FINDING: fragmentation Λ on both sides (~23.6–23.9 per call); reaching the BIT bar needs −5.8 (≈21%); what does not help

[Matarener]

FINDING / no new κ / main 32890bc5d

This is a short orientation note for anyone working on fragmentation. Everything below was recomputed with separately written code on the frozen public words. All of it is negative or descriptive; nothing here changes the selected record.

Both sides carry the same fragmentation

With Λ = Σ_f (Σ climbs r^t − R_f^t) (#5226, Theorem B):

word	t	Λ per call	intermediate levels	helper share
complex #4610 (kappa-round3-warp-20261022)	1 − 0.0030218	23.86	2,658 (2,742 distinct frames incl. warp frames)	68%
BIT PR589 reverse (source of #5421)	1 − 0.0034441	23.62	2,250	77%
BIT #5421 word (nine owned pivots)	1 − 0.0034441	23.12	2,200	76%

#5226 and #5426 already showed that Λ is the whole gap to Λ = 0 on the complex side.

What reaching the BIT bar would take

At t = 1 − 0.00344412, the #4610 word has Λ = 27.18, and the kap-family threshold is Λ* ≈ 21.43 (best controller p14 a7; 21.39 at p16 a7). A complex word in this family needs about −5.8 per call (≈ 21%). Note that Λ and Λ* must be taken at the same t.

Where the levels sit (complex word)

• No climb happens in the first 29% of events, which form a free frame-0 prelude.
• About 33% of levels touch a B field (centre distributions, B↔B exchanges, deliveries into B). About 9–15% carry warped content, depending on labelling. 0.3% are the eight centre reads. The remaining ~58% assemble sums for those.
• The eight centres deliver exactly ±1/2 of the needed B_j += A_j diagonal on 88 fields and 0 on 24. The rest of the coupling arrives through warps and B↔B exchanges, so the whole word is the decoder network.
• Contact contents alone force about 18–47% of Λ (birth-frame window bound 4.23, value-level upper estimate 11.13). The rest is held by the partners' chains (Certify that the fragmentation of the #4610 word is rigid under free scalar reordering (negative result, no new kappa) #5426).

What does not help, measured on this word

• Restoration (uncompute) batching. Helpers already cancel on the full frame (24,846 cancellations versus 4,148 on intermediate frames). Removing all 180 pure-restore levels at once would save 1.59.
• Low "bond dimension". The cross-content rank rises 112 → 503 → 112. The final operator's cross rank is 112, so the peak is data transport, not information.
• Product-plane pairing tariffs (Discussion CLAIM: where does the one unpaid coordinate per live family come from, and is it forced? (ctoth, expires 07:30Z) #1587). The tariffs 15,232 and 15,552 reproduce exactly, but the rank of F gives no tariff floor (single-entry terms cost 0). The binding issue there is the physical plane program, not the tariff.

Open question: is there a floor on Λ over all words in this family, for example via a minimum decoder network with warps? We do not have one.

Prepared by Mark (@Matarener) with Claude Code assistance. Scripts are local and can be shared on request.

[Matarener]

Partial answer to the open question above (is there a floor on О› over all words in this family?). Short version: the simple floors we can prove are about 20 times too weak to matter, so this question will not be settled by counting. No new Оє, main 32890bc5d.

Model. This is the warp-free formal model of the complex P112 word, with the selected centre code fixed (pinned research/coded-native-caller/NORMALIZERS.json). Fields walk nested chains of subspaces of F2^9, and scalar contacts happen at a common current frame. As in #4610, A_p starts on its port line l_p, and B fields and helpers start at frame 0. О› is #5226's quantity, taken at the BIT bar t = 1 в€’ 0.00344412.

1. The junk is forced. B_q has to end with A_q, and it ends at P_q = q^вЉҐ. On the 60 ports not orthogonal to q, the local part is zero: those lines never lie in a frame inside P_q. With the eight selected reads, the centre part is then uniquely row q of C = MMбµЂ/4. So every word in this family must deliver the local part в€’J_q into B_q. J_q has coefficients В±1/2 on exactly the 52 ports orthogonal to q, and these lines span P_q. The centre weights M_q/4 are therefore forced too: with P112 fixed there is no "other decoder".

2. Capacity of a hyperplane. For every q, each 7-dimensional subspace of P_q contains at most 32 of the 52 lines of J_q. This was checked exhaustively for all 112 ports, and the maximum 32 is attained for every q. All frames of B_q before P_q lie inside one subspace of dimension at most 7. So at least 20 lines must reach B_q at the frame P_q itself.

3. A rigorous but weak floor. Content for such a line arrives from a partner standing at P_q, which is not B_q. Follow that content back in time. Some field other than B_q stands at P_q and also has at least one intermediate level: it either passes through P_q, or it brought the line up from a lower frame. Each field has only one 8-dimensional frame, so these fields are distinct for different q. The cheapest field with one intermediate level costs 0.01033 (rank split 1+7). Hence

Λ ≥ 112 × 0.01033 ≈ 1.16 per call.

For comparison, the #4610 word has О› = 27.18 at this t, and the BIT bar needs О› в‰І 21.4. This floor neither closes nor opens the route.

4. Frame sharing is not the bottleneck. Call a pair (p, H) an arrival if local A_p content is delivered into some B field at frame H. A pair at H can serve only receivers q with H вЉ† P_q. Weighting each (line, receiver) incidence by the number of receivers its pair can serve gives a lower bound per port. We enumerated all subspaces exactly, and dropping nestedness only weakens the bound. The bound is f(q) ≥ 2689/78 ≈ 34.47 arrivals, and all 112 ports give exactly this value (exact rational arithmetic; updated: earlier this said 11 of 112). Reproduction: Actions run on fork branch gn-arrivals-lb, 28 jobs x 4 ports. So sharing frames between receivers saves at most about 33.7% of the 52 Г— 112 = 5,824 incidences. Greedy hub-sharing trees reach about 38.5 per line. The #4610 word uses 9,824 arrival pairs (87.7 per line) and reaches 8,324 incidences, so it delivers and then cancels extra content. Arrivals are therefore not what makes it expensive.

Where О› actually sits. It sits in assembling exact sums of pieces through nested chains. LP relaxations that share pieces fractionally lose exactly this constraint. A strong floor would be a restricted linear-circuit lower bound, and we have no method for that. This matches open question 2 of research/format-limits-2 ("no skeleton-free bound"). It is related to F1 there (cap-aware port-quad coverings, 14.29 carriers per source with lower bound 13), but it is not the same count.

Two side checks:

• Inside the E8 line system, no pair of orthogonal lines has nested orthogonal neighbourhoods. This holds for E8, P112, P112 with up to 12 lines removed, and E7. Nesting appears only for D-type systems (D8: 28 pairs, 56 ports), which have too few ports.
• A from-scratch greedy and annealing synthesizer for this gathering network got nowhere near Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610: О› ≈ 186–229 against 23.9.

Prepared by Mark (@Matarener) with Claude Code assistance. Scripts are local and can be shared on request.