Re-instantiated atom exponent at the toll edge on the searched pair module: conditional κ = 5.557315e-4

[geckods]

Conditional improvement

$$T(n)=O\big(n(\log n)^{1-\kappa}\big),\qquad \kappa=\frac{1111463}{2000000000}=\mathbf{5.557315}\times10^{-4}.$$

Stacked on #157 (6741112, eumemic's paired cubes at p = 11 with a searched pair module, physical frames and a paired-cube bit word). Please review/merge that dependency first; its changes are not claimed here. The only files of this branch are research/paired-cube-toll-edge/ (README, verifier). The branch carries a single squashed commit on main because the push credential lacks workflow scope; relative to #157 the diff is exactly those two files.

Change

One assembly instantiation constant is re-chosen. #157 inherits the merged #144 atom exponent 1/1000. The atom exponent theta is a free positive constant of the retained assembly: it enters only through ab = (1-theta)*COARSE + theta*OLD and the strict toll ab < theta < 1-ab. #157's bit side binds (COARSE = 5566382/10^10 below the complex saving 5622769/10^10), so raising ab onto the binding face moves kappa directly. ab strictly increases in theta while kappa strictly decreases across the acceptance window, so the grid-maximal instantiation is the first 10^-12-grid point where the strict toll holds — exactly one step above theta* = COARSE/(1+COARSE-OLD):

	atom exponent	kappa
#157 (inherited)	1/1000	5555021/10^10
this branch	556349912/10^12	1111463/2*10^9 (+2294 grid points)

The phase-stop stays at #157's inherited 1/10^6: it is slack at this face (the binding bit value sits below the (1-beta)*AC - 10^-10 ceiling). The same re-instantiation genre, at the same constants, is #148's toll tightening and #153's phase-stop re-instantiation. No word, frame, pair, certificate or interface changes; every inherited interface remains an assumption exactly as in #157.

Verification

python3 research/paired-cube-toll-edge/verify.py (stdlib, ~2 min; do not use -O): reproduces #157's published 5555021/10^10 exactly at the inherited constants (running #157's own producer, physical record, certificates, bridge and assembly), checks one 10^-12 step below the new theta violates the strict toll, then re-runs #157's 47-constraint assembly at theta = 556349912/10^12: kappa = 1111463/2000000000, next 10^-10 grid point rejected, all 47 strict constraints positive, 7 margins.

Credits

• eumemic: Paired cubes at p = 11 with a searched pair module, physical frames and a paired-cube bit word: conditional κ = 5.555021e-4 #157, Paired cubes at p = 11 with an h = 21 bit word: conditional κ = 5.108289e-4 #152, Deferred bit word and a 28,705-role complex producer under stopped-product accounting: conditional κ = 1.044939e-4 #117 (construction this branch re-prices), with Anthropic Claude assistance.
• gupt1156: Raise conditional kappa to 4.611281e-4 with tightened adapter toll #148 (tightened adapter toll lineage).
• icekylinx: Paired cubes and shared completed cores for conditional kappa = 4.609169e-4 #144/Three-stage Cayley covers for conditional kappa = 3.146011e-4 > 2^-12 #130 (paired cubes, shared cores, retained assembly).
• Zhihao Chen (Deferred endpoints and signed phase correction for a conditional 63965813/10^12 > 2^-14 bound #97), Swapnil Jain (round-seven checkers), an664 (Completed-core sharing: conditional kappa > 2^-13 #128), DaysSky (Recycled bit registers on #147's reduced word: conditional κ = 4.71809569e-4 #150), hpst3r (Terminal elimination on #144's shared bit cores: conditional κ = 4.646633e-4 #147), jamesyc (Birth-read slot recycling: conditional kappa 1.1038332094346e-4 #124), SovereignSteak (Terminal accumulator elimination with exact majorization audit #122).
• Re-instantiation and this package: Abhinav Ramachandran with Hermes (Nous Research) assistance.

Conditional; no unconditional multiplication theorem is claimed. Failed next-grid upper-enclosure tests mean only the failure of that certificate, not a mathematical optimality result.

[Mister-Meeseeks]

Thank you @geckods. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Atom-toll tightening on the searched-pair PR157 construction.

Review and limits: Fresh unchanged patch replay on parent 6741112. Correct the prose monotonicity: ab decreases with theta. Next-grid rejection is certificate failure, not global optimality.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 6e086603f936dbb95c6d7171132ed943cc024db9. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.