Add combined E8 witness at κ = 0.001307044366925301

.github/workflows/e8-combined.yml

@@ -0,0 +1,39 @@
+name: Verify combined E8 witness
+
+on:
+  push:
+    paths:
+      - 'research/e8-combined/**'
+      - '.github/workflows/e8-combined.yml'
+      - 'Makefile'
+  pull_request:
+    paths:
+      - 'research/e8-combined/**'
+      - '.github/workflows/e8-combined.yml'
+      - 'Makefile'
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  verify:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 20
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Install the exact geometry dependency
+        run: python3 -m pip install -r research/e8-combined/requirements.txt
+      - name: Rebuild and check both suppliers and their composition
+        run: make e8-combined-verify
+      - name: Require unchanged tracked files
+        run: git diff --exit-code

Makefile

@@ -21,6 +21,7 @@ verify:
 	$(MAKE) verify-recycled-bit
 	$(MAKE) verify-entrance-banks
 	$(MAKE) gen4-bank-verify
+	$(MAKE) e8-combined-verify
 	$(MAKE) verify-certificates
 	$(MAKE) verify-ternary
 	$(MAKE) verify-research
@@ -547,3 +548,7 @@ recycled-bit-full-verify:
 
 verify-recycled-bit: recycled-bit-full-verify
 	python3 -m unittest discover -s tests -p 'test_recycled_bit_integration.py' -v
+
+.PHONY: e8-combined-verify
+e8-combined-verify:
+	python3 -B research/e8-combined/verify.py --output "$$(mktemp -d)"

research/e8-combined/NOTICE

@@ -0,0 +1,89 @@
+Attribution and assistance
+
+Ali Romman prepared the two changed words, their composition, and this
+compact replay package with OpenAI Codex assistance. The fixed witness is
+not a claim of a global optimum. Source code is under Apache-2.0; see the
+repository LICENSE. The source manifests record original file hashes,
+extractions, and path changes. Generated output is not an input premise.
+
+The bit construction builds on DreamingOfClouds' R715 word and XOR hierarchy
+(PR463), rohanarun's response kernel and whole-residual pooling (PR467/462),
+DaysSky's idle hosting (PR442), and the E8 producer and geometry lineage of
+Dylan Crow, Jacob Sussman, Zhihao Chen, eumemic, and other contributors.
+The complex construction retains DreamingOfClouds' PR449 kernel scripts,
+chafreaky's PR458 paid adapters and distinct-host construction, icekylinx's
+PR433 split centres, and the prior supplier and geometry work.
+The comparison is with pinned main, not every open PR.
+
+Pinned source records
+
+  Main: 12a4934d15006e8c10f9b2264f13f8f341be802b.
+  PR467 cached head: f92d6a7ffcf74553522e967c2600e30c05ffe938.
+  PR462 cached tree: c4df4fdbf411f047ff7d63c73c432e828e94d924.
+  PR449: 7fac5292693bf738ff5eed6ec754f8ec22ebcf8e.
+  PR451: f3a38153678b55063fc005901cc968266166507d.
+  PR442: 4e16d001d505d4f21dd2bcfc44591f9be4c279bd.
+
+The vendor/bit/SOURCES.json and vendor/complex/SOURCES.json files identify
+all imported files and their changes. Paths in those manifests name the
+cached source packages. The two final words and compact schedule are
+included, so replay does not require those caches or pending PR branches.
+
+Retained upstream notices
+
+The notices below are copied without changes from the source packages.
+Their paths and descriptions refer to those packages.
+
+
+Source: research/e8-complex-adapted-kernel/NOTICE.md
+
+# Attribution and assistance
+
+The shared-donor complex kernel selection and three generalized local
+checker/emitter/phase scripts come from DreamingOfClouds PR449 at
+`7fac5292693bf738ff5eed6ec754f8ec22ebcf8e`, prepared with Claude Opus 5.5
+through Claude Code. The original PROOF and NOTICE are retained verbatim
+under `upstream449`. That draft's entrance-gauge construction and numerical
+claims are not adopted here.
+
+Exact frozen reconstruction, the paid two-sided residual construction,
+distinct-host assignment, independent local/global audits, new cost invoice
+and this release were prepared by chafreaky with OpenAI Codex assistance.
+They build on PR454 target mixing, PR451's eleven-stage complex geometry,
+PR443–445 kernels and row bounds, DaysSky's PR442 idle-data hosting,
+icekylinx's PR433 split centres, Zhihao Chen's PR425 common-frame realization
+and PR377/386 geometry, and the inherited E8 work by Jacob Sussman, Dylan
+Crow, eumemic and other contributors. All parent notices and licenses remain
+in their source-pinned packages.
+
+Apache License 2.0; see LICENSE. This is a conditional research artifact.
+
+
+Source: research/e8-complex-adapted-kernel/upstream449/NOTICE.md
+
+Apache-2.0. `code/audit_cxg.py`, `emit_cxg.py`, `phases_cxg.py` are modified copies of PR #433's `complex_audit.py`,
+`complex_emit.py`, `complex_phases_full.py` (icekylinx with Codex; adapted from PR #425 by jacklightChen / Zhihao Chen).
+`code/hostmodel.py` replicates the profile arithmetic of PR #442's `complex_hosted.py` (DaysSky) for mixed widths.
+Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+
+
+Source: work/next-track/bit/pinned467/research/r715-kernel-pool/NOTICE.md
+
+R715 producer and carrier/XOR construction: DreamingOfClouds, with Claude Opus 5.5, PR463. Shared-donor kernel mechanism and exact engines: chafreaky and the PR417/443 lineage. Whole hosting: DaysSky PR442. Target mixing: DreamingOfClouds PR450/463; no cross-family split used. New selection, 74 retiming moves and four-replica pooling: Rohan Arun with OpenAI Codex. Complex supplier: chafreaky PR458, unchanged and cited only at this stage. The inherited repository package notices and Apache-2.0 licenses remain applicable. See the retained PR463 README and source pins.
+
+
+Source: work/next-track/bit/pinned462/research/bit-whole-pool4/NOTICE.md
+
+Apache-2.0. Original upstream notices and licenses are retained.
+Four-replica selection, changed owner mapping, physical-family prefix audit,
+finite repricing and composition by Rohan Arun with OpenAI Codex assistance.
+
+Unchanged bit word: Rohan Arun with Claude Opus5.5, PR459, including DreamingOfClouds'
+target-prefix mixing from PR450. Unchanged complex supplier: chafreaky with
+OpenAI Codex, PR458, building on DreamingOfClouds' PR449 kernel.
+Parent composition: Rohan Arun with OpenAI Codex, PR460.
+Hosting and solver lineage: DaysSky PR442; chafreaky PR443/445/451/454;
+DreamingOfClouds' whole-hosting solver; Rohan Arun PR456 whole-replica pooling.
+Retained geometry and local tools: icekylinx, eumemic, Zhihao Chen, Dylan Crow,
+Jacob Sussman, Henry Grant and all authors in the upstream source-pinned notices.
+No independent discovery of either unchanged local supplier is claimed.

research/e8-combined/PROOF.txt

@@ -0,0 +1,184 @@
+Conditional composition of two E8 suppliers
+
+Claim. In the interfaces stated in the conditions below, this construction gives
+T(n) = O(n (log n)^(1-kappa)), where
+
+  kappa = 1307044366925301 / 10^18 = 0.001307044366925301.
+
+This is a conditional exponent bound. It is 80.6378781278% above the selected
+bound on main at 12a4934d15006e8c10f9b2264f13f8f341be802b. It is not a
+runtime measurement or a proof of the full multiplication theorem.
+
+1. Local identities
+
+The bit word uses 120 source and 120 target streams and 705 dirty helper
+streams. Its 111 response-kernel pivots, 45 target receivers and 72 frame
+changes are fixed data. Both directions act correctly on all 945 independent
+input columns. The emitted 19,550 records include nine COPY/ERASE windows.
+Each common-frame ADD is the scalar XOR on the normalized streams. Each MOVE
+is the exact frame change. COPY/ERASE uses a fresh temporary and an unchanged
+source. Thus the common-frame induction gives the full local address operator
+on arbitrary streams. The helper endpoint is its charged residual projector,
+not an identity inferred from scalar restoration.
+
+The complex word uses 240 data and 560 dirty helper streams. Retiming changes
+728 frame labels and preserves every scalar operation and both endpoints.
+The same common-frame induction preserves the local physical operator.
+Both scalar directions check all 800 input columns. Exact Clifford checks
+include phase. For a helper entrance E, its residual is T_FULL T_E^-1.
+The paid pre/post monomials convert this residual into the exact coordinate
+Fourier factor of width 8 or 9. No isotropic entrance is treated as a
+coordinate entrance by an orthogonal conjugation.
+
+2. Complete finite templates
+
+Each supplier uses the signed eleven-stage layout. The bit template has four
+replicas, 4,800 live families and 4,809 physical families. Each of the 31,020
+whole helper occurrences has an exact owner route. Concrete routes are
+injective within a call, even when several helpers use one family. The routes
+use at most 106 factors; the invoice allows 144. All 8,955 unbanked scalar
+columns and all 1,320 data routes pass. The complex template has eight
+replicas, 9,600 live families and 10,178 physical families. Its 49,280 whole
+helper occurrences use distinct families within each call. All 8,520 phase
+equalities and 1,320 data routes pass.
+
+Each host lies in a checked idle window. Whole helper residuals occupy
+disjoint pending coordinates. The complementary idle child completes the
+original idle operator. No residual is split across physical families.
+Bridges, terminal shifts, exchanges and the complex cover correction remain
+in the emitted program and paid profile.
+
+3. Finite costs and moments
+
+The independent invoice uses the emitted counts, actual physical stock and
+fresh prefix bounds. The bit prefix has 610 bits and is charged at 640 bits.
+Its three-level coefficient has 1,883 bits and is charged at 1,920 bits.
+Its row slope is 19,428. The ceiling term adds 9,267. At log2(e) >= 32,
+the 20,161 reserve covers both terms, with margin 14,189 at the threshold. The complex invoice charges 788,480 restored
+rank-zero wrapper groups. Its router, literal and precision charges fit the
+retained bounds. Since 89^9 > 16*64^9, nine recursive steps give four binary
+halvings. The row cost is at most (9/4*3929+9909+252)*log2(e)+9*3929. For
+log2(e) >= 32 it is strictly below 20,161*log2(e), with margin 1,751 at the
+threshold. Smaller widths use the retained ordinary leaf interface.
+
+For profile H, dimension m and live stock W, the moment is
+
+  sum_r H_r*(r/m)^(1-z)/W
+    + 10^-16 * 32*m^2*sum_r H_r * m^(z-1)/W.
+
+The second term charges the retained bad-class fallback. Rational intervals
+give the strict lower endpoints
+
+  bit c = 1308789373120170692069 / 10^24,
+  complex b = 654377483866781393011 / (5*10^23).
+
+The moment is below 1 at each lower endpoint and above 1 one 10^-24 step
+later. The complex supplier is the smaller endpoint.
+
+4. Outer composition
+
+Set eta = beta = 10^-18 and s = min(c,(1-beta)*b)-10^-24. Start at
+a_0 = 384599/10^10 and apply the completed-ordinary-leaf bound six times:
+a_j = s-s^j*(s-a_0). Each step has a strictly positive gap. The cutoff constants are chosen
+sequentially, as stated below. Put a=a_6, q=a*(1-2*eta) and epsilon=(1-eta)/(1+q).
+The seven cost margins have minimum mu=epsilon*q. All 47 strict outer
+constraints hold. The largest 10^-18 grid point strictly below mu is the
+claimed kappa. The adjacent grid point fails. Every calculation uses exact
+rational arithmetic; finite certificates do not discharge the conditions below.
+
+
+Proof boundary and source map
+
+This submission is a finite E8 construction and an exact arithmetic
+composition. It does not complete the all-size multiplication proof.
+Its bound is conditional on the interfaces below. The finite checks do
+not establish these interfaces for every input size.
+
+Conditions already stated on main
+
+Main at 12a4934d15006e8c10f9b2264f13f8f341be802b states conditions for:
+the all-size weighted compiler; common-ancestor and weighted charts;
+complete-stream movement; restored rows and selectors; exact-tape routing;
+prime supply; precision and exact recovery; complex symbolic correctness;
+and the analytic reduction from transforms to multiplication. The source
+proof's normalization by live stock W, rather than physical stock, is part
+of this contract. The complete finite router uses physical stock.
+
+The initial saving a_0=384599/10^10 and the completed-ordinary-leaf bootstrap
+theorem are inherited. Their application uses the fresh finite coefficients.
+The checks establish the six positive gaps. They do not prove the initial
+theorem or the uniform recursion schedule. The supplier coefficient alone
+does not bound the constants of each earlier ordinary algorithm.
+
+Conditions from pending E8 work
+
+These are extensions of the main framework. They must not be described as
+unchanged hypotheses already reviewed on main.
+
+1. The E8 bit compiler must realize the projector label on mutable source,
+helper and target streams, including target-to-target additions, at all
+address widths. Common-frame induction proves the finite operator identity.
+It does not prove the compiler cost, fine grid or uniform exact-tape lowering.
+The relevant pending lineage includes PR370, PR386, PR433 and PR450/454.
+
+2. The whole-call hosting interface must allow arbitrary correlated dirty
+streams on idle data families. The finite argument checks distinct owners,
+unused windows, whole residuals, disjoint coordinates and exact complements.
+It does not enumerate the complete cover or establish the all-size compressed
+window, row restoration and selector implementation. PR442 introduces this
+extension; PR456/462 applies whole-residual pooling. No PR448 split-residual
+compiler premise is used.
+
+3. The complex Gaussian compiler must implement canonical Clifford frame
+changes and the paid two-sided monomial adapters on arbitrary streams, with
+restored work and the stated fixed-group costs at all sizes. Exact tableau
+and state checks establish the local phases and operator factors. They do
+not establish the full all-size compiler or analytic precision routine.
+The relevant lineage includes PR433, PR449/458 and their source notices.
+
+4. The bad-prime and bad-class envelope must apply to these new E8 frames,
+covers and compiler operations. The moment retains the same 10^-16 envelope
+and 32*m^2 rank-one fallback calls per paid child. This is an admitted
+framework condition, not a consequence of the moment's rational signs.
+
+5. The restored-row interface must charge stock once per actual recursion
+level and use the retained small ordinary leaves. The new complex nine-step
+row estimate proves that the numerical 20,161 reserve suffices under that
+contract. It does not construct a new tape machine or row divisor.
+
+
+Sequential ordinary bootstrap
+
+Let A_0 be a finite constant for the admitted initial ordinary algorithm with
+saving a_0. At level j, the earlier algorithm and its constant A_(j-1) have
+already been fixed. The admitted ordinary-wrapper theorem supplies the new
+construction and its finite coefficients. Choose an integer C_j >= 1 which
+bounds all its primitive, recurrence, wrapper, row, setup and padding
+constants, its large-width constant, and the earlier ordinary constant. Such
+a C_j exists under the admitted interfaces. It need not be at most 2^1920.
+
+The saving is a_j = s*(1-s+a_(j-1)). Let delta_j be the minimum of its four
+exact positive exponent gaps. Set B_j = ceil(log2(C_j)) and choose
+
+  L_j = ceil(max(1, 36/delta_j^2, 2*(4+B_j)/delta_j)),
+  e_j = 2^L_j.
+
+For t = log2(e) >= L_j, use
+
+  log2(1+t) <= 3*sqrt(t) <= delta_j*t/2,
+  log2(C_j)+4 <= delta_j*t/2.
+
+Then C_j*(1+log2(e))*e^(-delta_j) <= 1/16. This pays the lower
+order terms and makes the high borrowed row fit. For e < e_j, use the earlier
+ordinary algorithm. A finite constant which covers both ranges is
+
+  A_j = C_j + A_(j-1)*2^ceil(L_j*(a_j-a_(j-1))).
+
+This choice is sequential. C_j uses A_(j-1), while A_j is assigned after L_j.
+No constant depends on itself. Six steps give the stated exponent a_6, with
+finite constants and cutoffs whose numerical values are not certified by this
+package. The exact saving exponents, their positive gaps, and the formula for
+sufficient cutoffs are certified. The source's ordinary algorithm and its
+uniform scheduling remain admitted interfaces.
+
+The 1,920-bit supplier charge is separate from the stage constants C_j.

research/e8-combined/README.md

@@ -0,0 +1,37 @@
+# Combined E8 witness
+
+This package checks the conditional bound
+
+\[
+\kappa=1307044366925301/10^{18}=0.001307044366925301.
+\]
+
+It is 80.637878% above the selected value on main at
+`12a4934d15006e8c10f9b2264f13f8f341be802b`. This compares exponent
+savings. It does not measure runtime. The complex supplier limits the bound.
+
+The package combines an R705 bit word with a retimed R560 complex word.
+It contains the fixed positive words, owner choices, finite schedule, and
+the source code needed to check them. It regenerates records, geometry,
+phases, templates, costs, and all 47 strict outer inequalities. It does
+not contain searches or the nested historical source packages.
+
+Use Python 3.11 or later. The replay was checked with Python 3.14 and
+SymPy 1.14.0. Install the one external dependency, then run:
+
+```sh
+python3 -m pip install -r research/e8-combined/requirements.txt
+python3 -B research/e8-combined/verify.py --output "$(mktemp -d)"
+```
+
+The command uses an empty output directory outside the source package. It checks the committed
+`certificate.json` against a fresh construction. The output includes
+the complete inequalities and component audit records. No network access,
+C++ compiler, old checkout, or saved build directory is needed.
+
+Read [PROOF.txt](PROOF.txt) for the dependency argument and assumptions.
+The finite checks do not prove all-size compilation, complete-cover
+execution, bad-class portability, or the analytic multiplication theorem.
+The pending E8 mutable-target, idle-hosting, and Gaussian-adapter interfaces
+extend the conditions stated on main. This package does not select a new
+main result. Source attribution and AI assistance are in [NOTICE](NOTICE).