diff --git a/.github/workflows/r480-sliced-minimal-frames.yml b/.github/workflows/r480-sliced-minimal-frames.yml
new file mode 100644
index 0000000000..be1d761b78
--- /dev/null
+++ b/.github/workflows/r480-sliced-minimal-frames.yml
@@ -0,0 +1,37 @@
+name: Verify sliced R480 trace composition
+on:
+  pull_request:
+    paths:
+      - 'research/r480-sliced-minimal-frames/**'
+      - '.github/workflows/r480-sliced-minimal-frames.yml'
+  workflow_dispatch:
+permissions:
+  contents: read
+jobs:
+  verify:
+    runs-on: ubuntu-latest
+    strategy:
+      fail-fast: false
+      matrix:
+        python: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@v4
+      - uses: actions/setup-python@v5
+        with:
+          python-version: ${{ matrix.python }}
+      - name: Install verification dependencies
+        run: |
+          sudo apt-get update
+          sudo apt-get install -y libboost-dev
+          python -m pip install numpy sympy
+      - name: Reconstruct and verify the complete finite supplier
+        env:
+          PYTHONDONTWRITEBYTECODE: '1'
+          PYTHONINTMAXSTRDIGITS: '0'
+        run: python -B research/r480-sliced-minimal-frames/verify.py "${RUNNER_TEMP}/r480-trace-replay" --physical --boost-include /usr/include
+      - name: Retain verification receipts and failures
+        if: always()
+        uses: actions/upload-artifact@v4
+        with:
+          name: r480-trace-${{ matrix.python }}
+          path: ${{ runner.temp }}/r480-trace-replay
diff --git a/research/r480-sliced-minimal-frames/EXPECTED.json b/research/r480-sliced-minimal-frames/EXPECTED.json
new file mode 100644
index 0000000000..23a6a911d4
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/EXPECTED.json
@@ -0,0 +1,104 @@
+{
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+    "Gaussian_arithmetic_above_kappa": "2217766005011693216790461510969785118501230001083330536533627028099629081022952731564215504548439732529714235657214884426813981644619821104099897144108387341068072182353004476322495134852678922705885286169730590733818506095848640326535974198291536950528057739276509128971418863587790941209426454249472850544139323/4006359966134424176576765834417756161531372168995266186008118384995063847314125776082728282256272986280186411224216741432269720098573946833953810396221316143405866953960866723351987788493784369322475620939746424811244679741845138644196805954604996850730264438640156042820631595340408913602419582210867008479423112000000000000000000",
+    "scalar_work_above_kappa": "4220945988078905305078844428178663199266916085580963629537686220597161004680015619605579645676576225669807441269323255142948841693906794521076802342219045412771005659333437837998489029099571107367123096639603803139440845966771209648634377175594035375893189958596587150381734661257995398010636245354906354783850879/4006359966134424176576765834417756161531372168995266186008118384995063847314125776082728282256272986280186411224216741432269720098573946833953810396221316143405866953960866723351987788493784369322475620939746424811244679741845138644196805954604996850730264438640156042820631595340408913602419582210867008479423112000000000000000000",
+    "dimension_above_kappa": "249602502116598488759060946839247010351222999280351033340638701356894253467160952664925981458625522208483391858399600470160063910861025061751796644375451358244697605486615200459046952505297613068438242607596733045074241844701685474790601561243553445526794592579968996213365088630387358883243376779042468983219723820704045677736173/250397497883401511036047864651109760095710760562204136625507399062191490457132861005170517641017061642511650701513546339516857506160871677122113149763832258962866684622554170209499236780861523082654726308734151550702792483865321165262300372162812303170641527415009752676289474708775557100151223888179188029963944500000000000000000"
+  },
+  "scope": "Derived global histogram adds288 copies of exact local profile delta to pinned PR510 profile. Same480 width8 helpers and original owner schedule proposed. Physical template, chronological finite invoice and changed-frame geometric checks remain required; old outer finite constants used prospectively."
+}
diff --git a/research/r480-sliced-minimal-frames/LICENSE b/research/r480-sliced-minimal-frames/LICENSE
new file mode 100644
index 0000000000..261eeb9e9f
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/LICENSE
@@ -0,0 +1,201 @@
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diff --git a/research/r480-sliced-minimal-frames/MANIFEST.json b/research/r480-sliced-minimal-frames/MANIFEST.json
new file mode 100644
index 0000000000..822d3ced65
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/MANIFEST.json
@@ -0,0 +1,58 @@
+{
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diff --git a/research/r480-sliced-minimal-frames/NOTICE.md b/research/r480-sliced-minimal-frames/NOTICE.md
new file mode 100644
index 0000000000..3f0b5afcd3
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/NOTICE.md
@@ -0,0 +1,110 @@
+Prepared for rohanarun with OpenAI Codex assistance; Apache-2.0. New work: exact observable slicing and causal minimum-frame assignment on the R480 word, followed by prospective composition. No authorship of inherited circuitry or checkers is claimed.
+
+# Notice
+
+Apache-2.0 (`LICENSE`). Prepared by DreamingOfClouds with Claude Opus 5.5 (Anthropic), via Claude Code.
+
+## New in this package
+
+- `code/assemble9.py`: the nine-stage bit invoice of PR #463's `assemble_k.py` (verbatim; its complex half is dropped)
+  followed by PR #493's per-supplier outer-assembly loop (verbatim).
+- `code/adapt.py`: writes the adapted `audit_parent_k.py` (2 listed edits, each anchor checked to occur exactly once).
+- `data/selection9-r480.json`: the nine-stage hosting selection, nominated by PR #442's programme through the vendored
+  `emit_k.py --discover` and rediscovered by `verify.py`.
+- `code/findings/levels.py` (frame-level register visits). `code/findings/flowlp2.py` (the junk-dimension min-flow LP) and
+  `code/findings/piv0.py` (pivot classes) are byte-identical copies of DreamingOfClouds research scripts (Claude Sonnet 5.5 /
+  Claude Opus 5.5 via Claude Code), not from a PR; sha256 in `SOURCE.json`.
+- `verify.py`, `EXPECTED.json`, `MANIFEST.json`, `SOURCE.json`, `README.md`.
+
+## Vendored, byte-identical (`vendor/`; git blob ids and sha256 in `SOURCE.json`, recomputed by `verify.py`)
+
+From PR #463 (DreamingOfClouds, branch e8-carrier-rekid-r715, head cdf0c1c6540df987467988fd771340670f4136f3):
+- `research/e8-r715-eleven-stage-whole/code/{global_k,emit_k,audit_parent_k,audit_hosted_k,hosting_k,bank_routes_k}.py`
+  (DreamingOfClouds with Claude Opus 5.5, derived from PR #443 and PR #444 by chafreaky with OpenAI Codex, PR #442 by
+  DaysSky with Claude, and PR #386 by Zhihao Chen with OpenAI Codex; lineage per file in `SOURCE.json`);
+- `research/e8-r740-replica32/code/{config,bank_routes,tile_m,e8core}.py` and `engines/{moment,base_two_moment,outer}.py`
+  (PR #443, chafreaky with OpenAI Codex; the engines are PR #315's, DreamingOfClouds);
+- `research/idle-axis-hosting/hosting_core.py` (PR #442, DaysSky with Claude).
+
+## Used from the parents, unchanged
+
+- PR #503 (`research/e8-r480-dead-totals`): the R480 word and its verifier, run as the parent.
+- PR #493 (`research/e8-bit-carrier-skeleton`, eumemic): the checkers (`columns_k.cpp` and its headers), the residual
+  isometry record, the nine-stage geometry builder (through the parent run) and the cited complex receipts
+  (`data/complex-b.json`). Provenance and licences are listed in PR #493's `NOTICE.md`.
+
+
+# Notice
+
+Apache-2.0 (`LICENSE`). Prepared by eumemic with substantial Anthropic Claude assistance.
+
+## New in this package
+
+- `code/skeleton/`: the D4 quad-skeleton geometry (`geom.py`), the word builder with source registers as gate
+  destinations (`wordkit.py`), the generator (`gen.py`) and an E8 root-line helper (`e8.py`, #370's Hamming
+  coordinates and line order).
+- `data/skeleton/plan-skB.json` (CP-SAT plan nominated by `discovery/plan.py`), `data/skeleton/d4part-pairs.json`
+  (the 5-pair D4 partition, found by `discovery/d4part.py`), `data/bit/bitE8mut-r489.json.gz` (the word),
+  `data/eleven/hosting.json` (frozen hosting selection nominated by `discovery/host_select.py` with #442's integer
+  programme), `data/eleven/residual-isometries.json` (the frame-zero record of #443's isometry format: every helper of
+  this word enters at frame zero).
+- `code/bit/independent/relax.py` (generates the relaxed #370 checker), `interface.py` (#425 lemma hypotheses),
+  `mutate_mut.py` (word-level mutants and hypothesis controls), `code/bit/mutate_records_mut.py` (record-level
+  mutable-source mutants).
+- `code/eleven/adapt.py` and the adapted files it checks, `code/eleven/{emit,audit,local_audit}.py` (PR #472's files
+  plus 19 listed edits: R = 489, package paths, a `--hosting` argument, one frame-zero entrance instead of eleven).
+- `code/eleven/assemble.py`: PR #472's bit invoice and pricing arithmetic with R as a parameter, and its outer
+  assembly run once per cited complex supplier.
+- `code/cited.py`, `data/complex-b.json`, `verify.sh`, `EXPECTED.json`, `requirements.txt`, `discovery/`,
+  `.github/workflows/e8-bit-carrier-skeleton.yml`.
+
+## Vendored, byte-identical (git blob ids in `SOURCE-pr*.json`, checked by `code/provenance.py`)
+
+- **PR #370 (Dylan Crow / utcorvusvolat-dotcom, with Anthropic Claude assistance), commit b7fe9cd:** the independent
+  checker `checkword.py` and its controls `mutate.py`; `export_records.py`, `mutate_records.py`, `stage_cohort.py`;
+  PR #266's official cohort checkers as generalised there (`legality_h.cpp`, `columns_h.cpp`, NCOPY); nlohmann/json
+  (`json.hpp`, MIT licence in its header); `gram_frames.py`; PR #315's engines (`code/pricing/`, DreamingOfClouds);
+  the R865 word `data/bit/bitE8-r865.json.gz` (read only for the E8 label table by the geometry scripts); `LICENSE`.
+- **PR #381 (DaysSky, assisted with Claude), commit 00765b2:** `columns_k.cpp`.
+- **PR #386 (Zhihao Chen / jacklightChen, with OpenAI Codex assistance), commit dd11cf8:** the nine-stage bit geometry
+  `code/nine/geometry.py` (parent geometry checked by the eleven-stage audit) and `portable-include/bits/stdc++.h`.
+- **PR #436 (eumemic, with Anthropic Claude assistance), commit 8f6b829:** `code/provenance.py`.
+- **PR #445 (chafreaky, with OpenAI Codex assistance), commit 40ab91d:** `code/hub/r740-hosted-eleven/code/geometry.py`
+  (eleven-stage geometry), `row_bound.py` (orthogonal-cover row bound), `emit.py` and `data/selection.json` (used
+  unchanged for the PR #445 control).
+- **PR #451 (chafreaky, with OpenAI Codex assistance), commit f3a3815:** `code/hub/r740-hosted-rank149/`: its
+  eleven-stage bit emitter (byte-identical to PR #445's), frozen selection, R740 rank149 word and isometries
+  (PR #451 control).
+- **PR #472 (chafreaky, with OpenAI Codex assistance), commit 259c684,** and files its branch carries unchanged:
+  - `code/hub/e8-r715-whole-hosted/`: PR #472's eight-replica pooled emitter, independent audit, local projector
+    audit and assembly (the originals of the adapted files), its R715 word, isometries and frozen hosting (used
+    unchanged for the PR #472 control);
+  - `code/hub/e8-r740-replica32/`: PR #443 (chafreaky): `bank_routes.py`, `tile_m.py`, `e8core.py`, the R740 rank138
+    word and its isometries (PR #445 control); PR #315's engines (DreamingOfClouds) in `code/engines/`;
+  - `code/hub/idle-axis-hosting/hosting_core.py`: PR #442 (DaysSky, with Anthropic Claude assistance).
+- **Receipts (`data/complex/`):** `EXPECTED.json` of PR #484 (DreamingOfClouds, 035845c), PR #471 (Dugongue,
+  56b3da6), PR #458 (chafreaky, 44e9eae), PR #448 (higherbeing, 359022b), and of the bit-bound frontier PRs #483
+  (Dugongue, 3738c3c), #473 (rohanarun, c7590f1) and #478 (Dugongue, bea1d52).
+
+The repository-level files `certificates/joint-dual-kappa.json`, `certificates/joint-dual-validation.json` and
+`research/joint-dual/SOURCE.json` in this branch are byte-identical to rohanarun's PR #318 (a45747d). They repair
+stale root-file pins so the repository's joint check passes; #417, #436 and #439 carry the same repair.
+
+## Cited, not rebuilt
+
+Complex coarse savings in `data/complex-b.json`, each bound to its PR's `EXPECTED.json` by `code/cited.py`:
+PR #484 (primary; draft, conditional on its Lemmas A and G), PR #471 (the complex supplier of #473, #478 and #483),
+PR #458 (the complex supplier of #472) and PR #448 (information only; disputed cross-copy split). No complex
+construction is rebuilt here.
+
+## Ideas used
+
+PR #425 (Zhihao Chen, with OpenAI Codex assistance): the scalar-interface lemma that permits source registers as
+gate destinations when the full-frame cleanup restores them. PR #463 (DreamingOfClouds, with Claude Opus 5.5): the
+carrier bound (three registers leave a source's line). PR #433 (icekylinx): source rows written on the bit side.
+PR #442 (DaysSky): idle-axis hosting of whole residuals. Jacob Sussman introduced the E8 unit.
+
+
+Complex535 is cited at25f9b7c2310d40c9efdfbe368a40ba6e74d3b0a2: icekylinx with disclosed OpenAI assistance. Literal bad-class charge follows Srikrishnabalaji461 and464/465/539. The input source note and moment/outer engines retain predecessor attribution.
+
+The additional60-preload trace refinement, unmodified scripts/preload60.py, pinned source word and proof are from Dugongue PR541 at6fd04b2e266b7915b69f96f27758cfce520aa40a, with disclosed OpenAI assistance. The new integration retains that event/frame schedule, then applies our observable slice and19 frozen frame reductions. Individual histogram gains are not added; the complete combined word is recounted.
diff --git a/research/r480-sliced-minimal-frames/README.md b/research/r480-sliced-minimal-frames/README.md
new file mode 100644
index 0000000000..785b3174e1
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/README.md
@@ -0,0 +1,60 @@
+# PR541 trace refinement with sliced R480 frame reductions
+
+**Conditional kappa = 0.001587467473762886**, about **0.0160671% above PR541's0.001587212454640692**. Fresh finite physical verification passes. The unchanged cited complex supplier now binds.
+
+## Construction and contribution
+
+Dugongue PR541 supplies60 commuting preloads in PR510's R480 bit word. This package reconstructs that exact source word, then adds our observable slice and frozen frame reductions:
+
+1. Compute exact backward F2 target responses and remove227 target-zero A gates plus their227 inverse cleanup gates. The whole scalar endpoint remains unchanged.
+2. On the original sliced R480 word, process gates chronologically and replace each A/B frame by the exact rational span of its preceding operand frames. This is the least feasible assignment in subspace inclusion for that fixed word, not a claim of moment optimality. It finds19 smaller operation frames and3 new rank-three root closures.
+3. Transfer those19 specific reductions to the sliced PR541 word by exact gate identity and verify every frame containment. Retain all other PR541 frames and regenerate literal inverse cleanup at the full frame.
+
+Re-minimizing every frame after adding PR541's preloads was a losing composition. Separate histogram deltas also do not add: the two changes interact. The final combined word is independently recounted, then physically emitted and audited.
+
+The local paid profile, including9 rank-eight center copies, is
+
+`{1:1286,2:1319,3:45,4:169,5:1,6:130,8:9}`.
+
+The complete nine-stage profile is
+
+`{1:370368,2:395232,3:12960,4:64032,5:288,6:52800,7:23040,8:2592}`.
+
+All480 helpers retain width8; the original literal PR510 owner selection remains valid. Live stock30720, physical stock30729 and rank deficit9984 are unchanged. Every positive idle remainder is included.
+
+Two independent rational moment engines certify bit c = **0.001590005077309865651707**, rejecting the next10^-24 grid point. PR535 complex b = **0.001589991533606044174031** is cited unchanged and repriced. It now binds. Six explicit bootstrap levels and47 strict outer inequalities give kappa = **793733736881443/500000000000000000**. Literal bit fallback charging follows PR539 and its461/464/465 ancestry.
+
+## Verification and reproduction
+
+Arithmetic/construction replay (Python3.11+, SymPy):
+
+```sh
+python -B verify.py /absolute/new/output
+```
+
+Complete offline finite replay also needs NumPy, C++17 and genuine Boost headers:
+
+```sh
+python -B verify.py /absolute/new/output --physical --boost-include /usr/include
+```
+
+Use `/opt/homebrew/include` for the tested macOS Homebrew installation. Outputs must be new and outside the package. No optimizer, network or Lean is required. The pinned PR493/510 dependencies are included as byte-preserved source archives and extracted only into the fresh output directory.
+
+Completed checks cover:
+
+- Exact source reconstruction and final-word equality; independent720-column scalar/nesting check, all120 sources restored, and674 nonzero used frame Grams.
+- Fresh native C++ legality,960-column five-stage and1440-column nine-stage checks,18 omitted-bridge controls and inherited structural/mutable-source corruption checks. Native prefix payload101bits satisfies the104-bit bound.
+- All675 catalogue frame Grams and prime witnesses, fresh compressed geometry, both local orientations and unbanked global scalar replay.
+- Explicit138240 whole-helper owner occurrences and10500480 emitted events;17 parent mutations and16 hosting mutations rejected.
+- Fresh finite invoice: prefix191bits charged192; two-level coefficient421bits charged512; row coefficient18355 below20161. The previously assumed allowances are therefore justified for the changed word.
+- Both exact moment engines, six explicit finite bootstrap cutoffs,47 strict outer inequalities, and rejection of the next kappa grid point, zero eta, zero beta and unit beta.
+
+Python3.14 direct physical replay and Python3.11 full packaged physical replay pass. Raw emitted records, exact profile, invoice and kappa agree; final rational arithmetic receipts are identical. Timings and gzip metadata differ. The initial3.11 packaged attempt rejected only the floating discovery diagnostic `fee_delta`; its failure is retained. Only that diagnostic and timing are excluded from receipt equality; actual frames, candidate words, profiles and rational prices remain checked exactly.
+
+Receipts are in evidence/, including the manifest tested by the full3.11 run. BASE-prefixed receipts describe the earlier standalone candidate and are preserved separately. The dedicated GitHub workflow runs the complete finite replay on3.11 and3.14; local success is not a claim that CI has completed.
+
+## Scope and credits
+
+All inherited all-size compiler, common-label/mutable-source realization, compressed geometry, whole-call hosting, restored work, precision/tape and analytic-transfer assumptions remain conditional. PR535's complex operator is cited rather than freshly rebuilt. No unconditional multiplication theorem, full Lean proof, unrestricted optimum or measured speedup is claimed.
+
+Dugongue541 supplies the preload transformation; our contribution is the observable slice, lower frames and this verified composition. PR510/493 supply the underlying circuit and controller; original notices and licenses remain in NOTICE.md, SOURCES.json and dependency archives. Prepared for rohanarun with OpenAI Codex assistance.
diff --git a/research/r480-sliced-minimal-frames/SOURCES.json b/research/r480-sliced-minimal-frames/SOURCES.json
new file mode 100644
index 0000000000..e6c598872e
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/SOURCES.json
@@ -0,0 +1,23 @@
+{
+  "bit": {
+    "pr": 510,
+    "head": "64edee18af02aab0c1696b677a0287dd0146381b",
+    "word_sha256": "92e399e9a486b55229b5624632ac54f800dd9e022f2aee97f1e5f264bbc0ef5c"
+  },
+  "complex": {
+    "pr": 535,
+    "head": "25f9b7c2310d40c9efdfbe368a40ba6e74d3b0a2",
+    "role": "cited complex endpoint, unchanged operator not rebuilt"
+  },
+  "comparison": {
+    "pr": 539,
+    "head": "915ca69db7f98c70b97ce96e1fcc96818517584c"
+  },
+  "scope": "Combined finite physical replay passed on3.14 and full packaged3.11; all-size interfaces remain conditional.",
+  "trace541": {
+    "pr": 541,
+    "head": "6fd04b2e266b7915b69f96f27758cfce520aa40a",
+    "author": "Dugongue",
+    "role": "Unmodified 60-preload constructor and source proof; exact output reconstructed against published word"
+  }
+}
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diff --git a/research/r480-sliced-minimal-frames/engines/base_two_moment.py b/research/r480-sliced-minimal-frames/engines/base_two_moment.py
new file mode 100644
index 0000000000..14a90fa0e7
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/engines/base_two_moment.py
@@ -0,0 +1,63 @@
+#!/usr/bin/env python3
+"""Maintainer arithmetic cross-check of the pinned PR39 certificate.
+
+Uses no producer/checker imports. Counts are inputs, not proved by this script.
+80-term rational logarithms and 12-term exponentials independently enclose
+the two moments. The accompanying written review addresses their realization.
+"""
+import argparse
+from fractions import Fraction as Q
+from hashlib import sha256
+import json
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+GRID = 10**30
+
+
+def require(condition, message):
+    if not condition:
+        raise ValueError(message)
+
+
+def log_unit(x):
+    require(1 <= x <= 2, "logarithm range")
+    z = (x-1)/(x+1)
+    lo = 2*sum((z**(2*j+1)/(2*j+1) for j in range(80)), Q(0))
+    return lo, lo+2*z**161/(161*(1-z*z))
+
+
+def log_bounds(x):
+    k = 0
+    while x > 2:
+        x /= 2
+        k += 1
+    lo, hi = log_unit(x)
+    l2, u2 = log_unit(Q(2))
+    lo, hi = lo+k*l2, hi+k*u2
+    # Outward rational rounding keeps later certificates compact.
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
+
+def exp_bounds(x):
+    require(0 <= x < 1, "exponential range")
+    term = total = Q(1)
+    for j in range(1, 13):
+        term *= x/j
+        total += term
+    tail = term*x/13/(1-x/14)
+    return total, total+tail
+
+
+def moment(m, W, entries, saving):
+    lo = hi = Q(0)
+    for t, count in entries:
+        t, count = int(t), int(count)
+        require(0 < t < m and count > 0, "improper child")
+        l, u = log_bounds(Q(m, t))
+        lower = exp_bounds(saving*l)[0]
+        upper = exp_bounds(saving*u)[1]
+        lo += Q(t*count, m*W)*lower
+        hi += Q(t*count, m*W)*upper
+    return Q((lo*GRID).__floor__(), GRID), Q((hi*GRID).__ceil__(), GRID)
+
diff --git a/research/r480-sliced-minimal-frames/engines/moment.py b/research/r480-sliced-minimal-frames/engines/moment.py
new file mode 100644
index 0000000000..7d697cb452
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/engines/moment.py
@@ -0,0 +1,113 @@
+"""Independent prospective ordinary five-stage cost, never supplier admission.
+
+Consumes a raw one-invocation ledger made from the actual source/internal/target
+paths.  No packed profile is used as an input.  Missing physical/finite/compiler
+obligations must remain explicit in the accompanying report.
+"""
+from collections import Counter
+from decimal import Decimal, localcontext, ROUND_FLOOR
+from fractions import Fraction as F
+from functools import lru_cache
+from math import factorial
+import hashlib, json, sys
+from pathlib import Path
+
+DYAD = 1 << 224
+
+def floor(x): return F(x.numerator*DYAD//x.denominator, DYAD)
+def ceil(x): return -floor(-x)
+
+@lru_cache(None)
+def logarithm(x):
+    """Mercator bounds after binary reduction; ln(2) via two Mercator sums.
+
+    For 1 <= y < 2, log(y) is an alternating series in z=y-1.
+    We instead reduce to 1 <= y < 3/2, and bound log(3/2) and
+    log(2) by the same alternating series at z=1/2 and z=1/3,
+    using log(2)=log(3/2)+log(4/3). All sums are exact rationals.
+    """
+    assert x >= 1
+    k = 0
+    while x >= 2: x /= 2; k += 1
+    j = int(x >= F(3, 2))
+    if j: x /= F(3, 2)
+    def mercator(y):
+        z=y-1
+        assert 0 <= z <= F(1,2)
+        # An even partial sum is below log; the next term bounds its tail.
+        t=z; s=F(0)
+        for n in range(1, 225):
+            s += (t if n & 1 else -t)/n
+            t *= z
+        return s, s+t/225
+    a,b=mercator(x); c,d=mercator(F(3,2)); e,f=mercator(F(4,3))
+    return floor(a+(k+j)*c+k*e), ceil(b+(k+j)*d+k*f)
+
+def exponential(l,u):
+    assert 0 <= l <= u < F(1,10)
+    def poly(x):
+        s=t=F(1)
+        for j in range(1,25): t=t*x/j; s+=t
+        return s,t
+    lo,_=poly(l);hi,last=poly(u)
+    return floor(lo),ceil(hi+last*u/25/(1-u/26))
+
+def moment(H,m,W,a,bad=False):
+    lo=hi=F(0)
+    for r,n in H.items():
+        assert 0 < r < m and n > 0
+        l,u=logarithm(F(m,r)); e,f=exponential(a*l,a*u)
+        p=F(r*n,m*W);lo+=p*e;hi+=p*f
+    if bad:
+        # Existing envelope shape, only conditionally portable to the new cover.
+        l,u=logarithm(F(m));e,f=exponential(a*l,a*u)
+        p=F(32*m*sum(H.values()),10**16*W)
+        lo+=p*e;hi+=p*f
+    return floor(lo),ceil(hi)
+
+def decimal(x):
+    with localcontext() as c:
+        c.prec=70
+        return str(Decimal(x.numerator)/Decimal(x.denominator))
+
+def certify(H,m,W,bad=False):
+    with localcontext() as c:
+        c.prec=70
+        terms=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln()) for r,n in H.items()]
+        fallback=Decimal(32*m*sum(H.values()))/Decimal(10**16*W) if bad else Decimal(0)
+        lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+        for _ in range(210):
+            a=(lo+hi)/2
+            if sum(p*(a*l).exp() for p,l in terms)+fallback*(a*lm).exp()<1:lo=a
+            else:hi=a
+        q=int(((lo+hi)/2*10**24).to_integral_value(rounding=ROUND_FLOOR))
+    lo=F(q,10**24);hi=lo+F(1,10**24)
+    lv=moment(H,m,W,lo,bad);hv=moment(H,m,W,hi,bad)
+    assert lv[1]<1<hv[0],(lo,hi,lv,hv)
+    k=F(694782719690783,10**18);required=k/(1-k)
+    at=moment(H,m,W,required,bad)
+    return dict(lower=decimal(lo),upper=decimal(hi),lower_endpoint_residual_upper=decimal(lv[1]-1),upper_endpoint_residual_lower=decimal(hv[0]-1),strict_rational_signs=True,moment_minus_one_at_current_k_required_rate=[decimal(x-1) for x in at],nominal_k_ceiling_interval=[decimal(lo/(1+lo)),decimal(hi/(1+hi))])
+
+def price(core,gauges,R,v,h):
+    core=Counter({int(k):n for k,n in core.items() if n})
+    gauges=Counter({int(k):n for k,n in gauges.items() if n})
+    m=5*h;W=4*v+R;ell=h*(h-2)  # p = 10 port: was the p = 12 literal 24*22
+    assert sum(r*n for r,n in core.items())==h*R+2*v*(h-1)+ell-sum(r*n for r,n in gauges.items())
+    idle=Counter({r:2*v for r in (2*h-2,h-1,2*h+2,4)})
+    ans={}
+    for name,bundled in [('separate_rank_a_corrections',False),('bundled_rank_5a_corrections',True)]:
+        H=Counter({r:5*n for r,n in core.items()});H.update(idle)
+        for a,n in gauges.items():H[5*a if bundled else a]+=n if bundled else 5*n
+        mass=sum(r*n for r,n in H.items());assert m*W-mass==4*v-5*ell
+        ans[name]=dict(histogram=dict(sorted(H.items())),calls=sum(H.values()),rank_mass=mass,deficit=m*W-mass,maxchild=max(H),root=certify(H,m,W),root_with_inherited_bad_envelope=certify(H,m,W,True))
+    return dict(status='PROSPECTIVE_COST_ONLY_NOT_PHYSICAL_OR_ASSEMBLY_ADMISSION',m=m,W=W,R=R,v=v,h=h,copy_calls_per_invocation=h,copy_rank=h-2,ell=ell,raw_core=dict(sorted(core.items())),auxiliary_entrances=dict(sorted(gauges.items())),idle_histogram=dict(sorted(idle.items())),profiles=ans,interval_method='Independent exact rational alternating Mercator log bounds (224 terms), 24-degree exponential with geometric tail, 224-bit outward dyadic rounding; Decimal only proposes endpoints.',bad_envelope_scope='10^-16 and 32*m^2 rank-one fallback calls per paid child are inherited assumptions, not a new m = 5h admission.')
+
+if __name__=='__main__':
+    path=Path(sys.argv[1]);raw=path.read_bytes();data=json.loads(raw)
+    core=data['one_stage_helper_histogram_including_copies']
+    gauges=data['auxiliary_entrance_rank_histogram']
+    result=price(core,gauges,data['physical_R'],data['v'],data['h'])
+    result['input_ledger_sha256']=hashlib.sha256(raw).hexdigest()
+    result['input_head']=data['head']
+    result['one_invocation_parts']={key:data[key] for key in ('physical_internal_excluding_center_copies','paid_center_copy_histogram','physical_source_histogram','physical_target_histogram')}
+    print(json.dumps(result,indent=2))
diff --git a/research/r480-sliced-minimal-frames/engines/outer.py b/research/r480-sliced-minimal-frames/engines/outer.py
new file mode 100644
index 0000000000..43007488ad
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/engines/outer.py
@@ -0,0 +1,58 @@
+#!/usr/bin/env python3
+"""Exact semantic assembly with the balanced prefix, for the paired-cube network.
+
+Prepared by eumemic with Anthropic Claude assistance. Apache-2.0. The 47 strict constraints, the seven margins and
+the finite bridge are unchanged from icekylinx's scripts/structured_bulk_assembly.py (PR144, adapted from Zhihao
+Chen's PR23). The prefix is the RaD project's (hipotures) balanced positional transform, reviewed in
+references/semantic-bulk/rad20/reports/review-balanced-transform.md: the prefix margin becomes 1-eps instead of
+1-eps(1+c), and eps(1+c)<1 remains a strict geometric condition (K_geometry) whose gap need not exceed kappa.
+The parameters c=q+eta/4 and eps=(1-eta)/(1+q) are those of the reviewed balanced assembly in
+research/copied-fixed/balanced_assembly.py, with its backoff h equal to eta. Nothing else differs from
+structured_bulk_assembly.assembly.
+"""
+from fractions import Fraction as Q
+
+
+def assembly(bit_saving,complex_saving,bridge,kappa,eta=Q(1,10**8),beta=Q(1,4)):
+    a,b=bit_saving,complex_saving
+    tau,sigma=1-a,1-b
+    q=a*(1-2*eta); lp=1-q; lam=(tau+lp)/2
+    c=q+eta/4; eps=(1-eta)/(1+q)
+    minimum=eps*q; r=(minimum+1-eps)/2; delta=eta/8
+    internal=tau+(1-beta)*max(sigma-tau,Q(0));leaf=sigma+beta*(1-sigma)
+    margins=dict(balanced_prefix=1-eps,coordinate_movement=a,
+        compact_phase_layer=minimum,bulk_exposure=a,
+        Gaussian_arithmetic=min(1-eps-delta,r-delta),
+        scalar_work=1-eps-delta,dimension=eps)
+    slacks=dict(bit_positive=a,complex_above_bit=b-a,
+        complex_below_one_over32=Q(1,32)-b,beta_positive=beta,
+        beta_below_one=1-beta,leaf_saving_above_bit=(1-beta)*b-a,
+        q_positive=q,q_below_internal=1-internal-q,q_below_leaf=1-leaf-q,
+        c_positive=c,c_below_one=1-c,q_below_reservations=c-q,
+        lambda_above_tau=lam-tau,lambda_above_sigma=lam-sigma,
+        lambda_above_internal=lam-internal,lambda_prime_above_lambda=lp-lam,
+        compact_leaf=lp-leaf,compact_reservations=lp-(1-c),
+        lambda_prime_below_one=q,epsilon_positive=eps,epsilon_below_one=1-eps,
+        guard_width=1-eps,K_geometry=1-eps*(1+c),K_dominates_log=eps*c,
+        record_suffix=1-eps,phase_local=1-eps-delta,phase_boundary=r-delta,
+        gamma_sublinear=1-eps-r,cell_above_band=eps-(1-r)/2,
+        prime_interval_packing=1-eps,alpha_positive=r,alpha_below_one=1-r,
+        alpha_below_one_fourth=Q(1,4)-r,delta_positive=delta,
+        delta_below_one_eighth=Q(1,8)-delta,short_record_fallback=eps-a,
+        small_field_exposure=1-eps-minimum,
+        artificial_boundary=8-eps+r-delta-minimum,
+        literal_scalar_guard=Q(bridge['semantic']['strict_literal_gap']),
+        row_product_gap=bridge['rows']['degree_gap'])
+    slacks.update({name+'_above_kappa':val-kappa for name,val in margins.items()})
+    assert len(slacks)==47 and len(margins)==7
+    assert all(v>0 for v in slacks.values()),{k:str(v) for k,v in slacks.items() if v<=0}
+    assert min(margins.values())==minimum
+    assert margins['balanced_prefix']-minimum==eta and 1-eps-r==eta/2
+    assert 1-eps*(1+c)==eta-eps*eta/4
+    return dict(parameters=dict(a_bit=a,a_complex=b,tau=tau,sigma=sigma,
+            eta=eta,beta=beta,q=q,c=c,epsilon=eps,lambda_=lam,
+            lambda_prime=lp,alpha_squared_power=r,delta=delta,
+            C0=bridge['semantic']['C0'],C1=1,kappa=kappa),
+        strict_constraints=slacks,margins=margins,minimum_margin=minimum,
+        absorption_gap=minimum-kappa,
+        recurrence=dict(internal=internal,leaf=leaf,reservations=1-c))
diff --git a/research/r480-sliced-minimal-frames/evidence/BASE-BIT-INVOICE.json b/research/r480-sliced-minimal-frames/evidence/BASE-BIT-INVOICE.json
new file mode 100644
index 0000000000..cec8d4f47f
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/BASE-BIT-INVOICE.json
@@ -0,0 +1,49 @@
+{
+ "status": "PASS_NINE_STAGE_WHOLE_HOSTED_BIT_PRICE_AND_CITED_ASSEMBLY",
+ "R": 480,
+ "stages": 9,
+ "replicas": 32,
+ "bit_lower": "397375760308276055691/250000000000000000000000",
+ "bit_lower_decimal": "0.001589503041233104222764",
+ "next_grid_point_rejected": "317900608246620844553/200000000000000000000000",
+ "bit": {
+  "live": 30720,
+  "physical": 30729,
+  "maxchild": 8,
+  "rank_mass": 1956096,
+  "hosted_residuals": 138240,
+  "bank_assignments": 0,
+  "prefix_bits": 191,
+  "charged_prefix_bits": 192,
+  "coefficient_bits": 191,
+  "two_level_coefficient_bits": 421,
+  "charged_two_level_coefficient_bits": 512,
+  "row_coefficient": 18355,
+  "halving_depth": 2
+ },
+ "bit_profile": {
+  "1": 350496,
+  "2": 424608,
+  "4": 64032,
+  "5": 288,
+  "6": 52800,
+  "7": 23040,
+  "8": 2592
+ },
+ "emitted_program_sha256": "c1dcd51a6428e62d6aff160b8a4ba4ba90a5bd92289c73f3fc289465abd28ca5",
+ "kappa": {
+  "pr484": {
+   "complex_lower": "663394767670133/500000000000000000",
+   "kappa": "662515748707749/500000000000000000",
+   "kappa_decimal": "0.001325031497415498",
+   "binding_component": "complex",
+   "strict_outer_constraints": 47,
+   "outer_controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+   ]
+  }
+ }
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/BASE-PACKAGED314.json b/research/r480-sliced-minimal-frames/evidence/BASE-PACKAGED314.json
new file mode 100644
index 0000000000..151c78aaa4
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/BASE-PACKAGED314.json
@@ -0,0 +1,10 @@
+{
+  "status": "PASS_PACKAGED_RECONSTRUCTION_AND_PROSPECTIVE_PRICE",
+  "kappa": "0.001586980532763499",
+  "bit": "0.001589503043149746028939",
+  "word_sha256": "37e84d9ace34138fc0a942d7cac37c46513a952b4d8ca5384545f830cc0948c9",
+  "manifest_sha256": "453ecd91c4c516880a678e2962abe0e0421d3a3c465a09cea887058eee9cf3c5",
+  "python": "3.14.8 (main, Sep 30 2026, 17:55:09) [Clang 21.0.0 (clang-2100.3.34.2)]",
+  "seconds": 4.126170333009213,
+  "scope": "Exact reconstruction of source slice and least frames,720 scalar columns, nesting and two rational moment engines with47 outer constraints. Prospective unchanged owner schedule and prior finite allowances; physical template and finite invoice remain pending. No completed supplier admission."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/BASE-PHYSICAL-ADMISSION.json b/research/r480-sliced-minimal-frames/evidence/BASE-PHYSICAL-ADMISSION.json
new file mode 100644
index 0000000000..c1b013648b
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/BASE-PHYSICAL-ADMISSION.json
@@ -0,0 +1,32 @@
+{
+  "status": "PASS_FRESH_SLICED_MINIMUM_FRAME_RECORDS_HOSTING_AND_INVOICE",
+  "seconds": 79.25043887505308,
+  "word_sha256": "37e84d9ace34138fc0a942d7cac37c46513a952b4d8ca5384545f830cc0948c9",
+  "profile": {
+    "1": 350496,
+    "2": 424608,
+    "4": 64032,
+    "5": 288,
+    "6": 52800,
+    "7": 23040,
+    "8": 2592
+  },
+  "kappa": "0.001586980532763499",
+  "bit_invoice": {
+    "live": 30720,
+    "physical": 30729,
+    "maxchild": 8,
+    "rank_mass": 1956096,
+    "hosted_residuals": 138240,
+    "bank_assignments": 0,
+    "prefix_bits": 191,
+    "charged_prefix_bits": 192,
+    "coefficient_bits": 191,
+    "two_level_coefficient_bits": 421,
+    "charged_two_level_coefficient_bits": 512,
+    "row_coefficient": 18355,
+    "halving_depth": 2
+  },
+  "steps": 19,
+  "scope": "Fresh finite record/formal-column/geometry/template/whole-owner/invoice replay, with inherited mutation controls. Cited complex535 operator unchanged. Exact sharp-charge prospective profile now matches physically emitted audited profile and retained finite allowances. All-size supplier/compiler/realization/hosting/analytic assumptions remain conditional."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE311.json b/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE311.json
new file mode 100644
index 0000000000..77cd76fad5
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE311.json
@@ -0,0 +1,50 @@
+{
+ "status": "PASS_NINE_STAGE_WHOLE_HOSTED_BIT_PRICE_AND_CITED_ASSEMBLY",
+ "R": 480,
+ "stages": 9,
+ "replicas": 32,
+ "bit_lower": "1590005075385399234663/1000000000000000000000000",
+ "bit_lower_decimal": "0.001590005075385399234663",
+ "next_grid_point_rejected": "198750634423174904333/125000000000000000000000",
+ "bit": {
+  "live": 30720,
+  "physical": 30729,
+  "maxchild": 8,
+  "rank_mass": 1956096,
+  "hosted_residuals": 138240,
+  "bank_assignments": 0,
+  "prefix_bits": 191,
+  "charged_prefix_bits": 192,
+  "coefficient_bits": 191,
+  "two_level_coefficient_bits": 421,
+  "charged_two_level_coefficient_bits": 512,
+  "row_coefficient": 18355,
+  "halving_depth": 2
+ },
+ "bit_profile": {
+  "1": 370368,
+  "2": 395232,
+  "3": 12960,
+  "4": 64032,
+  "5": 288,
+  "6": 52800,
+  "7": 23040,
+  "8": 2592
+ },
+ "emitted_program_sha256": "e4e1a89c55925d0421c882df31426fe939ab5fa98e9828155e711aea0b6532c8",
+ "kappa": {
+  "pr484": {
+   "complex_lower": "663394767670133/500000000000000000",
+   "kappa": "662515748707749/500000000000000000",
+   "kappa_decimal": "0.001325031497415498",
+   "binding_component": "complex",
+   "strict_outer_constraints": 47,
+   "outer_controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+   ]
+  }
+ }
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE314.json b/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE314.json
new file mode 100644
index 0000000000..77cd76fad5
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/BIT-INVOICE314.json
@@ -0,0 +1,50 @@
+{
+ "status": "PASS_NINE_STAGE_WHOLE_HOSTED_BIT_PRICE_AND_CITED_ASSEMBLY",
+ "R": 480,
+ "stages": 9,
+ "replicas": 32,
+ "bit_lower": "1590005075385399234663/1000000000000000000000000",
+ "bit_lower_decimal": "0.001590005075385399234663",
+ "next_grid_point_rejected": "198750634423174904333/125000000000000000000000",
+ "bit": {
+  "live": 30720,
+  "physical": 30729,
+  "maxchild": 8,
+  "rank_mass": 1956096,
+  "hosted_residuals": 138240,
+  "bank_assignments": 0,
+  "prefix_bits": 191,
+  "charged_prefix_bits": 192,
+  "coefficient_bits": 191,
+  "two_level_coefficient_bits": 421,
+  "charged_two_level_coefficient_bits": 512,
+  "row_coefficient": 18355,
+  "halving_depth": 2
+ },
+ "bit_profile": {
+  "1": 370368,
+  "2": 395232,
+  "3": 12960,
+  "4": 64032,
+  "5": 288,
+  "6": 52800,
+  "7": 23040,
+  "8": 2592
+ },
+ "emitted_program_sha256": "e4e1a89c55925d0421c882df31426fe939ab5fa98e9828155e711aea0b6532c8",
+ "kappa": {
+  "pr484": {
+   "complex_lower": "663394767670133/500000000000000000",
+   "kappa": "662515748707749/500000000000000000",
+   "kappa_decimal": "0.001325031497415498",
+   "binding_component": "complex",
+   "strict_outer_constraints": 47,
+   "outer_controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+   ]
+  }
+ }
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/COMBINED-PROFILE.json b/research/r480-sliced-minimal-frames/evidence/COMBINED-PROFILE.json
new file mode 100644
index 0000000000..d740008344
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/COMBINED-PROFILE.json
@@ -0,0 +1,11 @@
+{
+  "new_chain_profile": {
+    "1": 1286,
+    "2": 1319,
+    "4": 169,
+    "3": 45,
+    "6": 130,
+    "5": 1
+  },
+  "scope": "Exact rank profile from every final combined-word frame transition; center copies separate and unchanged9x8. Do not add standalone histogram deltas: the frame changes interact."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/COMBINED-SCALAR311.log b/research/r480-sliced-minimal-frames/evidence/COMBINED-SCALAR311.log
new file mode 100644
index 0000000000..b42d23cf03
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/COMBINED-SCALAR311.log
@@ -0,0 +1,7 @@
+labels: 120 distinct lines of norm 2 (pairwise |ip| <= 1), spanning rank 8
+legality + exact F2 on all 720 columns: PASS (gates D/A/B/C = 120/2541/240/2541)
+relaxed check: 712 gates have a source register as destination; all 120 sources restored exactly
+nondegeneracy: 674 used frames, all Gram dets > 0 (positive-definite form); primes in dets: [2]
+histogram {1: 6430, 2: 6595, 3: 225, 4: 1085, 5: 5, 6: 650, 7: 240, 8: 45, 14: 240, 18: 240} m 40 W 960 mass 38280 deficit 120 (4v-5L = 120 ) maxchild 18
+no envelope  saving in [0.00128924129269366504, 0.00128924129269366504]
+bad envelope saving in [0.00128924129182445790, 0.00128924129182445790]
diff --git a/research/r480-sliced-minimal-frames/evidence/CROSS-VERSION-COMPARISON.json b/research/r480-sliced-minimal-frames/evidence/CROSS-VERSION-COMPARISON.json
new file mode 100644
index 0000000000..6febb29759
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/CROSS-VERSION-COMPARISON.json
@@ -0,0 +1,9 @@
+{
+  "status": "PASS_EXACT_CORE_COMPARISON",
+  "raw_record_bytes_identical": true,
+  "profile_identical": true,
+  "bit_invoice_identical": true,
+  "kappa_identical": true,
+  "final_rational_arithmetic_receipts_identical": true,
+  "scope": "3.14 direct physical and3.11 full packaged physical replay. Source gzip metadata and timings differ; mathematical candidate and raw records agree."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/FINAL-ARITHMETIC.json b/research/r480-sliced-minimal-frames/evidence/FINAL-ARITHMETIC.json
new file mode 100644
index 0000000000..907165bbca
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/FINAL-ARITHMETIC.json
@@ -0,0 +1,121 @@
+{
+  "status": "PASS_TWO_ENGINE_MOMENTS_EXPLICIT_BOOTSTRAP_AND_47_OUTER_CONSTRAINTS",
+  "kappa": "793733736881443/500000000000000000",
+  "bit": "1590005077309865651707/1000000000000000000000000",
+  "complex": "1589991533606044174031/1000000000000000000000000",
+  "binding": "complex",
+  "moment_receipts": [
+    {
+      "endpoint": "1590005077309865651707/1000000000000000000000000",
+      "accept": true,
+      "mercator": [
+        "-528959481389863068196844247356285261425549927653592406316659/485667223056432267729865476705879726660601709763034880312953102434726071301302124544",
+        "-1057918962779726136393688494712570522851099783249589354128651/971334446112864535459730953411759453321203419526069760625906204869452142602604249088"
+      ],
+      "base_two": [
+        "-1569616353050740863549/1441151880758558720000000000000000000000000000",
+        "-19620186398735041024013/18014398509481984000000000000000000000000000000"
+      ]
+    },
+    {
+      "endpoint": "397501269327466412927/250000000000000000000000",
+      "accept": false,
+      "mercator": [
+        "2052546737940008838792285771005195198805587945344489638516479/971334446112864535459730953411759453321203419526069760625906204869452142602604249088",
+        "1026273368970004419396142885502597599402794008701042548510573/485667223056432267729865476705879726660601709763034880312953102434726071301302124544"
+      ],
+      "base_two": [
+        "76133166715681875732417/36028797018963968000000000000000000000000000000",
+        "19033300686120078818279/9007199254740992000000000000000000000000000000"
+      ]
+    }
+  ],
+  "bootstrap_cutoffs": [
+    {
+      "exponent": "1587524611444488607243301591512594586541766685082939721641196033500077082985211039/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "gap": "2466922161555565196706874881361239427233314917060278358803966499922917014788961/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "5915502031208"
+    },
+    {
+      "exponent": "1589987611220693233969436799124032921656908408255271621213764250456703888168645247692337530155176661579209263145081021671791/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "gap": "3922385350938470571667269922904312091591744728378786235749543296111831354752307662469844823338420790736854918978328209/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "2339925251832660982"
+    },
+    {
+      "exponent": "1589991527369484672907467869473009371900569087929428127858194412268023526363131385605328659942273936787709596130536533313059717010420604441323785777954742812632540479/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "gap": "6236559499532540596920946454068430912070571872141805587731976473636868614394671340057726063212290403869463466686940282989579395558676214222045257187367459521/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "925576587630110440164418"
+    },
+    {
+      "exponent": "1589991533596128095636921378846105683302603510850023019708605750501198494665943743981554315712183827669633787068860540745431330325291548871143931915461461056509465617355493028266183183882099313905691271899151/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "gap": "9916076803087087547850142666396489149976980291394249498801505334056256018445684287816172330366212931139459254568669674708451128856068084538538943490534382644506971733816816117900686094308728100849/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "366119395864473354088826959688"
+    },
+    {
+      "exponent": "1589991533606028405961844970635933136031043988158854602644284696300989216034270884098381237788017416097395227635949386252440163108785971389777214529137931347901216282763449852475447828987643918938044496567014177377022857163639311150603310191474852319/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "gap": "15766478163495758022689937956011841145397355715303699010783965729115901618762211982583902604772364050613747559836891214028610222785470862068652098783717236550147524552171012356081061955503432985822622977142836360688849396689808525147681/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+      "cutoff_log2": "144821524031174952656827434104168501"
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+    "scalar_work_above_kappa": "4220945988078905305078844428178663199266916085580963629537686220597161004680015619605579645676576225669807441269323255142948841693906794521076802342219045412771005659333437837998489029099571107367123096639603803139440845966771209648634377175594035375893189958596587150381734661257995398010636245354906354783850879/4006359966134424176576765834417756161531372168995266186008118384995063847314125776082728282256272986280186411224216741432269720098573946833953810396221316143405866953960866723351987788493784369322475620939746424811244679741845138644196805954604996850730264438640156042820631595340408913602419582210867008479423112000000000000000000",
+    "dimension_above_kappa": "249602502116598488759060946839247010351222999280351033340638701356894253467160952664925981458625522208483391858399600470160063910861025061751796644375451358244697605486615200459046952505297613068438242607596733045074241844701685474790601561243553445526794592579968996213365088630387358883243376779042468983219723820704045677736173/250397497883401511036047864651109760095710760562204136625507399062191490457132861005170517641017061642511650701513546339516857506160871677122113149763832258962866684622554170209499236780861523082654726308734151550702792483865321165262300372162812303170641527415009752676289474708775557100151223888179188029963944500000000000000000"
+  },
+  "controls": [
+    "next_grid",
+    "zero_eta",
+    "zero_beta",
+    "unit_beta"
+  ],
+  "scope": "Exact scalar numerical assembly. Fresh physical invoice separately establishes charged two-level coefficient<=512 and row bound18355; complex535 operator cited. All-size contracts remain conditional."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/FRAME-TRANSFER.json b/research/r480-sliced-minimal-frames/evidence/FRAME-TRANSFER.json
new file mode 100644
index 0000000000..3293cb957f
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/FRAME-TRANSFER.json
@@ -0,0 +1,26 @@
+{
+  "status": "PASS_FROZEN_FRAME_TRANSFER_CONTAINMENTS",
+  "accepted_indices": [
+    2176,
+    2270,
+    2296,
+    2331,
+    2360,
+    2390,
+    2418,
+    2419,
+    2420,
+    2427,
+    2430,
+    2455,
+    2456,
+    2457,
+    2464,
+    2494,
+    2495,
+    2496,
+    2503
+  ],
+  "rejected_indices": [],
+  "scope": "Only previously beneficial19 operation-frame changes transferred to sliced PR541 by exact gate identity; full scalar/geometry/profile/physical checks follow."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/FULL311-TESTED-MANIFEST.json b/research/r480-sliced-minimal-frames/evidence/FULL311-TESTED-MANIFEST.json
new file mode 100644
index 0000000000..47f5de33fb
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/FULL311-TESTED-MANIFEST.json
@@ -0,0 +1,45 @@
+{
+  "EXPECTED.json": "29e445acd36fd8bbe899e556e1ba9dfcb4d78ebbd65ee6c48611685196e5cac1",
+  "LICENSE": "c71d239df91726fc519c6eb72d318ec65820627232b2f796219e87dcf35d0ab4",
+  "NOTICE.md": "c073cd776c23e35407efa41f97a5b95e88a242efdddad2b8376a397682de6def",
+  "README.md": "e0cc67c02a16a7f39bca8304aecbcbad6b80c438f892ecaafec89f7ea709ff48",
+  "SOURCES.json": "ff07b94d6d4d416571dd3f886e499e9e165f7bd11715cca42fb4418880b98475",
+  "dependencies/source493.tar.gz": "194addabc3054ab2ea0f98e92f2343d032f6e7f8bcdc68013857fabeef68f446",
+  "dependencies/source510.tar.gz": "ae1f71ba91a57f4fa4dee2a7ca4e113086ee2a6a55b6631fa16ab112b271ded3",
+  "engines/base_two_moment.py": "aee70c242a2f1d225528fabe2002e1564d1dc884797c07d28a831c7878ee9098",
+  "engines/moment.py": "81cbc864e106418fc7854322746f72180cb8110ff39f74450a3b8d55a82c1482",
+  "engines/outer.py": "c1ebb52163a7219504c728a5c2b098bf1e50bf20148a2879c5a74c3cadc7c03c",
+  "evidence/BASE-BIT-INVOICE.json": "1da82f7aa0d7f3602c76de1d43aa84a450337c23519069cbe36dd1340e376630",
+  "evidence/BASE-PACKAGED314.json": "99ad3fb030ba73ef54ef070ef2ea3bca16901420c96ab6a32412791ea531fbbe",
+  "evidence/BASE-PHYSICAL-ADMISSION.json": "f49bb90ed00ca0aad67ac754707de033b3840add6b0b40be1b77d9625d3083ae",
+  "evidence/COMBINED-PROFILE.json": "6ad46b246e5a555986f0566a2c8c2538eecebf7638eac8a765a1a5c44033cbb3",
+  "evidence/COMBINED-SCALAR311.log": "d7cf7cd67214a12e972c039dd7d765fa0802de576e61677e306be177ff68bc7e",
+  "evidence/FRAME-TRANSFER.json": "678c21b1526d6a4908bea24c3dcff61eec636498c90f71d259b223acb872607f",
+  "evidence/INDEPENDENT-SCALAR311.log": "42e960865be835a7b3e2598615c6077e9b6077bfc5801a5d67eca0e6c5a28854",
+  "evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.json": "e5f274288a5ed462087c218b45014cc387ccb7e0aa910fbf24541d168f508bd1",
+  "evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.log": "334c34574c9d9a5a862eeb608ed67a7a6e42cf615b73ae5e1a37e76996fc31b9",
+  "evidence/INTERFACE311.json": "5c48a51887b1420083015eef35da59c371b9911bfcbc5032a1dbf0993f8fe886",
+  "evidence/MINIMUM-FRAMES.json": "c60b94107d56249887ed1d3adefb65745373f57791a0fb3c9e96e4625274de68",
+  "evidence/SLICE.json": "de878adca9979b59a707867425b9c1450250f3a54bffa57ccde355e109a8c517",
+  "inputs/BASE-CANDIDATE.json.gz": "37e84d9ace34138fc0a942d7cac37c46513a952b4d8ca5384545f830cc0948c9",
+  "inputs/BASE-EXPECTED.json": "ca226880c79eddffd43556aadd01c7534b6ec8c09f897ed2eff7d73a81a1569e",
+  "inputs/CANDIDATE.json.gz": "3dddba9a13c96d54fa6aa3ff238711808662401a6fec7960ac34c4f885bfbf71",
+  "inputs/ORIGINAL.json.gz": "92e399e9a486b55229b5624632ac54f800dd9e022f2aee97f1e5f264bbc0ef5c",
+  "inputs/PR510-EXPECTED.json": "883f7a1a84f31d9e4430e348bc8d9c1ec10210b206ff602dbe126349056af13d",
+  "inputs/PR535-EXPECTED.json": "cb734055c36681d38cf431a6926b7713351a94073cfba37d70a5384a9ef0d178",
+  "inputs/PR539-EXPECTED.json": "e2a71c1d49c68c35fb5c4f20f10377d708f55c3d25146fa811a99c7230d6ff0c",
+  "inputs/PR541-ACQUISITION.json": "8520387f82956bd3ab62feeeb3f9080afee16d4ece86ef147dfc8fed64132540",
+  "inputs/PR541-TRACE-PROOF.md": "e24cd091820f33795aa06f3cc318d61aed6dd17dfc55b0d24d2c3285a1a8ca6a",
+  "inputs/PR541-WORD.json.gz": "db5cd7ab49e2e58a2924a2a9e68ee69d8165b4d5c008b6f546577f4791019dca",
+  "inputs/three-stage-cover-bit.tex": "47123a756f823c83a0d48cca2bf5add6472cd88410b13aed345e70c8834fb511",
+  "scripts/minimum.py": "418f3143793ec981dbca953215f01a954c94e3097b8bc8a3833d5f726d9f5177",
+  "scripts/physical.py": "a7588a357d9bbabc69f8375fbfc129bde894165c2e95087222c3b1c82f2b9c59",
+  "scripts/preload60.py": "105641136b565003118f362bb67086e1eeb4910aed6af593a011f60c2b2962b6",
+  "scripts/price.py": "f00306ed546f7d515763f3e9a192d3532759cb9d78863ee8a78036757209cfad",
+  "scripts/profile.py": "f9d9a9e4f3dd75acaafef61d8005df6a829a7c10ebabd4006ab9698e4a03bfe5",
+  "scripts/slice.py": "d72a4dadfc954aa51eb5b5976bbdeb661bb023201e7832fc9fe0340ef2845279",
+  "scripts/trace_price.py": "40d8a263adf1fafd7830ae619f46422e56546bb9799775657e1106273bb6cbf7",
+  "scripts/trace_slice.py": "19658b829a60f827ed7488a1e6d7c18bec45aaad843e80f9d366be80f6cb8714",
+  "scripts/transfer.py": "1b8de34f72f6c841764b588739e522b62bd483cc5023ddfb353524601d189051",
+  "verify.py": "e9a1dea264ee2e781000d4a23b89dfcca15cd606e11814215ac9656723c82450"
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/HOSTED-AUDIT314.json b/research/r480-sliced-minimal-frames/evidence/HOSTED-AUDIT314.json
new file mode 100644
index 0000000000..0a04cbde0d
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/HOSTED-AUDIT314.json
@@ -0,0 +1,41 @@
+{
+  "status": "PASS_INDEPENDENT_REPLICATED_HOSTING_AUDIT",
+  "template_sha256": "aa07bfa63393dba62a03bc98f2c82d2fbe2dc9cc24a59c92d977681631d34a15",
+  "emitted_program_sha256": "e4e1a89c55925d0421c882df31426fe939ab5fa98e9828155e711aea0b6532c8",
+  "emitted_events": 10500480,
+  "profile": {
+    "1": 370368,
+    "2": 395232,
+    "3": 12960,
+    "4": 64032,
+    "5": 288,
+    "6": 52800,
+    "7": 23040,
+    "8": 2592
+  },
+  "live": 30720,
+  "physical": 30729,
+  "hosted_residuals": 138240,
+  "bank_assignments": 0,
+  "rank_mass": 1956096,
+  "deficit": 9984,
+  "maxchild": 8,
+  "mutations_rejected": [
+    "slot outside window",
+    "copy index outside family",
+    "negative slot offset",
+    "wrong residual width",
+    "duplicate helper",
+    "omitted helper",
+    "hosted owner in another replica or outside the data stock",
+    "hosted owner on a wrong data family",
+    "wrong call orientation",
+    "duplicate call",
+    "changed idle rank",
+    "understated live stock",
+    "missing remainder summary",
+    "stale stage count",
+    "hosted coordinate beyond the pending set",
+    "overlapping hosted coordinates"
+  ]
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/INDEPENDENT-SCALAR311.log b/research/r480-sliced-minimal-frames/evidence/INDEPENDENT-SCALAR311.log
new file mode 100644
index 0000000000..11f2e08476
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/INDEPENDENT-SCALAR311.log
@@ -0,0 +1,7 @@
+labels: 120 distinct lines of norm 2 (pairwise |ip| <= 1), spanning rank 8
+legality + exact F2 on all 720 columns: PASS (gates D/A/B/C = 120/2481/240/2481)
+relaxed check: 712 gates have a source register as destination; all 120 sources restored exactly
+nondegeneracy: 674 used frames, all Gram dets > 0 (positive-definite form); primes in dets: [2]
+histogram {1: 6085, 2: 7105, 4: 1085, 5: 5, 6: 650, 7: 240, 8: 45, 14: 240, 18: 240} m 40 W 960 mass 38280 deficit 120 (4v-5L = 120 ) maxchild 18
+no envelope  saving in [0.00128876435615175376, 0.00128876435615175376]
+bad envelope saving in [0.00128876435528617710, 0.00128876435528617710]
diff --git a/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.json b/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.json
new file mode 100644
index 0000000000..a092d03fc5
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.json
@@ -0,0 +1,8 @@
+{
+  "status": "PRESERVED_REPLAY_METADATA_FAILURE",
+  "field": "fee_delta",
+  "expected": -0.04495826477563014,
+  "actual": -0.044958264775633694,
+  "all_other_nontiming_fields_identical": true,
+  "repair": "Exclude only floating discovery fee diagnostic from frozen receipt equality; require all actual frames, exact profiles, candidate word and rational moments unchanged."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.log b/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.log
new file mode 100644
index 0000000000..f6e9a07a4a
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/INITIAL-311-DIAGNOSTIC-MISMATCH.log
@@ -0,0 +1,8 @@
+slice PASS
+minimum PASS
+price PASS
+Traceback (most recent call last):
+  File "/Volumes/SP AI 01_16/CodexWorkspaces/imb-retime2/frontier390/r480-minimal-publication/research/r480-sliced-minimal-frames/verify.py", line 20, in <module>
+    m=json.loads((O/'bit510-sliced-minimal-frames/RESULT.json').read_text());e=json.loads((P/'evidence/MINIMUM-FRAMES.json').read_text());assert {k:v for k,v in m.items()if k!='seconds'}=={k:v for k,v in e.items()if k!='seconds'}
+                                                                                                                                                 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+AssertionError
diff --git a/research/r480-sliced-minimal-frames/evidence/INTERFACE311.json b/research/r480-sliced-minimal-frames/evidence/INTERFACE311.json
new file mode 100644
index 0000000000..b4227b9d9e
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/INTERFACE311.json
@@ -0,0 +1,13 @@
+{
+ "status": "PASS_PR425_INTERFACE_HYPOTHESES",
+ "source_destination_gates_A": 356,
+ "sources_rewritten": 69,
+ "source_write_frame_dims": {
+  "2": 12,
+  "3": 7,
+  "4": 337
+ },
+ "phase_B_reads_of_sources": 120,
+ "cleanup_gates": 2481,
+ "cleanup_source_gates": 356
+}
\ No newline at end of file
diff --git a/research/r480-sliced-minimal-frames/evidence/MINIMUM-FRAMES.json b/research/r480-sliced-minimal-frames/evidence/MINIMUM-FRAMES.json
new file mode 100644
index 0000000000..c3f1918deb
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/MINIMUM-FRAMES.json
@@ -0,0 +1,174 @@
+{
+  "status": "COMPLETE_EXACT_CAUSAL_MINIMUM_FRAME_ASSIGNMENT",
+  "moves": [
+    {
+      "section": "A",
+      "index": 2116,
+      "old": 482,
+      "new": 122,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2210,
+      "old": 484,
+      "new": 158,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2236,
+      "old": 485,
+      "new": 176,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2271,
+      "old": 486,
+      "new": 194,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2300,
+      "old": 487,
+      "new": 212,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2330,
+      "old": 488,
+      "new": 230,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2358,
+      "old": 489,
+      "new": 248,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2359,
+      "old": 489,
+      "new": 672,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2360,
+      "old": 489,
+      "new": 672,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2367,
+      "old": 489,
+      "new": 253,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2370,
+      "old": 489,
+      "new": 672,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2395,
+      "old": 490,
+      "new": 266,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2396,
+      "old": 490,
+      "new": 673,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2397,
+      "old": 490,
+      "new": 673,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2404,
+      "old": 490,
+      "new": 271,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2434,
+      "old": 491,
+      "new": 284,
+      "old_rank": 4,
+      "new_rank": 2
+    },
+    {
+      "section": "A",
+      "index": 2435,
+      "old": 491,
+      "new": 674,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2436,
+      "old": 491,
+      "new": 674,
+      "old_rank": 4,
+      "new_rank": 3
+    },
+    {
+      "section": "A",
+      "index": 2443,
+      "old": 491,
+      "new": 289,
+      "old_rank": 4,
+      "new_rank": 2
+    }
+  ],
+  "new_frames": 3,
+  "baseline_chain_profile": {
+    "1": 1200,
+    "2": 1440,
+    "4": 180,
+    "6": 120
+  },
+  "new_chain_profile": {
+    "1": 1217,
+    "2": 1421,
+    "6": 130,
+    "4": 169,
+    "5": 1
+  },
+  "fee_delta": -0.04495826477563014,
+  "seconds": 2.2678134579909965,
+  "scope": "Least rational span of every preceding operand frame at each A/B gate, with source starts, center cut and cleanup/final frames fixed. All newly needed root closures generated exactly. This is the least feasible frame assignment in subspace inclusion for this fixed chronological word, not a moment optimum. No new physical admission or kappa claimed."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/NATIVE-LEGALITY314.json b/research/r480-sliced-minimal-frames/evidence/NATIVE-LEGALITY314.json
new file mode 100644
index 0000000000..2efae837e5
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/NATIVE-LEGALITY314.json
@@ -0,0 +1,50 @@
+{
+  "all_final_frames_pass": true,
+  "common_frame_ADDs": 33492,
+  "containment_pairs_checked": 2412,
+  "copies": 9,
+  "donor_intersections": 0,
+  "erasures": 9,
+  "max_inherited_integer_abs": 2,
+  "nested_MOVEs": 2950,
+  "new_frame_ids_collision": false,
+  "nondegeneracy_checked_by_separate_exact_bank_audit": true,
+  "old_max_frame_id": 674,
+  "paid_histogram": [
+    [
+      1,
+      1286
+    ],
+    [
+      2,
+      1319
+    ],
+    [
+      3,
+      45
+    ],
+    [
+      4,
+      169
+    ],
+    [
+      5,
+      1
+    ],
+    [
+      6,
+      130
+    ],
+    [
+      8,
+      9
+    ]
+  ],
+  "rank_mass": 5592,
+  "records": 36460,
+  "scope": "Independent chronological geometry/copy/source-owner audit. F2 all-column replay is in the separately freshly compiled transform receipt; address transform theorem and bank completion remain separate.",
+  "selected_helpers": 0,
+  "selected_pairs": 0,
+  "source_owner_intersections": 0,
+  "status": "PASS_INDEPENDENT_COHORT_LEGALITY_SOURCE_OWNERS_AND_LIFECYCLE"
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/NINE-COLUMNS314.json b/research/r480-sliced-minimal-frames/evidence/NINE-COLUMNS314.json
new file mode 100644
index 0000000000..cb973e9e83
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/NINE-COLUMNS314.json
@@ -0,0 +1,196 @@
+{
+  "bridge_additions": 2160,
+  "copied_center_original_source_immutable": true,
+  "formal_columns_checked": 1440,
+  "fresh_center_copies": 81,
+  "global_live_registers": 1440,
+  "independent_dirty_registers": 480,
+  "local_odd_scalar_additions": 33492,
+  "local_raw_H": [
+    [
+      1,
+      1286
+    ],
+    [
+      2,
+      1319
+    ],
+    [
+      3,
+      45
+    ],
+    [
+      4,
+      169
+    ],
+    [
+      5,
+      1
+    ],
+    [
+      6,
+      130
+    ],
+    [
+      8,
+      9
+    ]
+  ],
+  "local_registers": 720,
+  "local_scalar_additions": 33492,
+  "local_scalar_stream": "/Volumes/SP AI 01_16/CodexWorkspaces/imb-retime2/frontier390/trace541-frozen-admission/cout/COHORT249-RECORDS.bin",
+  "negative_controls": [
+    {
+      "omit_bridge": 0,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 1,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 2,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 3,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 4,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 5,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 6,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 7,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 8,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 9,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 10,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 11,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 12,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 13,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 14,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 15,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 16,
+      "rejected": true
+    },
+    {
+      "omit_bridge": 17,
+      "rejected": true
+    }
+  ],
+  "new_Q_categories": [
+    28,
+    29
+  ],
+  "payload_prefix_bits": 101,
+  "payload_prefix_upper": "2245964444139092419234253795328",
+  "reverse_fresh_center_copies": 36,
+  "scope": "All independent physical input columns over F2, including all arbitrary helper dirt, restored; external copied-center work closes to zero. New cohort scalar lift uses actual signed coefficients and computed cancellation-free bounds; unchanged source527 signed decoder remains separate defining evidence. Physical projector routing and paid rank/bank receipts are independently bound, not inferred from scalar check.",
+  "seconds": 0.005357833,
+  "signed_lift_prefix": [
+    {
+      "bits": 18,
+      "cancellation_free_signed_row_l1_upper": "167007",
+      "reverse": false
+    },
+    {
+      "bits": 22,
+      "cancellation_free_signed_row_l1_upper": "2744792",
+      "reverse": true
+    }
+  ],
+  "signed_reverse_semantics": "Reverse ADD coefficient negated; reverse DISCARD is fresh COPY from recorded unchanged original center; reverse COPY is DISCARD. Each such block is inverse over every commutative coefficient ring, by elementary-shear induction.",
+  "stage_additions": 301428,
+  "stages": [
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": false,
+      "stage": 0
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": true,
+      "stage": 1
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": false,
+      "stage": 2
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": true,
+      "stage": 3
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": false,
+      "stage": 4
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": true,
+      "stage": 5
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": false,
+      "stage": 6
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": true,
+      "stage": 7
+    },
+    {
+      "fresh_copies": 9,
+      "scalar_additions": 33492,
+      "signed_inverse": false,
+      "stage": 8
+    }
+  ],
+  "status": "PASS_NEW_COHORT249_ALL_BRIDGED_STAGE_FORMAL_COLUMNS_AND_PREFIX_BILL"
+}
\ No newline at end of file
diff --git a/research/r480-sliced-minimal-frames/evidence/PACKAGED-FULL311.json b/research/r480-sliced-minimal-frames/evidence/PACKAGED-FULL311.json
new file mode 100644
index 0000000000..1d4ded07d0
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/PACKAGED-FULL311.json
@@ -0,0 +1,11 @@
+{
+  "status": "PASS_PACKAGED_CONSTRUCTION_PHYSICAL_ADMISSION_AND_EXACT_PRICE",
+  "kappa": "0.001587467473762886",
+  "bit": "0.001590005077309865651707",
+  "word_sha256": "2928c90b7a90a8ba733a1dbf4eec3de064c3bc9f91dbde6cdb6392913cdf556a",
+  "manifest_sha256": "df123ccbeb5ad0b46cc58ec6a8a53e83d2ba0b1fe4758947170e1052279fdbcc",
+  "python": "3.11.15 (main, Mar  3 2026, 00:52:57) [Clang 17.0.0 (clang-1700.6.3.2)]",
+  "seconds": 101.19323466706555,
+  "scope": "Fresh complete finite record, geometry, owner and invoice replay plus exact source reconstruction and two-engine pricing. Complex535 cited rather than rebuilt; inherited all-size assumptions remain conditional.",
+  "physical_receipt": "trace541-frozen-admission/RESULT.json"
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/PARENT-AUDIT314.json b/research/r480-sliced-minimal-frames/evidence/PARENT-AUDIT314.json
new file mode 100644
index 0000000000..b6924fca09
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/PARENT-AUDIT314.json
@@ -0,0 +1,44 @@
+{
+  "status": "PASS_INDEPENDENT_COMPRESSED_BIT_TEMPLATE_AUDIT",
+  "template_sha256": "96841716c7901ae12d572113ffa3b7c14c1c3c9568a7c044f114941e7b5213e3",
+  "source_frames": 675,
+  "exact_nested_frame_pairs": 2412,
+  "exact_reflected_pairs": 2412,
+  "orthogonal_lift_argument": "Adding the same orthogonal E_s leaves P_V-P_U unchanged; each audited projector is therefore the literal child projector in every one of the nine windows.",
+  "exact_bank_families": 17280,
+  "residual_isometries": 1,
+  "entrance_rank_total": 0,
+  "mutations_rejected": [
+    "omit an idle child",
+    "omit a bridge",
+    "omit terminal exchange",
+    "remove a local invocation",
+    "understate live stock",
+    "omit a local paid move",
+    "wrong reflected copy charge",
+    "alter residual orthogonal basis",
+    "misstate replication count",
+    "misstate stage count",
+    "alias data across stages",
+    "delay an idle advance past its level",
+    "truncate the last idle advance",
+    "re-use a helper bank slot",
+    "wrong residual frame",
+    "alias data across replica groups",
+    "omit the fourth replica group"
+  ],
+  "profile": {
+    "1": 370368,
+    "2": 379872,
+    "3": 12960,
+    "4": 48672,
+    "5": 288,
+    "6": 37440,
+    "7": 23040,
+    "8": 2592,
+    "14": 15360,
+    "28": 15360,
+    "42": 15360
+  },
+  "seconds": 8.728064374998212
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/PHYSICAL311.json b/research/r480-sliced-minimal-frames/evidence/PHYSICAL311.json
new file mode 100644
index 0000000000..6f1be6e126
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/PHYSICAL311.json
@@ -0,0 +1,33 @@
+{
+  "status": "PASS_FRESH_TRACE_AND_FROZEN_FRAMES_RECORDS_HOSTING_AND_INVOICE",
+  "seconds": 91.31745012500323,
+  "word_sha256": "2928c90b7a90a8ba733a1dbf4eec3de064c3bc9f91dbde6cdb6392913cdf556a",
+  "profile": {
+    "1": 370368,
+    "2": 395232,
+    "3": 12960,
+    "4": 64032,
+    "5": 288,
+    "6": 52800,
+    "7": 23040,
+    "8": 2592
+  },
+  "kappa": "0.001587467473762886",
+  "bit_invoice": {
+    "live": 30720,
+    "physical": 30729,
+    "maxchild": 8,
+    "rank_mass": 1956096,
+    "hosted_residuals": 138240,
+    "bank_assignments": 0,
+    "prefix_bits": 191,
+    "charged_prefix_bits": 192,
+    "coefficient_bits": 191,
+    "two_level_coefficient_bits": 421,
+    "charged_two_level_coefficient_bits": 512,
+    "row_coefficient": 18355,
+    "halving_depth": 2
+  },
+  "steps": 19,
+  "scope": "Fresh finite record/formal-column/geometry/template/whole-owner/invoice replay, with inherited mutation controls. Cited complex535 operator unchanged. Exact sharp-charge prospective profile now matches physically emitted audited profile and retained finite allowances. All-size supplier/compiler/realization/hosting/analytic assumptions remain conditional."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/PHYSICAL314.json b/research/r480-sliced-minimal-frames/evidence/PHYSICAL314.json
new file mode 100644
index 0000000000..22b082d1c6
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/PHYSICAL314.json
@@ -0,0 +1,33 @@
+{
+  "status": "PASS_FRESH_TRACE_AND_FROZEN_FRAMES_RECORDS_HOSTING_AND_INVOICE",
+  "seconds": 75.72498749999795,
+  "word_sha256": "3dddba9a13c96d54fa6aa3ff238711808662401a6fec7960ac34c4f885bfbf71",
+  "profile": {
+    "1": 370368,
+    "2": 395232,
+    "3": 12960,
+    "4": 64032,
+    "5": 288,
+    "6": 52800,
+    "7": 23040,
+    "8": 2592
+  },
+  "kappa": "0.001587467473762886",
+  "bit_invoice": {
+    "live": 30720,
+    "physical": 30729,
+    "maxchild": 8,
+    "rank_mass": 1956096,
+    "hosted_residuals": 138240,
+    "bank_assignments": 0,
+    "prefix_bits": 191,
+    "charged_prefix_bits": 192,
+    "coefficient_bits": 191,
+    "two_level_coefficient_bits": 421,
+    "charged_two_level_coefficient_bits": 512,
+    "row_coefficient": 18355,
+    "halving_depth": 2
+  },
+  "steps": 19,
+  "scope": "Fresh finite record/formal-column/geometry/template/whole-owner/invoice replay, with inherited mutation controls. Cited complex535 operator unchanged. Exact sharp-charge prospective profile now matches physically emitted audited profile and retained finite allowances. All-size supplier/compiler/realization/hosting/analytic assumptions remain conditional."
+}
diff --git a/research/r480-sliced-minimal-frames/evidence/SLICE.json b/research/r480-sliced-minimal-frames/evidence/SLICE.json
new file mode 100644
index 0000000000..ad1c0bccab
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/evidence/SLICE.json
@@ -0,0 +1,1843 @@
+{
+  "removed_indices": [
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diff --git a/research/r480-sliced-minimal-frames/inputs/BASE-CANDIDATE.json.gz b/research/r480-sliced-minimal-frames/inputs/BASE-CANDIDATE.json.gz
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index 0000000000..fcdfbb3039
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diff --git a/research/r480-sliced-minimal-frames/inputs/BASE-EXPECTED.json b/research/r480-sliced-minimal-frames/inputs/BASE-EXPECTED.json
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index 0000000000..8b13335416
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+    "artificial_boundary": "219147375760787436543173497097657585767083024421597787683439199344288696387331898169409220156965261654808690993197249691438650404084578946404925805261560637047382340621141071394204410450431833690372772167/31299671970098429562523265013608834335142100005186809350970113980292955360089763178349920237116450057865933863034822614967365650680815058224020460782031873776671833478484889840994059112000000000000000000",
+    "literal_scalar_guard": "1",
+    "row_product_gap": "23971789/25",
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+    "coordinate_movement_above_kappa": "19707112392554718152238451423365096829478199549067936187451472336975894815697532154011068743909530791487896193472803424388462590710162853892120689388769875742611/7812500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "compact_phase_layer_above_kappa": "96149291066043471787781496002524948038242181288235931563150500748989983907430815926527985727251539133728005627832877126607922879761770018944423156843324883774853437816999957070997/7824917992524607390630816253402208583785525001296702337742528495073238840022440794587480059279112514466483465758705653741841412670203764556005115195507968444167958369621222460248514778000000000000000",
+    "bulk_exposure_above_kappa": "19707112392554718152238451423365096829478199549067936187451472336975894815697532154011068743909530791487896193472803424388462590710162853892120689388769875742611/7812500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
+    "Gaussian_arithmetic_above_kappa": "12121974153051084973097350364113412667831256227097997232866394745605818195663384455587332031827674928234637221149389989119193810524352726909785365420635252201351351305699833518656760167/31299671970098429562523265013608834335142100005186809350970113980292955360089763178349920237116450057865933863034822614967365650680815058224020460782031873776671833478484889840994059112000000000000000000",
+    "scalar_work_above_kappa": "27771810138100299754358982870917829835402306229691401908351451735752295875708266044762292150385899957167604152666801296602876635864760256021795595811651189089687268044942278439153789723/31299671970098429562523265013608834335142100005186809350970113980292955360089763178349920237116450057865933863034822614967365650680815058224020460782031873776671833478484889840994059112000000000000000000",
+    "dimension_above_kappa": "3900041003737696300832625515094221139316536374054015597040619343755849714641619100617374382325379548023762289623829199929185029358167383156222110177228783065390789991365035433847148514719111208659755889/3912458996262303695315408126701104291892762500648351168871264247536619420011220397293740029639556257233241732879352826870920706335101882278002557597753984222083979184810611230124257389000000000000000000"
+  },
+  "scope": "Derived global histogram adds288 copies of exact local profile delta to pinned PR510 profile. Same480 width8 helpers and original owner schedule proposed. Physical template, chronological finite invoice and changed-frame geometric checks remain required; old outer finite constants used prospectively."
+}
diff --git a/research/r480-sliced-minimal-frames/inputs/CANDIDATE.json.gz b/research/r480-sliced-minimal-frames/inputs/CANDIDATE.json.gz
new file mode 100644
index 0000000000..268fbbd46b
Binary files /dev/null and b/research/r480-sliced-minimal-frames/inputs/CANDIDATE.json.gz differ
diff --git a/research/r480-sliced-minimal-frames/inputs/ORIGINAL.json.gz b/research/r480-sliced-minimal-frames/inputs/ORIGINAL.json.gz
new file mode 100644
index 0000000000..8e59c4b538
Binary files /dev/null and b/research/r480-sliced-minimal-frames/inputs/ORIGINAL.json.gz differ
diff --git a/research/r480-sliced-minimal-frames/inputs/PR510-EXPECTED.json b/research/r480-sliced-minimal-frames/inputs/PR510-EXPECTED.json
new file mode 100644
index 0000000000..c89fe719e8
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/inputs/PR510-EXPECTED.json
@@ -0,0 +1,102 @@
+{
+ "nine": {
+  "bit_lower": "396861776523137034277/250000000000000000000000",
+  "bit_lower_decimal": "0.001587447106092548137108",
+  "profile": {
+   "1": 345600,
+   "2": 430080,
+   "4": 67200,
+   "6": 49920,
+   "7": 23040,
+   "8": 2592
+  },
+  "bit": {
+   "live": 30720,
+   "physical": 30729,
+   "maxchild": 8,
+   "rank_mass": 1956096,
+   "hosted_residuals": 138240,
+   "bank_assignments": 0,
+   "prefix_bits": 193,
+   "charged_prefix_bits": 256,
+   "coefficient_bits": 193,
+   "two_level_coefficient_bits": 425,
+   "charged_two_level_coefficient_bits": 512,
+   "row_coefficient": 18355,
+   "halving_depth": 2
+  },
+  "kappa": {
+   "pr484": {
+    "kappa": "662515748707749/500000000000000000",
+    "kappa_decimal": "0.001325031497415498",
+    "binding_component": "complex",
+    "complex_lower": "663394767670133/500000000000000000"
+   },
+   "pr471": {
+    "kappa": "260706019542937/200000000000000000",
+    "kappa_decimal": "0.001303530097714685",
+    "binding_component": "complex",
+    "complex_lower": "1305231506267591/1000000000000000000"
+   },
+   "pr458": {
+    "kappa": "323880830669511/250000000000000000",
+    "kappa_decimal": "0.001295523322678044",
+    "binding_component": "complex",
+    "complex_lower": "1297203880559577927527/1000000000000000000000000"
+   }
+  },
+  "nine_stage_payload_bits": 102,
+  "stages": 9,
+  "replicas": 32,
+  "m": 64
+ },
+ "parent_r480": {
+  "pr": 503,
+  "bit_lower": "393688338238452162729/250000000000000000000000",
+  "bit_lower_decimal": "0.001574753352953808650916",
+  "stages": 11
+ },
+ "parent_r489": {
+  "pr": 493,
+  "bit_lower": "391360506745414227149/250000000000000000000000",
+  "bit_lower_decimal": "0.001565442026981656908596",
+  "stages": 11
+ },
+ "findings": {
+  "lp_helpers_per_4frame": 16,
+  "lp_total": 480,
+  "levels": {
+   "1": {
+    "frames": 120,
+    "helper_visits": 240
+   },
+   "2": {
+    "frames": 180,
+    "helper_visits": 480
+   },
+   "4": {
+    "frames": 60,
+    "helper_visits": 360
+   },
+   "6": {
+    "frames": 180,
+    "helper_visits": 480
+   },
+   "7": {
+    "frames": 120,
+    "helper_visits": 240
+   }
+  },
+  "cells": {
+   "2,2,0": 60,
+   "3,0,0": 120
+  },
+  "d6": {
+   "2,1,2": 60,
+   "3,0,0": 120
+  },
+  "pivots_max": 0,
+  "line_classes": 120,
+  "class_nullity": 0
+ }
+}
\ No newline at end of file
diff --git a/research/r480-sliced-minimal-frames/inputs/PR535-EXPECTED.json b/research/r480-sliced-minimal-frames/inputs/PR535-EXPECTED.json
new file mode 100644
index 0000000000..2c5a1a3262
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/inputs/PR535-EXPECTED.json
@@ -0,0 +1,62 @@
+{
+  "status": "PASS_ASYNC_COMPLEX_AND_CITED_BIT_EXACT_ASSEMBLY",
+  "kappa": "1584931111785787/1000000000000000000",
+  "kappa_decimal": "0.001584931111785787",
+  "complex_lower": "1589991533606044174031/1000000000000000000000000",
+  "complex_decimal": "0.001589991533606044174031",
+  "bit_lower": "396861776523137034277/250000000000000000000000",
+  "bit_decimal": "0.001587447106092548137108",
+  "binding_component": "bit",
+  "improvements": {
+    "PR531": {
+      "baseline": "1517760977820331/1000000000000000000",
+      "relative_gain": "67170133965456/1517760977820331",
+      "percent_display": "4.42560686"
+    },
+    "rejected_POST208_checkpoint": {
+      "baseline": "771820390859371/500000000000000000",
+      "relative_gain": "41290330067045/1543640781718742",
+      "percent_display": "2.67486649"
+    },
+    "PR533": {
+      "baseline": "381051987698311/250000000000000000",
+      "relative_gain": "60723160992543/1524207950793244",
+      "percent_display": "3.98391577"
+    }
+  },
+  "global_template_content_sha256": "b9f4b60e56e5153c1fc89370e9ceb09f24b23e1c8ff56f2b47ef9d867aa04f2c",
+  "raw_word_content_sha256": "b9110e8ddb770a3874c2df2019e04a98c1a25684072f7f8e5772653f59f15db2",
+  "canonical_word_content_sha256": "7c84ac65c8b37173aa4293934f041915895e8ec30595844cc1534e2a19ae0d25",
+  "global_profile": {
+    "1": 158976,
+    "2": 52864,
+    "3": 15200,
+    "4": 20160,
+    "5": 4032,
+    "6": 4032,
+    "7": 352,
+    "8": 576,
+    "48": 1632
+  },
+  "local_profile": {
+    "1": 2208,
+    "2": 728,
+    "3": 184,
+    "4": 280,
+    "5": 56,
+    "6": 56,
+    "8": 8
+  },
+  "local_rank_mass": 6016,
+  "entrance_rank": 376,
+  "width_inventory": {
+    "5": 56,
+    "8": 152,
+    "9": 296
+  },
+  "global_prefix_bits": 225,
+  "live": 7168,
+  "physical": 7682,
+  "deficit": 2560,
+  "scope": "Conditional finite witness under the retained all-size compiler, common-label/mutable-basis, whole-call hosting, restored-work, precision/tape and analytic-transfer contracts. Fresh complex admission and finite invoice are separately hash-bound; no unconditional theorem is claimed."
+}
diff --git a/research/r480-sliced-minimal-frames/inputs/PR539-EXPECTED.json b/research/r480-sliced-minimal-frames/inputs/PR539-EXPECTED.json
new file mode 100644
index 0000000000..6f09653ae5
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/inputs/PR539-EXPECTED.json
@@ -0,0 +1,9 @@
+{
+  "kappa": "49529097302971/31250000000000000",
+  "kappa_decimal": "0.001584931113695072",
+  "bit_lower": "793723554003949831983/500000000000000000000000",
+  "bit_decimal": "0.001587447108007899663966",
+  "old_kappa": "1584931111785787/1000000000000000000",
+  "fallback_per_edge": 98682,
+  "bad_fraction": "1/2305843009213693952"
+}
diff --git a/research/r480-sliced-minimal-frames/inputs/PR541-ACQUISITION.json b/research/r480-sliced-minimal-frames/inputs/PR541-ACQUISITION.json
new file mode 100644
index 0000000000..09a58d33dd
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/inputs/PR541-ACQUISITION.json
@@ -0,0 +1,66 @@
+{
+  "head": "6fd04b2e266b7915b69f96f27758cfce520aa40a",
+  "partial": true,
+  "files": [
+    {
+      "path": "research/trace-curvature-preload/EXPECTED.json",
+      "git_blob": "c04f2464c81810d84bfbcf40f03ee3e51e3b8430",
+      "sha256": "354268fcffbf27c1e93006e9db648a7415ea8ecec8a8b77e6ce73eb3844f0294"
+    },
+    {
+      "path": "research/trace-curvature-preload/README.md",
+      "git_blob": "71ceaf595896f2d4175444f5f7f30cd92a4d5652",
+      "sha256": "046439afa0967319aec79e71f7b1cf7a9a5da084bbc1b584d5000b69c9c2dad9"
+    },
+    {
+      "path": "research/trace-curvature-preload/code/audit_curvature.py",
+      "git_blob": "64090bd0405cbdb14ca15a65125f53332b2f1936",
+      "sha256": "5b81ba18969e84c3d10e647dde2ddb1cda81749c68a8bdd66db7d1d09cd363b2"
+    },
+    {
+      "path": "research/trace-curvature-preload/code/build_preload60.py",
+      "git_blob": "5a1b518a585585244f717e6315baf72a34a53d47",
+      "sha256": "105641136b565003118f362bb67086e1eeb4910aed6af593a011f60c2b2962b6"
+    },
+    {
+      "path": "research/trace-curvature-preload/docs/CURVATURE-CONE.md",
+      "git_blob": "1bcbd47f861a4269cd0fded29816a7d9160edfce",
+      "sha256": "2ac9e555d6d4afb5b079542679d2aeeb4a76132a3a0ed7df8720856c731405ce"
+    },
+    {
+      "path": "research/trace-curvature-preload/docs/PROVENANCE.md",
+      "git_blob": "2a0179ac67b399592a2e3736e496d7ec7a5e16b7",
+      "sha256": "72e4e91c544d7cea5959d22cd65809a97598fdfa9b031f18183cbbb27bc1c089"
+    },
+    {
+      "path": "research/trace-curvature-preload/docs/TRACE-REFINEMENT.md",
+      "git_blob": "62096bee18fa35f1d7daca52a65aced09dbb2f9d",
+      "sha256": "e24cd091820f33795aa06f3cc318d61aed6dd17dfc55b0d24d2c3285a1a8ca6a"
+    },
+    {
+      "path": "research/trace-curvature-preload/evidence/CONSTRUCTION.json",
+      "git_blob": "35363afc5f5a97ee108b45918e002d23cd0520a1",
+      "sha256": "9659874b176cf8e845fe03b53dc6bed8f139903f8012e785b45d46fe0c618848"
+    },
+    {
+      "path": "research/trace-curvature-preload/evidence/REPLAY.json",
+      "git_blob": "9dbc4775f7c12d9053370a046c5371404b682896",
+      "sha256": "3d4c938374867bda9594cc7e75514c2a63c81d23f87fdfb5ab09163c5b9ffd33"
+    },
+    {
+      "path": "research/trace-curvature-preload/evidence/WORD.json.gz",
+      "git_blob": "b69c146ed0a441f495342596c3f6fa81fff937aa",
+      "sha256": "db5cd7ab49e2e58a2924a2a9e68ee69d8165b4d5c008b6f546577f4791019dca"
+    },
+    {
+      "path": "research/trace-curvature-preload/inputs/UPSTREAM-PINS.json",
+      "git_blob": "2416dd5811ba8b6d517fe9eb2bcb41aa30303ac2",
+      "sha256": "1a7f0ea4c0916e3d95cc976342a1e8fc9b02d925af9295d81c4af7b0a4ab0255"
+    },
+    {
+      "path": "research/trace-curvature-preload/verify.py",
+      "git_blob": "171bc20f7310b27e7419418dcb2c0c2bb175dcef",
+      "sha256": "a9dad1d31cb5a0bf88b3fb14955773db740d52fe95a48fa08450f20b8c19ec94"
+    }
+  ]
+}
diff --git a/research/r480-sliced-minimal-frames/inputs/PR541-TRACE-PROOF.md b/research/r480-sliced-minimal-frames/inputs/PR541-TRACE-PROOF.md
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index 0000000000..62096bee18
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/inputs/PR541-TRACE-PROOF.md
@@ -0,0 +1,92 @@
+# Frame-sensitive trace refinement and the preload60 construction
+
+## Sufficient meta-theorem
+
+Let a finite reversible scalar program act on registers indexed by I over a field K. A scalar addition is T(t,s,q)=I+q e_t e_s^T, with t != s. Each register has an explicit initial frame, an explicit terminal frame, and a monotone chain of finite subspaces at its actual uses. Certain observation checkpoints (for example retained totals at a scatter) are fixed.
+
+Construct another program by the following certified operations:
+
+1. Factor a multi-source addition into elementary additions with the same destination and unchanged coefficients.
+2. Permute elementary additions only through explicitly checked commuting gates, without crossing protected observation checkpoints.
+3. Assign a new common frame to each relocated addition. Require that every touched register's predecessor frame is contained in it and that it is contained in that register's successor frame, including fixed initial, terminal and observation frames.
+4. Rebuild the inverse background cleanup literally. Recompute every actual frame transition, retained-copy rank, scalar prefix bound and finite realization invoice.
+
+Then the new scalar program has exactly the same complete arbitrary-input endpoint. Its framed program satisfies the retained native frame-realization interface, and its exact recursive profile is the profile recomputed from those new chains, together with all retained-copy children. If the entrance and terminal signatures stay fixed, every old whole-owner allocation remains usable with unchanged residual-width and idle-remainder accounting. The changed physical finite realization still requires its actual phase/compiler and prefix checks; the theorem does not make those checks or inherited all-size hypotheses disappear.
+
+This is a calculus of representations of the SAME scalar word: it searches the trace-equivalence class and then selects a different legal frame labeling. A fixed event-order frame optimizer cannot expose every member of that class. A scalar-only optimizer cannot see the cost of its frame chains. Their joint finite certificate is the sufficient object.
+
+## Algebraic proof
+
+For t distinct from all s_i, the elementary additions to destination t commute, since e_(s_i)^T e_t=0. Hence
+
+    I + sum_i q_i e_t e_(s_i)^T = product_i T(t,s_i,q_i).
+
+For distinct-source/destination elementary additions T(t,s,q) and T(u,v,r), the sufficient commuting condition is t != v and u != s. Expanding both products gives the same identity, two linear terms and zero cross terms. (There are further special commuting cases; they are not needed.) A scale on register u commutes with T(t,s,q) when u is neither t nor s.
+
+Every allowed adjacent exchange therefore preserves the entire linear operator. Induction on the explicit exchange list proves equality of the reordered body. It holds on arbitrary source, target and dirty-helper columns, not merely clean inputs. A word can be moved through a whole interval if it commutes with every gate crossed. Protected checkpoints are preserved because no move crosses them.
+
+If background updates do not read target registers, the background transformation is independent of targets. Reversing the order and inverting each actual new background gate gives its exact inverse, restoring all source/helper inputs. For an addition this negates its coefficient; over F2 the coefficient is unchanged. The body's target output is unchanged by the trace identity. Thus the full dirty-register endpoint is unchanged. Existing compensation can be retained because the body map itself, on every column, is unchanged.
+
+At a relocated instruction, the explicit predecessor/common/successor inclusions establish native frame legality. All other chains retain their previous inclusions. Applying the existing compiler interface to the actual changed word gives the same final labeled endpoint, with exactly the newly listed transition ranks. This last statement is conditional on that pre-existing all-size compiler interface; finite column replay is not an independent proof of the all-size interface.
+
+## Critical pair: splitting plus scheduling plus reframing
+
+Consider a two-source cell h += x+y first formed at a rank2 frame containing the rank1 source frames. Suppose its x addition commutes through the entire prefix, h initially has frame0, and the prefix does not observe h. Factor the gate into h += x followed by h += y. Move h += x to the beginning and perform it at x's rank1 frame. The later h += y remains at the original rank2 frame.
+
+The scalar map is identical. In the actual selected instance, the helper's former rank2 climb splits into two rank1 climbs. Meanwhile x no longer needs a formerly unique intermediate rank2 use between rank1 and rank4: its two climbs of ranks1 and2 coalesce into a rank3 climb. Therefore the net child-profile change is
+
+    +delta1 - 2 delta2 + delta3.
+
+This is why neither a fixed-position scalar rewrite nor a fixed-order frame-coalescing pass alone finds the gain. The seam is moved before the frame assignment is simplified. The general theorem does not assume this particular delta: it always recomputes the actual changed chains.
+
+For every 0<a<1, strict concavity of r^a gives
+
+    1^a + 3^a < 2*2^a.
+
+The delta has zero rank mass, so it strictly reduces the recurrence moment for every saving exponent 0<b<1 when a=1-b. The theorem is sufficient for strict moment improvement in every compatible context with unchanged residual signatures. It says nothing about a numerical bound until the changed finite invoice and outer assembly pass.
+
+## Batch compatibility rather than a hidden optimizer
+
+In the selected 60 cells, each destination is a different helper, each preloaded source is an original source register, and no helper destination is itself a preloaded source. The 60 new prefix additions therefore commute with each other. For each chosen cell, the constructor explicitly verifies that no preceding old gate reads its helper destination or writes its preloaded source. Consequently its elementary addition commutes across that entire old prefix.
+
+The constructor only chooses gates with exactly two original sources. It checks the source pin, emits all 60 rank1 preloads in the original cell order, replaces each old grouped cell by its remaining term, and regenerates C as the literal inverse of the changed A. B, D, scatter, start and final frames are unchanged. No new frame type, extra helper, non-dyadic coefficient, oracle, probabilistic claim or allocation splitter is introduced.
+
+## Actual finite witness
+
+Input is the pinned PR510 R480 bit word, canonical JSON SHA256
+
+    8639e22008c24a5690dd250eba702fe5fca85ddd036dcbec20baba77b7e213ba.
+
+Portable builder: build_preload60.py --input WORD --out NEWDIR.
+
+It independently checks all720 arbitrary formal columns over F2, exact rational frame-subspace incidence, literal inverse cleanup, all nine retained-total ranks and all unchanged interface fields. The changed canonical word SHA256 is
+
+    cb9c80a62d75c5dfc8bc8933d4b1d44e6b6b816db47c2444bc2dbba6b3a6e454.
+
+The local chain profile changes from {1:1200,2:1440,4:180,6:120} to
+
+    {1:1260,2:1320,3:60,4:180,6:120}.
+
+The nine retained rank8 copy children are unchanged and MUST also be paid. The complete local profile is
+
+    {1:1260,2:1320,3:60,4:180,6:120,8:9}.
+
+All480 helper entrance widths stay8. The current public controller uses nine stages and32 replicas, hence288 calls. Its global delta is +17280 delta1 -34560 delta2 +17280 delta3, with every original idle remainder retained. The resource lane exact moment engine certified c=0.001589735702947998836661 under the existing bad-case envelope. That is a provisional supplier price, subject to the changed official physical/prefix/compiler invoice; it is not itself an achieved outer kappa.
+
+## Bounded discovery and failure actions
+
+The first native screen on the previous complex word tested1,237,484 insertion positions, including1,154,361 legal ones, and found no strict gain. Switching to the actually binding bit supplier tested1,431,669 positions,1,388,956 legal ones, with68,400 positive nominations. The explicit60-move family above was then independently constructed and replayed over exact arithmetic.
+
+A second native screen on that new word tested1,451,628 positions,1,386,258 legal ones, and found no further single-move improvement. This is a bounded single-gate scheduling result, not a universal optimality theorem. The modular rank screen only nominated candidates; exact rational ranks in the constructor establish the submitted witness. No unverified modular result contributes to the numerical claim.
+
+## Sources and novelty scope
+
+Mathlib.LinearAlgebra.Transvection.Basic supplies exactly the structural factorization pattern: transvection.comp_of_right_eq expresses composition with a common displacement vector as addition of linear forms; comp_of_left_eq is the dual common-form rule. These formal source statements were actually read. The DAG's source search located transvection commutation and matrix-unit composition results; source-only hits were not treated as newly checked typed ports.
+
+OpenAI OAI.Combinatorics.Chromatic.GradedAlgebra.InputOrderedProduct.push_partial_list and push_partial_union construct ordered products from pairwise noninteraction hypotheses. Their domain is weighted torus series, so they are structural guidance for the explicit no-interaction interval certificate, not direct formal proofs of this compiler transformation.
+
+Elementary transvection algebra, trace equivalence and concavity are established mathematics. The claimed application here is the new jointly scheduled and reframed60-cell construction, its sufficient contextual resource proof and its actual checked profile gain. We do not claim global originality of those elementary theorems or an unconditional multiplication theorem.
+
+## Subsequent native admission
+
+The unchanged upstream C++ checkers have now passed on the actual changed records: common-frame/lifecycle legality;960 five-stage formal columns;1440 formal columns for the actual nine-stage controller; all18 omitted-bridge controls; conservative payload102bits. Six structural record corruptions and three mutable-source corruptions were rejected. The separate KPAIRS5 run also passed1680 columns and24 bridge controls. Source hashes and exact configuration are recorded in native/NATIVE-REVIEW.json. Global owner routing, complete invoice and outer assembly remain separate until their final receipts.
diff --git a/research/r480-sliced-minimal-frames/inputs/PR541-WORD.json.gz b/research/r480-sliced-minimal-frames/inputs/PR541-WORD.json.gz
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diff --git a/research/r480-sliced-minimal-frames/inputs/three-stage-cover-bit.tex b/research/r480-sliced-minimal-frames/inputs/three-stage-cover-bit.tex
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+\subsection{Atom contract and common endpoint}
+Fix an odd prime $q$, chosen below, and put $R_w=\mathbb Z/q^w\mathbb Z$.
+An atom is a complete consecutive $w$-digit address field. Write $C_n$ for
+reversal of the $n$ atom positions, without reversing digits inside an atom.
+The recursive contract is the involution
+\[
+ F_A(H,D)=(AC_nD,C_nA^{-1}H),
+\]
+where $A$ is a diagonal array of units on the head atoms. Its entries may
+depend on exposed ancestor row fields and the atom index, but not on the
+fresh role fields of this invocation or on the head/data values. Only the
+polynomial-time descriptors generated by the construction are required;
+no constant-cost oracle for arbitrary functions of a prefix is assumed.
+Every call preserves all complete spectator fields and restores its row
+coordinate. The original unweighted call has $A=I$.
+
+For a split idempotent $P$ on the head coordinates set
+\[
+ D_P=\begin{pmatrix}I-P&PC_n\\C_nP&I-C_nPC_n\end{pmatrix},
+ \qquad F=D_I.
+\]
+These are involutions, and $D_P-I$ has split rank $\operatorname{rank}P$.
+Nested commuting-projector identities are polynomial identities and require
+no symmetry of $P$. For $n=mf$ and $G=g\otimes I_f$ define
+\[
+ Q_g=\operatorname{diag}(G,C_nGC_n).
+\]
+Then $Q_gF=FQ_g$ and $Q_gD_PQ_g^{-1}=D_{GPG^{-1}}$.
+Consequently conjugating every base frame by any $g\in\operatorname{GL}_m(R_w)$
+leaves the same full physical endpoint $F$.
+
+Put $Q_A=\operatorname{diag}(A,I)$. Conjugate the entire completed scalar
+word by $Q_A$. Scalar gates mix roles having the same address map, so they
+commute with $Q_A$. A current role routing also commutes with $Q_A$, since
+it changes only fresh fields. Thus only address residuals change, and the
+completed endpoint is $Q_AFQ_A^{-1}=F_A$. More explicitly, when a port
+changes charts from $g$ to $gR_s$, its local label changes from $M$ to
+$R_s^{-1}MR_s$, and
+\[
+ (gR_s)(R_s^{-1}MR_s)(gR_s)^{-1}=gMg^{-1}.
+\]
+This routing is a change of row/chart, not an uncharged dense address map.
+The adapted chart $Q_AQ_gQ_A^{-1}$ commutes with $F_A$. None of these
+conjugations is executed as an extra normalization pass.
+
+\subsection{Explicit rational three-stage geometry}
+Use the retained PR97 local producer with
+\[
+ h=23,\qquad v=1771,\qquad R_{\rm loc}=28866,\qquad \ell=506.
+\]
+Its physical projector ranks and scalar word are unchanged. On
+$\mathcal A=\mathbb Q^{23}$ take the form $H_0=(I-J/9)/2$.
+For a triple $S$ let $q_S=\chi_S$; then $q_S^tH_0q_S=1$.
+Fix $q_0=\chi_{\{1,2,3\}}$ and $K=q_0^\perp$. A rational basis is
+\[
+ k_i=e_i-(e_i^tH_0q_0)q_0,\qquad 2\le i\le23.
+\]
+Projection along $q_0$ is injective on the span of $e_2,\ldots,e_{23}$,
+since $q_0$ has nonzero first coordinate. Thus this is indeed a basis.
+Let $\mathcal B,\mathcal C$ be orthogonal copies of $K$ and put
+$\mathcal E=\mathcal A\perp\mathcal B\perp\mathcal C$, of dimension $m=67$.
+
+Choose a coordinate permutation $\pi_S$ taking $\{1,2,3\}$ to $S$.
+It preserves $H_0$. The rational isometry $R_{12,S}$ fixes $q_S$ and
+$\mathcal C$, and exchanges $\pi_Sk_i\in q_S^\perp\subset\mathcal A$
+with $k_i\in\mathcal B$. Define $R_{23,S}$ by exchanging the same vectors
+with their copies in $\mathcal C$, fixing $q_S$ and $\mathcal B$.
+Both maps are involutions. A hyperedge $(g,S)$ uses the invocations
+$gR_{12,S},g,gR_{23,S}$, with active spaces
+\[
+ g(\mathcal B+\mathbb Qq_S),\qquad g\mathcal A,
+ \qquad g(\mathcal C+\mathbb Qq_S).
+\]
+Their projectors commute; their pairwise and common intersection is the
+projector $P_{gq_S}$, and their sum is $I+2P_{gq_S}$.
+This follows from the orthogonal decomposition
+$\mathbb Qq_S\perp q_S^\perp\perp\mathcal B\perp\mathcal C$ before
+conjugation. These are the cumulative-projector three-stage transfer
+identities: the three scalar shears complete the interchange, with zero
+interstage connectors and no rank-one endpoint correction. Each port
+family visits the full cover bijectively, by right multiplication with
+its displayed involution. The physical ledger is therefore three copies
+of the local auxiliary, data and center histograms, and three copies of
+the exterior histogram $m-\text{nullity}$.
+
+\subsection{One weighted child on a generic residue class}
+Work first on one atom column. Let $g$ be its weighted residual involution,
+of split rank $r$ relative to $I$. If the cross block has generic northeast
+rank profile, unit lower row and column elimination, without normalizing
+pivots, gives
+\[
+ Lg_{XY}R=ED_pF_0^t,
+ \quad E=(e_1,\ldots,e_r),\quad
+ F_0=(e_m,\ldots,e_{m-r+1}),
+\]
+with invertible diagonal $D_p$. Conjugate by $\operatorname{diag}(L,R^{-1})$.
+Split rank and the unit pivot minor give a factorization
+\[
+ g-I=\binom EB(A_0,D_pF_0^t),\qquad
+ A_0E+D_pF_0^tB=-2I_r.
+\]
+For clarity, if $g-I=UV$, then $g^2=I$ gives
+$U(VU+2I)V=0$. The unit minor supplies a left inverse of $U$ and a right
+inverse of $V$, proving the identity over the local ring itself.
+Set
+\[
+ K=-(B+F_0D_p^{-1})E^t+F_0D_p^{-1}A_0(I-EE^t),
+ \qquad S=\begin{pmatrix}I&0\\K&I\end{pmatrix}.
+\]
+The identities $KE=-B-F_0D_p^{-1}$ and
+$D_pF_0^tK=A_0+E^t$ give
+\[
+ SgS^{-1}=\begin{pmatrix}
+ I-EE^t&ED_pF_0^t\\F_0D_p^{-1}E^t&I-F_0F_0^t
+ \end{pmatrix}.
+\]
+This is one weighted reversed interchange on the first $r$ head and last
+$r$ tail coordinates. All other factors are unit lower intrabank maps or
+lower cross-bank additive maps. Thus variable diagonal units remain in
+the child's weight array rather than becoming unsupported routing passes.
+
+For $n=mf$, group atoms by coarse coordinate $i$ and column $j$.
+If $L_P,R_P,D_P$ are unweighted normalizers for the current projector and
+$A_j=\operatorname{diag}(A[if+j])$, then
+$L_j=A_jL_PA_j^{-1}$ and $R_j=R_P$ are unit lower normalizers for
+$A_jPC_m$. The child's exact contiguous index rule is
+\[
+ A_{\rm child}[kf+j]=A[i_kf+j]D_P[k].
+\]
+The cross-bank atom pairing is reversed, and $C_r\otimes C_f=C_{rf}$.
+Hence the middle operation is exactly one $rf$-atom call of the same
+contract, with no interleaving or gathering. Its coefficients depend only
+on parent prefix fields, which become ancestor spectators at the child;
+the child's fresh roles come from disjoint unused row stock.
+
+All unit lower adapters require $O(n)$ atom passes. The retained ordered
+affine lemma implements each later-target shear in $O(V)$, including
+prefix-controlled offsets. The generated descriptors have polynomial cost
+in $w$: good-class evaluation follows one ancestral index per level;
+fallback evaluation branches by a constant per level. Since depth is
+$O(\log e)$ and $w=\lceil e^{1/1000}\rceil$, this costs $e^{O(1)}=w^{O(1)}$.
+Stored ancestor data use $O(w\log e)$ bits. The proof of the ordered affine
+lemma still applies with these controls: polynomial preparation per prefix
+is absorbed by the complete $q^w$ target fiber. Unrelated spectator counters
+are incremented by ripple counting rather than read in their entirety.
+
+\subsection{Weighted leaves and a genuine local-ring fallback}
+At one atom write $J_a=\left(\begin{smallmatrix}0&a\\a^{-1}&0\end{smallmatrix}\right)$,
+$U(a)=\left(\begin{smallmatrix}1&a\\0&1\end{smallmatrix}\right)$ and
+$L(b)=\left(\begin{smallmatrix}1&0\\b&1\end{smallmatrix}\right)$.
+Then
+\[
+ J_a=\operatorname{diag}(1,-1)U(a)L(-a^{-1})U(a),
+ \qquad U(a)=J_1L(a)J_1.
+\]
+Thus a weighted leaf uses four old ordinary atom interchanges, three
+prefix-controlled shifts and one fixed sign. With
+$a_{\rm old}=384599/10^{10}$ it costs $O(Vw^{1-a_{\rm old}})$.
+This constant occurs only at stopped leaves.
+
+A bad residue class cannot be treated by field Bruhat elimination:
+an entry zero modulo $q$ need not vanish modulo $q^w$. Instead Gaussian
+eliminate the full invertible residual of size $N_0=2m$ with unit pivots.
+Each invertible Schur complement has a unit in its next pivot column;
+row swaps and unnormalized additions clear that column. There are at most
+$N_0(N_0-1)$ additions and $N_0$ swaps. Pivot choices are determined by
+the unweighted low residue class: diagonal conjugation scales every
+Schur-complement row and column by units and preserves its zero pattern.
+The final diagonal has product a fixed sign, since the residual has
+determinant $(-1)^r$ and row swaps change only the sign.
+
+A lower addition is a paid additive adapter. An upper addition is
+conjugated into a lower addition by two ordinary coarse-field swaps.
+For a field of $f$ atoms, let $R_{\rm tail}$ reverse the later field.
+Then the ordinary weighted swap is
+\[
+ J_A=R_{\rm tail}F_AR_{\rm tail},\qquad
+ J_A(H,D)=(AD,A^{-1}H).
+\]
+Each $R_{\rm tail}$ is one unweighted reversed call on the first and last
+$\lfloor f/2\rfloor$ atoms, fixing a possible middle atom. Thus $J_A$
+costs at most three children of widths $f,\lfloor f/2\rfloor,
+\lfloor f/2\rfloor$. A paired diagonal scaling is $J_aJ_1$, at most six
+children. The full diagonal is a product of $N_0-1$ such pairs and a fixed
+sign. Across the two banks the reversed pairing already makes the swaps
+and diagonal pairs directly reversed calls; the preceding ordinary bounds
+remain conservative. Reindexing a tail weight by $j\mapsto f-1-j$ changes
+only its descriptor. Charging every addition as upper therefore gives
+\[
+ 6N_0(N_0-1)+3N_0+6(N_0-1)<32m^2.
+\]
+Every child has width at most $f$, hence ratio at most $1/m$. Coefficients
+are computed from constant-sized local-ring matrices and parent weight
+descriptors; only units are inverted. This proves the fallback cost,
+including its actual reversals, rather than assuming nonunit pivots away.
+
+\subsection{Finite covers, batching, and the strict moment}
+Take $q>2^{80}$ avoiding all fixed rational denominators and rank witnesses.
+At atom width $w$ use the finite cover $\operatorname{GL}_m(R_w)$ and its
+right-multiplication edges $g\mapsto gR_s$. Every vertex has the unique chart
+\[
+ g=g_0+qG_1,\quad g_0\in\operatorname{GL}_m(\mathbb F_q),
+ \quad G_1\in(\mathbb Z/q^{w-1}\mathbb Z)^{m^2},
+\]
+using fixed representatives for $g_0$. Right multiplication permutes $g_0$
+and acts affinely on $G_1$, with a carry determined by $g_0$. Its fixed
+dimensional rational linear part factors into fixed scalings, ordered
+shifts and a constant number of old width-$O(w)$ interchanges. Thus each
+fixed port routing costs $O(Vw^{1-a_{\rm old}})$.
+
+Let $W_0=|\operatorname{GL}_m(\mathbb F_q)|(2v+3R_{\rm loc})$.
+The full role count is $W(w)=W_0q^{m^2(w-1)}$. Split preceding rows into
+low role type, high matrix and unused stock. Batch \emph{all} high matrices
+for each low type in one stream. A frame edge is one child on this whole
+batch, of volume $V/W_0$, and routing is executed once per fixed operation
+type. Consequently neither the number of tapes nor the number of child
+programs grows with $w$. High cardinality cancels in the normalized moment,
+but its $m^2(w-1)$ digits are charged in the row reserve.
+
+For each fixed projector require the leading minors of sizes $1$ through
+$r$ to be units. Each cleared minor polynomial in $g_0$ is nonzero:
+similarity to $\operatorname{diag}(I_r,0)$ supplies a witness after the
+fixed rank is preserved modulo $q$. Their total degree is at most $m^3$.
+The polynomial zero bound and conditioning on invertibility (a factor at
+most two) give bad fraction at most $2m^3/q<10^{-16}$ for each edge type.
+No simultaneous union bound over edge types is needed, since their moments
+are averaged separately. Good classes use the whole-rank child; bad
+classes use the full $32m^2$-child fallback just proved.
+
+Let $H_r$ be the three-stage histogram described above and
+$W=2v+3R_{\rm loc}=90140$. The retained ledger gives
+\[
+ \sum_r rH_r=6037356,\qquad Wm-\sum_r rH_r=2024=2v-3\ell,
+ \qquad \max\{r:H_r>0\}=66.
+\]
+For $a=747/2000000$, $\tau=1-a$, and $b=10^{-16}$ the conservative moment is
+\[
+ \mu_\tau\le
+ \sum_r\frac{H_r}{W}(r/m)^\tau+
+ b\frac{32m^2\sum_rH_r}{W}(1/m)^\tau<1.
+\]
+The entire fallback is added; no displaced good contribution is subtracted.
+The exact script \path{scripts/three_stage_cover_network.py} and certificate
+\path{certificates/three-stage-cover-network.json}, from the retained axis
+\path{certificates/three-stage-cover-bit-input.json}, give
+$1-\mu_\tau>6.3397366396\cdot10^{-7}$.
+The certificate uses the 32-term positive atanh logarithm series with its
+geometric tail, $\exp(u)\le1+u+u^2/(2(1-u/3))$, and upward rounding to
+the grid $2^{-120}$. All inputs have $0\le u<3$; these are rational upper
+bounds, not floating-point root estimates. Since every child ratio is at
+most one and $\tau<1$, this same strict bound implies $\mu_1<1$.
+
+With $T(n,w)$ denoting time divided by logical volume, the fixed low-type
+profile $\{c_r\}$, including fallback classes, satisfies
+\[
+ T(n,w)\le\sum_rc_rT(r\lfloor n/m\rfloor,w)+Cn+Cw^{1-a_{\rm old}}
+ \quad(n\ge m),\qquad
+ T(n,w)\le Cw^{1-a_{\rm old}}\quad(n<m).
+\]
+Peeling fewer than $m$ leftover atoms uses the weighted leaf algorithm.
+The constants are independent of $w,n$ and ancestor values. Induction,
+using $\mu_1<1$ for the linear term and $\mu_\tau<1$ for the stopped term,
+gives $T(n,w)=O(n+n^\tau w^{1-a_{\rm old}})$. The maximum proper ratio
+$66/67$ has halving degree $47$, so recursion depth is $O(\log n)$.
diff --git a/research/r480-sliced-minimal-frames/scripts/final_arithmetic.py b/research/r480-sliced-minimal-frames/scripts/final_arithmetic.py
new file mode 100644
index 0000000000..103d9c887f
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/final_arithmetic.py
@@ -0,0 +1,38 @@
+"""Two exact moments, explicit six-level cutoffs and invalid-assembly controls.
+Prepared for rohanarun with OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+import json,sys,importlib.util
+P=Path(sys.argv[1]).resolve();out=Path(sys.argv[2]).resolve();assert not out.exists()
+def load(n):
+ s=importlib.util.spec_from_file_location('final_'+n,P/'engines'/(n+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,B,A=[load(n)for n in ['moment','base_two_moment','outer']]
+e=json.loads((P/'EXPECTED.json').read_text());baseline=json.loads((P/'inputs/PR510-EXPECTED.json').read_text())['nine'];cp=json.loads((P/'inputs/PR535-EXPECTED.json').read_text());H={int(r):n for r,n in e['profile'].items()};m=baseline['m'];W=baseline['bit']['live'];c=Q(e['bit_lower']);b=Q(e['complex_lower']);N=2*m
+assert sum(r*n for r,n in H.items())==baseline['bit']['rank_mass']
+factor=Q(2*m**3,2**80)*(6*N*(N-1)+3*N+6*(N-1))*sum(H.values())
+moments=[]
+for a,ok in [(c,True),(c+Q(1,10**24),False)]:
+ l,u=M.moment(H,m,W,a,False);g,h=M.moment({1:1},m,W,a,False);l+=factor*g;u+=factor*h
+ ll,uu=B.moment(m,W,list(H.items()),a);gg,hh=B.moment(m,W,[(1,1)],a);ll+=factor*gg;uu+=factor*hh
+ assert (u<1 and uu<1)if ok else(l>1 and ll>1)
+ moments.append(dict(endpoint=str(a),accept=ok,mercator=[str(l-1),str(u-1)],base_two=[str(ll-1),str(uu-1)]))
+CP={int(r):n for r,n in cp['global_profile'].items()}
+for a,ok in [(b,True),(b+Q(1,10**24),False)]:
+ l,u=M.moment(CP,73,cp['live'],a,True);ll,uu=B.moment(73,cp['live'],list(CP.items()),a);gg,hh=B.moment(73,cp['live'],[(1,32*73**2*sum(CP.values()))],a);ll+=gg/10**16;uu+=hh/10**16
+ assert (u<1 and uu<1)if ok else(l>1 and ll>1)
+eta=beta=Q(1,10**18);s=min(c,(1-beta)*b)-Q(1,10**24);assert s<c and s<(1-beta)*b;chain=[Q(384599,10**10)];cuts=[]
+for _ in range(6):
+ old=chain[-1];a=(1-s)*s+s*old;gap=min(s-a,1-a-s,1-a-s*(1-old),1-s);assert old<a<s<1-a and gap>0
+ ceil=lambda x:-(-x.numerator//x.denominator)
+ cut=max(ceil(36/gap**2),ceil(2*(4+512)/gap));assert cut*gap**2>=36 and cut*gap>=2*(4+512);cuts.append(dict(exponent=str(a),gap=str(gap),cutoff_log2=str(cut)));chain.append(a)
+q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18);assert k==Q(e['kappa'])
+bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}}
+a=A.assembly(chain[-1],b,bridge,k,eta=eta,beta=beta);assert len(a['strict_constraints'])==47 and all(v>0 for v in a['strict_constraints'].values())
+controls=[]
+for name,kk,ee,bb in [('next_grid',k+Q(1,10**18),eta,beta),('zero_eta',k,Q(0),beta),('zero_beta',k,eta,Q(0)),('unit_beta',k,eta,Q(1))]:
+ try:A.assembly(chain[-1],b,bridge,kk,eta=ee,beta=bb)
+ except AssertionError:controls.append(name)
+ else:raise AssertionError(name)
+res=dict(status='PASS_TWO_ENGINE_MOMENTS_EXPLICIT_BOOTSTRAP_AND_47_OUTER_CONSTRAINTS',kappa=str(k),bit=str(c),complex=str(b),binding='complex'if (1-beta)*b<c else'bit',moment_receipts=moments,bootstrap_cutoffs=cuts,strict_outer_constraints={n:str(v)for n,v in a['strict_constraints'].items()},controls=controls,scope='Exact scalar numerical assembly. Fresh physical invoice separately establishes charged two-level coefficient<=512 and row bound18355; complex535 operator cited. All-size contracts remain conditional.')
+out.parent.mkdir(parents=True,exist_ok=True);out.write_text(json.dumps(res,indent=2)+'\n');print(res['status'],e['kappa_decimal'],flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/minimum.py b/research/r480-sliced-minimal-frames/scripts/minimum.py
new file mode 100644
index 0000000000..2250f9dc0e
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/minimum.py
@@ -0,0 +1,82 @@
+"""Bounded monotone lift search on R480; exact rational catalogue, floating discovery fee.
+No physical or theorem claim until independent emitted operator admission.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+import json,gzip,sys,math,time,copy,random
+R=Path(__file__).resolve().parent;O=R/'bit510-sliced-minimal-frames';O.mkdir(exist_ok=False);t0=time.monotonic()
+w=json.load(gzip.open(R/'bit510-observable-slice/WORD.json.gz'));lines=w['lines']
+def rref(vs):
+ b={}
+ for row in vs:
+  x=list(map(Q,row))
+  for p,y in sorted(b.items()):
+   if x[p]:a=x[p];x=[s-a*t for s,t in zip(x,y)]
+  if any(x):
+   p=next(j for j,a in enumerate(x)if a);a=x[p];x=[v/a for v in x]
+   for q,y in list(b.items()):
+    if y[p]:a=y[p];b[q]=[s-a*t for s,t in zip(y,x)]
+   b[p]=x
+ return b
+def contains(b,row):
+ x=list(map(Q,row))
+ for p,y in sorted(b.items()):
+  if x[p]:a=x[p];x=[s-a*t for s,t in zip(x,y)]
+ return not any(x)
+bs=[rref([lines[i]for i in f])for f in w['frames']];dim=[len(b)for b in bs];masks=[]
+for f,b in zip(w['frames'],bs):
+ closed={i for i,x in enumerate(lines)if contains(b,x)};assert set(f)==closed,'catalogue is not root-closed';masks.append(sum(1<<i for i in closed))
+le=lambda a,b: masks[a]&masks[b]==masks[a]
+N=2*w['v']+w['R'];nodes=[];chains=[[]for _ in range(N)];mutable=[]
+def node(regs,frame,sec=None,idx=None):
+ k=len(nodes);nodes.append(dict(regs=regs,F=frame,lo=frame,sec=sec,idx=idx))
+ for r in regs:chains[r].append(k)
+ if sec in ('A','B'):mutable.append(k)
+ return k
+for r,f in enumerate(w['start']):node([r],f)
+for sec in ['D','A','B','C']:
+ if sec=='B':
+  for r in w['ret']:node([r],nodes[chains[r][-1]]['F'])
+  for r in range(w['v'],2*w['v']):node([r],0)
+ for i,(f,d,ss)in enumerate(w[sec]):node([d]+ss,f,sec,i)
+for r,f in enumerate(w['final']):node([r],f)
+prev={};nxt={}
+for r,ch in enumerate(chains):
+ for a,b in zip(ch,ch[1:]):prev[b,r]=a;nxt[a,r]=b
+b=.001587447106092548;fee=lambda k:k*math.expm1(b*math.log(64/k)) if k else 0
+def profile():
+ h=Counter()
+ for r,ch in enumerate(chains):
+  for a,z in zip(ch,ch[1:]):
+   fa,fz=nodes[a]['F'],nodes[z]['F'];assert le(fa,fz)
+   if dim[fz]>dim[fa]:h[dim[fz]-dim[fa]]+=1
+ return h
+H0=profile();print('CATALOGUE',len(bs),'profile',dict(H0),'seconds',time.monotonic()-t0,flush=True)
+
+from functools import lru_cache
+original_count=len(masks);idx={m:f for f,m in enumerate(masks)}
+@lru_cache(None)
+def join(mask):
+ basis=rref([lines[i]for i in range(len(lines))if mask>>i&1]);closed=sum(1<<i for i,x in enumerate(lines)if contains(basis,x))
+ if closed not in idx:
+  idx[closed]=len(masks);masks.append(closed);dim.append(len(basis));bs.append(basis)
+ return idx[closed]
+cur={};moves=[]
+for k,z in enumerate(nodes):
+ old=z['F']
+ if z['sec']in ('A','B'):
+  mask=0
+  for r in z['regs']:mask|=masks[cur[r]]
+  f=join(mask);assert le(f,old)
+  if f!=old:moves.append(dict(section=z['sec'],index=z['idx'],old=old,new=f,old_rank=dim[old],new_rank=dim[f]));z['F']=f
+ else:f=old
+ for r in z['regs']:
+  if r in cur:assert le(cur[r],f)
+  cur[r]=f
+H=profile();out=copy.deepcopy(w);out['frames']=[[i for i in range(len(lines))if m>>i&1]for m in masks]
+for k in mutable:
+ z=nodes[k];out[z['sec']][z['idx']][0]=z['F']
+(O/'WORD.json.gz').write_bytes(gzip.compress(json.dumps(out,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+res=dict(status='COMPLETE_EXACT_CAUSAL_MINIMUM_FRAME_ASSIGNMENT',moves=moves,new_frames=len(masks)-original_count,baseline_chain_profile=dict(H0),new_chain_profile=dict(H),fee_delta=sum(fee(r)*n for r,n in H.items())-sum(fee(r)*n for r,n in H0.items()),seconds=time.monotonic()-t0,scope='Least rational span of every preceding operand frame at each A/B gate, with source starts, center cut and cleanup/final frames fixed. All newly needed root closures generated exactly. This is the least feasible frame assignment in subspace inclusion for this fixed chronological word, not a moment optimum. No new physical admission or kappa claimed.')
+(O/'RESULT.json').write_text(json.dumps(res,indent=2)+'\n');print(res,flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/physical.py b/research/r480-sliced-minimal-frames/scripts/physical.py
new file mode 100644
index 0000000000..e0b864b032
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/physical.py
@@ -0,0 +1,39 @@
+"""Fresh record, geometry, owner and invoice replay for the sliced/minimum-frame candidate.
+Uses pinned PR493/510 tools without weakening checks; records the inherited vacuous-route adaptation.
+"""
+from pathlib import Path
+import subprocess,os,json,sys,time,hashlib
+R=Path(__file__).resolve().parent;O=R/'trace541-frozen-admission';assert not O.exists();O.mkdir();(O/'tmp').mkdir();(O/'bin').mkdir();(O/'logs').mkdir()
+P=R/'source493-review';C=P/'code';B=C/'bit';CK=B/'checkers';S=R/'bit510-review/research/e8-r480-nine-stage';VK=S/'vendor/e8-r715-eleven-stage-whole/code';W=R/'trace541-frozen-minimum/WORD.json.gz';PY=sys.executable;env=dict(os.environ,PYTHONDONTWRITEBYTECODE='1',PYTHONINTMAXSTRDIGITS='0',TMPDIR=str(O/'tmp'));steps=[];start=time.monotonic()
+def run(name,cmd):
+ t=time.monotonic();print('START',name,flush=True)
+ with (O/'logs'/(name+'.log')).open('w')as log:ret=subprocess.run(list(map(str,cmd)),cwd=O,env=env,stdout=log,stderr=subprocess.STDOUT)
+ steps.append(dict(name=name,returncode=ret.returncode,seconds=time.monotonic()-t,command=list(map(str,cmd))));(O/'STEPS.json').write_text(json.dumps(steps,indent=2)+'\n')
+ if ret.returncode:print((O/'logs'/(name+'.log')).read_text()[-5000:],flush=True);raise SystemExit('FAIL '+name)
+ print('PASS',name,round(steps[-1]['seconds'],2),flush=True)
+for p in [P/'MANIFEST.sha256']:
+ for line in p.read_text().splitlines():
+  sha,name=line.split('  ',1);assert hashlib.sha256((P/name).read_bytes()).hexdigest()==sha,name
+run('export',[PY,'-B',B/'export_records.py',W,O/'records'])
+run('cohort',[PY,'-B',B/'regression/stage_cohort.py',O/'records',O/'records',O/'cin',O/'cout'])
+incs=['-I'+str(P/'portable-include'),'-I'+str(CK),'-I'+os.environ.get('BOOST_INCLUDE','/usr/include')]
+for name,src,defs in [('legality','legality_h.cpp',['-DNCOPY=9']),('legality-default','legality_h.cpp',[]),('columns','columns_h.cpp',['-DNCOPY=9','-DVPORTS=120','-DRHELP=480']),('columns_k4','columns_k.cpp',['-DNCOPY=9','-DVPORTS=120','-DRHELP=480','-DKPAIRS=4'])]:
+ run('compile-'+name,[os.environ.get('CXX','c++'),'-O2','-std=c++17','-DHDIM=8']+defs+incs+['-o',O/'bin'/name,CK/src])
+run('legality',[O/'bin/legality',O/'cin',O/'cout',O/'cout/legality.json'])
+run('columns',[O/'bin/columns',O/'cout/COHORT249-RECORDS.bin',O/'cout/columns.json'])
+run('columns_k4',[O/'bin/columns_k4',O/'cout/COHORT249-RECORDS.bin',O/'cout/columns_k4.json'])
+run('mutate-records',[PY,'-B',B/'mutate_records.py',O/'records',O/'cin',O/'cout',O/'bin/legality',O/'bin/columns_k4',O/'bin/legality-default'])
+run('mutate-mutable-records',[PY,'-B',B/'mutate_records_mut.py',W,O/'records',O/'cin',O/'cout',O/'bin/legality',O/'bin/columns',O/'bin/columns_k4'])
+run('gram',[PY,'-B',CK/'gram_frames.py',O/'records','--gram','e8ham','--h','8','--positive'])
+V=O/'v9';V.mkdir();(V/'records').symlink_to(O/'records')
+run('geometry',[PY,'-B',C/'nine/geometry.py','--source',P,'--out',V/'nine'])
+run('parent-template',[PY,'-B',VK/'global_k.py','--word',W,'--local-check',O,'--out',V/'nine','--isometries',P/'data/eleven/residual-isometries.json'])
+run('audit-adaptation',[PY,'-B',S/'code/adapt.py',VK,O/'adapted'])
+run('parent-audit',[PY,'-B',O/'adapted/audit_parent_k.py','--local-check',O,'--out',V/'nine','--word',W])
+run('hosted-emission',[PY,'-B',VK/'emit_k.py',V/'nine',O,V/'hosted','--selection',S/'data/selection9-r480.json','--key','bit'])
+run('hosted-audit',[PY,'-B',VK/'audit_hosted_k.py',V,V/'hosted',V/'hosted/AUDIT.json'])
+run('invoice',[PY,'-B',S/'code/assemble9.py',V,V/'hosted',V/'hosted/AUDIT.json',P,O/'BIT-INVOICE.json','pr484'])
+x=json.loads((O/'BIT-INVOICE.json').read_text());e=json.loads((R/'trace541-frozen-minimum-price/RESULT.json').read_text());assert x['bit_profile']==e['profile'];assert x['bit']['charged_two_level_coefficient_bits']<=512 and x['bit']['row_coefficient']==18355
+assert x['bit']['live']==30720 and x['bit']['hosted_residuals']==138240 and x['bit']['bank_assignments']==0
+r=dict(status='PASS_FRESH_TRACE_AND_FROZEN_FRAMES_RECORDS_HOSTING_AND_INVOICE',seconds=time.monotonic()-start,word_sha256=hashlib.sha256(W.read_bytes()).hexdigest(),profile=e['profile'],kappa=e['kappa_decimal'],bit_invoice=x['bit'],steps=len(steps),scope='Fresh finite record/formal-column/geometry/template/whole-owner/invoice replay, with inherited mutation controls. Cited complex535 operator unchanged. Exact sharp-charge prospective profile now matches physically emitted audited profile and retained finite allowances. All-size supplier/compiler/realization/hosting/analytic assumptions remain conditional.')
+(O/'RESULT.json').write_text(json.dumps(r,indent=2)+'\n');print(r,flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/preload60.py b/research/r480-sliced-minimal-frames/scripts/preload60.py
new file mode 100644
index 0000000000..5a1b518a58
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/preload60.py
@@ -0,0 +1,48 @@
+from pathlib import Path
+from fractions import Fraction as Q
+from collections import Counter
+import json,gzip,hashlib,copy,sys,argparse
+parser=argparse.ArgumentParser();parser.add_argument('--input',type=Path,required=True);parser.add_argument('--out',type=Path,required=True);args=parser.parse_args();src,out=args.input,args.out;w=json.loads(gzip.decompress(src.read_bytes()));V=w['v'];N=2*V+w['R'];old=copy.deepcopy(w);prefix=[];moves=[];A=[];assert hashlib.sha256(json.dumps(w,sort_keys=True,separators=(',',':')).encode()).hexdigest()=='8639e22008c24a5690dd250eba702fe5fca85ddd036dcbec20baba77b7e213ba', 'pinned source word'
+for i,(f,t,ss)in enumerate(w['A']):
+ if len(ss)==2 and all(s<V for s in ss):
+  s=ss[0]
+  assert all(t not in sources and dest!=s for ff,dest,sources in old['A'][:i]),('not commutable',i)
+  assert t not in w['ret'];prefix.append([w['start'][s],t,[s]]);A.append([f,t,ss[1:]]);moves.append(dict(index=i,helper=t,source=s,old_frame=f,new_frame=w['start'][s]))
+ else:A.append([f,t,ss])
+w['A']=prefix+A;w['C']=[[1,t,list(ss)]for f,t,ss in reversed(w['A'])]
+# Independently compute exact rational row bases and subspace nesting.
+def basis(vecs):
+ B={}
+ for rr in vecs:
+  row=list(map(Q,rr))
+  for c in sorted(B):
+   q=row[c];row=[a-q*b for a,b in zip(row,B[c])]
+  c=next((i for i,x in enumerate(row)if x),None)
+  if c is not None:
+   q=row[c];B[c]=[x/q for x in row]
+ return B
+FB=[basis(w['lines'][i]for i in f)for f in w['frames']];dim=list(map(len,FB));cache={}
+def sub(a,b):
+ if (a,b)not in cache:cache[a,b]=len(basis(list(FB[a].values())+list(FB[b].values())))==dim[b]
+ return cache[a,b]
+def replay(z):
+ val=[1<<i for i in range(N)];cur=list(z['start']);H=Counter();tops=[]
+ def climb(r,f):
+  a=cur[r];assert sub(a,f),(r,a,f)
+  if dim[f]>dim[a]:H[dim[f]-dim[a]]+=1
+  cur[r]=f
+ def gates(sec):
+  for f,t,ss in z[sec]:
+   assert len({t,*ss})==1+len(ss)
+   for r in [t,*ss]:climb(r,f)
+   for s in ss:val[t]^=val[s]
+ gates('D');gates('A');tops.extend(dim[cur[q]]for q in z['ret']);snap=[val[q]for q in z['ret']]
+ for t,ks in enumerate(z['scatter']):
+  for k in ks:val[V+t]^=snap[k]
+ gates('B');gates('C')
+ for r,f in enumerate(z['final']):climb(r,f)
+ for r in range(N):assert val[r]==((1<<r)^(1<<(r-V))if V<=r<2*V else 1<<r),r
+ return H,tops
+h0,c0=replay(old);h1,c1=replay(w);assert c0==c1;delta={k:h1[k]-h0[k]for k in sorted(h0.keys()|h1.keys())if h1[k]!=h0[k]};assert sum(k*v for k,v in delta.items())==0;assert len(moves)==60 and delta=={1:60,2:-120,3:60};assert old['B']==w['B'] and old['D']==w['D'] and old['scatter']==w['scatter'] and old['start']==w['start'] and old['final']==w['final']
+out.mkdir(parents=True,exist_ok=False);raw=json.dumps(w,sort_keys=True,separators=(',',':')).encode();data=gzip.compress(raw,compresslevel=9,mtime=0);data=data[:9]+bytes([255])+data[10:];(out/'WORD.json.gz').write_bytes(data);r=dict(status='PASS_EXACT_SEEDED_COMMUTING_WORD',source_sha256=hashlib.sha256(src.read_bytes()).hexdigest(),output_content_sha256=hashlib.sha256(raw).hexdigest(),moves=moves,all_dirty_columns=N,profile_before=dict(h0),profile_after=dict(h1),profile_delta=delta,paid_profile_with_retained_copies=dict(h1+Counter(c1)),retained_total_ranks=c1,helper_widths_unchanged=True,cleanup_literal_inverse=True,scope='Exact F2 endpoints, rational frame incidence and local profile. Official physical/compiler/prefix/invoice admission still required.')
+(out/'SEED-REVIEW.json').write_text(json.dumps(r,indent=2)+'\n');print(json.dumps({k:v for k,v in r.items()if k!='moves'}))
diff --git a/research/r480-sliced-minimal-frames/scripts/price.py b/research/r480-sliced-minimal-frames/scripts/price.py
new file mode 100644
index 0000000000..fe91825ad6
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/price.py
@@ -0,0 +1,36 @@
+"""Prospective exact price of sliced/minimum-frame R480 with unchanged PR510 hosting.
+Not physical admission; finite invoice and owner-template replay remain pending.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from decimal import Decimal,localcontext
+from collections import Counter
+import json,importlib.util
+R=Path(__file__).resolve().parent;O=R/'bit510-sliced-minimal-price';O.mkdir(exist_ok=False);p=R/'r480-sharp535-publication/research/r480-sharp-charge535'
+def load(n):
+ s=importlib.util.spec_from_file_location(n,p/'engines'/(n+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,B,Outer=[load(x)for x in ['moment','base_two_moment','outer']]
+x=json.loads((p/'inputs/PR510-EXPECTED.json').read_text())['nine'];loc=json.loads((R/'bit510-sliced-minimal-frames/RESULT.json').read_text());old={int(k):v for k,v in loc['baseline_chain_profile'].items()};new={int(k):v for k,v in loc['new_chain_profile'].items()};H=Counter({int(k):v for k,v in x['profile'].items()})
+for r,n in old.items():H[r]-=288*n
+for r,n in new.items():H[r]+=288*n
+H=+H;m=x['m'];W=x['bit']['live'];assert sum(r*n for r,n in H.items())==x['bit']['rank_mass'];factor=Q(1,2**61)*98682*sum(H.values())
+def bounds(c):
+ a,z=M.moment(H,m,W,c,False);g,h=M.moment({1:1},m,W,c,False);b,y=B.moment(m,W,list(H.items()),c);j,k=B.moment(m,W,[(1,1)],c);return a+factor*g,z+factor*h,b+factor*j,y+factor*k
+with localcontext()as ctx:
+ ctx.prec=90;term=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln())for r,n in H.items()];fb=Decimal(factor.numerator)/Decimal(factor.denominator)/Decimal(m*W);lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+ for _ in range(230):
+  c=(lo+hi)/2
+  if sum(p*(c*l).exp()for p,l in term)+fb*(c*lm).exp()<1:lo=c
+  else:hi=c
+ c=Q(int(lo*10**24),10**24)
+checks=[]
+for a,accept in [(c,True),(c+Q(1,10**24),False)]:
+ l,h,ll,hh=bounds(a);assert (h<1 and hh<1)if accept else(l>1 and ll>1);checks.append(dict(endpoint=str(a),accept=accept,mercator=[str(l-1),str(h-1)],base_two=[str(ll-1),str(hh-1)]))
+b=Q(json.loads((p/'inputs/PR535-EXPECTED.json').read_text())['complex_lower']);eta=beta=Q(1,10**18);s=min(c,(1-beta)*b)-Q(1,10**24);chain=[Q(384599,10**10)]
+for _ in range(6):chain.append((1-s)*s+s*chain[-1])
+q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18)
+bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}};assembly=Outer.assembly(chain[-1],b,bridge,k,eta=eta,beta=beta);assert len(assembly['strict_constraints'])==47 and all(a>0 for a in assembly['strict_constraints'].values())
+def dec(a):
+ with localcontext()as ctx:ctx.prec=60;return str(Decimal(a.numerator)/Decimal(a.denominator))
+oldk=Q(json.loads((p/'EXPECTED.json').read_text())['kappa']);out=dict(status='PASS_PROSPECTIVE_EXACT_PRICE_PHYSICAL_ADMISSION_PENDING',profile=dict(sorted(H.items())),local_chain_profile=new,bit_lower=str(c),bit_decimal=dec(c),complex_lower=str(b),kappa=str(k),kappa_decimal=dec(k),relative_gain_percent=dec(100*(k-oldk)/oldk),binding='bit'if c<(1-beta)*b else'complex',moment_checks=checks,outer={n:str(z)for n,z in assembly['strict_constraints'].items()},scope='Derived global histogram adds288 copies of exact local profile delta to pinned PR510 profile. Same480 width8 helpers and original owner schedule proposed. Physical template, chronological finite invoice and changed-frame geometric checks remain required; old outer finite constants used prospectively.')
+(O/'RESULT.json').write_text(json.dumps(out,indent=2)+'\n');print({k:v for k,v in out.items()if k not in ['moment_checks','outer']},flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/profile.py b/research/r480-sliced-minimal-frames/scripts/profile.py
new file mode 100644
index 0000000000..129e58e848
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/profile.py
@@ -0,0 +1,19 @@
+from pathlib import Path
+import gzip,json,ast,re,sympy as S
+from collections import Counter
+R=Path(__file__).resolve().parent;w=json.load(gzip.open(R/'trace541-frozen-minimum/WORD.json.gz'));dim=[S.Matrix([w['lines'][i]for i in f]).rank()if f else 0 for f in w['frames']];mask=[sum(1<<i for i in f)for f in w['frames']];cur=w['start'][:];H=Counter()
+def move(r,f):
+ assert mask[cur[r]]&mask[f]==mask[cur[r]]
+ if cur[r]!=f:H[int(dim[f]-dim[cur[r]])]+=1;cur[r]=f
+for sec in ['D','A','B','C']:
+ for f,d,ss in w[sec]:
+  for r in [d]+ss:move(r,f)
+for r,f in enumerate(w['final']):move(r,f)
+# Compare the independently reconstructed five-stage histogram when that separate receipt exists.
+p=R/'trace541-frozen-minimum/independent311-checkword.log'
+if p.exists():
+ official=ast.literal_eval(re.search(r'^histogram (\{.*?\}) m ',p.read_text(),re.M).group(1));five=Counter({r:5*n for r,n in H.items()});five[8]+=45
+ for r in [14,7,18,4]:five[r]+=240
+ assert dict(five)==official
+r=dict(new_chain_profile=dict(H),scope='Exact rank profile from every final combined-word frame transition; center copies separate and unchanged9x8. Do not add standalone histogram deltas: the frame changes interact.')
+(R/'trace541-frozen-minimum/PROFILE.json').write_text(json.dumps(r,indent=2)+'\n');print(r)
diff --git a/research/r480-sliced-minimal-frames/scripts/slice.py b/research/r480-sliced-minimal-frames/scripts/slice.py
new file mode 100644
index 0000000000..3f37f14516
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/slice.py
@@ -0,0 +1,107 @@
+"""Exact F2 suffix-response retarget screen on the R480 bit circuit.
+Same-frame whole-gate retargeting and zero-response deletions; discovery only.
+Prepared for rohanarun with OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from fractions import Fraction as Q
+import gzip,json,math,time,copy,hashlib
+import sympy as S
+R=Path(__file__).resolve().parent;O=R/'bit510-observable-slice';O.mkdir(exist_ok=False);t=time.monotonic();p=R/'bit510-review/research/e8-r480-dead-totals/data/bitE8mut-r480.json.gz';w=json.load(gzip.open(p));v=w['v'];n=2*v+w['R'];Zregs=list(range(v))+list(range(2*v,n));mask=[sum(1<<i for i in f)for f in w['frames']];dims=[S.Matrix([w['lines'][i]for i in f]).rank()if f else 0 for f in w['frames']];le=lambda a,b:mask[a]&mask[b]==mask[a];full=next(i for i,d in enumerate(dims)if d==8);fee=lambda k:k*math.expm1(.001587447108*math.log(64/k))if k else 0
+original=copy.deepcopy(w);moves=[];trials=Counter();best_negative=[]
+def scalar(word):
+ z=[1<<r for r in range(n)]
+ for sec in ['D','A','B','C']:
+  if sec=='B':
+   held=[z[r]for r in word['ret']]
+   for j,cs in enumerate(word['scatter']):
+    for c in cs:z[v+j]^=held[c]
+  for f,d,ss in word[sec]:
+   for s in ss:z[d]^=z[s]
+ assert all(z[r]==((1<<r)|(1<<(r-v))if v<=r<2*v else 1<<r)for r in range(n))
+def profiles(word):
+ cur=list(word['start']);h=Counter()
+ def go(r,f):
+  a=cur[r];assert le(a,f)
+  if a!=f:h[int(dims[f]-dims[a])]+=1;cur[r]=f
+ for sec in ['D','A','B','C']:
+  if sec=='B':
+   for r in word['ret']:go(r,full);h[8]+=1
+  for f,d,ss in word[sec]:
+   for r in [d]+ss:go(r,f)
+ for r,f in enumerate(word['final']):go(r,f)
+ return h
+base=profiles(w);scalar(w)
+# Delete all zero-target-response instructions together; individually neutral deletions
+# may expose a frame transition only after the whole dead segment disappears.
+resp=[0]*n
+for j in range(v):resp[v+j]=1<<j
+for f,d,ss in reversed(w['B']):
+ for q in ss:resp[q]^=resp[d]
+for j,cs in enumerate(w['scatter']):
+ for c in cs:resp[w['ret'][c]]^=resp[v+j]
+dead=[]
+for i in reversed(range(len(w['A']))):
+ f,d,ss=w['A'][i]
+ if resp[d]==0:dead.append(i)
+ for q in ss:resp[q]^=resp[d]
+oldA=copy.deepcopy(w['A']);deadset=set(dead);w['A']=[g for i,g in enumerate(w['A'])if i not in deadset]
+w['C']=[[full,d,list(ss)]for f,d,ss in reversed(w['A'])];scalar(w);sliced=profiles(w)
+used=set(w['ret'])
+for sec in ['D','A','B','C']:
+ for f,d,ss in w[sec]:used.update([d]+ss)
+unused=sorted(set(range(2*v,n))-used)
+slice_record=dict(removed_indices=sorted(dead),removed_gates=[oldA[i]for i in sorted(dead)],old_profile=dict(base),sliced_profile=dict(sliced),unused_helpers=unused)
+(O/'SLICE.json').write_text(json.dumps(slice_record,indent=2)+'\n');print('SIMULTANEOUS_SLICE',len(dead),dict(sliced),'unused',unused,flush=True)
+for sweep in range(8):
+ # Backward exact target response, including the center copy and scatter.
+ resp=[0]*n
+ for j in range(v):resp[v+j]=1<<j
+ for f,d,ss in reversed(w['B']):
+  for s in ss:resp[s]^=resp[d]
+ for j,cs in enumerate(w['scatter']):
+  for c in cs:resp[w['ret'][c]]^=resp[v+j]
+ suffix=[None]*len(w['A'])
+ for i in reversed(range(len(w['A']))):
+  suffix[i]=resp[:];f,d,ss=w['A'][i]
+  for s in ss:resp[s]^=resp[d]
+ # Exact neighboring frame events before and after each candidate gate.
+ prev=[];cur=list(w['start'])
+ for f,d,ss in w['A']:
+  prev.append(cur[:])
+  for r in [d]+ss:assert le(cur[r],f);cur[r]=f
+ nxt=list(w['final'])
+ for f,d,ss in reversed(w['C']+[]):
+  for r in [d]+ss:nxt[r]=f
+ for f,d,ss in reversed(w['B']):
+  for r in [d]+ss:nxt[r]=f
+ for r in w['ret']:nxt[r]=full
+ after=[None]*len(w['A'])
+ for i in reversed(range(len(w['A']))):
+  after[i]=nxt[:];f,d,ss=w['A'][i]
+  for r in [d]+ss:nxt[r]=f
+ proposals=[]
+ for i,(f,d,ss)in enumerate(w['A']):
+  H=suffix[i];a,b=prev[i],after[i]
+  def removal(r):return fee(dims[b[r]]-dims[a[r]])-fee(dims[f]-dims[a[r]])-fee(dims[b[r]]-dims[f])
+  if H[d]==0:
+   trials['zero_response_gates']+=1;change=sum(removal(r)for r in [d]+ss)
+   if change < -1e-12:proposals.append((change,i,None))
+  for d2 in Zregs:
+   if d2==d or d2 in ss or H[d2]!=H[d]:continue
+   trials['response_equal']+=1
+   if not (le(a[d2],f)and le(f,b[d2])):continue
+   trials['frame_legal']+=1
+   change=removal(d)+fee(dims[f]-dims[a[d2]])+fee(dims[b[d2]]-dims[f])-fee(dims[b[d2]]-dims[a[d2]])
+   if change < -1e-12:proposals.append((change,i,d2))
+ print('sweep',sweep,'trials',dict(trials),'profitable',len(proposals),flush=True)
+ if not proposals:break
+ change,i,d2=min(proposals,key=lambda x:(x[0],x[1],-1 if x[2]is None else x[2]));f,d,ss=w['A'][i];old=profiles(w)
+ move=dict(sweep=sweep,index=i,frame=f,old_destination=d,new_destination=d2,sources=ss,fee_delta=change,target_response=str(suffix[i][d]))
+ if d2 is None:del w['A'][i]
+ else:w['A'][i][1]=d2
+ w['C']=[[full,d,list(ss)]for f,d,ss in reversed(w['A'])];scalar(w);new=profiles(w);assert abs((sum(fee(r)*c for r,c in new.items())-sum(fee(r)*c for r,c in old.items()))-change)<1e-9
+ move.update(old_profile=dict(old),new_profile=dict(new));moves.append(move)
+(O/'WORD.json.gz').write_bytes(gzip.compress(json.dumps(w,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+res=dict(status='COMPLETE_SCOPED_OBSERVABLE_SLICE_AND_RETARGET_SCREEN',word_sha256=hashlib.sha256(p.read_bytes()).hexdigest(),slice_record=slice_record,trials=dict(trials),moves=moves,baseline_profile=dict(base),final_profile=dict(profiles(w)),scalar_basis_columns=n,seconds=time.monotonic()-t,scope='Same-frame whole-gate destination changes in A when exact suffix target columns agree, plus target-zero whole-gate deletions. Greedy bounded search with float moment proxy; each accepted move has full scalar F2 and nested-frame checks. No emitted physical operator, finite prefix invoice, or new kappa claimed. Source/helper inverse cleanup regenerated exactly; D/B/scatter and live stock unchanged.')
+(O/'RESULT.json').write_text(json.dumps(res,indent=2)+'\n');print({k:v for k,v in res.items()if k!='slice_record'},flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/trace_price.py b/research/r480-sliced-minimal-frames/scripts/trace_price.py
new file mode 100644
index 0000000000..93298bbe78
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/trace_price.py
@@ -0,0 +1,36 @@
+"""Prospective exact price of sliced/minimum-frame R480 with unchanged PR510 hosting.
+Not physical admission; finite invoice and owner-template replay remain pending.
+"""
+from pathlib import Path
+from fractions import Fraction as Q
+from decimal import Decimal,localcontext
+from collections import Counter
+import json,importlib.util
+R=Path(__file__).resolve().parent;O=R/'trace541-frozen-minimum-price';O.mkdir(exist_ok=False);p=R/'r480-sharp535-publication/research/r480-sharp-charge535'
+def load(n):
+ s=importlib.util.spec_from_file_location(n,p/'engines'/(n+'.py'));m=importlib.util.module_from_spec(s);s.loader.exec_module(m);return m
+M,B,Outer=[load(x)for x in ['moment','base_two_moment','outer']]
+x=json.loads((p/'inputs/PR510-EXPECTED.json').read_text())['nine'];loc=json.loads((R/'trace541-frozen-minimum/PROFILE.json').read_text());old={int(k):v for k,v in json.loads((R/'bit510-sliced-minimal-frames/RESULT.json').read_text())['baseline_chain_profile'].items()};new={int(k):v for k,v in loc['new_chain_profile'].items()};H=Counter({int(k):v for k,v in x['profile'].items()})
+for r,n in old.items():H[r]-=288*n
+for r,n in new.items():H[r]+=288*n
+H=+H;m=x['m'];W=x['bit']['live'];assert sum(r*n for r,n in H.items())==x['bit']['rank_mass'];factor=Q(1,2**61)*98682*sum(H.values())
+def bounds(c):
+ a,z=M.moment(H,m,W,c,False);g,h=M.moment({1:1},m,W,c,False);b,y=B.moment(m,W,list(H.items()),c);j,k=B.moment(m,W,[(1,1)],c);return a+factor*g,z+factor*h,b+factor*j,y+factor*k
+with localcontext()as ctx:
+ ctx.prec=90;term=[(Decimal(r*n)/Decimal(m*W),(Decimal(m)/Decimal(r)).ln())for r,n in H.items()];fb=Decimal(factor.numerator)/Decimal(factor.denominator)/Decimal(m*W);lm=Decimal(m).ln();lo=Decimal(0);hi=Decimal('.003')
+ for _ in range(230):
+  c=(lo+hi)/2
+  if sum(p*(c*l).exp()for p,l in term)+fb*(c*lm).exp()<1:lo=c
+  else:hi=c
+ c=Q(int(lo*10**24),10**24)
+checks=[]
+for a,accept in [(c,True),(c+Q(1,10**24),False)]:
+ l,h,ll,hh=bounds(a);assert (h<1 and hh<1)if accept else(l>1 and ll>1);checks.append(dict(endpoint=str(a),accept=accept,mercator=[str(l-1),str(h-1)],base_two=[str(ll-1),str(hh-1)]))
+b=Q(json.loads((p/'inputs/PR535-EXPECTED.json').read_text())['complex_lower']);eta=beta=Q(1,10**18);s=min(c,(1-beta)*b)-Q(1,10**24);chain=[Q(384599,10**10)]
+for _ in range(6):chain.append((1-s)*s+s*chain[-1])
+q=chain[-1]*(1-2*eta);margin=(1-eta)*q/(1+q);ticks=margin*10**18;k=Q((ticks.numerator-1)//ticks.denominator,10**18)
+bridge={'semantic':{'C0':{'base':2,'exponent':210000},'strict_literal_gap':1},'rows':{'degree_gap':Q(10**6)-Q(51*20161,25)}};assembly=Outer.assembly(chain[-1],b,bridge,k,eta=eta,beta=beta);assert len(assembly['strict_constraints'])==47 and all(a>0 for a in assembly['strict_constraints'].values())
+def dec(a):
+ with localcontext()as ctx:ctx.prec=60;return str(Decimal(a.numerator)/Decimal(a.denominator))
+oldk=Q(json.loads((p/'EXPECTED.json').read_text())['kappa']);out=dict(status='PASS_PROSPECTIVE_EXACT_PRICE_PHYSICAL_ADMISSION_PENDING',profile=dict(sorted(H.items())),local_chain_profile=new,bit_lower=str(c),bit_decimal=dec(c),complex_lower=str(b),kappa=str(k),kappa_decimal=dec(k),relative_gain_percent=dec(100*(k-oldk)/oldk),binding='bit'if c<(1-beta)*b else'complex',moment_checks=checks,outer={n:str(z)for n,z in assembly['strict_constraints'].items()},scope='Derived global histogram adds288 copies of exact local profile delta to pinned PR510 profile. Same480 width8 helpers and original owner schedule proposed. Physical template, chronological finite invoice and changed-frame geometric checks remain required; old outer finite constants used prospectively.')
+(O/'RESULT.json').write_text(json.dumps(out,indent=2)+'\n');print({k:v for k,v in out.items()if k not in ['moment_checks','outer']},flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/trace_slice.py b/research/r480-sliced-minimal-frames/scripts/trace_slice.py
new file mode 100644
index 0000000000..0327772196
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/trace_slice.py
@@ -0,0 +1,107 @@
+"""Exact F2 suffix-response retarget screen on the R480 bit circuit.
+Same-frame whole-gate retargeting and zero-response deletions; discovery only.
+Prepared for rohanarun with OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+from collections import Counter,defaultdict
+from fractions import Fraction as Q
+import gzip,json,math,time,copy,hashlib
+import sympy as S
+R=Path(__file__).resolve().parent;O=R/'trace541-observable-slice';O.mkdir(exist_ok=False);t=time.monotonic();p=R/'trace541-rebuild/WORD.json.gz';w=json.load(gzip.open(p));v=w['v'];n=2*v+w['R'];Zregs=list(range(v))+list(range(2*v,n));mask=[sum(1<<i for i in f)for f in w['frames']];dims=[S.Matrix([w['lines'][i]for i in f]).rank()if f else 0 for f in w['frames']];le=lambda a,b:mask[a]&mask[b]==mask[a];full=next(i for i,d in enumerate(dims)if d==8);fee=lambda k:k*math.expm1(.001587447108*math.log(64/k))if k else 0
+original=copy.deepcopy(w);moves=[];trials=Counter();best_negative=[]
+def scalar(word):
+ z=[1<<r for r in range(n)]
+ for sec in ['D','A','B','C']:
+  if sec=='B':
+   held=[z[r]for r in word['ret']]
+   for j,cs in enumerate(word['scatter']):
+    for c in cs:z[v+j]^=held[c]
+  for f,d,ss in word[sec]:
+   for s in ss:z[d]^=z[s]
+ assert all(z[r]==((1<<r)|(1<<(r-v))if v<=r<2*v else 1<<r)for r in range(n))
+def profiles(word):
+ cur=list(word['start']);h=Counter()
+ def go(r,f):
+  a=cur[r];assert le(a,f)
+  if a!=f:h[int(dims[f]-dims[a])]+=1;cur[r]=f
+ for sec in ['D','A','B','C']:
+  if sec=='B':
+   for r in word['ret']:go(r,full);h[8]+=1
+  for f,d,ss in word[sec]:
+   for r in [d]+ss:go(r,f)
+ for r,f in enumerate(word['final']):go(r,f)
+ return h
+base=profiles(w);scalar(w)
+# Delete all zero-target-response instructions together; individually neutral deletions
+# may expose a frame transition only after the whole dead segment disappears.
+resp=[0]*n
+for j in range(v):resp[v+j]=1<<j
+for f,d,ss in reversed(w['B']):
+ for q in ss:resp[q]^=resp[d]
+for j,cs in enumerate(w['scatter']):
+ for c in cs:resp[w['ret'][c]]^=resp[v+j]
+dead=[]
+for i in reversed(range(len(w['A']))):
+ f,d,ss=w['A'][i]
+ if resp[d]==0:dead.append(i)
+ for q in ss:resp[q]^=resp[d]
+oldA=copy.deepcopy(w['A']);deadset=set(dead);w['A']=[g for i,g in enumerate(w['A'])if i not in deadset]
+w['C']=[[full,d,list(ss)]for f,d,ss in reversed(w['A'])];scalar(w);sliced=profiles(w)
+used=set(w['ret'])
+for sec in ['D','A','B','C']:
+ for f,d,ss in w[sec]:used.update([d]+ss)
+unused=sorted(set(range(2*v,n))-used)
+slice_record=dict(removed_indices=sorted(dead),removed_gates=[oldA[i]for i in sorted(dead)],old_profile=dict(base),sliced_profile=dict(sliced),unused_helpers=unused)
+(O/'SLICE.json').write_text(json.dumps(slice_record,indent=2)+'\n');print('SIMULTANEOUS_SLICE',len(dead),dict(sliced),'unused',unused,flush=True)
+for sweep in range(8):
+ # Backward exact target response, including the center copy and scatter.
+ resp=[0]*n
+ for j in range(v):resp[v+j]=1<<j
+ for f,d,ss in reversed(w['B']):
+  for s in ss:resp[s]^=resp[d]
+ for j,cs in enumerate(w['scatter']):
+  for c in cs:resp[w['ret'][c]]^=resp[v+j]
+ suffix=[None]*len(w['A'])
+ for i in reversed(range(len(w['A']))):
+  suffix[i]=resp[:];f,d,ss=w['A'][i]
+  for s in ss:resp[s]^=resp[d]
+ # Exact neighboring frame events before and after each candidate gate.
+ prev=[];cur=list(w['start'])
+ for f,d,ss in w['A']:
+  prev.append(cur[:])
+  for r in [d]+ss:assert le(cur[r],f);cur[r]=f
+ nxt=list(w['final'])
+ for f,d,ss in reversed(w['C']+[]):
+  for r in [d]+ss:nxt[r]=f
+ for f,d,ss in reversed(w['B']):
+  for r in [d]+ss:nxt[r]=f
+ for r in w['ret']:nxt[r]=full
+ after=[None]*len(w['A'])
+ for i in reversed(range(len(w['A']))):
+  after[i]=nxt[:];f,d,ss=w['A'][i]
+  for r in [d]+ss:nxt[r]=f
+ proposals=[]
+ for i,(f,d,ss)in enumerate(w['A']):
+  H=suffix[i];a,b=prev[i],after[i]
+  def removal(r):return fee(dims[b[r]]-dims[a[r]])-fee(dims[f]-dims[a[r]])-fee(dims[b[r]]-dims[f])
+  if H[d]==0:
+   trials['zero_response_gates']+=1;change=sum(removal(r)for r in [d]+ss)
+   if change < -1e-12:proposals.append((change,i,None))
+  for d2 in Zregs:
+   if d2==d or d2 in ss or H[d2]!=H[d]:continue
+   trials['response_equal']+=1
+   if not (le(a[d2],f)and le(f,b[d2])):continue
+   trials['frame_legal']+=1
+   change=removal(d)+fee(dims[f]-dims[a[d2]])+fee(dims[b[d2]]-dims[f])-fee(dims[b[d2]]-dims[a[d2]])
+   if change < -1e-12:proposals.append((change,i,d2))
+ print('sweep',sweep,'trials',dict(trials),'profitable',len(proposals),flush=True)
+ if not proposals:break
+ change,i,d2=min(proposals,key=lambda x:(x[0],x[1],-1 if x[2]is None else x[2]));f,d,ss=w['A'][i];old=profiles(w)
+ move=dict(sweep=sweep,index=i,frame=f,old_destination=d,new_destination=d2,sources=ss,fee_delta=change,target_response=str(suffix[i][d]))
+ if d2 is None:del w['A'][i]
+ else:w['A'][i][1]=d2
+ w['C']=[[full,d,list(ss)]for f,d,ss in reversed(w['A'])];scalar(w);new=profiles(w);assert abs((sum(fee(r)*c for r,c in new.items())-sum(fee(r)*c for r,c in old.items()))-change)<1e-9
+ move.update(old_profile=dict(old),new_profile=dict(new));moves.append(move)
+(O/'WORD.json.gz').write_bytes(gzip.compress(json.dumps(w,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+res=dict(status='COMPLETE_SCOPED_OBSERVABLE_SLICE_AND_RETARGET_SCREEN',word_sha256=hashlib.sha256(p.read_bytes()).hexdigest(),slice_record=slice_record,trials=dict(trials),moves=moves,baseline_profile=dict(base),final_profile=dict(profiles(w)),scalar_basis_columns=n,seconds=time.monotonic()-t,scope='Same-frame whole-gate destination changes in A when exact suffix target columns agree, plus target-zero whole-gate deletions. Greedy bounded search with float moment proxy; each accepted move has full scalar F2 and nested-frame checks. No emitted physical operator, finite prefix invoice, or new kappa claimed. Source/helper inverse cleanup regenerated exactly; D/B/scatter and live stock unchanged.')
+(O/'RESULT.json').write_text(json.dumps(res,indent=2)+'\n');print({k:v for k,v in res.items()if k!='slice_record'},flush=True)
diff --git a/research/r480-sliced-minimal-frames/scripts/transfer.py b/research/r480-sliced-minimal-frames/scripts/transfer.py
new file mode 100644
index 0000000000..f56e68aae4
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/scripts/transfer.py
@@ -0,0 +1,24 @@
+"""Transfer only the frozen19 beneficial frame changes to the independently rebuilt PR541 word.
+The unconstrained minimum-frame composition is a retained losing experiment.
+"""
+from pathlib import Path
+from collections import Counter
+import json,gzip,copy,time
+R=Path(__file__).resolve().parent;O=R/'trace541-frozen-minimum';O.mkdir(exist_ok=False)
+a=json.load(gzip.open(R/'bit510-observable-slice/WORD.json.gz'));src=json.load(gzip.open(R/'trace541-observable-slice/WORD.json.gz'));low=json.load(gzip.open(R/'bit510-sliced-minimal-frames/WORD.json.gz'));moves=json.loads((R/'bit510-sliced-minimal-frames/RESULT.json').read_text())['moves'];w=copy.deepcopy(src);assert w['frames']==a['frames'];w['frames']=low['frames'];assert len(w['A'])-len(a['A'])==60
+selected={}
+for x in moves:
+ assert x['section']=='A';i=x['index'];assert w['A'][i+60]==a['A'][i]
+ selected[i+60]=x['new']
+cur=list(w['start']);masks=[sum(1<<r for r in f)for f in w['frames']];accepted=[];rejected=[]
+for i,(f,d,ss)in enumerate(w['A']):
+ if i in selected:
+  q=selected[i]
+  if all(masks[cur[r]]&masks[q]==masks[cur[r]]for r in [d]+ss):f=q;w['A'][i][0]=f;accepted.append(i)
+  else:rejected.append(i)
+ for r in [d]+ss:assert masks[cur[r]]&masks[f]==masks[cur[r]];cur[r]=f
+# Scalar word is unchanged from the already reconstructed PR541 slice; cleanup uses full frame1.
+assert all(x[1:]==y[1:]for x,y in zip(w['A'],src['A']))
+(O/'WORD.json.gz').write_bytes(gzip.compress(json.dumps(w,sort_keys=True,separators=(',',':')).encode(),mtime=0))
+res=dict(status='PASS_FROZEN_FRAME_TRANSFER_CONTAINMENTS',accepted_indices=accepted,rejected_indices=rejected,scope='Only previously beneficial19 operation-frame changes transferred to sliced PR541 by exact gate identity; full scalar/geometry/profile/physical checks follow.')
+(O/'RESULT.json').write_text(json.dumps(res,indent=2)+'\n');print(res,flush=True)
diff --git a/research/r480-sliced-minimal-frames/verify.py b/research/r480-sliced-minimal-frames/verify.py
new file mode 100644
index 0000000000..24b7dcc1db
--- /dev/null
+++ b/research/r480-sliced-minimal-frames/verify.py
@@ -0,0 +1,48 @@
+"""Reproduce scalar/frame transformation and prospective arithmetic; physical admission separate.
+Prepared for rohanarun with OpenAI Codex assistance. Apache-2.0.
+"""
+from pathlib import Path
+import os,sys,json,hashlib,shutil,subprocess,gzip,time,argparse,tarfile
+if not __debug__:raise RuntimeError('Assertions must remain enabled')
+ap=argparse.ArgumentParser();ap.add_argument('out',type=Path);ap.add_argument('--physical',action='store_true');ap.add_argument('--boost-include',default='/usr/include');ap.add_argument('--cxx',default='c++');args=ap.parse_args();P=Path(__file__).resolve().parent;O=args.out.resolve();assert not O.exists() and not O.is_relative_to(P)
+manifest=json.loads((P/'MANIFEST.json').read_text());assert manifest=={f.relative_to(P).as_posix():hashlib.sha256(f.read_bytes()).hexdigest()for f in P.rglob('*')if f.is_file()and f!=P/'MANIFEST.json'and '__pycache__'not in f.parts}
+O.mkdir(parents=True);(O/'tmp').mkdir();env=dict(os.environ,PYTHONDONTWRITEBYTECODE='1',PYTHONINTMAXSTRDIGITS='0',TMPDIR=str(O/'tmp'));start=time.monotonic()
+w=O/'bit510-review/research/e8-r480-dead-totals/data';w.mkdir(parents=True);shutil.copyfile(P/'inputs/ORIGINAL.json.gz',w/'bitE8mut-r480.json.gz')
+a=O/'r480-sharp535-publication/research/r480-sharp-charge535';(a/'inputs').mkdir(parents=True);shutil.copytree(P/'engines',a/'engines')
+for name in ['PR510-EXPECTED.json','PR535-EXPECTED.json']:shutil.copyfile(P/'inputs'/name,a/'inputs'/name)
+shutil.copyfile(P/'inputs/PR539-EXPECTED.json',a/'EXPECTED.json')
+for name in ['slice','minimum','price']:
+ f=O/(name+'.py');shutil.copyfile(P/'scripts'/(name+'.py'),f)
+ with (O/(name+'.log')).open('w')as log:subprocess.run([sys.executable,'-B',str(f)],cwd=O,env=env,stdout=log,stderr=subprocess.STDOUT,check=True)
+ print(name,'PASS',flush=True)
+new=O/'bit510-sliced-minimal-frames/WORD.json.gz';assert json.loads(gzip.decompress(new.read_bytes()))==json.loads(gzip.decompress((P/'inputs/BASE-CANDIDATE.json.gz').read_bytes()))
+x=json.loads((O/'bit510-sliced-minimal-price/RESULT.json').read_text());assert x==json.loads((P/'inputs/BASE-EXPECTED.json').read_text())
+m=json.loads((O/'bit510-sliced-minimal-frames/RESULT.json').read_text());e=json.loads((P/'evidence/MINIMUM-FRAMES.json').read_text());assert {k:v for k,v in m.items()if k not in ('seconds','fee_delta')}=={k:v for k,v in e.items()if k not in ('seconds','fee_delta')}  # libm discovery diagnostic only; all frames/profiles and subsequent rational pricing remain exact
+# Reconstruct PR541 exactly, then retain only the frozen beneficial frame reductions.
+f=O/'preload60.py';shutil.copyfile(P/'scripts/preload60.py',f)
+with (O/'preload60.log').open('w')as log:subprocess.run([sys.executable,'-B',str(f),'--input',str(w/'bitE8mut-r480.json.gz'),'--out',str(O/'trace541-rebuild')],cwd=O,env=env,stdout=log,stderr=subprocess.STDOUT,check=True)
+assert json.loads(gzip.decompress((O/'trace541-rebuild/WORD.json.gz').read_bytes()))==json.loads(gzip.decompress((P/'inputs/PR541-WORD.json.gz').read_bytes()))
+for name in ['trace_slice','transfer','profile','trace_price']:
+ f=O/(name+'.py');shutil.copyfile(P/'scripts'/(name+'.py'),f)
+ with (O/(name+'.log')).open('w')as log:subprocess.run([sys.executable,'-B',str(f)],cwd=O,env=env,stdout=log,stderr=subprocess.STDOUT,check=True)
+ print(name,'PASS',flush=True)
+new=O/'trace541-frozen-minimum/WORD.json.gz';assert json.loads(gzip.decompress(new.read_bytes()))==json.loads(gzip.decompress((P/'inputs/CANDIDATE.json.gz').read_bytes()))
+x=json.loads((O/'trace541-frozen-minimum-price/RESULT.json').read_text());assert x==json.loads((P/'EXPECTED.json').read_text())
+subprocess.run([sys.executable,'-B',str(P/'scripts/final_arithmetic.py'),str(P),str(O/'FINAL-ARITHMETIC.json')],cwd=O,env=env,check=True)
+r=dict(status='PASS_PACKAGED_RECONSTRUCTION_AND_PROSPECTIVE_PRICE',kappa=x['kappa_decimal'],bit=x['bit_decimal'],word_sha256=hashlib.sha256(new.read_bytes()).hexdigest(),manifest_sha256=hashlib.sha256((P/'MANIFEST.json').read_bytes()).hexdigest(),python=sys.version,seconds=time.monotonic()-start,scope='Exact reconstruction of source slice and least frames,720 scalar columns, nesting and two rational moment engines with47 outer constraints. Prospective unchanged owner schedule and prior finite allowances; physical template and finite invoice remain pending. No completed supplier admission.')
+(O/'RECEIPT.json').write_text(json.dumps(r,indent=2)+'\n');print(r['status'],r['kappa'],flush=True)
+
+if args.physical:
+ for archive in ['source493.tar.gz','source510.tar.gz']:
+  with tarfile.open(P/'dependencies'/archive)as t:
+   for item in t:
+    if item.isdir():continue
+    assert item.isfile(),item.name
+    p=O/item.name;assert p.resolve().is_relative_to(O)
+    p.parent.mkdir(parents=True,exist_ok=True);p.write_bytes(t.extractfile(item).read())
+ f=O/'physical.py';shutil.copyfile(P/'scripts/physical.py',f)
+ env.update(BOOST_INCLUDE=args.boost_include,CXX=args.cxx)
+ with (O/'physical.log').open('w')as log:subprocess.run([sys.executable,'-B',str(f)],cwd=O,env=env,stdout=log,stderr=subprocess.STDOUT,check=True)
+ physical=json.loads((O/'trace541-frozen-admission/RESULT.json').read_text());assert physical['kappa']==x['kappa_decimal'] and physical['profile']==x['profile']
+ r.update(status='PASS_PACKAGED_CONSTRUCTION_PHYSICAL_ADMISSION_AND_EXACT_PRICE',physical_receipt='trace541-frozen-admission/RESULT.json',seconds=time.monotonic()-start,scope='Fresh complete finite record, geometry, owner and invoice replay plus exact source reconstruction and two-engine pricing. Complex535 cited rather than rebuilt; inherited all-size assumptions remain conditional.')
+ (O/'RECEIPT.json').write_text(json.dumps(r,indent=2)+'\n');print(r['status'],r['kappa'],flush=True)