.py changes exactly one check in #144's code: the bit dimension check accepts R < 28866.

Before this PR was opened, the following were checked:

• a fresh-archive run of verify.py (exit 0, tree unchanged);
• a full make verify of this exact commit, split into per-group fresh clones;
• a separate adversarial review session.

Known gaps from that review (none affects κ):

• The tamper controls cover a missing redirected update and a sign-flipped one, but not a wrong-target or wrong-time update.
• The replay runs one core on its own bank. By Completed-core sharing: conditional kappa > 2^-13 #128 §2, that is the premise the shared-core argument needs.
• elim144.py picks each target's subset by comparing float gains, with a 1e-12 margin. verify.py then requires the regenerated plan to equal plan.json, so a platform whose log differs by more than that margin could disagree.

PROOF.md gives the argument and its scope. Every interface #144 retains stays an assumption.

Credits

• icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144 and Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130 (paired-cube and three-stage cover framework, selected gauges, pricing and assembly), and Stopped product-ring interchange and rational centers for conditional kappa = 7.79476e-5 #104.
• Swapnil Jain: the round-seven deferred bit word and its exact frame tools.
• Zhihao Chen: the PR97 ledger.
• an664 / Andrey Mas: Completed-core sharing: conditional kappa > 2^-13 #128 completed-core sharing.
• SovereignSteak: Terminal accumulator elimination with exact majorization audit #122 terminal accumulator elimination.
• jamesyc: Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124 birth-read reuse.

Applying elimination to this bit word, and re-certifying it, were prepared by William Porter with Anthropic Claude assistance (Claude Opus 5.5). No priority over concurrent work is claimed.

[Th0rgal]

Technical Review & Composition with Exact Telescoping Interval-Min-Cut (#146)

We reviewed elim144.py, row144.py, replay144.py, and price144.py:

• Eliminating $1,549$ deferred never-source output accumulators from Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144's $9,543$-slot PR Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97 bit selection reduces $R$ from $28,866$ to $27,317$ ($W_b = 2v + R = 30,859$, rank sum $2,127,247$, deficit $2,024$ unchanged) while re-certifying exact $\mathbb{Q}$-subspace nesting along the $944$ target chains that receive redirected forward updates, lifting $a_0$ from $4617656 / 10^{10}$ to $4655227 / 10^{10}$ and $\kappa$ to $4646633 / 10^{10} = 4.646633 \times 10^{-4}$.

Orthogonal Composition with #146's Exact Telescoping Interval-Min-Cut

In #146, we showed that the bit entrance-gauge subset selection problem reduces via summation by parts over consecutive active target ranks to an exact integer $s$-$t$ minimum cut / maximum flow with strictly positive interval capacities $c_{t,p,q} &gt; 0$.

Applying the exact telescoping interval-min-cut from #146 on top of your $1,549$ eliminated terminal slots (while restricting candidate gauges on the $944$ redirected targets to subsequences of #144's selection so that every $\mathbb{Q}$-nested redirected target chain certified by elim144.py is preserved without modification, and optimizing freely over the remaining $827$ unredirected targets):

• Retains $8,010$ non-eliminated entrance gauges (vs $7,994$ in row-elim.json),
• Lifts coarse bit saving from $4655227 / 10^{10}$ to $a_0 = 232763446903 / (5 \times 10^{14}) = 4.65526893806 \times 10^{-4}$ (4655268 / 10^10),
• Lifts stopped bit saving (at ATOM = 1/1000) to $a_b = 232549913406097 / (5 \times 10^{17}) \approx 4.650998268122 \times 10^{-4}$ and certified saving to $\kappa = 464667579244 / 10^{15} \approx 4.64667579244 \times 10^{-4}$ (4646675 / 10^10),
• And if also combined with ATOM = 1/2000 (Raise conditional kappa to 4.611281e-4 with tightened adapter toll #148, since $a_b = 4.6531336 \times 10^{-4} &lt; 5 \times 10^{-4}$), reaches $\kappa = 464880715940 / 10^{15} \approx 4.64880715940 \times 10^{-4}$ (4648807 / 10^10).