[Matarener]

Requested by Mark

Change and mathematical scope

Negative result, no new κ, no changed selected record. Adds research/fragmentation-reorder-rigidity. #5226 prices the whole gap between Cao Yifan's #4610 word and the +0.05 % bar as fragmentation loss Λ = 23.86 per call and asks which scalar-program changes can merge climbs; #4993 and #4610's own MILP re-chose frames at a fixed chronological order; #4610 lists a changed chronological order as open. This package frees the order: scalar operations may be rescheduled anywhere their value dependencies allow (ADDs into one destination commute; write/read and write/write conflicts are kept; the zero_frame_precondition, full_frame_cleanup and done markers are barriers; warps stay anchored between their two frames), all 632 frame chains and all contact frames are re-chosen jointly under the retained contract (nested chains, one common frame per ADD inside S_d + S_s ≤ F ≤ T_d ∩ T_s, the field at the frame of every C0 contact), and the eight owned-centre reads may move along their source chains.

Exact certificate (verify.py, standard library, about 2 minutes): for every one of the 2,526 droppable intermediate levels (the 12 frame-revisiting fields and the warp frames excluded), every admissible re-routing of its contacts is enumerated (518,315 in all) and tested by an exact acyclicity check of the operation/level graph (Pearce–Kelly dynamic order). No level can be dropped at a gain: 322 levels have no admissible re-routing, 2,196 close a cycle under every re-routing, and the 8 remaining levels are the centre-read levels themselves, droppable only by reading the centre in the full frame (transport rank 9 instead of 8, +0.983 per call). The fragmentation of #4610 is therefore a property of its operation set and centre interface, not of its schedule. Supporting census: endpoints alone force 12 of 2,754 levels, the partners' exact chains force 2,437, the dependency DAG has critical path 694 of 18,159 operations. What Λ would buy (kap-family formula of #5226): word 0.0030217759 (p16 a7); mains one climb each 0.0037103 (p13); helpers one climb each 0.0051074 (p9); Λ = 0 0.0081386 (p5 a2).

Status: finite exact certificate on the frozen word (operation set, coefficients, native maps, endpoints fixed) for single-level merges; multi-step moves were explored by 37 million recorded annealing moves only (search/, not replayed). It is not a floor for the family. Remaining obligations: none for this claim.

Task and overlap

Base 1a3ced60b2f7a46bc70280a04f2eeb8ca027c700 (current main, selected PR5413). Inputs pinned by SHA256: research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz, ARITHMETIC.json. Task card complex-rank-shape-decoder, priority revision 7, scope changed (recorded in TASK.json): instead of synthesizing a block-climb decoder, the package asks exactly whether rescheduling can coarsen the climbs of the current word. Nearest work and added value: #5226 (fragmentation-ceiling: Λ, ship test, kap formula reused), #4993 and #4610's MILP (frames at fixed order), cao-trace-retiming-20261023 (ADD compression at fixed chronology; its bounded searches keep the scalar order), fusion-pair-scalar-rewiring (#5422: no consecutive rank-1 pair fusable by free rewiring with frames fixed), owned-centre-wire-floor (#5060: why the delivery frame stays 0). None frees the chronological order or enumerates the joint re-routings; a check of open PRs and discussions on 2026-10-24 found no package on reordering.

Validation

python3.12 -I -B research/fragmentation-reorder-rigidity/verify.py
python3.12 -I -B research/fragmentation-reorder-rigidity/test_verify.py
python3.12 -B scripts/research_package.py --package research/fragmentation-reorder-rigidity

Marker PASS_FRAGMENTATION_REORDER_RIGIDITY; the registered runner prints PASS_REGISTERED_RESEARCH_PACKAGE … outcome=negative-result locally (CPython 3.12, Windows). verify.py pins both inputs, rebuilds the word model (climb profile + 8 centre jobs = ARITHMETIC.local_paid, Λ = 23.8626 at the word's root), runs the enumeration and compares RESULT.json exactly. Controls: ignoring the centre interface makes the same enumeration find 51 improving drops (the test is live); test_verify.py rejects a drifted input hash, a corrupted climb rank, a contact frame mismatch and a tampered result. Pruning used in the enumeration is the necessary condition that an operation which must precede another cannot sit at a higher frame of the same field. Not run: Lean, native/phase/compiler replays, hosted CI; no word is proposed, so no admission.

Credit

Cao Yifan (intqwq, #4610) for the word and its exact interpreters; DreamingOfClouds (#5226) for the fragmentation decomposition, ship test and kap-family formula; ahrazzle and Cinna Davis (#4692) for the fixed-chronology compression line; #4993 for the fixed-order frame search; Dugongue (#4534) and owned-centre-wire-floor for the centre wire; the fusion-pair-scalar-rewiring package (#5422); zxv, Jacob Alan Sussman and the retained native/controller lineage. Prepared by Mark (Matarener) with substantial Claude Code (Claude) assistance. Apache-2.0; author-checked, not independent acceptance.

Open next questions

TASK.json: (1) a skeleton-free fragmentation floor for the #4610 endpoint identity (first step: the chain-cover LP over the contact hypergraph with free frames; the one-field relaxations force nothing, so a floor must use the joint chains and the centre interface); (2) the same enumeration on the ADD-compressed word of cao-trace-retiming-20261023 (fewer contacts may unpin levels; blocked by taking that word as a frozen input); (3) a changed centre delivery frame is a changed mechanism, outside this package.

🤖 Generated with Claude Code

[Matarener]

Follow-up evidence for this package (no new claim, no κ change, main 32890bc5d, same frozen word research/kappa-round3-warp-20261022/ACTUAL-WORD.json.gz).

1. How much of Λ do contact contents alone force? For each two-field contact, I took the frame window that any nested-chain word with the same contact contents must use: [join(S_f, S_p, birth frames of the moved terms), meet(T_f, T_p)]. A term's birth frame is S_r for a raw input, the new frame after a warp of its carrier, and the distribution frame after it passes through C0. Contacts of fields that warp are left unconstrained. A maximum clique of pairwise-incompatible windows that miss both endpoints lower-bounds each field's intermediate levels, and the most uneven split then lower-bounds Λ_f.

• Birth-frame window bound: Λ ≥ 4.23 (461 forced levels). The actual word violates no window.
• If you use S_r for every term instead of birth frames, 246 actual contacts fall outside their window. The C0 distributions put A-content into B fields at frame 0. So that simpler window is not a valid constraint for this word.
• With frames only (common frame plus nesting, contents ignored), the bound drops to 0.11.
• Upper side: dropping every level whose contacts could each move to an endpoint by value leaves Λ = 11.13, counting the eight centre-read levels.

So the contents force between about 18% and 47% of Λ = 23.86. The rest is held by the partners' chains, which matches this package's census (2,437 levels forced by exact partner chains). These are value-level relaxations, not schedules.

2. The threshold at the BIT bar. With this package's kap-family formula (research/fragmentation-ceiling/verify.py run with BAR = 0.00344412), the #4610 word has Λ(t) = 27.18, and the threshold is Λ* = 21.39 for p16 a7, or 21.43 for the best controller of the family at that t (p14 a7). A word in this family would need about −5.8 per call (≈ 21% of Λ) to reach the current BIT saving. Comparing 23.86 (taken at the complex t) with a threshold at the BIT t understates this by about half.

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