Conditional κ = 0.0030272882389 (+3.9e-12 over #5413, numerical headroom only); exact owned-relation optimum closes the relation lever

research/owned-relation-exact-optimum-r14/README.md

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+# Exact owned-relation optimum on the #4610 word, and the #5413 caller re-funded at κ = 0.0030272882389
+
+This package has two parts.
+
+**Part A (numerical; compiler headroom only).** The complete #5413 caller, its invoice, its height carrier and
+its precision proof stay unchanged. Re-funding it inside its own reserve gives conditional
+**κ = 30272882389/10¹³ = 0.0030272882389**, tested at **0.0030272882489**. This is
+**3.9 × 10⁻¹²** above the selected #5413 value 0.003027288235. The recursive economic gain is
+**zero**. No paid job, target, word or compiler parameter changes, and the raw moment is identical. The
+result uses part of the 1.5 × 10⁻¹¹ gap between #5413 and the fixed-profile ceiling U = 0.00302728825
+(#5409). It stays below U.
+
+**Part B (exact; negative for further relation gains).** We solve exactly for the best owned
+helper-relation subspace on Cao Yifan's #4610 word, in both orientations. The search space covers:
+
+- relation sets that omit #5383's eight relations, which is #5433's open question;
+- 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) = 0.07570927 raw moment per backward local**, and #5383's span
+attains it. It is an optimum, not the unique one. Replacing component 194 by 207 (the relation
+h288 − h294 − h296 − h297 + h300 + h303 = 0) gives a different eight-dimensional space with the same
+value; zxv found this tie in review, and `relations.py` checks it as a regression. The forward optimum
+is 0. No ninth relation and no better joint set exists. The 144 exchanges of the
+#5398/#5413 controller are the maximum.
+
+## Part B: model and certificate (`relations.py`)
+
+The model is #5377/#5383's, as generalised by #5433. Helpers starting in the common input frame 0 are free
+inputs. There are 358 of them: 326 have rank-9 targets, 26 have rank-4 targets and 6 have rank-8 targets. A
+relation subspace V consists of linear forms in these inputs. Each dimension owns one pivot helper, which
+loses its productive target.
+
+- A paid climb is free iff its row vanishes on the restricted domain.
+- Rows are replayed exactly over Q in the free group ring of warp words. A row therefore vanishes iff, for
+  every warp word, its coefficient vector (its *component*) lies in V.
+- A centre job is free iff `row(centre) − row(snapshot)` lies in V.
+
+| Check | backward | forward |
+|---|---:|---:|
+| helper-only paid climbs | 224 (166 word-free, 50 two-word, 8 three-word) | 358 |
+| distinct components, linearly independent over Q | 225 | 358 |
+| centre snapshot options (all in the component span, rank-9 support) | 56 (8 centres) | 0 |
+| connected pieces (components joined by climbs/options) | 110×1, 1×3, 8×7, 4×14 | 358×1 |
+| exact optimum per local call | **8(8^t+1−9^t)**, attained by the span of #5383's relations (not uniquely; see the 194→207 tie) | **0** |
+
+The certificate has four steps:
+
+1. WLOG V is the span of a set S of components. An option outside span(S) adds a dimension with a rank-9
+   pivot, which costs 9^t, more than the 8^t it can save.
+2. Since S is independent, any pivot set has a perfect matching to S inside the supports (Laplace expansion).
+   The lost target moment is therefore at least Σ_{c∈S} min-target(c)^t.
+3. Each centre's seven options are supported on its zero-site rank-1 climb plus at most one private climb.
+   These supports are disjoint across centres.
+4. Each piece (at most 14 components) is enumerated exactly with 60-digit Decimal powers.
+
+This extends #5433 in three ways. It covers sets without #5383's relations, cheaper pivots and word-carrying
+rows. The ILP cross-check below also covers off-frame helpers, which are outside the admissible construction.
+A side check with HiGHS on all 408 helpers, including helpers in non-common start frames (332 rows, 4 disjoint
+circuits), gives the same optimum. That check is not part of the registered replay.
+
+## Part A: what changes and what is reused (`endpoint.py`)
+
+1. Authenticate all 63 capsule files of `research/coupled-helper-neutral-completion` against its
+   `SOURCE-BINDINGS.json`, whose SHA256 its `verify.py` pins. Copy them into private scratch.
+2. Rerun #5413's invoice/funding engine with only its two endpoint literals substituted. The regenerated
+   invoice equals #5413's expected invoice. Funding passes all endpoint predicates and the 54 strict plus 4
+   functional outer predicates.
+   - The gap stays 1/7·10¹¹, the height grid stays j = 55099 and H = 38569300000000000.
+   - The fields that downstream checks read are identical: paid and target profiles, reserve 10⁻¹², tuple,
+     BIT stock and controls.
+3. Rerun #5413's precision checker on a manifest whose funding row points to the new receipt. Its output
+   equals #5413's expected precision receipt except the manifest hash. The written precision proof is
+   symbolic in N = 2^H and uses only H, j and the gap, so it applies unchanged. Its §8 sentence naming the old
+   pair is the one textual difference.
+4. Rerun #5413's 25-premise conjunction with only the endpoint-gate literals moved to the new pair: PASS,
+   25/25.
+5. Controls:
+   - #5413's unsubstituted conjunction rejects the new funding.
+   - Tested values .00302728825 and .00302728824898 fail `funded_alpha_reserve`.
+   - A non-unit gap fails the height grid.
+   - Brackets, computed with #5413's directed engine:
+     - raw root ∈ (.0030272882499, .00302728825);
+     - reserve-limited tested saving ∈ (.00302728824897, .00302728824898).
+
+At the same carrier, the reserve-limited maximum is κ < 0.00302728823898. The chosen κ is 8×10⁻¹⁴ below that
+bound; against the certified bracket it sits between 7×10⁻¹⁴ and 8×10⁻¹⁴ below the actual reserve-limited
+maximum (corrected from 8×10⁻¹⁶ after review). A smaller unit gap, such as 1/7·10¹² (κ ≈ 0.0030272882479), also passes the endpoint arithmetic. It
+would change H, however, and #5413's precision checker pins the gap. It is not claimed here.
+
+**Assumptions.** The κ is conditional on exactly the five retained selected premises of #5413:
+
+- native all-column operation;
+- owned erasure;
+- the compiler contract;
+- precision/tape;
+- analytic recovery.
+
+No unconditional theorem, formal-kernel acceptance or κ-maximality is claimed. On macOS, `endpoint.py` skips
+the precision checker's `RLIMIT_AS` call, which Darwin cannot lower. Linux runs that checker byte-unchanged.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/owned-relation-exact-optimum-r14
+python3.12 -I -B research/owned-relation-exact-optimum-r14/relations.py      # PASS_OWNED_RELATION_EXACT_OPTIMUM_EQUALS_5383 (~3 s)
+python3.12 -I -B research/owned-relation-exact-optimum-r14/endpoint.py       # PASS_5413_CALLER_REFUNDED_AT_KAPPA_0.0030272882389 (~10 s)
+python3.12 -I -B research/owned-relation-exact-optimum-r14/test_controls.py  # PASS_OWNED_RELATION_EXACT_OPTIMUM_CONTROLS
+```
+
+The registered replay also reruns #5413's full `verify.py`, which Part A depends on.
+
+## Open next questions
+
+- Can a changed paid profile on the #4610 word fund κ above the fixed-profile ceiling U = .00302728825, rather
+  than a relation exchange? Relations are now exhausted exactly. The endpoint headroom left at this carrier is
+  below 1.1×10⁻¹¹. First step: price one candidate histogram change with `scripts/compare_recursive_invoices.py`
+  before building a word.
+- Can a precision proof that is parametric in the gap admit a finer unit gap? A gap of 1/7·10¹² would add about
+  9×10⁻¹² more, and it is numerical only.
+
+## Credit
+
+- Cao Yifan (intqwq) supplied the #4610 word.
+- Ali Romman (aliromman) supplied the owned-relation method and its successors:
+  - #5377, the first owned relation;
+  - #5383, eight joint relations;
+  - #5392, owned circuit transposition;
+  - #5398, the direct backward caller;
+  - #5421, nine BIT relations.
+- Dillon Amburgey (dillona) supplied the #5402 coupled provider.
+- Chris R (Clear1nsight) supplied:
+  - the #5413 complete caller, its funding engine and its conjunction;
+  - the reviews #5384 (owned-relation price ceiling), #5395 (eight-relation price plane) and
+    #5396/#5399/#5400/#5407/#5414.
+- Dugongue supplied #5387/#5390, the reserve refinements.
+- Mark (Matarener) supplied #5433, the relation-subspace certificate and open question that Part B answers.
+- #5409 supplied the fixed-profile ceiling U.
+- Lineage credit is retained for Cinna Davis, zxv, Jacob Alan Sussman (mixed-target paradigm, #4359) and the
+  native, compiler, BIT and analytic predecessors.
+
+Prepared by Joel Pulikkan (GamingPuzzled). AI disclosure: Claude Opus 5.5 (Anthropic) via Claude Code,
+directed by Joel Pulikkan (GamingPuzzled). New material is Apache-2.0.

research/owned-relation-exact-optimum-r14/RESULT-ENDPOINT.json

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+{
+ "authenticated_capsule_files": 63,
+ "carrier": {
+  "H": 38569300000000000,
+  "gap": "1/700000000000",
+  "j": 55099
+ },
+ "conjunction": {
+  "gain_over_prior_comparison": "89/10000000000000",
+  "premise_count": 25,
+  "status": "PASS_CONDITIONAL_MATHEMATICAL_KAPPA_CONJUNCTION"
+ },
+ "controls": {
+  "non_unit_gap_rejected": true,
+  "tested_.00302728824898_rejected_alpha_reserve": true,
+  "tested_.00302728825_rejected_alpha_reserve": true,
+  "unsubstituted_5413_conjunction_rejects_new_funding": true
+ },
+ "fixed_profile_ceiling_U": ".00302728825 (#5409); kappa stays below it",
+ "funding_changed_fields": [
+  "alpha_upper",
+  "delta_upper",
+  "eta_lower",
+  "gain_over_comparison",
+  "gamma_upper",
+  "kappa",
+  "outer",
+  "raw_tested_margin",
+  "reserve_slack",
+  "tested"
+ ],
+ "gain_over_5398": "89/10000000000000",
+ "gain_over_5413": "39/10000000000000",
+ "kappa": ".0030272882389",
+ "precision_receipt": "equal to #5413 expected receipt except input_manifest_sha256",
+ "previous": {
+  "kappa": ".003027288235",
+  "tested": ".003027288245"
+ },
+ "raw_root_bracket": [
+  ".0030272882499",
+  ".00302728825"
+ ],
+ "reserve_limited_tested_bracket": [
+  ".00302728824897",
+  ".00302728824898"
+ ],
+ "scope": "numerical compiler headroom only: same complete #5413 caller, invoice, carrier and precision proof",
+ "tested": ".0030272882489"
+}

research/owned-relation-exact-optimum-r14/RESULT-RELATIONS.json

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+{
+ "C": "0.003027288235",
+ "V0_gain_per_backward_local": "0.075709271555986216366649243749",
+ "conclusion": "No owned-relation subspace beats #5383; the 18 backward locals carry the maximum 144 exchanges.",
+ "frame0_helpers": 358,
+ "frame0_target_ranks": {
+  "4": 26,
+  "8": 6,
+  "9": 326
+ },
+ "model": "owned helper-relation subspaces over the 358 frame-0 helpers, any pivots, rows with warp words",
+ "orientations": {
+  "backward": {
+   "centre_options": 56,
+   "centres_with_options": 8,
+   "climbs_by_word_count": {
+    "1": 166,
+    "2": 50,
+    "3": 8
+   },
+   "components_independent": true,
+   "distinct_components": 225,
+   "helper_only_climbs": 224,
+   "optimum_attained_by": "span of the eight #5383 relations (an optimum; not unique)",
+   "optimum_per_local": "0.075709271555986216366649243749",
+   "piece_sizes": {
+    "1": 110,
+    "3": 1,
+    "7": 8,
+    "14": 4
+   },
+   "tied_optimum_regression": {
+    "original": [
+     56,
+     65,
+     66,
+     194,
+     200,
+     203,
+     205,
+     212
+    ],
+    "relation": "h288 - h294 - h296 - h297 + h300 + h303 = 0",
+    "swapped": "194 -> 207",
+    "tied": [
+     56,
+     65,
+     66,
+     200,
+     203,
+     205,
+     207,
+     212
+    ]
+   }
+  },
+  "forward": {
+   "centre_options": 0,
+   "centres_with_options": 0,
+   "climbs_by_word_count": {
+    "1": 358
+   },
+   "components_independent": true,
+   "distinct_components": 358,
+   "helper_only_climbs": 358,
+   "optimum_per_local": "0E-30",
+   "piece_sizes": {
+    "1": 358
+   }
+  }
+ }
+}