Conditional κ = 0.0030272882489783: gap-parametric precision for the #5413 caller (numerical headroom only, +1.0e-11 over #5462)

research/gap-parametric-precision-r15/README.md

@@ -0,0 +1,134 @@
+# Gap-parametric precision for the #5413 caller: κ = 0.0030272882489783 (numerical headroom only)
+
+**Scope.** This package gains numerical headroom only, from the compiler gap and precision carrier. The recursive economic
+gain is **zero**. The complete #5413 caller is unchanged: the same word, invoice, paid and target profiles, reserve
+10⁻¹², compiler tuple, BIT stock and 25-premise conjunction. No paid job, target or compiler parameter other than the
+strict gap and its recomputed height carrier changes, and the raw moment is identical.
+
+| | κ (conditional) | tested | gap | j | H |
+|---|---:|---:|---:|---:|---:|
+| #5413 (Clear1nsight) | .003027288235 | .003027288245 | 1/7·10¹¹ | 55099 | 38569300000000000 |
+| #5462 (ours, same carrier) | .0030272882389 | .0030272882489 | 1/7·10¹¹ | 55099 | 38569300000000000 |
+| **this package** | **.0030272882489783** | **.0030272882489784** | **1/7·10¹⁶** | **72096** | **5046720000000000000000** |
+
+The gain is **1.00783 × 10⁻¹¹** over #5462 and **1.39783 × 10⁻¹¹** over #5413. κ stays below the fixed-profile
+ceiling U = .00302728825 (#5409) and below the raw root, which lies in (.0030272882499, .00302728825).
+
+## Why the gap was pinned, and why it need not be
+
+#5413's numerical funding engine (`fund_uniform_endpoint.py`) already accepts any **unit** gap g = (tested − κ)/7.
+It derives the least height grid j with (H+1)¹⁰⁰⁰ < 2ʲ, where H = j·den(g). The gap was pinned only as literals:
+
+1. Precision checker `check_v16_precision_v3.py`, one line:
+   `need(gap==(tested-kappa)/7==Q(1,700000000000), 'actual strict compiler gap')`.
+   The carrier obligations next to it, `H==j*gap.denominator and (H+1)**1000<2**j`, are already parametric.
+2. Conjunction `check_neutral_closure.py`:
+   - the endpoint gate (κ, tested, gap and gain literals);
+   - the height gate (H and j literals);
+   - the hash of the accepted written proof.
+3. Written proof `V16-APPLICATION-PROOF-v3.md`. The argument is symbolic in N = 2^H and M = F16(N). The carrier
+   appears only as recorded values at four sites:
+   - §5, the H line;
+   - §5, the gap line;
+   - §6, a **non-claim** ("D0 < N^gap would fail: log₂D0 = 1048576 while H·gap = j, H·r0 = j/10");
+   - §8, the funded (K, C) pair.
+
+   §6/§7 explicitly avoid any absorption at the fixed carrier ("no C0bar<N^gap is required"). The stop, layout and d
+   parameters (all equal to the gap) enter only as positive rationals in outer domain-entry thresholds. A finer gap
+   moves those thresholds; it does not invalidate them. The §6 non-claim stays true because j = 72096 < 1048576.
+
+The outer 54 strict and 4 functional predicates have minimum margins proportional to the gap (minimum strict margin
+gap/10), so they stay positive for every positive gap. **The only binding constraint is `funded_alpha_reserve`.**
+#5413's directed moment engine puts the reserve-limited tested saving in (.003027288248978444, .003027288248978445).
+Every κ < tested below that bound is therefore fundable with its own least carrier. There is no finest admissible gap.
+This package certifies the 16-digit point, and its sweep funds the gaps 1/(7·10^m) for m = 12…18. At m = 18 that gives
+κ = .003027288248978443, with arithmetic only.
+
+## What `gap_endpoint.py` does (about 5 s)
+
+1. Authenticates all 63 capsule files of `research/coupled-helper-neutral-completion` against its `SOURCE-BINDINGS.json`,
+   whose SHA256 its `verify.py` pins. Copies them into private scratch.
+2. Reruns #5413's invoice/funding engine with only its two endpoint literals substituted:
+   - the regenerated invoice is byte-identical to #5413's;
+   - paid, targets, reserve, tuple, invoice hash, BIT stock, predicates and controls are unchanged;
+   - j and H are recomputed independently: the least grid point, with the preceding grid point failing.
+3. Generalises the precision checker. The single pinned line becomes "positive **unit** gap"; nothing else changes.
+   - **Conservativity:** on #5413's own manifest, the generalised checker reproduces #5413's expected precision receipt.
+   - At the new carrier, its receipt differs from #5413's only in `carrier_H`, `carrier_j`, `carrier_gap`,
+     `expanded.N`, `expanded.M` and `proof_sha256` (plus timing and hashes).
+   - The written proof is instantiated by exactly four unique literal substitutions, and the script asserts that no
+     old carrier or endpoint literal remains. The result is stored here as `V16-APPLICATION-PROOF-v3-R15.md`.
+     `diff` against #5413's file shows four changed lines.
+4. Reruns #5413's conjunction with only the endpoint/carrier literals moved (κ, tested, gap, gain, H, j and the
+   accepted-proof hash): PASS, 25/25, the same premise names.
+5. Controls (all must reject):
+   - #5413's **pinned-gap precision checker** rejects the new funding (`actual strict compiler gap`);
+   - #5413's unsubstituted conjunction rejects it (`exact_new_positive_endpoint_gap`);
+   - the generalised checker rejects the old H/j with the new gap (`actual funded carrier rank`);
+   - the generalised checker rejects a non-unit gap 3/7·10¹⁶;
+   - tested .00302728825 and .0030272882489785 fail `funded_alpha_reserve`;
+   - a non-unit gap fails the engine's unit height grid.
+
+   The script also checks the raw-root and reserve brackets, and the sweep.
+
+## Assumptions and non-claims
+
+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.
+
+The instantiated proof is #5413's reviewed proof with its carrier literals replaced. No new mathematical argument is
+added, and it has not been reviewed independently in its own right. No unconditional theorem, formal-kernel
+acceptance or κ-maximality is claimed. On macOS the script skips the precision checker's `RLIMIT_AS` call, which Darwin
+cannot lower; Linux runs the checker text unchanged apart from the one gap line.
+
+## Reproduce
+
+```sh
+python3.12 -B scripts/research_package.py --package research/gap-parametric-precision-r15
+python3.12 -I -B research/gap-parametric-precision-r15/gap_endpoint.py   # PASS_5413_CALLER_GAP_PARAMETRIC_PRECISION_AT_KAPPA_0.0030272882489783
+diff research/coupled-helper-neutral-completion/capsule/workspace/swarm-coupled-caller-sol-high-20261024/agents/precision/V16-APPLICATION-PROOF-v3.md \
+     research/gap-parametric-precision-r15/V16-APPLICATION-PROOF-v3-R15.md
+```
+
+`logs/` holds `.txt` transcripts of these runs, made on macOS:
+
+- `gap_endpoint.txt`: PASS.
+- `check_only.txt`: PASS_PACKAGE_METADATA_AND_PINS.
+- `research_package_replay.txt`: the full registered replay. Our command passes. In the dependency command, #5413's own
+  `verify.py`, runs 0–5 pass and run 6 (its precision checker) stops at `resource.setrlimit(RLIMIT_AS)`, which Darwin
+  cannot lower. That command is meant for Linux CI. Docker was unavailable locally.
+
+## Open next questions
+
+- **Does a changed profile clear U?** Can a changed paid or target profile on the #4610 word fund κ above the
+  fixed-profile ceiling U = .00302728825?
+  - At the #5413 profile, the endpoint headroom left by the alpha reserve is now below 1.5 × 10⁻¹⁶, and the raw root is
+    below U. Any further gain must be economic.
+  - First step: price one candidate histogram change with `scripts/compare_recursive_invoices.py`.
+- **Can the reserve itself move?** The rounding minimum is about 9.1 × 10⁻¹³, and #5413 fixes the reserve at 10⁻¹².
+  Lowering the reserve would change the compiler tuple, which this package keeps fixed. 1 − α falls about one-for-one
+  with the tested saving here, so the reserve change would add at most about 9 × 10⁻¹⁴.
+
+## Credit
+
+- Chris R (Clear1nsight) supplied:
+  - the #5413 complete caller;
+  - its funding engine, precision checker, written proof and conjunction;
+  - #5400, #5407 and #5414.
+- Ali Romman (aliromman) supplied the owned-relation chain #5377, #5383, #5392 and #5398.
+- Dillon Amburgey (dillona) supplied the #5402 coupled provider.
+- zxv supplied the physical/compiler/semantic precision and higher-moment machinery that the proof specialises.
+- Further contributors:
+  - Dugongue: reserve refinements;
+  - Cao Yifan (intqwq): the #4610 word;
+  - Cinna Davis: controller and helpers;
+  - Jacob Alan Sussman: integer flow.
+- #5409 supplied the ceiling U. Our #5462 is the pinned-gap predecessor of this package.
+
+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/gap-parametric-precision-r15/RESULT-GAP-PARAMETRIC.json

@@ -0,0 +1,137 @@
+{
+ "authenticated_capsule_files": 63,
+ "carrier": {
+  "H": "5046720000000000000000",
+  "gap": "1/70000000000000000",
+  "j": 72096
+ },
+ "conjunction": {
+  "gain_over_prior_comparison": "189783/10000000000000000",
+  "premise_count": 25,
+  "status": "PASS_CONDITIONAL_MATHEMATICAL_KAPPA_CONJUNCTION"
+ },
+ "controls": {
+  "engine_non_unit_gap_rejected": true,
+  "parametric_checker_rejects_non_unit_gap": true,
+  "parametric_checker_rejects_old_carrier_with_new_gap": true,
+  "pinned_gap_5413_precision_checker_rejects_new_funding": true,
+  "tested_.0030272882489785_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": [
+  "H",
+  "alpha_upper",
+  "delta_upper",
+  "eta_lower",
+  "gain_over_comparison",
+  "gamma_upper",
+  "gap",
+  "j",
+  "kappa",
+  "outer",
+  "raw_tested_margin",
+  "reserve_slack",
+  "tested"
+ ],
+ "gain_over_5413": "139783/10000000000000000",
+ "gain_over_5462": "100783/10000000000000000",
+ "gap_sweep": [
+  {
+   "H": "409549000000000000",
+   "gap": "1/7000000000000",
+   "j": 58507,
+   "kappa": ".003027288247",
+   "min_strict_outer": "1/70000000000000",
+   "tested": ".003027288248"
+  },
+  {
+   "H": "4333770000000000000",
+   "gap": "1/70000000000000",
+   "j": 61911,
+   "kappa": ".0030272882488",
+   "min_strict_outer": "1/700000000000000",
+   "tested": ".0030272882489"
+  },
+  {
+   "H": "45717000000000000000",
+   "gap": "1/700000000000000",
+   "j": 65310,
+   "kappa": ".00302728824896",
+   "min_strict_outer": "1/7000000000000000",
+   "tested": ".00302728824897"
+  },
+  {
+   "H": "480935000000000000000",
+   "gap": "1/7000000000000000",
+   "j": 68705,
+   "kappa": ".003027288248977",
+   "min_strict_outer": "1/70000000000000000",
+   "tested": ".003027288248978"
+  },
+  {
+   "H": "5046720000000000000000",
+   "gap": "1/70000000000000000",
+   "j": 72096,
+   "kappa": ".0030272882489783",
+   "min_strict_outer": "1/700000000000000000",
+   "tested": ".0030272882489784"
+  },
+  {
+   "H": "52839500000000000000000",
+   "gap": "1/700000000000000000",
+   "j": 75485,
+   "kappa": ".00302728824897843",
+   "min_strict_outer": "1/7000000000000000000",
+   "tested": ".00302728824897844"
+  },
+  {
+   "H": "552090000000000000000000",
+   "gap": "1/7000000000000000000",
+   "j": 78870,
+   "kappa": ".003027288248978443",
+   "min_strict_outer": "1/70000000000000000000",
+   "tested": ".003027288248978444"
+  }
+ ],
+ "kappa": ".0030272882489783",
+ "old_carrier": {
+  "H": "38569300000000000",
+  "gap": "1/700000000000",
+  "j": 55099
+ },
+ "precision_checker_change": {
+  "parametric": "need(gap==(Q(fund['tested'])-Q(fund['kappa']))/7 and gap>0 and gap.numerator==1,'actual strict compiler gap (positive unit gap; gap-parametric)')",
+  "pinned": "need(gap==(Q(fund['tested'])-Q(fund['kappa']))/7==Q(1,700000000000),'actual strict compiler gap')"
+ },
+ "precision_receipt_differs_only_in": [
+  "carrier_H",
+  "carrier_gap",
+  "carrier_j",
+  "expanded",
+  "mathematical_application"
+ ],
+ "previous": {
+  "pr5413": {
+   "kappa": ".003027288235",
+   "tested": ".003027288245"
+  },
+  "pr5462": {
+   "kappa": ".0030272882389",
+   "tested": ".0030272882489"
+  }
+ },
+ "proof_literal_sites": 4,
+ "raw_root_bracket": [
+  ".0030272882499",
+  ".00302728825"
+ ],
+ "reserve_limited_tested_bracket": [
+  ".003027288248978444",
+  ".003027288248978445"
+ ],
+ "scope": "numerical compiler headroom only: same complete #5413 caller, invoice and profiles; zero economic gain",
+ "supremum_note": "every positive unit gap is admissible; kappa < tested < .003027288248978445 is the only bound",
+ "tested": ".0030272882489784"
+}