Stopped pair-star trees: conditional κ = 7.808981744031e-5

.gitattributes

@@ -2,9 +2,14 @@
 *.pdf binary
 # Unified diffs require a leading space on blank context lines.
 *.patch -whitespace
+*.diff -whitespace
 # Preserve the exact upstream bytes, including original whitespace.
 upstream/** -whitespace
 # Preserve the exact pinned third-party proof snapshots and notices.
 references/copied-centers/pr29/** -whitespace
 references/copied-centers/pr33/** -whitespace
 references/semantic-bulk/** -whitespace
+# Immutable signed-recursion imports retain their original Git bytes.
+references/signed-recursion/** -text -whitespace
+# Immutable stopped-recursion inputs retain their original Git bytes.
+references/stopped-recursion/** -text -whitespace

.github/workflows/verify.yml

@@ -11,12 +11,12 @@ permissions:
 jobs:
   verify:
     runs-on: ubuntu-24.04
-    timeout-minutes: 20
+    timeout-minutes: 40
     strategy:
       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, rank-pair, ordered-frames, balanced-split, aligned-exchange, deferred-span, merged-span, saturated-deferred, stopped-pairtree]
     steps:
       - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
         with:
@@ -31,6 +31,9 @@ jobs:
             sudo apt-get update
             sudo apt-get install -y libboost-dev
           fi
+      - name: Install pinned symbolic endpoint-check dependencies
+        if: matrix.group == 'saturated-deferred'
+        run: python -m pip install sympy==1.14.0 mpmath==1.3.0
       - name: Check arithmetic, source hashes, finite identities, and patches
         run: make verify-${{ matrix.group }}
       - name: Require reproducible certificates and patches

Makefile

@@ -24,6 +24,14 @@ verify:
 	$(MAKE) verify-joint
 	$(MAKE) verify-pair
 	$(MAKE) verify-tests
+	$(MAKE) verify-rank-pair
+	$(MAKE) verify-ordered-frames
+	$(MAKE) verify-balanced-split
+	$(MAKE) verify-aligned-exchange
+	$(MAKE) verify-deferred-span
+	$(MAKE) verify-merged-span
+	$(MAKE) verify-saturated-deferred
+	$(MAKE) verify-stopped-pairtree
 
 verify-community: community-audit-check community-followup-check copied-reversed-producer copied-reversed-check copied-fixed-reversed-producer copied-fixed-reversed-check
 	$(MAKE) copied-fixed-verify
@@ -428,3 +436,57 @@ 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: verify-rank-pair rank-pair-verify rank-pair-refinement-verify
+verify-rank-pair: rank-pair-verify rank-pair-refinement-verify
+rank-pair-verify:
+	python3 research/rank-pair/frame_compile.py
+	python3 research/rank-pair/screen.py
+	python3 -m unittest discover -s tests -p 'test_rank_pair.py' -v
+
+rank-pair-refinement-verify:
+	python3 research/rank-pair-refinement/refine.py
+
+.PHONY: verify-ordered-frames ordered-frames-verify
+verify-ordered-frames: ordered-frames-verify
+ordered-frames-verify:
+	python3 research/ordered-frames/producer.py
+	python3 research/ordered-frames/graph_audit.py
+	python3 research/ordered-frames/check.py
+	python3 research/ordered-frames/independent_check.py
+
+.PHONY: verify-balanced-split balanced-split-frames-verify
+verify-balanced-split: balanced-split-frames-verify
+balanced-split-frames-verify:
+	python3 research/balanced-split-frames/producer.py
+	python3 research/balanced-split-frames/graph_audit.py
+	python3 research/balanced-split-frames/check.py
+	python3 research/balanced-split-frames/independent_check.py
+
+.PHONY: verify-aligned-exchange aligned-exchange-frames-verify
+verify-aligned-exchange: aligned-exchange-frames-verify
+aligned-exchange-frames-verify:
+	python3 research/aligned-exchange-frames/producer.py
+	python3 research/aligned-exchange-frames/graph_audit.py
+	python3 research/aligned-exchange-frames/check.py
+	python3 research/aligned-exchange-frames/independent_check.py
+
+.PHONY: verify-deferred-span deferred-span-frames-verify
+verify-deferred-span: deferred-span-frames-verify
+deferred-span-frames-verify:
+	python3 research/deferred-span-frames/verify.py
+
+.PHONY: verify-merged-span merged-span-frames-verify
+verify-merged-span: merged-span-frames-verify
+merged-span-frames-verify:
+	python3 research/merged-span-frames/verify.py
+
+.PHONY: verify-saturated-deferred saturated-deferred-verify
+verify-saturated-deferred: saturated-deferred-verify
+saturated-deferred-verify:
+	python3 research/saturated-deferred/verify.py
+
+.PHONY: verify-stopped-pairtree stopped-pairtree-verify
+verify-stopped-pairtree: stopped-pairtree-verify
+stopped-pairtree-verify:
+	python3 research/stopped-pairtree/verify.py

NOTICE

@@ -302,3 +302,107 @@ 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.
+
+Rank-first reclamation on interval-strip pair assembly, 8 October 2026
+Composition and exact profile witness: Dominik Scholz, with substantial OpenAI
+GPT-6 Astra / Codex assistance. The PR #60 descending-frame-rank compiler by
+Chafik Boukhalfa is reused unchanged on Avi Eisenberg's PR #62 scalar graph.
+The joint frame compiler and paid reclamation are eumemic's PR #57, developed
+with OpenAI Codex assistance. PR #62 records Anthropic Claude assistance.
+This composition improves actual fixed-basis block profiles with unchanged
+role counts; no new slot-priority rule or scalar graph is claimed. All prior
+contributors, licenses, compiler arguments and conditional analytic/tape
+interfaces remain credited. See research/rank-pair/PROOF.md.
+
+Ordered scalar frames, minimum-raise reclamation and coordinate relabeling
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+The new finite construction and exact arithmetic certificates are in
+research/ordered-frames, with an earlier parameter refinement in
+research/rank-pair-refinement. These build on Dominik Scholz's PR #63
+composition, Chafik Boukhalfa's PR #60 reclamation compiler, Avi Eisenberg's
+PR #62 scalar graph and eumemic's PR #57 reversible frame compiler.
+Concurrent PR #65/#67 work is acknowledged in the proof. All inherited
+licenses, author notices and conditional proof boundaries remain in force.
+
+Balanced split frames with core node ordering and coordinate relabeling
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+This composition combines eumemic's PR69 coarse column sums, Chafik Boukhalfa's
+PR71 anchored split/next-use-live compiler, and the earlier PR74 node ordering
+and coordinate relabeling. Rohan Garg's PR59 split operation, Avi Eisenberg's
+PR62 scalar graph, Rohan Arun's PR65/67 schedules and profile-cost selection,
+Dominik Scholz's PR63/68 composition and pending controls, and eumemic's PR57
+compiler remain credited. Original source closures and notices are preserved
+under references/frame-compiler/pr69 and pr71; all preceding licenses apply.
+See research/balanced-split-frames/PROOF.md for the finite conditional scope.
+
+Aligned multi-pair recursion and coordinated carrier exchanges
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+This finite construction splits and aligns six or nine top-level pairs,
+adds cardinality-preserving two-edge carrier cycles, and composes future-unit
+basis completion and coordinate-aware pricing with the inherited compiler.
+Chafik Boukhalfa's PR79 future completion, Rohan Arun's PR82 composition and
+geometric pricing, Alejandro Zarzuelo Urdiales's PR70 carrier exchanges,
+Dominik Scholz's PR76 profile pricing, eumemic's PR69 balanced sums, Rohan
+Garg's PR59 split operation, and the PR71/PR74 sources remain credited.
+All earlier source-specific licenses and notices apply. Selected exact PR82
+Git blobs supplement the preserved PR69/PR71 archives. See
+research/aligned-exchange-frames/PROOF.md for the finite conditional scope.
+
+The selected aligned-recursion update also preserves Chafik Boukhalfa's PR84
+indexed engine and its outgoing-use routing. Explicit coordinate pricing
+and the new deeper aligned/half-plane scalar graphs compose with that source;
+PR84's original source, license, notices and strategy audits are archived
+unchanged under references/frame-compiler/pr84.
+
+Deferred reconstruction from reserved retired spans
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+This storage extension reserves compatible retired carriers, clears a pending
+value using an explicit linear dependence, and reconstructs it before its next
+use. The selected witness includes 23 three-carrier reconstructions and pays
+all added operations and frame changes. It composes with the aligned scalar
+graphs, coordinate pricing and archived PR84 engine described above. All
+inherited contributors, licenses, notices and conditional proof boundaries
+remain credited. See research/deferred-span-frames/PROOF.md.
+
+Merged auxiliary endpoints with deferred span storage
+Composition prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+Merged auxiliary exterior profiles and the exact complementary-projector
+profiler are eumemic's PR96 contribution, prepared with Anthropic Claude
+assistance. Chafik Boukhalfa's PR91 supplies the smaller 23-dimensional axis;
+Rohan Arun's PR93 and Maxime Fleury's PR94 supply further coordinate choices.
+PR94 records Codebuff assistance. This composition adds an explicit
+multiplicative endpoint argument, fresh combined reproduction, independent
+audits, and the previously credited deferred storage extension. All original
+source notices, licenses, conditional hypotheses and contributor credits
+remain in force. See research/merged-span-frames/PROOF.md.
+
+Saturated deferred readouts with balanced transfer
+Integrated reproduction prepared for Thomas DiFiore with substantial OpenAI
+Codex assistance. Swapnil Jain supplies the round-seven deferred bit and
+round-six complex networks, lifted frames, late copies, V leaves and common
+flag/staircase basis, with the original Claude assistance disclosure.
+Zhihao Chen / jacklightChen supplies PR97's reflected ledgers, commuting fan
+order, nonzero endpoint gauges, signed complex correction and main-interface
+integration, with Codex assistance. djsmanchanda supplies PR99's nondegenerate
+readout saturation and selected changed frames, with Codex assistance.
+Rohan Arun supplies PR100's exact moment refinement and balanced transfer,
+with Codex assistance. Paureel, Avi Eisenberg, RaD, icekylinx, James Chang,
+Dominik Scholz, Douglas Colkitt, OpenAI and all earlier contributors retain
+the credits in the preserved notices. Original Git bytes and licenses are
+archived under references/signed-recursion. See
+research/saturated-deferred/PROOF.md for the finite conditional scope.
+
+Stopped pair-star trees and integer-weighted carrier exchanges
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+The new binary exclusion tree and deterministic cardinality-preserving
+integer-weighted exchanges refine the complete paid complex child moment.
+icekylinx supplies PR104's opposite-bank factorization, stopped atom interface,
+rational centers and retained ordinary leaf lineage. Rohan Arun supplies PR107
+and its comparison producer, PR100's balanced transfer and exact refinement
+precedents. Zhihao Chen, RaD / hipotures, James Chang, Aurel Prosz / Paureel,
+eumemic, Swapnil Jain, Dominik Scholz and all preceding contributors retain
+the credits in the original notices. Douglas Colkitt, OpenAI, David Harvey
+and Joris van der Hoeven retain their framework and analytic-work credits.
+The complete selected PR107 source closure is archived as original Git blobs
+under references/stopped-recursion/pr107, including original AI disclosures
+and licenses. See research/stopped-pairtree/PROOF.md for conditional scope.

README.md

@@ -1,3 +1,33 @@
+# Proposed stopped pair-tree refinement: conditional κ = 7.808981744031e-5
+
+This branch proposes **κ = 7808981744031/100000000000000000**, combining a
+new binary pair-star exclusion tree and deterministic integer-weighted carrier
+exchanges with icekylinx's PR104 stopped product-ring/rational-center interface
+and Rohan Arun's PR107 comparison and PR100 balanced transfer. It improves the
+pinned PR107 witness by approximately **0.13553%**. The complete paid recursive
+moment improves despite a larger auxiliary role count.
+
+[Construction and reproduction](research/stopped-pairtree/README.md) ·
+[Proof and credits](research/stopped-pairtree/PROOF.md) ·
+[Exact certificate](research/stopped-pairtree/certificate.json)
+
+Run `make stopped-pairtree-verify` with Python 3.11+ and a C++17 compiler.
+The gate freshly reproduces the complex producers and matching, checks every
+signed scalar coefficient, actual binary frame, carrier edge and literal role
+operation, and independently checks the recurrence, three row reserves,
+47 strict constraints and seven margins. The unchanged coarse bit producer
+and ordinary leaf are inherited, source-pinned inputs. Finite interface
+controls accompany the written stopped-recursion and balanced-layout review.
+The result retains all inherited and newly proposed analytic, streaming and
+all-size tape hypotheses; it is not a measured runtime speedup.
+
+Earlier constructions remain available, including
+[the saturated deferred composition](research/saturated-deferred/README.md),
+`research/merged-span-frames/` and `research/deferred-span-frames/`.
+The inherited reviewed result below is separate from this proposal.
+
+Prepared for Thomas DiFiore with substantial OpenAI Codex assistance.
+
 # A sharper exponent for integer multiplication
 
 **Community research maintained by Douglas Colkitt — conditional on the original

certificates/joint-dual-kappa.json

@@ -842,7 +842,7 @@
     "scripts/experiments/pin_joint_dual_sources.py": "4d31cc0e38b7902709ca5f8f714cba109760d2592346dd7f8dc0b900a2d9fe00",
     "scripts/experiments/verify_joint_dual.py": "7a31bba1e9c0ded32ae285df2ca9bc91ef20c330f6789fdba361876f3c7a1985"
   },
-  "source_manifest_sha256": "fd1aab7d75753b4b55b2377f26119dae74f367b0b4fab66cba42d944b3042cc9",
+  "source_manifest_sha256": "332e2f9a72f191b02edb10d33f58e2bd269f45fd28b3726c2489c641cd073d06",
   "sources": [
     {
       "files": {

certificates/joint-dual-validation.json

@@ -226,7 +226,7 @@
     }
   },
   "bit_saving": "2382370177/50000000000000",
-  "certificate_sha256": "2ed658f05365b28825e24a6e7bac9fc9cc015d5a5d63d9579f4ac6e0640ae559",
+  "certificate_sha256": "b2e0da4544dd256074c7151a8a3066cdf8e1ed5415069bdb252396b8bd5fe83c",
   "kappa": "4764513337/100000000000000",
   "margins": 7,
   "producer_words_regenerated_and_matched": true,