Mister-Meeseeks · Oct 13, 2026 · Oct 8, 2026 · Oct 8, 2026 · Oct 8, 2026
diff --git a/Makefile b/Makefile
@@ -1,6 +1,8 @@
-.PHONY: verify note audit-note tuned-note reuse-note incidence-note dag-note shared-point-note paired-note compact-note fetch
+.PHONY: verify note audit-note tuned-note reuse-note incidence-note dag-note shared-point-note paired-note compact-note retained-complex-note fetch
 
 verify:
+	python3 scripts/retained_complex.py
+	python3 scripts/make_retained_complex_patch.py
 	python3 scripts/certify.py
 	python3 scripts/search_network.py
 	python3 scripts/make_patch.py
@@ -54,6 +56,7 @@ verify:
 	git apply --check --directory=upstream patches/h46-shared-point.patch
 	git apply --check --directory=upstream patches/h50-paired-59.patch
 	git apply --check --directory=upstream patches/compact-control-34.patch
+	git apply --check --directory=upstream patches/retained-complex-31.patch
 
 note:
 	mkdir -p artifacts
@@ -93,3 +96,7 @@ compact-note:
 
 fetch:
 	python3 scripts/fetch_upstream.py
+
+retained-complex-note:
+	mkdir -p artifacts
+	tectonic --outdir artifacts notes/retained-complex-note.tex
diff --git a/NOTICE b/NOTICE
@@ -14,3 +14,10 @@ as separate files under patches/; upstream source files remain unchanged.
 The new research note, scripts, certificates, patches, and documentation
 were prepared by Douglas Colkitt with assistance from OpenAI Codex.
 This project is not an official OpenAI release or an OpenAI endorsement.
+
+scripts/complex_circuit.py is imported unchanged from eumemic's
+Claude-assisted PR #3 to this repository:
+https://github.com/CrocSwap/integer-mult-bounds/pull/3
+Commit: dfe5b818aad4d386cb5dd7d76df108088107765d
+The retained exclusion-total and stage-sharing composition is by dleen,
+prepared with substantial OpenAI Codex assistance.
diff --git a/README.md b/README.md
@@ -10,56 +10,48 @@ fixed number of one-dimensional tapes, the strongest supplied witness is
 
 $$
 T(n)=O\!\left(n(\log n)^{1-\kappa}\right),\qquad
-\boxed{\kappa=\frac{83}{10^{12}}=8.3\times10^{-11}>2^{-34}}.
+\boxed{\kappa=\frac{591}{10^{12}}=5.91\times10^{-10}>2^{-31}}.
 $$
 
-The simpler **`kappa = 2^-34`** is a corollary. The witness remains below
-`2^-33`. It increases the exponent saving by approximately **47.85 million
-fold** over our preceding published `2^-59` witness. The original manuscript
-uses `2^-182`. These compare asymptotic exponents, not practical runtimes.
+The simpler **`kappa = 2^-31`** is a corollary. The exact saving is
+**591/83 ≈ 7.12 times** the preceding compact-control witness `83/10^12`.
+These compare asymptotic exponents, not practical runtimes.
 
-**[Read the compact-control proof note (PDF)](artifacts/compact-control-note.pdf)** ·
-[Review the combined source patch](patches/compact-control-34.patch) ·
-[Inspect the exact certificate](certificates/compact-control-layer.json) ·
-[Review guide and dependencies](docs/research/compact-control-review.md)
+**[Read the combined construction](docs/research/shared-retained-complex.md)** ·
+[Review the source patch](patches/retained-complex-31.patch) ·
+[Inspect the exact certificate](certificates/retained-complex-layer.json)
 
-This is a research claim supported by written proofs and reproducible checks.
-The complete upstream theorem is assumed; the new arguments have not received
-independent mathematical review or formal verification.
+The shared-exclusion builder is imported verbatim from [PR #3](https://github.com/CrocSwap/integer-mult-bounds/pull/3)
+by `eumemic` (Claude-assisted), at commit `dfe5b818aad4d386cb5dd7d76df108088107765d`.
+The retained-total and stage-sharing extensions are by `dleen` with substantial
+OpenAI Codex assistance.
 
 ## What changed
 
-The new construction moves **compact control fields instead of entire spaced
-windows**. For `f` selected axes, it replaces the layer's movement cost
-`O(V*((f*K)^tau+1))` by
+The existing compact-control movement and paired-bit network are retained.
+The `h=24` complex producer combines PR #3's shared-exclusion DAG with
+retained one-point exclusion totals and stage-1/3 bank sharing. Grouped scatter
+at a common low/full frame gives return loss `(h-1)²+h`, restores arbitrary
+initial scratch, and preserves both phase paths and all-role endpoints.
+The larger global coefficient is split into half-sized updates, with every
+scalar gate included in the precision guard. The bit and complex arities
+remain independent.
 
-$$
-O\!\left(V\bigl((f\log p)^\tau+1\bigr)\right).
-$$
-
-The proof reserves temporary fields from existing address coordinates,
-allows arbitrary initial temporary values, restores them exactly, and charges
-exceptional-address repair at every recursion node. The temporary ranges
-remain complete through padding and recursive row splitting.
-
-Removing `K^tau` removes the restriction responsible for the preceding
-quadratic dependence on the finite-network saving. The bit network stays at
-`h=50`. The original complex network is separately instantiated at `h=25`,
-and a generalized stopping-depth guard completes the new parameter witness.
-This is a change to the movement construction and its proof, beyond parameter
-tuning of the preceding algorithm.
-
-The exact minimum assembly margin is
+The exact complex saving is `1-sigma=2970/10^11=2.97e-8`, about **2.12 times**
+PR #3's certified `1.4e-8`. The bit saving remains `296/10^11`, so the combined
+headline stays `591/10^12 > 2^-31`. The complete compact consumer
+uses `c=1`, with exact minimum assembly margin
 
 $$
-G_* = \frac{333833}{4\cdot10^{15}}
-    = 8.345825\times10^{-11} > \kappa.
+G_* = \frac{2956521}{5\cdot10^{15}} > \kappa,
+\qquad G_*-\kappa=\frac{1521}{5\cdot10^{15}}>0.
 $$
 
-The remaining bottleneck is the complex layer's saving. With the **fixed
-`h=25` complex motif and retained Gaussian/leaf inequalities**, the scoped
-ceiling is below `8.369598075e-11`, hence below `2^-33`. This is not a ceiling
-for other networks or integer multiplication in general.
+The complete upstream theorem, compact-control tape proofs and analytic
+interfaces remain assumptions. Exact producer and parameter checks support
+the supplied written argument; they do not formally verify the full machine.
+The [preceding compact-control witness](artifacts/compact-control-note.pdf)
+and its [review guide](docs/research/compact-control-review.md) remain available.
 
 ## Evidence and scope
 
@@ -69,7 +61,7 @@ for other networks or integer multiplication in general.
 | Dirty-control identities, inverses and repair | Finite exhaustive cases and seeded tests |
 | Wider-control tape bound, reservations and recursion | Written general proofs |
 | Separate complex arity and precision guard | Written proofs and exact accounting |
-| Source integration | Combined patch, reference checks and manuscript build |
+| Source integration | Combined patch, reference checks and patch applicability |
 | Full upstream multiplication theorem | Assumed |
 | Independent review / full formalization | Not supplied |
 
@@ -92,13 +84,14 @@ They regenerate the certificates and patches, run the tests, verify upstream
 hashes, and check each patch against the pinned manuscript. The second command
 checks exact regeneration on a clean checkout.
 
-With Tectonic installed, rebuild the latest note using:
+The preceding rectangle variant has a preserved PDF. With Tectonic installed,
+rebuild that earlier note using:
 
 ```sh
-make compact-note
+make retained-complex-note
 ```
 
-The output is `artifacts/compact-control-note.pdf`. The first PDF build may
+The output is `artifacts/retained-complex-note.pdf`. The first PDF build may
 download TeX resources. See [reproducibility instructions](docs/reproducibility.md)
 for applying the combined patch in a disposable copy and building older notes.
 [GitHub Actions](.github/workflows/verify.yml) runs the arithmetic and patch checks.

diff --git a/artifacts/retained-complex-note.pdf b/artifacts/retained-complex-note.pdf
diff --git a/certificates/retained-complex-layer.json b/certificates/retained-complex-layer.json
@@ -0,0 +1,138 @@
+{
+  "frames": {
+    "both_physical_frame_directions": true,
+    "pivot_and_fanout_roles_distinct": true,
+    "roles": 90950
+  },
+  "imported_builder": {
+    "author": "eumemic",
+    "commit": "dfe5b818aad4d386cb5dd7d76df108088107765d",
+    "pr": "https://github.com/CrocSwap/integer-mult-bounds/pull/3"
+  },
+  "kappa": "591/1000000000000",
+  "main": {
+    "absorption_gap": "1521/5000000000000000",
+    "constraint_slacks": {
+      "K_dominates_log_p": "1999/10000",
+      "K_smaller_than_ell": "3001/5000",
+      "alpha_below_sqrt_p": "8001/40000",
+      "beta_below_one": "999/1000",
+      "beta_positive": "1/1000",
+      "c_positive": "1",
+      "crt_layout": "296037/125000000000000",
+      "delta_below_one_eighth": "124999/1000000",
+      "delta_positive": "1/1000000",
+      "dimension_upper_bound": "4003/30000",
+      "epsilon_positive": "1999/10000",
+      "gamma_sublinear": "4003/20000",
+      "gaussian_cost": "31/250000",
+      "guard_width": "127961/100000000",
+      "kappa_positive": "591/1000000000000",
+      "lambda_above_sigma": "26741/1000000000000",
+      "lambda_above_tau": "1/1000000000000",
+      "lambda_below_one": "2959/1000000000000",
+      "lambda_prime_above_lambda": "1/1000000000000",
+      "lambda_prime_below_one": "1479/500000000000",
+      "leaf_cost": "267123/10000000000000",
+      "packed_overhead": "1/1000000000000",
+      "prefix_cost": "3001/5000",
+      "prime_interval_growth": "3001/5000",
+      "r_superpolynomial": "8001/10000",
+      "reserved_axes": "499999998521/500000000000",
+      "scalar_cost": "800099/1000000",
+      "sigma_below_one": "297/10000000000",
+      "sigma_positive": "9999999703/10000000000",
+      "tau_below_one": "37/12500000000",
+      "tau_positive": "12499999963/12500000000"
+    },
+    "counts": {
+      "D": 2990500480,
+      "L": 6796219584,
+      "N": 8291469824,
+      "W": 761750114048,
+      "additions": 66638,
+      "base_additions": 66518,
+      "eta": "365/1285272576",
+      "h": 24,
+      "m": 13824,
+      "new_ancestors": 120,
+      "retained_totals": 24,
+      "roles": 90950,
+      "s": 10530430586099072,
+      "side_outputs": 24288,
+      "v": 2024
+    },
+    "deficit_slack": "406952569/627574500000000000",
+    "exponents": {
+      "internal": "12499999963/12500000000",
+      "layer": "12499999963/12500000000",
+      "leaf": "9999999703297/10000000000000",
+      "preprocessing": "0"
+    },
+    "guard": {
+      "B": 28288999069398280989732240071385306560,
+      "C0": 1991520678995416584576942400432086873082564105960158546941478349741058754716840755200,
+      "C1": "49961/10000",
+      "E": 28288999069398280989721709640799207488,
+      "scalar_gates": 3475338442752,
+      "zeta": "1/10000"
+    },
+    "log_interval": [
+      "372392895082043730940457417056458792833039283878610311327817326284330812291611608520371253254839720156679896368576/39058798766085586528329820824800339900814251284792067297869901316943575749030397635346352668587566697127586464525",
+      "1324063626958377710010515607074581250389044502746244297906400428566771042174638328403966817444807792795396071539823/138875728946082085434061585154845652980672893457038461503537426904688269329885858259009253932755792700898085207200"
+    ],
+    "log_upper": "477/50",
+    "margins": {
+      "g1": "3001/5000",
+      "g2": "73963/125000000000000",
+      "g3": "2956521/5000000000000000",
+      "g4": "296037/125000000000000",
+      "g5": "31/250000",
+      "g6": "800099/1000000",
+      "g7": "1999/10000"
+    },
+    "minimum_margin": "2956521/5000000000000000",
+    "parameters": {
+      "C1": "49961/10000",
+      "beta": "1/1000",
+      "c": "1",
+      "delta": "1/1000000",
+      "epsilon": "1999/10000",
+      "kappa": "591/1000000000000",
+      "lam": "999999997041/1000000000000",
+      "lamp": "499999998521/500000000000",
+      "sigma": "9999999703/10000000000",
+      "tau": "12499999963/12500000000"
+    }
+  },
+  "producer": {
+    "activation_and_retirement_checked": true,
+    "additions": 66638,
+    "all_residuals_nonalt": true,
+    "circuit_sha256": "626a119365a66a9a04cf252be7a734ea3631fc8e035b0a9d5b9eecfd81890db9",
+    "exact_total_multiplicities": true,
+    "full_coefficient_map_exact": true,
+    "h": 24,
+    "inputs": 2024,
+    "output_residual_checks": 24288,
+    "partial_outputs": 24288,
+    "retained_totals": 24,
+    "roles": 90950,
+    "two_output_uses": 12144,
+    "zero_output_uses": 12144
+  },
+  "proof_sha256": {
+    "docs/research/shared-retained-complex.md": "4ac503e20d5da12c8fd0ea3c6c4a398ac690a7e3d1d05ce37cbe2b88a882da91",
+    "notes/compact-control-guard.tex": "38eac1e967ca30dd7df111943e41e8fbb93cafbe0356e8ebc8cb2db78089067f",
+    "notes/compact-control-layout.tex": "2bc28e50f3eb092195294cafa81213e520f6d42e7935e9905ad8a6199e2a7c87",
+    "notes/compact-control-movement.tex": "b963c3d2f4e1b7059aa289bb4b4ac89b688920bcf42f5b20d7698cdef0132cc6",
+    "notes/retained-complex-construction.tex": "c69be034d19f85e25ee14438d26bbcf0fb4d24be780e7449dcacc6df77d6e0c2",
+    "scripts/complex_circuit.py": "d623617f7c5cb3189580883a8ab30c0bd3832688ae3f1f0c1f881bbbbff8e0c1",
+    "scripts/make_retained_complex_patch.py": "0bbc009e31245461ace9d86b06d2cc3226fd7a63db0e0e152b53555d5f50eb00",
+    "scripts/retained_complex.py": "d360b5b1ecd2be98d19156961849a69f66f1a982b891e5b05f5d9c94e910856e"
+  },
+  "ratio_over_compact_witness": "591/83",
+  "scope": "Full h24 scalar coefficients, total multiplicities and compiled frame inclusions; small dirty-scratch and projector/phase controls in tests. The general tensor phase, fixed-tape, compact-control and analytic transfer use the written proof and retained upstream interfaces. This is not a runtime measurement.",
+  "status": "CONDITIONAL RETAINED-TOTAL COMPLEX WITNESS; NOT FULL FORMAL VERIFICATION",
+  "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a"
+}
diff --git a/docs/reproducibility.md b/docs/reproducibility.md
@@ -176,3 +176,18 @@ rebuilt locally using the command above.
 
 Build the latest note with `make compact-note`. The earlier notes and
 patches remain available as independent witnesses.
+
+## Retained-total complex witness
+
+`make verify` also regenerates `certificates/retained-complex-layer.json` and
+`patches/retained-complex-31.patch` using `scripts/retained_complex.py` and
+`scripts/make_retained_complex_patch.py`. Full h24 coefficients, integer total
+multiplicities and both compiled frame directions are checked; h8 tests cover
+arbitrary dirty scratch and every projector/phase edge. The combined tensor and
+consumer transfer is in `docs/research/shared-retained-complex.md`, with a
+separate proposition in the generated patch. The shared-exclusion builder is
+imported verbatim from PR #3 at `dfe5b818aad4d386cb5dd7d76df108088107765d`.
+Tests include the normalized half-sized scatter passes and their phase edges.
+The preceding rectangle variant remains in `notes/retained-complex-construction.tex`;
+build that historical PDF with `make retained-complex-note`. The bound `591/10^12 > 2^-31` remains
+conditional on the existing fixed-tape and analytic interfaces.
diff --git a/docs/research/current-status.md b/docs/research/current-status.md
@@ -4,12 +4,26 @@ Updated October 7, 2026. Author: Douglas Colkitt. All results remain conditional
 on the pinned upstream algorithmic interfaces and the identified written
 extensions. Nothing in this page asserts formal or independent verification.
 
+## Retained-total continuation
+
+The latest conditional witness is `591/10^12 > 2^-31`, a factor `591/83`
+over the compact-control witness below. See the [combined proof](shared-retained-complex.md),
+[certificate](../../certificates/retained-complex-layer.json) and
+[patch](../../patches/retained-complex-31.patch). The fixed-tape and analytic
+interfaces remain assumptions. The combined producer imports PR #3
+(commit `dfe5b818aad4d386cb5dd7d76df108088107765d`) and adds retained exclusion
+totals and stage-1/3 sharing. Its complex saving is `2970/10^11`, with
+`W=761750114048`, `s=10530430586099072`, and the unchanged bit saving
+`296/10^11`. The headline remains bit-limited. The rectangle proof/PDF is a
+preserved preceding variant. The following record preserves the earlier
+compact-control construction and its parameters.
+
 ## Current and earlier results
 
 | State | Exponent saving kappa | Artifacts |
 | --- | --- | --- |
 | Published baseline | `2^-59` | [paired note](../../artifacts/paired-note.pdf), [certificate](../../certificates/paired-network.json), [patch](../../patches/h50-paired-59.patch) |
-| Current conditional research draft | `83/10^12 = 8.3e-11 > 2^-34` | [proof note](../../artifacts/compact-control-note.pdf), [source](../../notes/compact-control-note.tex), [certificate](../../certificates/compact-control-layer.json), [patch](../../patches/compact-control-34.patch) |
+| Preceding compact-control witness | `83/10^12 = 8.3e-11 > 2^-34` | [proof note](../../artifacts/compact-control-note.pdf), [source](../../notes/compact-control-note.tex), [certificate](../../certificates/compact-control-layer.json), [patch](../../patches/compact-control-34.patch) |
 
 The new witness increases kappa by approximately 47,846,242 times over the
 published `2^-59`. This compares asymptotic exponent savings, not practical