[jacobalansussman]

Change and mathematical scope

Adds one package, research/complex-m4-time-unrolled-cuts: a counting rule for small complex units at block 4,
the cut-set bound of linear network coding applied to the program unrolled in time, and two exclusions it gives in
the engine game without address maps. No new κ; no record, target or frozen package changes.

Finding	Game	Status
The end-state chain rows: Σ_t rank π_(w_t)[U_t, U_(t−1)^c] ≥ W · rank R_K[U_s, U_0^c] for every shrinking chain of plane sets	engine game without address maps	argued by hand (one AI reviewer re-derived it); soundness tested by computer on random and real programs
No 3 in 10 at block 4 (1.32e-1), in any order; nothing below 10	same	checked by computer, given the rule and the reduction to one shape per orbit, both argued by hand (121,951 certificates over all 80 orbits that plane rows allow; n0's 324 are in this package)
No 2 in 7 at block 4 (9.63e-2), in any order	same	checked by computer here, given the rule and the reduction to one shape per orbit (3,098 certificates over 38 orbits)
With address maps every direction is equivalent and the rule is false (it would forbid the plain 12-call program)	engine game	checked by computer
A 3 in 10 or 2 in 7 in the engine game needs a cross-frame addition or a live array relabelled at the end	engine game	argued by hand (one AI reviewer)

Remaining obligations: no person has read the proof of the rule or the frame argument; the exclusions say nothing
about the engine game with address maps; 79 of the 80 certificate files for 3 in 10 (17 MB) are author evidence in
my repository, checked there by two independent checkers.

Task and overlap

• Base 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8. Card scope-navigation (method "Index one
  tightly related family of obstructions with explicit excluded and open hypotheses"), scope_status: changed: the
  package also adds results that are not on main (the rule and the two exclusions). Reason in TASK.json.
• No live inputs. Provenance rows: the task catalogue and the three predecessor texts credited below.
• Nearest work: eumemic's certificate method and description of these units as network codes on the graph of
  frames (Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ) #288); my earlier note (Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628: the kept-subspace bound, 41W/13 at block 4, the 640 margin for 2 in 7, and the
  exclusions with address maps and phases only at the ends); the block-four coordinate-face obstruction by Nick
  Petalas (Prove a sharp block-four direction-weight obstruction (no new κ) #4855; a different game: diagonal phases and coordinate permutations free, cost by Hamming weight); the
  temporal rank cuts on the frozen P104 itinerary by David Arias (Prove sharp P104 temporal rank bound and correction-budget obstruction #2082) and Adam Gall (Sharp eight-query temporal cut for the frozen P104 itinerary #2090), and Michael Yee's
  max-flow audit of that cut family (Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163; a time-expanded network of a different model, where each node carries
  one array); Shike Chen's coding-gap note (Research notes (no κ claim): #109 and lower bounds, the coding gap, a Lorentzian minrank obstruction, and a loss chain #634: a unit's rank saving is a gap between coding and routing, which
  bounds routing, not coding). A 3 in 10 at block 4 was proposed in feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818 and withdrawn by its author on 23 October
  2026, with feat(research): exact 5-array 14-call block-3 unit schedule via 14-neighbor symplectic cascade (κ <= 0.0628) #4813, feat(research): exact 13-array 41-call block-4 boundary schedule on C^16 (κ = 0.1714) #4819, feat(research): exact 3-array 9-call block-5 unit schedule on C^32 (κ = 0.3174) #4826, feat(research): exact 3-array 9-call block-6 unit schedule on C^64 (κ = 0.3869) #4934, feat(research): exact 3-array 9-call block-7 complex unit schedule on C^128 (κ = 0.4354) #4937 and feat(research): universal parametric block cascade theorem for all m >= 4 (κ(m) = 1 - ln 3 / ln m) #4941, because none of their checkers counted paid calls; nothing
  here relies on that withdrawal. This package adds the time-unrolled rank cut for every program of the game and
  its two exclusions, and links the earlier work instead of repeating it. Details in Sources.

Validation

python3 scripts/research_package.py --package research/complex-m4-time-unrolled-cuts --check-only
python3.12 -B scripts/research_package.py --package research/complex-m4-time-unrolled-cuts
# = python3 -B research/complex-m4-time-unrolled-cuts/verify.py --part main       (PASS_COMPLEX_M4_TIME_UNROLLED_CUTS_MAIN)
#   python3 -B research/complex-m4-time-unrolled-cuts/verify.py --part recount    (PASS_COMPLEX_M4_TIME_UNROLLED_CUTS_RECOUNT)
#   python3 -B research/complex-m4-time-unrolled-cuts/verify.py --part controls   (PASS_COMPLEX_M4_TIME_UNROLLED_CUTS_CONTROLS)
python3 -B scripts/check_research_commit.py --package research/complex-m4-time-unrolled-cuts

Standard library only. verify.py passes with Python 3.9.6 on an Apple M4 (macOS) in under 1, about 4 and under 1
seconds and at most 32 MB, and leaves no files behind; with Python 3.12.13 the three parts take about 0.4, 2.5 and
0.1 seconds and under 40 MB, with the same output. In a clone of the hub's main at 7c49fe32c5,
scripts/research_package.py passes the metadata check with Python 3.9.6 and 3.12.13
(PASS_PACKAGE_METADATA_AND_PINS) and runs the three registered commands with Python 3.12.13 in about 3 seconds
(PASS_REGISTERED_RESEARCH_PACKAGE), and scripts/check_research_commit.py passes on the commit that adds the
package, also with --check-source-history (PASS_RESEARCH_COMMIT_BYTES). It builds the call and target matrices on the
35 planes from their definitions (they equal those of both checkers in my repository), checks n0's 324 and all
3,098 two-in-seven certificates exactly with coverage, recounts the 1,097 and 398 shapes and their orbits, and
rejects six corrupted files or false claims. Not run here: the other 79 ten-call files and both checkers
(run_all.sh --full in my repository, about 7 minutes and 800 MB), and the soundness tests.

Credit

Ahlswede, Cai, Li and Yeung (the cut-set bound of network coding); Avestimehr, Diggavi and Tse (the linear
deterministic model); A. W. Ingleton, R. Dougherty, C. Freiling, K. Zeger and R. Kinser (the rank inequalities the
scrambled program satisfies). On the hub: eumemic (#288); Maximilian Lutz, whose review (#4643) led to the block-4
certificates published on 23 October, the state this work started from; Nick Petalas (#4855), David Arias (#2082),
Adam Gall (#2090), Michael Yee (#2163) and Shike Chen (#634). OpenAI for the cost model, the call and the kernel.

Substantial AI assistance: I did this work with the help of AI agents (Anthropic's Claude). The model, the
searches, the checkers, the arguments and the text were written by the agents under my direction. Two AI reviewers checked the work with their
own code. No independent human review is claimed. Apache-2.0.

Open next questions

In README (## Open next questions) and TASK.json. The smallest useful follow-ups: route (B) on 2 in 7 (plane
sets chosen for each relabelled target, then the chain search); a rule that survives cross-frame additions; 3 in 11
without address maps with the same search; the game with only the 24 bit permutations free, where 6 pieces survive.

Workflow feedback (optional)

None.

Sources

• eumemic: Negative results from a search for a new coding mechanism: m = 2 closed on both sides, flag theorems, closed routes (no new κ) #288. My earlier note: Research: lower bounds for small complex units: m = 2 never saves, m = 3 needs 14W/5 calls (Lean, engine game), m = 4 needs 41W/13, and the smallest open cases (no new κ) #4628 (research/complex-small-m-lower-bounds), and Maximilian Lutz's review of
  it, Independently check #4628's m ≥ 4 projection bound: 2.99172, 2.97290, 2.9447 reproduce; tail proved for all k; two uncertified block-4 inputs isolated (no new κ) #4643. Nick Petalas: Prove a sharp block-four direction-weight obstruction (no new κ) #4855, research/block4-coordinate-face-tariff (commit 8b8de0281f). Michael Yee:
  Independent max-flow audit of the P104 rank-eight cut family (no new κ) #2163, research/p104-cut-maxflow-audit (commit 9f6e1f2783), auditing Prove sharp P104 temporal rank bound and correction-budget obstruction #2082, Sharp eight-query temporal cut for the frozen P104 itinerary #2090, Prove P104 rank-eight cut under arbitrary frame-path interleaving #2113, Prove P104 transport floor with arbitrary added storage #2127 and Rule out identity from nested P104 transport refinements #2136.
  Shike Chen: Research notes (no κ claim): #109 and lower bounds, the coding gap, a Lorentzian minrank obstruction, and a loss chain #634. The withdrawn 3 in 10: feat(research): exact 3-array 10-call block-4 unit schedule on C^16 (κ = 0.1315 > 0.10) #4818.
• My repository: https://github.com/jacobalansussman/wht-power-saving-lean, folder
  research/block4-without-address-maps/, at commit 0788bc6568 (3 November 2026; the folder is byte-identical
  at 9c7f66f024, where SOURCES.json pins the copied data).

🤖 Generated with Claude Code