[GamingPuzzled]

This package has two parts.

Part A (numerical; compiler headroom only). The complete #5413 caller is re-funded inside its own reserve at
κ = 30272882389/10¹³ = 0.0030272882389 (tested 0.0030272882489). This is +3.9 × 10⁻¹² over the
selected #5413 value 0.003027288235.

• Zero recursive economic gain. The B/F packets, controller, invoice (regenerated byte-identically), paid and
  target profiles, compiler tuple, reserve 10⁻¹², BIT stock and height carrier are all unchanged. The carrier is
  gap 1/7·10¹¹, j = 55099, H = 38569300000000000.
• Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413's precision checker reproduces Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413's expected receipt except for the manifest hash.
• Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413's 25-premise conjunction passes, with only the endpoint literals moved.
• κ stays below Independently verify the controller-family ceiling by convexity (no new κ) #5409's fixed-profile ceiling U = 0.00302728825.
• Raw root ∈ (.0030272882499, .00302728825). The reserve limits the tested saving to the interval
  (.00302728824897, .00302728824898).

Part B (exact; closes the relation lever). This part answers #5433's open question and widens it. Over every
owned helper-relation subspace on Cao Yifan's #4610 word, the optimum value is attained by #5383's span (an optimum, not the unique one: replacing component 194 by 207 gives a tied eight-dimensional space, kept as a regression after review). The search covers:

• relation sets without Conditional κ = 0.00302666 (+4.31e-6 over #5377): eight joint helper relations #5383's eight;
• pivots on the 26 rank-4 and 6 rank-8 frame-0 helpers;
• rows that carry warp words.

The optimum is 8(8^t+1−9^t) = 0.0757092716 raw moment per backward local; the forward optimum is 0.

• Backward: 224 helper-only climbs give 225 distinct components, all linearly independent over Q.
• All 56 centre snapshot options lie in the components' span. Each centre's options are supported on its
  zero-site rank-1 climb plus at most one private climb, and the supports are disjoint across centres.
• Exact enumeration over the connected pieces (at most 14 components each) finds the optimum.
• No ninth complex relation exists. BIT saving 0.00344412 from nine owned helper relations #5421's BIT ninth relation (COPY H7 + MOVE H1) has the complex analogue
  centre H8 + H1, and the eight existing relations already cover all eight centres.

Reproduce

python3.12 -B scripts/research_package.py --package research/owned-relation-exact-optimum-r14

• Local run on macOS / CPython 3.12: relations.py, endpoint.py and test_controls.py pass (about 3 s, 10 s
  and 2 s).
• The fourth command reruns Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413's own verify.py. Locally it passes runs 0–5: source-loading, aliases, local
  649 columns, global, binding and invoice. It then fails only on Darwin, where setrlimit(RLIMIT_AS) is
  rejected; the same limitation applies to Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413 itself.
• endpoint.py uses a Darwin-only shim around that call. Linux runs the frozen checker byte-unchanged.
• Linux CI has not been run by us.
• check_research_commit.py passes.

Scope

• Conditional on exactly the five retained selected premises of Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413.
• No unconditional theorem, formal-kernel acceptance or κ-maximality.
• A finer unit gap, such as 1/7·10¹² (κ ≈ 0.0030272882479), passes the endpoint arithmetic. It would change H,
  and the precision checker pins the gap, so it is not claimed.

Credit

• Cao Yifan (Conditional κ = 0.0030217758: joint helper interfaces and Lean compiler tightening #4610 word).
• Ali Romman / aliromman:
  • Conditional κ = 0.00302235 (+5.742e-7 over #4610): one owned helper relation #5377 and Conditional κ = 0.00302666 (+4.31e-6 over #5377): eight joint helper relations #5383 (owned relations);
  • Conditional κ = 0.00302725 (+5.75235e-7 over #5387): owned circuit transposition #5392 (owned circuit transposition);
  • Conditional κ = 0.00302728823 (+3.823e-8 over #5392): direct backward caller #5398 (direct backward caller);
  • BIT saving 0.00344412 from nine owned helper relations #5421 (nine BIT relations).
• Dillon Amburgey (Preserve all 632 native inputs in an eight-mode coupled helper provider #5402 coupled provider).
• Chris R / Clear1nsight:
  • Complete coupled-output caller: conditional κ = 0.003027288235 (+5e-12 over #5400) #5413 caller, funding and conjunction engines;
  • Independently validate κ = 0.00302235 and the owned-relation price ceiling #5384 owned-relation price ceiling;
  • Independently validate κ = 0.003026674765 and the eight-relation price plane #5395 eight-relation price plane;
  • validations Independently validate joint relations and conditional κ = 0.003026674765 #5396/Independently validate κ = 0.00302725 and the owned-transpose phase/cost proof #5399/Independently validate κ = 0.00302728823 for the direct backward caller #5400/Independently audit the coupled provider and conditional closure budget (no new κ) #5407/Independently verify PR5413 neutrality, necessary pricing and source application #5414.
• Dugongue (Raise conditional κ to 0.003026674765 with a smaller completion reserve #5387/Certify κ = 0.0030266747680744 and a smaller matched compiler using direct tails #5390 reserve refinements).
• Mark / Matarener (Certify that no owned-helper relation subspace pays after #5421 (BIT) or #5383 (complex) (negative result, no new kappa) #5433 relation-subspace certificate).
• Independently verify the controller-family ceiling by convexity (no new κ) #5409 (fixed-profile ceiling).
• Cinna Davis, zxv (Conditional κ = 0.0023 (+5.690e-4, +32.87% over #4181): productive helper jobs compiled into a uniform supplier #4359), Jacob Alan Sussman (mixed-target accounting), and the retained native, compiler, BIT
  and analytic lineage.

AI disclosure: Claude Opus 5.5 (Anthropic) via Claude Code, directed by Joel Pulikkan (GamingPuzzled).

🤖 Generated with Claude Code

[zxv]

Reviewed 9cfe834fe796d34a7ede5c2c2d360faadd6c9d44.

The conditional κ re-funding checks out within the retained #5413 assumptions. The three new replay commands pass locally, including authentication of the 63 capsule files, the unchanged invoice/carrier, precision receipt and 25-premise conjunction. I also checked the successful full Linux registered replay, run 40140121823, whose checkout explicitly merges this head into 32890bc5…; it completes the unchanged #5413 replay as well. This supports κ = 0.0030272882389, tested 0.0030272882489, as numerical compiler headroom only, with the five retained premises and no recursive economic gain.

The Part B optimum value also checks out in the stated relation model. As an additional check, I replaced Decimal scoring with exact integer concave-order comparisons on every enumerated subset: 66,788 backward and 716 forward cases. At every positive-rank breakpoint, the hinge sums for “advertised optimum minus candidate” are nonnegative. The backward optimum has signed score vector {1:+8, 8:+8, 9:−8} and the forward optimum has score zero. I also checked that the centre-option supports are disjoint as required by the WLOG reduction. Thus the enumerated bound has an exact integer certificate, rather than depending on a near-equality of rounded powers.

Two corrections remain:

1. [P2] The optimal subspace is not unique. relations.py:23 says the optimum is attained only by the eight existing relations. The g > best enumeration retains one maximizer and does not establish uniqueness.

   There is an exact tie. In the component numbering returned by optimum() at this pinned head, the original optimal set is
   {56,65,66,194,200,203,205,212}. Replace 194 with 207, the relation

   h288 − h294 − h296 − h297 + h300 + h303 = 0.
   

   The resulting space is independent, eight-dimensional, and different from the original span. One valid pivot set is {225,227,229,230,241,245,257,288}, all rank-nine targets. Replaying the row-membership tests saves exactly eight rank-one climbs and eight rank-eight centres, with target debit eight rank-nine jobs. The freed backward climb events change from
   {15584,15870,15876,17233,17245,17256,17264,17722} to
   {15584,15870,15876,17245,17256,17264,17580,17722}; all eight centre sites remain free. These are zero-based source-event indices.

   This gives the same 8(1+8^t−9^t) for every t, so it does not improve the bound or supply a ninth profitable exchange. Please state that the original span is an optimum, remove the uniqueness claim, and retain this tied space as a regression.

2. [P3] The endpoint-distance exponent is off by a factor of 100. README:83–84 says the chosen κ is 8×10⁻¹⁶ below the quoted bound. In fact
   0.00302728823898 − 0.0030272882389 = 8×10⁻¹⁴.
   Given the certified bracket, the distance to the actual reserve-limited maximum is between 7×10⁻¹⁴ and 8×10⁻¹⁴. The selected endpoint and its funding checks are unaffected.

Scope: local checks used CPython 3.12.8 / Darwin arm64 with explicit timeouts and a sampled 1 GiB descendant-RSS limit; the endpoint check's documented Darwin resource shim remains explicit. The full unmodified foundation replay is covered by the matched Linux CI run, and I independently authenticated all 142 frozen foundation files locally. The optimality check shares the submitted rational replay and adds an independent integer pricing certificate; it concerns the stated fixed-word relation model, not arbitrary new native words or an unconditional multiplication theorem. No selected-record change or merge is proposed here.

Review performed with OpenAI Codex assistance.

[GamingPuzzled]

Thanks for the careful review. Both points are addressed in eb289efb4:

1. [P2] Uniqueness: removed. relations.py, the README and the PR description now say Conditional κ = 0.00302666 (+4.31e-6 over #5377): eight joint helper relations #5383's span attains the optimum (an optimum, not the unique one). relations.py adds your tie as a regression. It checks that the chosen set is {56, 65, 66, 194, 200, 203, 205, 212} at this head. It then checks that swapping 194 → 207 (h288 − h294 − h296 − h297 + h300 + h303 = 0) gives an independent eight-dimensional set, outside the Conditional κ = 0.00302666 (+4.31e-6 over #5377): eight joint helper relations #5383 span, with the same exact score 8(8^t+1−9^t). The result is recorded in RESULT-RELATIONS.json as tied_optimum_regression.
2. [P3] Exponent: corrected to 8×10⁻¹⁴, noting the 7×10⁻¹⁴ to 8×10⁻¹⁴ distance to the actual reserve-limited maximum.

relations.py (rebuild and compare) passes, the 5 corruption controls still reject, and --check-only and check_research_commit pass. Part A, the κ, its endpoint and its funding, is unchanged.