Indexed carry two-cycles and coordinate flags: conditional κ = 5.22712890914e-5

.gitattributes

@@ -8,3 +8,16 @@ upstream/** -whitespace
 references/copied-centers/pr29/** -whitespace
 references/copied-centers/pr33/** -whitespace
 references/semantic-bulk/** -whitespace
+
+# Immutable contributor archives retain original CRLF and whitespace bytes.
+references/frame-compiler/pr69/** -text -whitespace
+references/frame-compiler/pr70/** -text -whitespace
+references/frame-compiler/pr71/** -text -whitespace
+
+# Preserve original contributor and frozen discovery bytes.
+references/frame-compiler/pr74/** -text -whitespace
+references/frame-compiler/pr78/** -text -whitespace
+references/frame-compiler/pr79/** -text -whitespace
+references/frame-compiler/indexed-r5/** -text -whitespace
+# Byte-identical credited PR74 extracts retain original trailing blank lines.
+scripts/experiments/indexed_cycle_nodeops.py -whitespace

.github/workflows/verify.yml

@@ -16,7 +16,7 @@ jobs:
       fail-fast: false
       matrix:
         python-version: ['3.11', '3.13', '3.14']
-        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair]
+        group: [community, producers, certificates, ternary, research, tests, strips, clones, positive, joint, pair, balanced, indexed]
     steps:
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
         with:

Makefile

@@ -23,6 +23,8 @@ verify:
 	$(MAKE) verify-positive
 	$(MAKE) verify-joint
 	$(MAKE) verify-pair
+	$(MAKE) verify-balanced
+	$(MAKE) verify-indexed
 	$(MAKE) verify-tests
 
 verify-community: community-audit-check community-followup-check copied-reversed-producer copied-reversed-check copied-fixed-reversed-producer copied-fixed-reversed-check
@@ -428,3 +430,31 @@ pair-assembly-verify: pair-assembly-producer pair-assembly-check
 .PHONY: verify-pair
 verify-pair: pair-assembly-verify
 	python3 scripts/audit_pair_candidate.py --check docs/research/community-pair-arithmetic.json
+
+.PHONY: split-pair-verify
+split-pair-verify:
+	python3 scripts/experiments/verify_split_pair.py
+	python3 -m unittest discover -s tests -p 'test_split_pair*.py' -v
+
+verify-joint: split-pair-verify
+
+.PHONY: balanced-split-verify verify-balanced
+balanced-split-verify:
+	python3 scripts/experiments/verify_balanced_split.py
+	python3 scripts/experiments/audit_balanced_split_completion.py
+	python3 scripts/experiments/audit_balanced_split_exchange.py
+	python3 -m unittest discover -s tests -p 'test_balanced_split*.py' -v
+
+verify-balanced: balanced-split-verify
+
+.PHONY: indexed-cycle-verify verify-indexed
+indexed-cycle-verify:
+	python3 scripts/experiments/verify_indexed_cycle.py
+	python3 scripts/experiments/audit_indexed_cycle_completion.py
+	python3 scripts/experiments/audit_indexed_cycle_exchange.py
+	python3 scripts/experiments/audit_indexed_cycle_cycles.py
+	python3 scripts/experiments/audit_indexed_cycle_retired.py
+	python3 scripts/experiments/audit_indexed_cycle_pending.py
+	python3 -m unittest discover -s tests -p 'test_indexed_cycle*.py' -v
+
+verify-indexed: indexed-cycle-verify

NOTICE

@@ -302,3 +302,38 @@ inherited PR #48 sources. The stacked witness in research/pair-assembly/frame
 uses eumemic's PR #57 joint frame compiler unchanged; its driver and verifier
 are adaptations of PR #57's skip_frame_compiler.py, skip_frame_compose.py and
 verify_skip_frame.py. Original notices are retained.
+
+Aligned split-pair composition (2026-10-08)
+------------------------------------------
+Avi Eisenberg: PR62 interval strips and core-aware pair assembly. Dominik
+Scholz with OpenAI assistance: PR63 ranked-frame composition and PR68 pending
+live-carrier controls. Rohan Arun with OpenAI assistance: PR65 region schedules
+and refined exact arithmetic, and PR67 all-rank profile-cost selection and
+its profile oracle. Rohan Garg / rohangar1
+with substantial OpenAI Codex assistance: PR59 split-pair operation. Eumemic
+with substantial OpenAI Codex assistance: PR57 joint reversible compiler and
+independent physical-word/profile checks. Chafik Boukhalfa with OpenAI Codex
+assistance: PR60 ranked reclamation and this selected [1,1,2] split vector,
+aligned common-point permutation, next-use profile score, composition and
+certification package.
+The complete PR65 baseline words, profiles and arithmetic, plus 43 original
+source/provenance files, are preserved under references/frame-compiler/pr65.
+The original PR59 split operation and notices are separately preserved.
+PR67's compiler/oracle and baseline profile, and PR68's compiler/driver/proof,
+retain their original bytes, licenses and notices in their own archives.
+All previous notices and mathematical dependencies remain applicable.
+
+Balanced coarse split composition (2026-10-08)
+---------------------------------------------
+Eumemic / @eumemic supplied PR69's balanced coarse sums, with OpenAI Codex
+assistance. Alejandro Zarzuelo Urdiales / @alejandrozu supplied PR70's bounded
+carry-exchange strategy with the AI disclosures retained in the source archive.
+Chafik Boukhalfa / @chafreaky, with OpenAI Codex assistance, selected the row/
+half-plane coarse rules, future-compatible unit completion, exact integer-price
+carry adaptation, composition and this certificate package. The PR71 anchored
+split vector, schedules, arithmetic, pending controls and their contributor
+chain are retained as credited above. Original PR69/70 sources, licenses and
+notices and the complete published PR71 baseline are preserved under
+references/frame-compiler. All earlier notices remain applicable.
+
+Indexed two-cycle composition (2026-10-08): Chafik Boukhalfa with OpenAI Codex assistance. Thomas DiFiore (@tomdif, PR74) supplied exact node ordering and coordinate relabeling, including the selected h25 permutation. Rohan Arun (@rohanarun, PR78) supplied the coordinate-flag search idea. Balanced multi-splits, exact retired/pending indices, bounded two-cycles, late-output routing and selected-coordinate certification compose inherited PR57/59/60/62/63/65/67/68/69/70/71/79 work. All original authorship, Apache-2.0 and AI disclosures remain in unchanged contributor archives.

README.md

@@ -21,6 +21,39 @@ speedup. The model and general reduction are inherited from OpenAI's
 [Selected parameter certificate](research/matrix-exponent-synthesis/candidate/arithmetic.json) ·
 [Independent arithmetic check](docs/research/community-pair-arithmetic.json)
 
+## Additional conditional construction: balanced coarse split frames
+
+The [balanced-split proof and reproduction guide](research/balanced-split/README.md)
+gives **κ=25872812736433/500000000000000000=5.1745625472866e-5**, approximately
+**0.6437453019850683%** above the complete published PR71 witness. It composes
+PR69 balanced coarse sums and a bounded PR70 carry exchange with PR71's
+anchored split graph, future unit completion and profile-cost pending controls.
+All actual words, dirty inputs, paid profiles and 47 strict assembly constraints
+are checked by `make balanced-split-verify`. This finite conditional construction
+retains the inherited all-size hypotheses and contributor notices. Chafik
+Boukhalfa prepared the selected composition with OpenAI Codex assistance.
+
+## Additional conditional construction: anchored split frames with live reclamation
+
+The [split-pair proof and reproduction guide](research/split-pair/README.md)
+gives **κ=51414646104039/10^18=5.1414646104039e-5**, approximately
+**0.7463402960%** above the complete pinned PR67 witness. It combines
+the PR62/63 interval pair graph, the PR59 split operation, PR65's schedules
+and exact arithmetic, PR67's profile-cost selection, and PR68's pending live
+controls on the PR60/57 reversible frame compiler.
+The new selection uses split vector `[1,1,2]` and aligns the first intact
+global pair across common-point permutations. A new integer score uses each
+live control's known next-use frame. Actual roles are R23=27455
+and R25=36015, with W=135075665.
+
+`make split-pair-verify` regenerates the words, checks arbitrary dirty scratch
+and actual fixed-I+J profiles, and certifies the complete recursive child
+list and strict assembly inequalities. This separately submitted finite
+conditional witness retains the inherited all-size hypotheses. Its namespace,
+sources and comparison remain distinct from the maintainer-reviewed statement
+above. Prepared by Chafik Boukhalfa with OpenAI Codex assistance; contributor
+credits and original notices are retained in the package.
+
 ## What changed
 
 **Avi Eisenberg's interval strips and core-aware pair assembly (#62)** arrange
@@ -149,3 +182,7 @@ The project is [Apache-2.0](LICENSE). Bundled RaD sources retain their separate
 CC0 license and notices. The pinned original OpenAI manuscript remains unchanged
 under `upstream/`; its source hashes are in [upstream/manifest.json](upstream/manifest.json).
 This project is not an official OpenAI release or endorsement.
+
+### Indexed two-cycle finite witness
+
+[Indexed-cycle proof and evidence](research/indexed-cycle/README.md) gives κ = 261356445457/5000000000000000 with W=132466108. It combines attributed balanced multi-splits, exact frame indexes, bounded carry two-cycles, PR74 node ordering and coordinate flags; all-size transfer remains inherited. Verify with `make indexed-cycle-verify`. The comparison is to pinned historical PR74/PR79 witnesses observed 2026-10-08.