dillona · Nov 10, 2026
diff --git a/research/p104-funded-cohort-split/APPLICATION.md b/research/p104-funded-cohort-split/APPLICATION.md
@@ -0,0 +1,29 @@
+# Conditional application to the funded native and signed caller
+
+This applies Rohan Arun's #7113 component proof and #7116's changed-native adaptation to the new funded native. The input closure and explicit construction are in PROOF.md. Both artifact hashes and public heads are in README.md. The five inherited assumptions and selected completed BIT predecessor are retained. The source-specific proof 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 repair. Its generic root/storage and scalar/physical arguments require the substitutions below; a predecessor's numeric receipt alone is insufficient.
+
+## Native and caller operator substitution
+
+The new exact constructor preserves every independent arbitrary-input native endpoint while replacing one rank-nine helper with rank-eight and rank-one helpers. Two rank-two moves are subdivided into four rank-one moves, with every new occurrence charged. It checks all three public retiming intervals, the entire rational prefix/suffix, both basis identities and the retained scalar alphabet. Its complete forward and adjoint replays include every owned row. All 3,047 current paid/helper chart pairs and 104 main boundary pairs are then rebuilt with the pinned compiler and distinct signed-Pauli/absolute-phase interpreter. For these Clifford maps, the signed Pauli action fixes the operator up to global phase, which the exact Gaussian check fixes. Thus every column of each chart map is determined. The fresh replay includes a deliberately wrong absolute phase.
+
+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, 110,672 paid positions and 22,206 productive fields. Main-owner aliasing is rejected. Allocate 18,462 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 1,486 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 24. Fresh helpers receive their own independent inputs. 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 405 and context mass exponent 1168, giving c=2336. This exceeds #7116's allowance 2328 and is recomputed rather than copied. 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=16480300000000000000000. The independent inequalities check every rounded caller bound as well. 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 777,561,336 includes the caller and native conversion services. It is distinct from the physical-machine bound. The fresh raw-occurrence counts give 1,144,287 native scalar macro inputs, 117,856 caller macro inputs and 58,012 conversions. The retained conservative factor 32 gives 42,244,960 scalar services; with the reviewed 2²⁰ service constant this is 44,297,051,176,960 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 298,604 monomial map slots. At actual ambient 145, the retained elementary bound 4(145+1)² is 85,264 records per map, hence at most 25,460,171,456 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 private 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 paid-cohort construction by dillona with substantial OpenAI Codex assistance. Apache-2.0.
diff --git a/research/p104-funded-cohort-split/DISCOVERY.json b/research/p104-funded-cohort-split/DISCOVERY.json
@@ -0,0 +1,165 @@
+{
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+  "pairs": 43638,
+  "hits": 3,
+  "classes": {
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+    "F2zero": 266
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+  "budget_reject": {
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+    "26": 240,
+    "30": 105,
+    "31": 39,
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+  "best_budget": {
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+    "gain": 0.011365787087289192,
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+        "neutral": false
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+        "neutral": true
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+          1,
+          1
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+        "cost": 0.005038210573221402,
+        "private": false,
+        "neutral": false
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+      {
+        "role": 299,
+        "lo": 43732,
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+        "old": 246,
+        "new": 127,
+        "moves": [
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+        "old_paid": [
+          1
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+        "new_paid": [
+          0,
+          1
+        ],
+        "cost": 0.0,
+        "private": false,
+        "neutral": true
+      },
+      {
+        "role": 1359,
+        "lo": 43757,
+        "hi": 47192,
+        "frame": 246,
+        "old": 246,
+        "new": 127,
+        "moves": [
+          44791
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+        "old_paid": [
+          1
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+        "new_paid": [
+          0,
+          1
+        ],
+        "cost": 0.0,
+        "private": false,
+        "neutral": true
+      }
+    ],
+    "paid_roles": 2
+  },
+  "scope": "One total-movement-budget census on exact7115, group3,127 and existing frames; complete kernels, individual basis pair nominations plus complete decision witness; not complete cost optimization over rational combinations.",
+  "selected_candidate": 2,
+  "selected_word_sha256": "954d64194222d5080cfc7ca860e78a59979c825e0c68632f5e4ce75ae496bab8",
+  "evidence": "Discovery floating cost screens only. Public verification trusts exact source/witness/invoice instead.",
+  "credit": "Ali Romman, CubeJerry, Matarener and full native lineage; new screen by dillona/OpenAI Codex."
+}
diff --git a/research/p104-funded-cohort-split/ENVIRONMENT.json b/research/p104-funded-cohort-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": "Pending publication; canonical base current-CI-navigation onboarding defect is known.",
+  "completed_utc": "2026-11-10T18:29:09.357193+00:00"
+}
diff --git a/research/p104-funded-cohort-split/INPUTS.json b/research/p104-funded-cohort-split/INPUTS.json
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