dillona · Nov 10, 2026
diff --git a/research/p104-nonunit-pivot-split/APPLICATION.md b/research/p104-nonunit-pivot-split/APPLICATION.md
@@ -0,0 +1,31 @@
+# Conditional application of the nonunit-pivot helper split
+
+This applies Rohan Arun's #7113 component proof and the retained changed-native adapters to the exact #7123 word plus one rank-nine split. The complete frozen source closure remains intact. PROOF.md supplies the source-specific cut, legal unit basis and arbitrary-input replacement. Both native and application invoices are recounted. The five assumptions and selected completed BIT predecessor remain. The template is the [#7000 application review](https://github.com/CrocSwap/integer-mult-bounds/blob/e54523517d0f03e58352e647c403a5af3e525a57/docs/research/pr-intake-evidence/pr7000-application/PROOF-REVIEW.md), including its BIT ambient 134 / log-power 617 repair. Current substitutions follow.
+
+## Native and caller operator substitution
+
+The new basis swaps cocharted boundary coordinates 639 and 802 using four charged scalar instructions, replacing a forbidden nonunit normalization by the legal unit 3. Complete boundary/cut bases 426/17 are applied in both directions. The cut after event 37583 is at chart 1027; the declared neutral placement leaves the source program byte-identical. One independent helper target 3→127 is split into 3→1027 (rank 5) and 1027→127 (rank 4). All 880 ADDs, 6 SCALEs and the fresh input are charged. Owned transitions and lifetimes are retained. The complete forward and adjoint replays include every owned row.
+
+All 3,048 current paid/helper chart pairs and 104 main boundary pairs are rebuilt with the pinned compiler and distinct signed-Pauli/absolute-phase interpreter. For these Clifford maps, the signed Pauli action determines the operator up to global phase, which the exact Gaussian check fixes. Thus every column of each map is determined. The native check additionally replays all 512 columns of the three target boundary pairs and rejects a deliberately wrong absolute phase. No new pair is introduced, but changed multiplicities and the entire catalogue are recounted.
+
+The unchanged 17-stage caller has five noncommuting quarter turns and twelve padding turns in inverse pairs. Its grouped idle bases lie over Z[1/30]. Its exact rational and integer Laurent engines agree on all 72 output fields for every formal central/body variable. In the physical substitution, seventeen disjoint eight-axis stage blocks and nine central axes use ambient 145. For each invocation, exchange the eight native complement axes with the fresh stage block while fixing the routed central axis and spectator axes. The phase-polynomial checks verify both directions. Literal A and B routes then apply commuting body operators on disjoint stage blocks.
+
+A signed rank-nine idle uses one positive rank-nine recursive request plus the charged ordinary conjugation/phase implementing the inverse on the protected central axis. Current finite signed-child checks cover widths 1,2,3; the symbolic phase geometry supplies the disjoint-axis substitution. The ordered occurrence digest is recomputed from the literal caller and current native geometry. It binds 13,600 native contact boundaries, 110,655 paid positions and 22,308 productive fields. Main-owner aliasing is rejected. Allocate 18,564 distinct (stage,helper) public inputs and 867 reserved native owned banks, plus the retained serial query/LAMBDA bank. The surplus is 1,424. The ownership discipline and owned-erasure hypothesis remain explicit.
+
+## Common-grid and fixed-root induction
+
+Scale each of the 3,744 raw caller main input fields by 675 and divide each completed caller main output by 675. The independent integer Laurent replay checks the necessary valuations 3³ and 5², rejects a lower exponent of either prime, and uses safe binary valuation 381. Every normalized main coefficient and pre-ADD product is integral on that odd grid. Substituting actual binary Clifford address operators introduces only powers of two, so the odd-divisibility invariant transfers to whole arrays and all spectators.
+
+Within each native invocation, scale all 1,492 public fields by 1125 and unscale the completed returns. The separate exact prefix/pre-product derivation and integer replay establish the native grid at fractional height 25. Fresh helpers receive their own independent inputs. The previous F24 grid is rejected on the new word. The forward integer replay changes both its scale to 2^25 and its measured trailing-bit offset to 26. Both native and caller precision adapters explicitly request frac=25 from the unchanged authenticated grid engine. The exact native endpoint ensures these helper inputs do not perturb the normalized main caller invariant. Both odd scales are included simultaneously in the unfinished-context mass allowance.
+
+The fresh forward/adjoint envelope has height 406 and context mass exponent 1181, giving c=2362. This is recomputed from the current word and caller; #7123's allowance was 2344. It remains below 4096. The same conservative ambient 153 compiler threshold funds it: Kstar has 14,093 bits and c·Kstar has 14,104 bits, below H=16495300000000000000000. The regenerated compiler gap is 1/100000000000000000. Every rounded caller bound, outer and fixed-tape gate is checked with these current values. Apply the retained full-root common-format induction with fixed padding, signed width p+h+2D+3, fractional width p+D, and 26 temporary bits. Completed child returns preserve this format. Exact finite-state odd division retains the reviewed carry/remainder invariant.
+
+## Physical upper-cost substitution
+
+The source-unit ordinary bound 779,122,208 includes the caller and native conversion services. It is distinct from the physical-machine bound. The fresh raw-occurrence counts give 1,161,423 native scalar macro inputs, 117,856 caller macro inputs and 58,216 conversions. The retained conservative factor 32 gives 42,799,840 scalar services; with the reviewed 2²⁰ service constant this is 44,878,885,027,840 times the root scalar-width envelope. Numerators of all actual native/caller instructions and both conversion scales fit 24 bits.
+
+The current occurrence count gives 298,774 monomial map slots. At actual ambient 145, the retained elementary bound 4(145+1)² is 85,264 records per map, hence at most 25,474,666,336 elementary records per raw occurrence. Use actual owner widths and these finite coefficients in the same fixed-tape stock and movement envelopes from the selected review. The positive compiler/outer slack permits enlarging the eventual cutoffs. This is a conditional asymptotic substitution, not an enumerated machine trace.
+
+The recomputed complete recurrence has the bracket and claim in README.md; both exact price engines, the compiler and 54 strict plus 4 functional outer inequalities pass. These checks establish the stated conditional component application under the five assumptions. Independent mathematical review, formal-kernel validation and maintainer admission are separate obligations.
+
+Credit Rohan Arun for the #7113 proof and implementation, Douglas Colkitt and the #7000 application reviewers, the retained compiler/precision/source authors, and the complete lineage in README.md. Adapted for this exact nonunit-pivot helper split, retaining all peer and predecessor credit, by dillona with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/p104-nonunit-pivot-split/DISCOVERY.json b/research/p104-nonunit-pivot-split/DISCOVERY.json
@@ -0,0 +1,246 @@
+{
+  "source_head": "020d581d9856b6db83e448c3f20418bc63b72c34",
+  "screen_kappa": ".003641239650184",
+  "scope": "Complete rational kernels followed by basis nominations in group 3\u2192127; no exhaustive rational support/cost optimization.",
+  "counts": {
+    "systems": 38473,
+    "saved": 10293,
+    "pairs": 44172,
+    "hits": 0,
+    "classes": {
+      "F1zero": 8286,
+      "witness": 1756,
+      "F2zero": 251
+    },
+    "rescued": 0,
+    "budget_reject": {
+      "8": 3240,
+      "10": 2865,
+      "6": 4231,
+      "13": 1545,
+      "12": 1926,
+      "4": 2297,
+      "9": 3954,
+      "7": 3679,
+      "14": 1738,
+      "15": 2006,
+      "5": 3213,
+      "3": 3031,
+      "2": 1236,
+      "18": 893,
+      "16": 1113,
+      "21": 466,
+      "24": 407,
+      "11": 2332,
+      "17": 1103,
+      "20": 469,
+      "19": 644,
+      "22": 322,
+      "25": 164,
+      "23": 530,
+      "28": 181,
+      "29": 109,
+      "27": 152,
+      "26": 167,
+      "30": 99,
+      "31": 34,
+      "33": 5,
+      "32": 10
+    },
+    "best_budget": {
+      "cut": 37622,
+      "frame": 1027,
+      "cost": 0.0,
+      "gain": 0.02236181478033039,
+      "paths": [
+        {
+          "role": 1298,
+          "lo": 36613,
+          "hi": 37622,
+          "frame": 1027,
+          "old": 1027,
+          "new": 525,
+          "moves": [
+            37621
+          ],
+          "old_paid": [
+            1
+          ],
+          "new_paid": [
+            0,
+            1
+          ],
+          "cost": 0.0,
+          "private": false,
+          "neutral": true
+        }
+      ],
+      "paid_roles": 0
+    }
+  },
+  "chosen_index": 10,
+  "constructed_indices": [
+    10
+  ],
+  "legal_basis_nominations": [
+    {
+      "index": 10,
+      "alpha": "1",
+      "cut": 37583,
+      "paths": 1,
+      "G": 426,
+      "J": 17,
+      "instructions": 886,
+      "operations": {
+        "ADD": 880,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/256"
+    },
+    {
+      "index": 8,
+      "alpha": "7",
+      "cut": 37585,
+      "paths": 0,
+      "G": 427,
+      "J": 17,
+      "instructions": 888,
+      "operations": {
+        "ADD": 882,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 9,
+      "alpha": "7",
+      "cut": 37584,
+      "paths": 0,
+      "G": 427,
+      "J": 17,
+      "instructions": 888,
+      "operations": {
+        "ADD": 882,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 7,
+      "alpha": "7",
+      "cut": 37586,
+      "paths": 0,
+      "G": 427,
+      "J": 18,
+      "instructions": 890,
+      "operations": {
+        "ADD": 884,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 5,
+      "alpha": "7",
+      "cut": 37588,
+      "paths": 0,
+      "G": 427,
+      "J": 19,
+      "instructions": 892,
+      "operations": {
+        "ADD": 886,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 6,
+      "alpha": "7",
+      "cut": 37587,
+      "paths": 0,
+      "G": 427,
+      "J": 19,
+      "instructions": 892,
+      "operations": {
+        "ADD": 886,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 0,
+      "alpha": "1",
+      "cut": 37622,
+      "paths": 1,
+      "G": 426,
+      "J": 20,
+      "instructions": 892,
+      "operations": {
+        "ADD": 890,
+        "SCALE": 2
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 2,
+      "alpha": "7",
+      "cut": 37591,
+      "paths": 0,
+      "G": 427,
+      "J": 20,
+      "instructions": 894,
+      "operations": {
+        "ADD": 888,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 3,
+      "alpha": "7",
+      "cut": 37590,
+      "paths": 0,
+      "G": 427,
+      "J": 20,
+      "instructions": 894,
+      "operations": {
+        "ADD": 888,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 4,
+      "alpha": "7",
+      "cut": 37589,
+      "paths": 0,
+      "G": 427,
+      "J": 20,
+      "instructions": 894,
+      "operations": {
+        "ADD": 888,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    },
+    {
+      "index": 1,
+      "alpha": "7",
+      "cut": 37592,
+      "paths": 0,
+      "G": 427,
+      "J": 21,
+      "instructions": 896,
+      "operations": {
+        "ADD": 890,
+        "SCALE": 6
+      },
+      "maximum_scalar": "1338551/384"
+    }
+  ],
+  "selection": "Try alpha=1 and the common nonunit denominator-clearing gauge, then order legal candidates by total scalar instruction count, path count, largest absolute scalar, index.",
+  "basis_repair": "Keep pivot only if its nonzero entry is a 2,3,5 unit; otherwise choose the same smallest smooth swap entry as #7112. Original d639=-7867/16, chosen d802=3.",
+  "corrected_screen": "An inherited discovery script still used .003640102187396 in two preliminary group screens. Both were rerun at the actual .003641239650184 and gave identical zero-hit counts; the group 3\u2192127 nominations and this result use the corrected value. Exact certificate and prices do not trust discovery floats.",
+  "local_only": "Discovery scripts, rejected nominations and earlier negative rounds remain local. Public WITNESS and source-bound constructor are sufficient to reproduce this result; broader search closure is not claimed.",
+  "stop_reason": "One legal candidate selected; full verification takes priority."
+}
diff --git a/research/p104-nonunit-pivot-split/ENVIRONMENT.json b/research/p104-nonunit-pivot-split/ENVIRONMENT.json
@@ -0,0 +1,14 @@
+{
+  "initial_python": "3.14.7",
+  "initial_platform": "macOS-27.0.1-arm64-arm-64bit-Mach-O",
+  "registered_replay": {
+    "python": "3.12.16",
+    "platform": "macOS-27.0.1-arm64-arm-64bit",
+    "result": "PASS both complete commands; every native and composition receipt field matched"
+  },
+  "dependencies": "Python standard library",
+  "interpreter_origin": "Official Astral python-build-standalone release20261026; isolated /tmp installation",
+  "interpreter_archive_sha256": "fbf4afef53c2a6b72bb741bc06d5671a9dfaf87bd84e5aa73ac9b3d0d589f14e",
+  "hosted_ci": "Not yet published; no hosted scientific pass claimed.",
+  "completed_utc": "2026-11-10T21:58:03.626947+00:00"
+}