Add an exact κ-ceiling check for the exponent assembly (no new κ)

[romainhedouin]

Change and mathematical scope

Adds an exact κ-ceiling check (scripts/assembly_ceiling.py), tests, and a short note for both the original-prefix assembly (structured_bulk_assembly.py) and the balanced-prefix one (research/copied-fixed/balanced_assembly.py). No new κ, no existing file changed. Both assemblies already compute this ceiling (minimum_margin); what is new is the closed form, a table of which constraints can fail, and the tests.

Result. With q = a(1−2η) (η is the backoff, h for balanced), the ceiling is q(1−η)/(1+q(2+η)) for the original prefix and q(1−h)/(1+q) for the balanced one. The assembly accepts κ exactly when κ is below the ceiling and nine simple conditions hold: a, β and η in range (five checks), b < 1/32, (1−β)b > a, and the two network constants positive (balanced also needs κ > 0). So the 47 constraints reduce to 11 conditions (10 for the original prefix); three carry the content: b < 1/32, (1−β)b > a, κ < ceiling.

Why it is the same answer as the 47 constraints.

• A failed condition makes the reference reject: eight of the 11 are constraints in the reference word for word; the other three equal one up to a positive factor.
• All 11 holding makes it accept: they imply 0 < a < (1−β)b < b < 1/32, 0 < q < a, 0 < ε < 1 and 0 < G < q (G is the ceiling). Each of the other 36 constraints is then positive because it is a positive multiple of a positive slack (e.g. q_below_internal = 2aη), a sum of positive terms (e.g. b−a = ((1−β)b−a) + βb; every margin is G plus a positive term), or a simple bound from a < 1/32 and η < 1/2 (e.g. 2−3η−q(1+2η) > 7/16). κ appears only in the seven margins, each G plus a positive term, so the accepted κ are exactly those below G.

The argument is elementary but not machine-checked; COVERAGE in the module has the row for every constraint.

Validation

• 22 tests (about 3 s) check the above at finitely many exact points: agreement with both assemblies at 30 certified points and 2,400 grid points per prefix, each condition violated once, and the identities.
• make verify-tests and make paired-cube-verify pass on Python 3.12, 3.13 and 3.14 on a clean clone (the last amend changed only the note); structured_bulk_assembly.py is unchanged. Python 3.11 was only syntax-checked.
• AI assistance: written with Claude Code (Claude Sonnet 5.5 coordinating). Other Claude models (Opus, Sonnet, Haiku, Fable) re-derived the implications in sympy and searched millions of exact points for counterexamples, finding none; those scripts are not committed and nothing relies on them. No human has independently verified the mathematics.
• Credit and overlap: RaD project (hipotures, Source-framed h53 conditional witness above 2^-20: attributed research record #20), James Chang (jamesyc, Reversed two-stage basis and balanced assembly for conditional κ = 1.639226629e-5 #34), Zhihao Chen (Semantic precision and bulk resampling for a conditional 1099/10^8 > 2^-17 bound #23, Two-stage controlled corners for a conditional 15536/10^9 > 2^-16 bound #29), icekylinx. PRs Annealed modules, terminal sinks and physical bit words on paired cubes: conditional κ = 6.558894e-4 #168, Bit operation-frame descent on #168's bit word: conditional κ = 6.105562e-4 #169, Causal bit-frame refinement of #169: conditional kappa = 6.11068410271618e-4 #171, Mixed-gap bit all-but-one module on #168: conditional κ = 6.129700e-4 #173 and Research/atom exponent 169 #174 add a paired_cube_assembly.py with the same balanced arithmetic; this PR does not import it. NOTICE, CONTRIBUTORS and a Make target are left out because Makefile, README.md and NOTICE are hash-pinned (editing them needs a re-pin); happy to add them.

🤖 Generated with Claude Code

[Mister-Meeseeks]

Integrated into main in the 100-PR reconciliation. Thank you, Romain, for the exact assembly-ceiling and reduced-feasibility tool.

All three submitted files are incorporated. The Python module and tests are unchanged; the note has a historical-status banner and a link to the domain audit. All 22 exact tests pass against the retained assembly modules on Python 3.12. Your original commit, attribution, predecessor credit and Claude assistance disclosure are preserved.

This is scoped arithmetic tooling, not a new κ or a universal 1/32 barrier. The original contribution was previously reviewed but had not actually been imported; that integration is now complete.