A κ ceiling of 7.010e-4 for every frame layout of #168's word (no new κ)

[DaysSky]

Adds research/kappa-ceiling/ and one workflow; no existing file changes and no new κ. #168's word is not on main, so the package carries the 48 files of #168 it reads (scripts, frozen frames, pairs, sinks, modules and certificates) as a pinned archive, baseline-pr168.tar.gz (0.6 MB). It rebuilds them in a temporary directory, following research/coordinated-frames-and-entrance-banks. Each file's sha256 is pinned in SOURCE.json to commit 4a3c769. Reviewers can confirm the pins with git show 4a3c769:<path> | sha256sum, or rebuild the archive byte for byte with git archive --format=tar 4a3c769 <paths> | gzip -n -9.

Result

No choice of operation frames for #168's complex word can certify more than

κ < 7010/10^7 = 7.010e-4.

#168's κ = 6.558894e-4 is 93.6% of that. So every frame refinement still possible on this word is worth at most +6.9% in total. Going further needs a different circuit, different reuse or gauges, or a different design.

The ceiling covers one word. Its DAG, operations, carrier arcs, 2,310 gauges and reuse pairs, and 47 terminal sinks stay fixed, and only the operation frames vary. It does not bound constructions that change any of those.

Argument

Write D = 2v − 3ℓ for the shared-core deficit and C = Σ n_r·r·ln(m/r) for the cost of the children.

1. Every certified saving satisfies a < D/C. This follows in two lines from e^x ≥ 1 + x. On Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 the certified savings sit within 0.09% of D/C.
2. κ < G(a). G is the assembly's closed-form acceptance ceiling from Add an exact κ-ceiling check for the exponent assembly (no new κ) #183 (romainhedouin), and it is increasing in a.
3. Splitting a frame climb into more steps only costs more, because r·ln(m/r) is subadditive.
4. No frame layout can push C below a floor C_floor. The floor comes from three ingredients:
   • frame bounds propagated jointly through both registers of every operation;
   • an exact per-register dynamic programme over frame dimensions;
   • Lagrangian coupling of the two registers of each operation. Each of the 400 integer rounds is itself a valid floor.

Hence κ < G(D/C_floor). Each stage of the floor lowers the ceiling:

Floor	Ceiling on κ
two steps per register	1.153e-3
joint bounds + per-register programme	7.158e-4
+ Lagrangian coupling	7.010e-4

PROOF.md has the statements and proofs. It also gives weaker design ceilings for any word with #168's ports and decoder, from 4.08e-3 to 1.25e-2 under stated hypotheses.

Verify

python3 research/kappa-ceiling/ceiling.py --check     # stdlib only, about 50 seconds; -O is refused

The script rebuilds the pinned #168 files after checking the archive and every file hash, then regenerates #168's complex word with #168's own producer. It reads the frozen physical layer and terminal sinks, computes every bound and compares the results with expected.json. A tampered archive is rejected. --tree PATH runs on a full #168 checkout instead. It checks itself:

• Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's own layout lies inside every derived frame bound;
• its slot chains reproduce Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's certified complex ledger child for child, and the sink record;
• every per-slot floor, and every Lagrangian round's total, lies below Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's actual cost.

All bounds use exact integers with rounded-down logarithms, so the printed ceiling is an upper bound.

Run before opening:

• --check passes on this branch (Python 3.14, macOS), from the archive alone and on a full Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 checkout.
• On this branch, research/bit-reuse-147/verify.py, research/bit-elim-144/verify.py and tests/test_recycled_bit_integration.py pass. They failed in an earlier version of this PR that carried Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's scripts.

Not run: the full make verify. The new workflow runs --check on Python 3.11 and 3.14.

Scope and limits

• The proofs are elementary but not machine-checked beyond the script's exact arithmetic and self-tests. No one else has reviewed them.
• The ceiling is about Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168's word only. Newer PRs (Shared-edge local circuit on PR168 v4 paired cubes (κ = 6.59180578557293e-4) #181 onward) change the word, and each would need its own run. Constructions kept in their own package format, such as Shared-edge supplier, coordinated frames and entrance banks: conditional kappa = 6.61885549259598e-4 #186's, would need an adapter.
• The pins refer to Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 at 4a3c769. If Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168 changes, the archive and expected.json stay valid for that commit and would need a new version for the new one.

Credits

• eumemic: Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168, its physical-layer word and checker, and the terminal-sink gate.
• icekylinx: the paired-cube framework, compiler, gauges, shared cores and assembly (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).
• romainhedouin: the closed-form assembly ceiling (Add an exact κ-ceiling check for the exponent assembly (no new κ) #183).
• jamesyc: the terminal-sink lemma (Add exact dirty-target terminal compiler and 67-sink witness #166).

The bounds, the frame floor and the script were prepared by DaysSky, assisted with Claude.

[ploniaaaaaaa]

zamn :(

[Mister-Meeseeks]

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

Contribution: Fixed-word frame-layout ceiling for PR168.

Review and limits: Prior exact-head scoped ceiling review. Applies to the specified word and frame-layout class; not a universal multiplication lower bound.

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: ba52fac95ebc7cd25d284888aefee2284b670fd6. Publication: 50d768818f579d4e165b4c257c6ee25bca300802.