diff --git a/.github/workflows/bit-reuse-147.yml b/.github/workflows/bit-reuse-147.yml
new file mode 100644
index 0000000000..6a8d3a27ec
--- /dev/null
+++ b/.github/workflows/bit-reuse-147.yml
@@ -0,0 +1,29 @@
+name: Verify recycled bit registers on the paired-cube bit word
+
+on:
+  push:
+  pull_request:
+  workflow_dispatch:
+
+permissions:
+  contents: read
+
+jobs:
+  certificate:
+    runs-on: ubuntu-24.04
+    timeout-minutes: 30
+    strategy:
+      fail-fast: false
+      matrix:
+        python-version: ['3.11', '3.14']
+    steps:
+      - uses: actions/checkout@3d3c42e5aac5ba805825da76410c181273ba90b1 # v7
+        with:
+          persist-credentials: false
+      - uses: actions/setup-python@5fda3b95a4ea91299a34e894583c3862153e4b97 # v7
+        with:
+          python-version: ${{ matrix.python-version }}
+      - name: Reproduce PR144, run the ledger, check exact nesting, replay and reprice
+        run: python3 research/bit-reuse-147/verify.py
+      - name: Require committed files to remain unchanged
+        run: git diff --exit-code
diff --git a/research/bit-reuse-147/NOTICE b/research/bit-reuse-147/NOTICE
new file mode 100644
index 0000000000..aaeb125cab
--- /dev/null
+++ b/research/bit-reuse-147/NOTICE
@@ -0,0 +1,21 @@
+Recycled bit registers on the paired-cube shared cores, 9 October 2026
+Copyright 2026 DaysSky. Apache-2.0.
+
+Prepared by DaysSky with Anthropic Claude assistance. The contribution is the late read timetable, the hand-off
+of dead bit registers to later deferred births on this word, and their verification. No priority over concurrent
+work is claimed.
+
+Credits for what this package uses:
+
+- icekylinx: #144 and #130 (paired-cube and three-stage cover framework, selected gauges, completed-core ledger,
+  pricing and assembly), and #104.
+- hpst3r (William Porter): #147, terminal elimination on this bit word. Its plan of 1,549 eliminated slots is used
+  unchanged, and price147.py follows its pricing approach.
+- jamesyc (James Chang): #124, the birth-read reuse operation.
+- SovereignSteak: #122, terminal accumulator elimination.
+- Swapnil Jain: the round-seven deferred bit word and its exact frame tools.
+- Zhihao Chen (jacklightChen): the PR97 ledger.
+- an664: #128, completed-core sharing.
+- gupt1156: #148, the atom exponent 1/2000 reported as a secondary value.
+
+All inherited licenses, notices and assistance disclosures remain in force and unchanged.
diff --git a/research/bit-reuse-147/PROOF.md b/research/bit-reuse-147/PROOF.md
new file mode 100644
index 0000000000..03bb202330
--- /dev/null
+++ b/research/bit-reuse-147/PROOF.md
@@ -0,0 +1,96 @@
+# Recycling dead bit registers at deferred births: argument and scope
+
+**Setting.** This is the bit branch of the merged #144 (icekylinx), after the terminal elimination of #147 (hpst3r).
+Its word is Swapnil Jain's round-seven PR97 word at h = 23:
+
+- non-deferred and omitted reads at frame 0;
+- early V gates, phase 1 and the copied centres;
+- deferred reads of the selected slots at their sigma frames, in the inherited readout order;
+- late V gates, the rest of L, the output reads, then L^-1 and V^-1 at the full frame.
+
+#144 charges the resulting chains per vertex in dimension m = 69: auxiliary chain steps, source and target chains and
+copied centres, each three times; one gauge child 3 dim(sigma) per selected slot; and 2v endpoint children.
+W = 2v + R and the deficit is 2,024 per vertex. #147 deletes 1,549 terminal output slots; that plan is used here
+unchanged, as is #144's choice of 9,543 selected gauges.
+
+**Two changes.** Nothing else in the word moves.
+
+1. *Reads run late.* The selected reads keep their inherited relative order, but each one is executed as late as
+   these constraints allow, instead of in one block after phase 1:
+   - a read precedes the first operation that touches its slot, and that slot's V gate follows the read;
+   - the V gates of one leaf keep their inherited chain order;
+   - every read on a target precedes the first redirected write into that target.
+2. *Dead registers are reused.* A slot D that is neither an output slot nor a retained centre is never touched
+   again after its last operation, where it sits at the frame U_D of that operation. For 3,338 pairs (B, D) with D's last operation
+   before B's read and U_D inside sigma_B, the register of D is moved from U_D to sigma_B and becomes B. B is then a
+   deferred slot without an entrance gauge. This is jamesyc's #124 birth-read operation, applied to the bit word.
+
+**Scalar identity.** The inherited identity holds for arbitrary start values of all slots: a slot s with start value
+z_s contributes C_{T,s} z_s to target T through L and the root reads, and its read subtracts exactly that, where C is
+the exact integer adjoint.
+
+- Delaying a read does not change what it reads. Until its first operation or V gate, the register holds its start
+  value, and a read only writes into targets, which nothing reads back.
+- For a recycled pair, B's start value is whatever D left behind, a linear form in the data and the other start
+  values. D's last operation is before the read and B's first operation is after it, so B's read sees that value
+  and B's own contribution carries the same value with the same coefficients. The two cancel as before.
+- D's remaining contributions are unchanged, because no operation reads D after its last operation.
+- The inverse pass undoes the register updates (operations and V gates) in exact reverse order, at the full frame.
+  On a recycled register this removes B's V gate before D's operations are undone. The inherited order, with every
+  V^-1 after L^-1, would not restore the registers; `verify.py` checks that it fails.
+
+`word147.complete` runs the literal word on the 23,979 physical registers with every data port, target and
+register as its own formal variable, over F2 (bitmasks) and over Z (integer linear forms with the exact integer
+coefficients). Every register must end as its own variable and every target as its start variable plus exactly
+its defining sum. This check is complete and uses no random vectors. `word147.replay` then runs the word on random
+scratch and data, where three tampered words must break the Z identity: a recycled read of the register's entrance
+value, a recycled read taken before its donor's last write, and the inherited inverse order.
+
+**Frames.** Every frame is an original frame key of the PR97 word, so nondegeneracy is inherited. A hand-off is the
+difference of two inherited projectors; it adds finitely many rank-witness primes for the fixed prime q of #130's
+weighted compiler to avoid, and nothing else. `word147.ledger`
+executes the forward word on frame keys. Each scalar gate must find all its registers at one common key, and no
+register may retreat in dimension. The check of actual nesting is separate:
+
+- A move between two keys of one inherited certified chain, in that chain's order, is nested by the pinned audits
+  (`check_lifted.py` D, `check_frames.py` X and Q) that #144 relies on. Those chains are the original slot chains,
+  the X chains in their original V order, and the target chains in readout order with positive Z support, ending
+  at t_T^perp. A subsequence of a nested chain is nested.
+- Every other move is checked exactly over Q with integer bases rebuilt by `check_lifted.py`, and both frames are
+  checked nondegenerate. There are 4,878 such moves:
+  - 3,338 hand-offs U_D -> sigma_B (993 of them have equal dimension, so the two frames are equal);
+  - 935 target moves from the last remaining level to a redirected V frame;
+  - 605 target moves from one eliminated slot's redirected frame to another's V frame. These arise because a
+    redirected V write now runs just before its slot's first operation rather than in the late-V block.
+
+Delaying the reads does not reorder any chain. With no elimination and no recycling, the delayed word reproduces
+every histogram of #144's row; `verify.py` checks this as a control.
+
+**Accounting.** #144's ledger (`notes/paired-cube-sharing.tex`) charges a physical role by its entrance gauge and
+its frame chain. A recycled register enters with its first occupant's gauge, follows one nested chain to the full
+frame, and has its raw value restored, so it satisfies the same completed-core contract. Relative to #147, one pair
+(B, D) with d = dim U_D and f = dim sigma_B
+
+- removes D's final step of rank 23 - d (three copies) and B's gauge child of width 3f;
+- adds the hand-off step of rank f - d (three copies);
+- lowers W by one.
+
+The removed rank is 3(23 - d) + 3f and the added rank is 3(f - d), so the mass drops by 69 = m and the deficit stays
+2,024. The row is rebuilt from the actual ledger moves (`word147.row`) and priced with #144's own functions
+(`price147.py`): `exact_moment`, the full rare-class fallback, the complex certificate, the finite bridge and the
+47-constraint assembly. The largest child falls from 66 to 60, which only shortens the recursion depth bound.
+
+**Scope.** This is a finite conditional construction. Every interface that #144 retains stays an assumption:
+
+- the unchanged complete scalar word and the inherited rational frame audit of PR97;
+- the paired source scheduling and completed-core sharing;
+- the uniform recursion, analytic and tape contracts.
+
+Not established here:
+
+- The replay runs one core on its own bank, as in #147. That is the premise the shared-core argument uses.
+- The pinned PR97 frame audits are relied on, not rerun.
+- The plan is one legal choice found by a matching heuristic (`make_plan.py`). No optimality is claimed, and the
+  certified value does not depend on how the plan was found.
+- The headline uses the merged atom exponent 1/1000. The value at 1/2000 (gupt1156's #148) is reported separately
+  and rests on the retained requirement a_bit < beta < 1 - a_bit.
diff --git a/research/bit-reuse-147/README.md b/research/bit-reuse-147/README.md
new file mode 100644
index 0000000000..44fe6de1da
--- /dev/null
+++ b/research/bit-reuse-147/README.md
@@ -0,0 +1,71 @@
+# Recycled bit registers on #144's shared cores
+
+Under the interfaces retained by the merged #144, this package certifies
+
+    T(n) = O(n (log n)^(1 - κ)),   κ = 471809569/10^12 = 4.71809569e-4
+
+That is +1.54% over #147 (4646633/10^10) and +2.36% over the merged #144 (4609169/10^10). It adds this directory
+and one workflow; no existing file changes.
+
+## What changes
+
+Only the bit word changes, and #144 is bit-limited. Starting from #147's reduced word:
+
+- The selected deferred reads keep their order but run as late as the word allows, instead of in one block.
+- 3,338 auxiliary registers that are dead (never touched again) are handed to a later deferred slot whose entrance
+  frame contains the dead register's frame. The recipient's inherited read cancels what the donor left. This is
+  jamesyc's #124 birth-read operation, applied to the bit word.
+
+Each hand-off removes one physical role, the donor's final step and the recipient's gauge child, and adds one
+shorter step. [PROOF.md](PROOF.md) gives the argument and its scope.
+
+| | roles R | W per vertex | coarse bit saving | κ |
+|---|---|---|---|---|
+| #144 (merged) | 28,866 | 32,408 | 4617656/10^10 | 4609169/10^10 |
+| #147 | 27,317 | 30,859 | 4655227/10^10 | 4646633/10^10 |
+| this package | 23,979 | 27,521 | 472689442/10^12 | **471809569/10^12** |
+
+The deficit per vertex (2,024) is unchanged. The complex side (4856569/10^10) still has about 2.9% headroom.
+
+The headline keeps the merged atom exponent 1/1000. At gupt1156's 1/2000 (#148) the same row gives
+472026274/10^12 = 4.72026274e-4; `verify.py` checks both.
+
+Not composed here: Th0rgal's exact min-cut gauge subset (#146). #144's own 9,543 selected gauges are used.
+
+## Verify
+
+    python3 research/bit-reuse-147/verify.py          # stdlib only, about 2 minutes; do not use -O
+    python3 research/bit-reuse-147/verify.py --all    # also re-checks every inherited frame move, about 9 minutes
+
+`verify.py` runs these checks:
+
+1. It reproduces #144's coarse saving, stopped saving and κ exactly on #144's own grid.
+2. With no elimination and no recycling, the delayed-read word reproduces every histogram of #144's row.
+3. It runs the frame ledger of the frozen plan: every scalar gate at one common frame, no register retreating. The
+   row rebuilt from the actual moves must equal `row.json`.
+4. It checks all 4,878 new adjacent frame pairs for exact nesting over Q, with bases rebuilt by `check_lifted.py`.
+   A control must reject mismatched hand-offs.
+5. It runs the literal word on the 23,979 physical registers with all 27,521 inputs as formal variables, over F2
+   and over Z, and requires the exact identity. This is complete, with no random vectors. A random replay then
+   confirms that three tampered words break the identity.
+6. It prices the row with #144's assembly (47 constraints, 7 margins); the next grid points are rejected.
+
+## Files
+
+- `word147.py`: timetable, literal events, scalar replay, frame ledger and row.
+- `frames147.py`: exact integer frame bases and nesting over Q.
+- `price147.py`: pricing with #144's `paired_cube_network.py` functions. As in #147, the one relaxed check is the
+  bit dimension test, which accepts R <= 28866.
+- `plan.json`: the eliminated slots (#147's plan, unchanged), the remaining gauge slots and the 3,338 pairs.
+- `row.json`, `expected.json`: the resulting row and the certified values.
+- `make_plan.py`: how the plan was found (needs numpy and scipy). Verification does not use it.
+
+## Limits
+
+- This is a finite conditional construction. Everything #144 retains stays an assumption, and the pinned PR97
+  frame audits are relied on, not rerun.
+- The replay runs one core on its own bank, as in #147.
+- The plan is one legal choice from a matching heuristic. No optimality is claimed.
+- The complete `make verify` and the formal targets were not run locally; the fork's CI runs them.
+
+Credits are in [NOTICE](NOTICE).
diff --git a/research/bit-reuse-147/expected.json b/research/bit-reuse-147/expected.json
new file mode 100644
index 0000000000..138271dc03
--- /dev/null
+++ b/research/bit-reuse-147/expected.json
@@ -0,0 +1,5 @@
+{
+ "coarse": "236344721/500000000000",
+ "kappa": "471809569/1000000000000",
+ "kappa_atom_2000": "236013137/500000000000"
+}
diff --git a/research/bit-reuse-147/frames147.py b/research/bit-reuse-147/frames147.py
new file mode 100644
index 0000000000..a336706b48
--- /dev/null
+++ b/research/bit-reuse-147/frames147.py
@@ -0,0 +1,72 @@
+"""Exact integer bases of the frames that the recycled word joins in a new order, and exact nesting over Q.
+
+Stdlib only.  Node frames, late-copy intersections and root frames are rebuilt from the pinned PR97 witness with
+Swapnil Jain's check_lifted.py machinery, exactly as check_frames.py derives them; sigma and V-leaf frames are the
+primitive integer bases of the pinned data file."""
+import sys
+from fractions import Fraction
+from math import gcd
+
+
+def prim(r):
+    r = [Fraction(x) for x in r]; den = 1
+    for x in r: den = den * x.denominator // gcd(den, x.denominator)
+    z = [int(x * den) for x in r]; g = 0
+    for x in z: g = gcd(g, x)
+    return [x // g for x in z] if g else z
+
+
+class Exact:
+    def __init__(self, S, W, folder):
+        sys.path.insert(0, str(folder))
+        import check_lifted as cl
+        from linalg import null_exact, rank_mod, Q31
+        self.S = S; self.h = h = S.h; self.cl = cl; self.null_exact = null_exact; self.rank_mod = rank_mod; self.Q31 = Q31
+        pr = cl.Prog(W); bas, dn = cl.spans(pr); self.users = cl.users_of(pr, dn)
+        fr = self.fr = cl.Frames(pr, dn, bas, self.users); fr.cobases(); d, _ = fr.dims(); self.ld, _ = cl.late_frames(fr, pr, self.users)
+        assert all(S.node_dims[n] == d[n] for n in pr.act) and len(S.node_dims) == len(pr.act), 'recorded node dimensions'
+        self.full = [[int(i == j) for j in range(h)] for i in range(h)]
+        self._basis = {}; self._null = {}
+
+    def ukey(self, n, k):
+        u = self.users[n][k]
+        return ('n', u[1]) if u[0] == 'gate' else self.cl.root_key(u)
+
+    def basis(self, key):
+        if key not in self._basis:
+            S, fr = self.S, self.fr; kind = key[0]
+            if kind == 'sigma': B = S.sigma[key[1]]
+            elif kind == 'v': B = S.vstart[key[1]]
+            elif kind == '0': B = []
+            elif kind == 'F': B = self.full
+            elif kind == 'out': B = [prim(r) for r in fr.exact_basis(('out', key[2][0], tuple(key[2])))]
+            elif kind == 'c':
+                n, lt = key[1], key[2]
+                B = self.basis(self.ukey(n, lt[-1]))
+                for kk in reversed(lt[:-1]): B = fr.intersect_exact(self.basis(self.ukey(n, kk)), B)
+                B = [prim(r) for r in B]
+            else: B = [prim(r) for r in fr.exact_basis(key)]
+            assert len(B) == S.dim(key), ('exact dimension', key)
+            self._basis[key] = B
+        return self._basis[key]
+
+    def null(self, key):
+        if key not in self._null:
+            B = self.basis(key)
+            self._null[key] = [prim(z) for z in self.null_exact(B, self.h)] if B else self.full
+        return self._null[key]
+
+    def inside(self, a, b):
+        """span(a) <= span(b), exactly over Q (integer dot products with an exact integer basis of b's annihilator)."""
+        A = self.basis(a)
+        if not A or len(self.basis(b)) == self.h: return True
+        if len(A) > len(self.basis(b)): return False
+        return all(sum(x * z for x, z in zip(ra, rz)) == 0 for ra in A for rz in self.null(b))
+
+    def nondegenerate(self, key):
+        """G = I - J/9 nondegenerate on the frame: det(9 B G B^T) nonzero mod 2^31-1, hence over Z."""
+        B = self.basis(key)
+        if not B or len(B) == self.h: return True
+        sm = [sum(b) for b in B]
+        gram = [[9 * sum(x * y for x, y in zip(a, b)) - sa * sb for b, sb in zip(B, sm)] for a, sa in zip(B, sm)]
+        return self.rank_mod(gram, self.Q31) == len(B)
diff --git a/research/bit-reuse-147/make_plan.py b/research/bit-reuse-147/make_plan.py
new file mode 100644
index 0000000000..2c927ee469
--- /dev/null
+++ b/research/bit-reuse-147/make_plan.py
@@ -0,0 +1,77 @@
+#!/usr/bin/env python3
+"""Regenerate plan.json: a heavy matching of dead registers to later deferred births.
+
+Needs numpy and scipy.  verify.py does not use this file: it checks the frozen plan directly, so the certified value
+does not depend on how the plan was found, and no optimality is claimed.
+
+  elim      taken unchanged from the existing plan.json (hpst3r's #147 plan at 29383a28d96b9880edbe59ed930f1da1dd6fdfa6);
+  level     #144's selected gauges that survive the elimination;
+  donors    every remaining slot that is neither an output slot nor a retained centre, at its last frame;
+  edge      donor D -> level slot B when D's last operation precedes B's read in the latest timetable and U_D is
+            inside sigma_B (screened modulo a prime here, verified exactly over Q by verify.py);
+  weight    the first-order moment saved, 3 phi(23 - d) + phi(3f) - 3 phi(f - d);
+  matching  maximum total weight (scipy.optimize.linear_sum_assignment).
+"""
+import json
+import sys
+from pathlib import Path
+
+import numpy as np
+from scipy.optimize import linear_sum_assignment
+
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE))
+import word147  # noqa: E402
+from frames147 import Exact  # noqa: E402
+
+PRIME = 1048573
+SAVING = 4.73e-4
+
+
+def main():
+    dr, W, D, S, record, folder = word147.load_schedule()
+    h, m = S.h, 3 * S.h
+    sel = json.loads((word147.ROOT / record['selection_file']).read_text())
+    elim = json.loads((HERE / 'plan.json').read_text())['elim']; gone = set(elim)
+    level = [s for s in sel['retained_readout_order'] if s not in gone]
+    word = word147.Word(S, elim, level, [])
+    exact = Exact(S, W, folder)
+    donors = [s for s in range(S.R) if s not in S.out and s not in S.ret and s not in gone]
+    last = {d: [k for k in S.chain_keys(d) if k[0] != 'F'][-1] for d in donors}
+    rows, owner = [], []
+    for i, d in enumerate(donors):
+        for r in exact.basis(last[d]): rows.append([x % PRIME for x in r]); owner.append(i)
+    V = np.array(rows, dtype=np.int64); owner = np.array(owner); size = np.bincount(owner, minlength=len(donors))
+    inside = {}
+    for b in level:
+        key = tuple(map(tuple, S.sigma[b]))
+        if key not in inside:
+            if len(key) == h: inside[key] = np.arange(len(donors))
+            else:
+                N = np.array([[x % PRIME for x in z] for z in exact.null(('sigma', b))], dtype=np.int64)
+                ok = (((V @ N.T) % PRIME) == 0).all(axis=1)
+                inside[key] = np.nonzero(np.bincount(owner[ok], minlength=len(donors)) == size)[0]
+    phi = lambda r: r * ((m / r) ** SAVING - 1) / SAVING if r else 0.0
+    edges = []
+    for j, b in enumerate(level):
+        f = S.f[b]
+        for i in inside[tuple(map(tuple, S.sigma[b]))]:
+            d = donors[int(i)]
+            if d == b or word.death[d] >= word.p[b]: continue
+            dd = S.dim(last[d])
+            edges.append((int(i), j, 3 * phi(h - dd) + phi(3 * f) - 3 * phi(f - dd)))
+    used_r = sorted({e[0] for e in edges}); used_c = sorted({e[1] for e in edges})
+    rr = {r: i for i, r in enumerate(used_r)}; cc = {c: i for i, c in enumerate(used_c)}
+    A = np.zeros((len(used_r), len(used_c)))
+    for r, c, x in edges: A[rr[r], cc[c]] = x
+    ri, ci = linear_sum_assignment(A, maximize=True)
+    pairs = sorted((level[used_c[c]], donors[used_r[r]]) for r, c in zip(ri, ci) if A[r, c] > 0)
+    taken = {b for b, _ in pairs}
+    plan = dict(elim=sorted(elim), retained=[s for s in level if s not in taken], pairs=[list(p) for p in pairs])
+    out = Path(sys.argv[1]) if len(sys.argv) > 1 else HERE / 'plan.json'
+    out.write_text(json.dumps(plan, separators=(',', ':')) + '\n')
+    print('edges %d, pairs %d, first-order gain %.0f -> %s' % (len(edges), len(pairs), sum(A[r, c] for r, c in zip(ri, ci)), out))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/bit-reuse-147/plan.json b/research/bit-reuse-147/plan.json
new file mode 100644
index 0000000000..6384777f51
--- /dev/null
+++ b/research/bit-reuse-147/plan.json
@@ -0,0 +1 @@
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diff --git a/research/bit-reuse-147/price147.py b/research/bit-reuse-147/price147.py
new file mode 100644
index 0000000000..adbe8342ea
--- /dev/null
+++ b/research/bit-reuse-147/price147.py
@@ -0,0 +1,104 @@
+"""Price a shared-core bit row with the merged #144 arithmetic (scripts/paired_cube_network.py, icekylinx).
+
+Stdlib only.  The functions used are #144's own: exact_moment(), the rare-class contamination of bit_certificate(),
+complex_certificate(), finite_bridge() and the 47-constraint assembly.  As in hpst3r's #147, the one relaxed check is
+the bit dimension test, which here accepts R <= 28866; every other check of shared_profile() is kept verbatim.  The
+coarse saving and kappa are the largest points of a 10^-12 grid that pass, and the next points are rejected.  On
+#144's own 10^-10 grid and row this reproduces 4617656/10^10, 4613422943/10^13 and 4609169/10^10 exactly.
+"""
+import json
+import sys
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+import paired_cube_network as pcn  # noqa: E402
+
+GRID = 10**12
+
+
+def bit_profile(row):
+    require = pcn.require
+    h, v, R, ell = (row[k] for k in ('h', 'v', 'R', 'loss'))
+    m, W, H = 3 * h, 2 * v + R, Counter()
+    selected = {int(r): n for r, n in row['selected_rank_histogram'].items()}
+    require(sum(selected.values()) == row['selected_roles'], 'Selected gauge count')
+    for r, n in selected.items():
+        require(0 < r < h and n > 0, 'Proper local gauges')
+        H[3 * r] += n
+    require((h, v, ell) == (23, 1771, 506) and 0 < R <= 28866, 'Retained bit local dimensions (R reduced)')
+    for name in ('auxiliary_histogram', 'source_data_histogram', 'target_data_histogram', 'copied_center_histogram'):
+        for r, n in row[name].items():
+            H[int(r)] += 3 * n
+    H[2] += 2 * v
+    H = pcn.clean(H)
+    require(all(0 < r < m and n > 0 for r, n in H.items()), 'Proper shared-core children')
+    mass = sum(r * n for r, n in H.items())
+    require(H == pcn.clean(row['child_histogram']), 'Saved complete child histogram')
+    require((m, W, mass, W * m - mass) == (row['m'], row['W_per_vertex'], row['rank_per_vertex'], row['deficit_per_vertex']),
+            'Saved sharing dimensions and rank')
+    require(W * m - mass == 2 * v - 3 * ell, 'Shared-core telescoping deficit')
+    return dict(m=m, local_dimension=h, W_per_vertex=W, rank_per_vertex=mass, deficit_per_vertex=W * m - mass,
+                child_multiplicities=H, maxchild=max(H), edge_count=sum(H.values()))
+
+
+def bit_gap(p, coarse):
+    """bit_certificate()'s strict moment gap at a coarse saving, or None if either strict inequality fails."""
+    try:
+        exact = pcn.exact_moment(p, coarse)
+    except (AssertionError, ValueError):
+        return None
+    m, W = p['m'], p['W_per_vertex']; fallback = 32 * m * m
+    added = pcn.BAD * Q(fallback * p['edge_count'], W * m) * pcn.exp_upper(coarse * pcn.log_upper(Q(m)))
+    rank_upper = Q(p['rank_per_vertex']) + pcn.BAD * fallback * p['edge_count']
+    gap = 1 - exact['moment_upper'] - added
+    return gap if gap > 0 and rank_upper < W * m else None
+
+
+def largest(ok, lo, hi):
+    assert ok(lo) and not ok(hi)
+    while hi - lo > 1:
+        mid = (lo + hi) // 2
+        lo, hi = (mid, hi) if ok(mid) else (lo, mid)
+    return lo
+
+
+def price(row, atom=None, grid=GRID):
+    """atom None: the merged #144 atom exponent 1/1000.  grid 10^10 is #144's own grid."""
+    atom = pcn.ATOM if atom is None else atom
+    p = bit_profile(row)
+    assert Q(2 * p['m']**3, 2**80) < pcn.BAD
+    k = largest(lambda k: bit_gap(p, Q(k, grid)) is not None, 4 * grid // 10**4, 6 * grid // 10**4)
+    coarse = Q(k, grid)
+    ab = (1 - atom) * coarse + atom * pcn.OLD
+    assert ab < atom < 1 - ab, 'Subordinate adapter and row tolls'
+    assembly_bit = min(ab, (1 - pcn.PHASE_STOP) * pcn.AC - Q(1, 10**10))
+    assert assembly_bit == ab, 'the bit side still binds'
+    crow = json.loads((ROOT / 'certificates' / 'paired-cube-complex-input.json').read_text())
+    phase = pcn.complex_certificate(crow)
+    bridge = pcn.finite_bridge(phase, None, crow)
+    assert Q(json.loads((ROOT / 'certificates' / 'copied-centers-network.json').read_text())['bit']['saving']) == pcn.OLD
+    def accepts(kk):
+        try: pcn.assembly(assembly_bit, pcn.AC, bridge, Q(kk, grid), beta=pcn.PHASE_STOP)
+        except (AssertionError, ValueError): return False
+        return True
+    kap = largest(accepts, 0, int(pcn.AC * grid) + 1)
+    res = pcn.assembly(assembly_bit, pcn.AC, bridge, Q(kap, grid), beta=pcn.PHASE_STOP)
+    assert len(res['strict_constraints']) == 47 and len(res['margins']) == 7
+    assert all(value > 0 for value in res['strict_constraints'].values())
+    return dict(R=row['R'], W_per_vertex=p['W_per_vertex'], maxchild=p['maxchild'], atom=atom,
+                coarse=coarse, coarse_gap=bit_gap(p, coarse), coarse_next_rejected=Q(k + 1, grid),
+                stopped=ab, complex=pcn.AC, kappa=Q(kap, grid), kappa_next_rejected=Q(kap + 1, grid),
+                absorption_gap=res['absorption_gap'])
+
+
+if __name__ == '__main__':
+    assert not sys.flags.optimize
+    row = json.loads(Path(sys.argv[1]).read_text())
+    for atom in (None, Q(1, 2000)):
+        r = price(row, atom)
+        print('atom %s: R %d W %d coarse %s stopped %.12e kappa %s = %.10e' % (
+            r['atom'], r['R'], r['W_per_vertex'], r['coarse'], float(r['stopped']), r['kappa'], float(r['kappa'])))
diff --git a/research/bit-reuse-147/row.json b/research/bit-reuse-147/row.json
new file mode 100644
index 0000000000..39ec6220bc
--- /dev/null
+++ b/research/bit-reuse-147/row.json
@@ -0,0 +1,99 @@
+{
+ "R": 23979,
+ "W_per_vertex": 27521,
+ "auxiliary_histogram": {
+  "1": 53031,
+  "10": 175,
+  "11": 338,
+  "12": 216,
+  "13": 501,
+  "14": 110,
+  "15": 628,
+  "16": 896,
+  "17": 1566,
+  "18": 635,
+  "19": 743,
+  "2": 64836,
+  "20": 2273,
+  "21": 262,
+  "3": 12632,
+  "4": 10831,
+  "5": 5848,
+  "6": 1619,
+  "7": 688,
+  "8": 1142,
+  "9": 497
+ },
+ "child_histogram": {
+  "1": 188241,
+  "10": 525,
+  "11": 1014,
+  "12": 648,
+  "13": 1536,
+  "14": 330,
+  "15": 1884,
+  "16": 3564,
+  "17": 5292,
+  "18": 3312,
+  "19": 9936,
+  "2": 200567,
+  "20": 6819,
+  "21": 789,
+  "22": 75,
+  "3": 38040,
+  "4": 33117,
+  "48": 72,
+  "5": 17577,
+  "51": 118,
+  "54": 806,
+  "57": 3651,
+  "6": 4857,
+  "60": 9,
+  "7": 2064,
+  "8": 3426,
+  "9": 1491
+ },
+ "copied_center_histogram": {
+  "22": 23
+ },
+ "deficit_per_vertex": 2024,
+ "eliminated_roles": 1549,
+ "h": 23,
+ "loss": 506,
+ "m": 69,
+ "omitted_roles": 2022,
+ "rank_per_vertex": 1896925,
+ "recycled_roles": 3338,
+ "selected_rank_histogram": {
+  "16": 72,
+  "17": 118,
+  "18": 806,
+  "19": 3651,
+  "20": 9
+ },
+ "selected_roles": 4656,
+ "source_data_histogram": {
+  "1": 5061,
+  "13": 11,
+  "16": 198,
+  "17": 22,
+  "19": 1540,
+  "3": 23,
+  "4": 208,
+  "5": 11
+ },
+ "source_manifest": "references/partial-gauge/pr97/SOURCE.json",
+ "source_manifest_sha256": "68fb539abcd6df21d142e4e204b6e9482eb7f9cd3907650f54bd058070266596",
+ "target_data_histogram": {
+  "1": 4655,
+  "16": 94,
+  "17": 176,
+  "18": 469,
+  "19": 1029,
+  "2": 839,
+  "21": 1,
+  "22": 2,
+  "3": 25
+ },
+ "v": 1771
+}
diff --git a/research/bit-reuse-147/verify.py b/research/bit-reuse-147/verify.py
new file mode 100644
index 0000000000..1408992788
--- /dev/null
+++ b/research/bit-reuse-147/verify.py
@@ -0,0 +1,123 @@
+#!/usr/bin/env python3
+"""Verify the recycled bit word from the pinned sources (Python stdlib only; a few minutes; do not use -O).
+
+  1. #144 control.  On #144's own 10^-10 grid, the merged #144 row prices to its coarse saving 4617656/10^10,
+     stopped saving 4613422943/10^13 and kappa 4609169/10^10 exactly, with the next grid points rejected.
+  2. Timetable control.  With no elimination and no recycling, the delayed-read word of this package reproduces every
+     histogram of #144's reconstruct() row: delaying reads in the inherited order changes no charge.
+  3. The frozen plan is legal, its timetable satisfies every ordering constraint, and the frame ledger runs: every
+     scalar gate finds its registers at one common frame key and no register ever retreats.  The row rebuilt from the
+     actual ledger moves equals row.json.
+  4. Every adjacent frame pair that is not an ordered pair of one inherited certified chain is nested exactly over Q
+     and both frames are nondegenerate.  Control: a recipient given a donor whose frame is not inside sigma is
+     rejected by the same test.
+  5. The literal scalar word on the physical registers.  First on formal variables, over F2 and over Z with the
+     exact integer read coefficients: every register must end as its own variable and every target as its start
+     variable plus exactly its defining sum.  This is complete and uses no random vectors.  Then with random
+     scratch and data, where three tampered words must break the Z identity: a recycled read of the register's
+     entrance value, a recycled read taken before its donor's last write, and the inherited inverse order.
+  6. #144's assembly on the new row: kappa and its next grid point, at the merged atom exponent 1/1000 and at 1/2000.
+Usage: python3 research/bit-reuse-147/verify.py [--all]
+  --all also checks exact nesting of every distinct ledger move, including those the pinned audits already certify
+  (about seven more minutes).
+"""
+import json
+import random
+import sys
+import time
+from collections import Counter
+from fractions import Fraction as Q
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+sys.path.insert(0, str(HERE))
+T0 = time.time()
+def log(*a): print('[%5.0fs]' % (time.time() - T0), *a, flush=True)
+
+EXPECT = json.loads((HERE / 'expected.json').read_text())
+
+
+def clean(H): return {str(k): n for k, n in H.items() if n}
+
+
+def main():
+    assert not sys.flags.optimize, 'run without -O'
+    import word147
+    import price147
+    from frames147 import Exact
+    from paired_cube_bit import reconstruct
+    ROOT = word147.ROOT
+
+    base = reconstruct()
+    r0 = price147.price(base, grid=10**10)
+    assert (r0['coarse'], r0['stopped'], r0['kappa']) == (Q(4617656, 10**10), Q(4613422943, 10**13), Q(4609169, 10**10)), r0
+    log('PASS #144 control: coarse 4617656/10^10, stopped 4613422943/10^13, kappa 4609169/10^10 (next rejected)')
+
+    dr, W, D, S, record, folder = word147.load_schedule()
+    outputs = {(c, tuple(T)): n for c, T, n in W['outputs']}
+    sel = json.loads((ROOT / record['selection_file']).read_text())
+    plain = word147.Word(S, [], sel['retained_readout_order'], [])
+    led0 = plain.ledger(); row0 = plain.row(led0, record)
+    for key in ('auxiliary_histogram', 'source_data_histogram', 'target_data_histogram', 'selected_rank_histogram',
+                'child_histogram'):
+        assert clean(row0[key]) == clean(base[key]), key
+    assert (row0['R'], row0['W_per_vertex'], row0['rank_per_vertex']) == (base['R'], base['W_per_vertex'], base['rank_per_vertex'])
+    assert plain.replay(2, random.Random(1), outputs) and plain.replay(0, random.Random(2), outputs)
+    log('PASS timetable control: #144 row reproduced with reads delayed to positions %d..%d of %d; replay holds' % (
+        min(plain.p.values()), max(plain.p.values()), plain.N))
+
+    plan = json.loads((HERE / 'plan.json').read_text())
+    assert set(plan['retained']) <= set(sel['retained_readout_order']), "level slots are among #144's selected gauges"
+    assert {b for b, _ in plan['pairs']} <= set(sel['retained_readout_order'])
+    word = word147.Word(S, plan['elim'], plan['retained'], plan['pairs'])
+    led = word.ledger(); row = word.row(led, record)
+    assert row == json.loads((HERE / 'row.json').read_text()), 'row differs'
+    occupants = Counter(led['phys'].values())
+    log('PASS ledger: %d eliminated, %d recycled, %d physical roles (longest occupant chain %d); row equals row.json' % (
+        len(plan['elim']), len(plan['pairs']), row['R'], max(occupants.values())))
+
+    exact = Exact(S, W, folder)
+    fresh = word.new_moves(led)
+    kinds = Counter((reg, a[0], b[0]) for reg, a, b in fresh)
+    for reg, a, b in fresh:
+        assert exact.inside(a, b), ('not nested', reg, a, b)
+        assert exact.nondegenerate(a) and exact.nondegenerate(b), ('degenerate', a, b)
+    handoffs = {(('n', 0), ('sigma', b)) for b in word.donor_of}
+    assert kinds['a', 'n', 'sigma'] + kinds['a', 'ret', 'sigma'] + kinds['a', 'c', 'sigma'] + kinds['a', 'v', 'sigma'] + \
+        kinds['a', 'sigma', 'sigma'] + kinds['a', 'out', 'sigma'] <= len(handoffs)
+    assert all(reg == 'y' or b[0] == 'sigma' for reg, a, b in fresh), 'only hand-offs and target chains are new'
+    rng = random.Random(20261009); rejected = 0; tried = 0
+    donors = sorted(word.recipient_of)
+    while tried < 200:
+        b = rng.choice(sorted(word.donor_of)); d = rng.choice(donors)
+        key = [k for k in S.chain_keys(d) if k[0] != 'F'][-1]
+        if S.dim(key) > S.f[b]: continue
+        tried += 1; rejected += not exact.inside(key, ('sigma', b))
+    assert rejected > 0, 'the nesting test never rejects'
+    log('PASS exact nesting over Q of all %d new adjacent pairs %s; control rejected %d of %d mismatched hand-offs' % (
+        len(fresh), dict(kinds), rejected, tried))
+    if '--all' in sys.argv:                                 # optional: every distinct move, inherited ones included
+        every = sorted(led['moves'], key=str)
+        for reg, a, b in every: assert exact.inside(a, b), ('not nested', reg, a, b)
+        log('PASS exact nesting over Q of all %d distinct ledger moves, inherited ones included' % len(every))
+
+    assert word.complete(2, outputs) and word.complete(0, outputs), 'scalar identity on formal variables'
+    log('PASS complete scalar identity on all %d formal variables, over F2 and over Z (no random vectors)' % (2 * S.v + row['R']))
+    rng = random.Random(20261009)
+    okF = all(word.replay(2, rng, outputs) for _ in range(3)); okZ = all(word.replay(0, rng, outputs) for _ in range(2))
+    assert okF and okZ, 'scalar identity'
+    for tamper in ('stale', 'early', 'vlast'):
+        assert not word.replay(0, rng, outputs, tamper), 'tamper accepted: ' + tamper
+    log('PASS literal replay on %d physical registers over Z and F2; stale read, premature read and inherited inverse '
+        'order all rejected' % row['R'])
+
+    for atom, name in ((None, 'kappa'), (Q(1, 2000), 'kappa_atom_2000')):
+        r = price147.price(row, atom)
+        assert str(r['kappa']) == EXPECT[name] and str(r['coarse']) == EXPECT['coarse'], (name, str(r['kappa']), str(r['coarse']))
+        log('PASS atom %s: kappa %s = %.10e (next %s rejected); coarse %s, stopped %.12e, R %d, W %d, largest child %d' % (
+            r['atom'], r['kappa'], float(r['kappa']), r['kappa_next_rejected'], r['coarse'], float(r['stopped']),
+            r['R'], r['W_per_vertex'], r['maxchild']))
+
+
+if __name__ == '__main__':
+    main()
diff --git a/research/bit-reuse-147/word147.py b/research/bit-reuse-147/word147.py
new file mode 100644
index 0000000000..bbad02df6e
--- /dev/null
+++ b/research/bit-reuse-147/word147.py
@@ -0,0 +1,438 @@
+"""The recycled bit word: #147's reduced PR97 word with dead auxiliary slots handed to later deferred births.
+
+Stdlib only.  A plan is three lists over the slots of Swapnil Jain's round-seven PR97 word (h = 23):
+
+  elim      terminal output slots deleted exactly as in #147 (hpst3r; SovereignSteak's #122 operation);
+  retained  first occupants that keep their entrance gauge sigma (as in #144);
+  pairs     [recipient B, donor D]: the physical register of D, dead after its last touch at frame U_D, is moved to
+            sigma_B and becomes B.  B's inherited deferred read then cancels the value D left behind (jamesyc's #124
+            birth-read operation, here on the bit word).
+
+Level slots (retained or recipients) are read at their sigma frames, in the inherited readout order, but each read is
+executed as late as the constraints below allow instead of in one block after phase one.  Everything else keeps its
+inherited place.  This module derives the timetable, the literal event list, the scalar replay, the frame ledger and
+the resulting row; verify.py drives it.
+"""
+import gzip
+import importlib.util
+import json
+import sys
+from collections import Counter, defaultdict
+from pathlib import Path
+
+HERE = Path(__file__).resolve().parent
+ROOT = HERE.parents[1]
+sys.path.insert(0, str(ROOT / 'scripts'))
+from partial_gauge_bit import frozen_sources  # noqa: E402
+
+
+def load_schedule():
+    record = json.loads((ROOT / 'certificates/paired-cube-bit-input.json').read_text())
+    folder, _ = frozen_sources(record)
+    spec = importlib.util.spec_from_file_location('pr97_deferred_reuse', folder / 'deferred.py')
+    dr = importlib.util.module_from_spec(spec); spec.loader.exec_module(dr)
+    W = json.load(gzip.open(folder / 'witness_23.json.gz', 'rt')); D = json.load(gzip.open(folder / 'deferred_23.json.gz', 'rt'))
+    return dr, W, D, dr.Schedule(W, D), record, folder
+
+
+class Word:
+    def __init__(self, S, elim, retained, pairs):
+        self.S = S; self.h = S.h; self.v = S.v; self.R = S.R
+        self.orig_sel = set(S.sel); self.readout = list(S.readout)
+        self.elim = set(elim); self.retained = set(retained)
+        self.donor_of = {b: d for b, d in pairs}; self.recipient_of = {d: b for b, d in pairs}
+        self.level = self.retained | set(self.donor_of)
+        ops = self.ops = S.ops
+        prev = {}; pred = defaultdict(list); last = {}; first = {}
+        for i, op in enumerate(ops):
+            for s in S.touch(op):
+                if s in prev: pred[i].append(prev[s])
+                prev[s] = i; last[s] = i; first.setdefault(s, i)
+        anc = set(); stack = [last[s] for s in S.ret]
+        while stack:
+            i = stack.pop()
+            if i in anc: continue
+            anc.add(i); stack.extend(pred[i])
+        assert not any(set(S.touch(ops[i])) & self.orig_sel for i in anc)
+        self.ph1 = [i for i in range(len(ops)) if i in anc and ops[i][0] != 'src']
+        self.rest = [i for i in range(len(ops)) if i not in anc and ops[i][0] != 'src']
+        self.pos = {i: k for k, i in enumerate(self.rest)}; self.N = len(self.rest)
+        self.first_pos = {s: (self.pos[first[s]] if first[s] in self.pos else -1) for s in first}
+        self.death = {s: (self.pos[last[s]] if last[s] in self.pos else -1) for s in last}
+        self.early_v = sorted(set(S.srcop) - self.orig_sel, key=lambda s: (S.srcop[s], len(S.vstart[s]), s))
+        src = Counter()
+        for op in ops:
+            if op[0] == 'add': src[op[2]] += 1
+            elif op[0] == 'fan': src[op[1]] += 1
+        self.never_source = {s for s in range(S.R) if src[s] == 0}
+        self.adj = S.adjoint()
+        self.target_of = {u: S.tid[S.out[u][1]] for u in self.elim}
+        self.check_plan()
+        self.timetable()
+
+    # ---------------------------------------------------------------------------------------- plan legality
+    def check_plan(self):
+        S = self.S
+        assert all(u in S.out and u in self.orig_sel and u in self.never_source for u in self.elim), 'eliminated slots'
+        assert self.retained <= self.orig_sel and set(self.donor_of) <= self.orig_sel, 'level slots are deferred slots'
+        assert not (self.retained & set(self.donor_of)), 'a recipient has no entrance gauge'
+        assert not (self.level & self.elim) and not (set(self.recipient_of) & self.elim)
+        assert len(self.recipient_of) == len(self.donor_of), 'one recipient per donor'
+        assert not (set(self.recipient_of) & set(S.out)), 'an output slot is read at the end and never dies'
+        assert not (set(self.recipient_of) & set(S.ret)), 'a retained centre is not used as a donor'
+        for b, d in self.donor_of.items(): assert b != d
+
+    # ---------------------------------------------------------------------------------------- timetable
+    def timetable(self):
+        """Latest read position p[B] (the read runs just before rest operation p[B]) and V time tau[s] of every
+        deferred V slot, as the greatest solution of
+          p[B] <= first operation touching B;            reads keep the inherited readout order;
+          a level V slot takes its V gate right after its read (tau = p);
+          tau[s] <= first operation touching s;          V gates of one leaf keep the inherited chain order;
+          every read on a target precedes the first redirected write into that target.
+        """
+        S, N = self.S, self.N
+        order = [b for b in self.readout if b in self.level]
+        p = {b: self.first_pos[b] for b in order}
+        late = [s for s in S.late_v]
+        tau = {s: self.first_pos[s] for s in late if s not in self.level}
+        def get(s): return p[s] if s in p else tau[s]
+        def cap(s, value):
+            if s in p:
+                if value < p[s]: p[s] = value; return True
+            elif value < tau[s]: tau[s] = value; return True
+            return False
+        chains = []
+        late_set = set(late)
+        for n, us in S.xs:
+            ch = [s for s in us if s in late_set]
+            if len(ch) > 1: chains.append(ch)
+        first_write = {}                               # target -> eliminated slots writing into it
+        for u in self.elim: first_write.setdefault(self.target_of[u], []).append(u)
+        readers = defaultdict(list)
+        for b in order:
+            for t in self.adj[b]:
+                if t in first_write: readers[t].append(b)
+        changed = True
+        while changed:
+            changed = False
+            for a, b in zip(reversed(order[:-1]), reversed(order[1:])):
+                changed |= cap(a, p[b])
+            for ch in chains:
+                for a, b in zip(reversed(ch[:-1]), reversed(ch[1:])):
+                    changed |= cap(a, get(b))
+            for t, us in first_write.items():
+                w = min((tau[u] if u in tau else self.first_pos[u]) for u in us)
+                for b in readers[t]: changed |= cap(b, w)
+        assert all(0 <= x <= N for x in p.values()) and all(0 <= x <= N for x in tau.values())
+        self.order = order; self.p = p; self.tau = tau
+        for b, d in self.donor_of.items():
+            assert self.death[d] < p[b], ('donor still live at the recipient read', b, d)
+
+    # ---------------------------------------------------------------------------------------- literal events
+    def events(self):
+        """Forward events of the middle section, in execution order:
+        ('read', B) | ('v', s) | ('vred', u) | ('op', i).  Reads and V events with time k run before rest op k."""
+        S = self.S
+        at = defaultdict(list)
+        rank = {b: i for i, b in enumerate(self.readout)}
+        lpos = {s: i for i, s in enumerate(S.late_v)}
+        for b in self.order:
+            at[self.p[b]].append((0, rank[b], 'read', b))
+            if b in S.srcop: at[self.p[b]].append((1, lpos[b], 'v', b))
+        for s, k in self.tau.items(): at[k].append((1, lpos[s], 'vred' if s in self.elim else 'v', s))
+        out = []
+        for k in range(self.N + 1):
+            out.extend((kind, s) for _, _, kind, s in sorted(at.get(k, ())))
+            if k < self.N: out.append(('op', self.rest[k]))
+        return out
+
+    def physical(self):
+        phys = {}
+        for s in range(self.R):
+            if s in self.elim: continue
+            r = s
+            while r in self.donor_of: r = self.donor_of[r]
+            phys[s] = r
+        return phys
+
+    # ---------------------------------------------------------------------------------------- scalar replay
+    def replay(self, ring, rng, outputs, tamper=None):
+        """Literal scalar word on physical registers with arbitrary scratch and data.  ring 0: exact integers with the
+        exact integer read coefficients; ring 2: the F2 word.  Returns True iff every physical register is restored
+        and every target receives exactly its defining sum (over F2 also exactly x_T)."""
+        S = self.S; trip, v = S.trip, S.v; ops = self.ops; srcop = S.srcop; elim = self.elim
+        mod = (lambda z: z & 1) if ring == 2 else (lambda z: z)
+        phys = self.physical()
+        x = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+        a0 = {r: rng.randrange(-10**6, 10**6) for r in set(phys.values())}
+        y0 = [rng.randrange(-10**6, 10**6) for _ in range(v)]
+        a = dict(a0); y = list(y0); done = []; count = [0]
+        def coefficient(c): return c & 1 if ring == 2 else c
+        def read(s):
+            value = a[phys[s]]
+            if tamper == 'stale' and s in self.donor_of and count[0] == 0:
+                count[0] = 1; value = a0[phys[s]]              # reads the register's entrance value, not what D left
+            for t, c in self.adj[s].items(): y[t] -= coefficient(c) * value
+        def redirected(t, value):
+            y[t] += value
+        def vgate(s, sign=1): a[phys[s]] += sign * x[srcop[s] - 1]
+        def runop(i, sign=1):
+            op = ops[i]
+            if op[0] == 'add':
+                if op[1] in elim:
+                    if sign == 1: redirected(self.target_of[op[1]], a[phys[op[2]]])
+                    return
+                a[phys[op[1]]] += sign * a[phys[op[2]]]
+            else:
+                for g in op[2]:
+                    if g in elim:
+                        if sign == 1: redirected(self.target_of[g], a[phys[op[1]]])
+                        continue
+                    a[phys[g]] += sign * a[phys[op[1]]]
+        for s in range(self.R):
+            if s not in self.level and s not in elim: read(s)
+        for s in self.early_v: vgate(s); done.append(('v', s))
+        for i in self.ph1: runop(i); done.append(('op', i))
+        for s, c in S.ret.items():
+            for t, T in enumerate(trip):
+                if c in T: y[t] += a[phys[s]]
+        events = self.events()
+        if tamper == 'early':                                 # one recycled read executed before its donor's last write
+            written = {}
+            for j, (kind, s) in enumerate(events):
+                if kind == 'v': written[s] = j
+                elif kind == 'op':
+                    op = ops[s]
+                    for g in ([op[1]] if op[0] == 'add' else op[2]): written[g] = j
+            b = next(e[1] for e in events if e[0] == 'read' and self.donor_of.get(e[1]) in written)
+            j = written[self.donor_of[b]]
+            assert events.index(('read', b)) > j
+            events = [e for e in events if e != ('read', b)]; events.insert(j, ('read', b))
+        for kind, s in events:
+            if kind == 'read': read(s)
+            elif kind == 'v': vgate(s); done.append(('v', s))
+            elif kind == 'vred': redirected(self.target_of[s], x[srcop[s] - 1])
+            else: runop(s); done.append(('op', s))
+        for s, (c, T) in S.out.items():
+            if s not in elim: y[S.tid[T]] += a[phys[s]]
+        if tamper == 'vlast':                                 # the inherited inverse order: every V^-1 after L^-1
+            done = [e for e in done if e[0] == 'v'] + [e for e in done if e[0] == 'op']
+        for kind, s in reversed(done):
+            if kind == 'v': vgate(s, -1)
+            else: runop(s, -1)
+        if any(mod(a[r] - a0[r]) for r in a0): return False
+        want = [0] * v
+        for (c, T), n in outputs.items():
+            A_, B_ = [q for q in T if q != c]
+            want[S.tid[T]] += sum(x[i] for i, Sx in enumerate(trip) if c in Sx and A_ not in Sx and B_ not in Sx)
+        for c in range(S.h):
+            tot = sum(x[i] for i, Sx in enumerate(trip) if c in Sx)
+            for t, T in enumerate(trip):
+                if c in T: want[t] += tot
+        if any(mod(y[t] - y0[t] - want[t]) for t in range(v)): return False
+        if ring == 2 and any(mod(y[t] - y0[t] - x[t]) for t in range(v)): return False
+        return True
+
+    def complete(self, ring, outputs):
+        """The same literal word on formal variables: every data port, target and physical register starts as its
+        own variable.  ring 2: bitmasks over the variables; ring 0: sparse integer linear forms.  Deterministic and
+        complete: returns True iff every register ends as its own variable and every target ends as its start
+        variable plus exactly its defining sum (over F2: plus x_T).  No random vectors are involved."""
+        S = self.S; trip, v = S.trip, S.v; ops = self.ops; srcop = S.srcop; elim = self.elim
+        phys = self.physical(); regs = sorted(set(phys.values()))
+        index = {r: 2 * v + k for k, r in enumerate(regs)}
+        want = [dict() for _ in range(v)]
+        for (c, T), n in outputs.items():
+            A_, B_ = [q for q in T if q != c]
+            row = want[S.tid[T]]
+            for i, Sx in enumerate(trip):
+                if c in Sx and A_ not in Sx and B_ not in Sx: row[i] = row.get(i, 0) + 1
+        for c in range(S.h):
+            star = [i for i, Sx in enumerate(trip) if c in Sx]
+            for t, T in enumerate(trip):
+                if c in T:
+                    for i in star: want[t][i] = want[t].get(i, 0) + 1
+        if ring == 2:
+            a = {r: 1 << index[r] for r in regs}; y = [1 << (v + t) for t in range(v)]; x = [1 << i for i in range(v)]
+            def into_y(t, value, c):
+                if c & 1: y[t] ^= value
+            def into_a(r, value, sign): a[r] ^= value
+        else:
+            a = {r: {index[r]: 1} for r in regs}; y = [{v + t: 1} for t in range(v)]; x = [{i: 1} for i in range(v)]
+            def accumulate(dst, value, c):
+                for k, z in value.items():
+                    z = dst.get(k, 0) + c * z
+                    if z: dst[k] = z
+                    else: del dst[k]
+            def into_y(t, value, c): accumulate(y[t], value, c)
+            def into_a(r, value, sign): accumulate(a[r], value, sign)
+        done = []
+        def read(s):
+            value = a[phys[s]]
+            for t, c in self.adj[s].items(): into_y(t, value, -c)
+        def vgate(s, sign=1): into_a(phys[s], x[srcop[s] - 1], sign)
+        def runop(i, sign=1):
+            op = ops[i]
+            if op[0] == 'add':
+                if op[1] in elim:
+                    if sign == 1: into_y(self.target_of[op[1]], a[phys[op[2]]], 1)
+                    return
+                into_a(phys[op[1]], a[phys[op[2]]], sign)
+            else:
+                for g in op[2]:
+                    if g in elim:
+                        if sign == 1: into_y(self.target_of[g], a[phys[op[1]]], 1)
+                        continue
+                    into_a(phys[g], a[phys[op[1]]], sign)
+        for s in range(self.R):
+            if s not in self.level and s not in elim: read(s)
+        for s in self.early_v: vgate(s); done.append(('v', s))
+        for i in self.ph1: runop(i); done.append(('op', i))
+        for s, c in S.ret.items():
+            for t, T in enumerate(trip):
+                if c in T: into_y(t, a[phys[s]], 1)
+        for kind, s in self.events():
+            if kind == 'read': read(s)
+            elif kind == 'v': vgate(s); done.append(('v', s))
+            elif kind == 'vred': into_y(self.target_of[s], x[srcop[s] - 1], 1)
+            else: runop(s); done.append(('op', s))
+        for s, (c, T) in S.out.items():
+            if s not in elim: into_y(S.tid[T], a[phys[s]], 1)
+        for kind, s in reversed(done):
+            if kind == 'v': vgate(s, -1)
+            else: runop(s, -1)
+        if ring == 2:
+            if any(a[r] != 1 << index[r] for r in regs): return False
+            if any({i for i, c in want[t].items() if c & 1} != {t} for t in range(v)): return False
+            return all(y[t] == (1 << (v + t)) ^ (1 << t) for t in range(v))
+        if any(a[r] != {index[r]: 1} for r in regs): return False
+        for t in range(v):
+            expected = dict(want[t]); expected[v + t] = 1
+            if y[t] != expected: return False
+        return True
+
+    # ---------------------------------------------------------------------------------------- frame ledger
+    def ledger(self):
+        """Run the forward word on frame keys.  Every scalar gate must find its registers at one common frame key;
+        every register only moves forward.  Returns the moves (register kind, old key, new key) in time order, the
+        key sequence of every register, and the rank histograms of the actual moves."""
+        S = self.S; h, v = S.h, S.v; ops = self.ops; elim = self.elim; phys = self.physical()
+        def norm(k): return ('out', 0, tuple(k[2])) if k[0] == 'out' else k
+        dim = S.dim
+        cur = {}; seq = {}
+        for r in set(phys.values()):
+            cur['a', r] = ('sigma', r) if r in self.retained else ('0',)
+        for n, _ in S.xs: cur['x', n] = ('n', n)
+        for t in range(v): cur['y', t] = ('0',)
+        for reg, k in cur.items(): seq[reg] = [k]
+        hist = {'a': Counter(), 'x': Counter(), 'y': Counter()}
+        moves = Counter()
+        def move(reg, dst):
+            dst = norm(dst); old = cur[reg]
+            if old == dst: return
+            r = dim(dst) - dim(old)
+            assert r >= 0, ('retreat', reg, old, dst)
+            moves[reg[0], old, dst] += 1
+            if r: hist[reg[0]][r] += 1
+            cur[reg] = dst; seq[reg].append(dst)
+        def gate(regs, frame):
+            for reg in regs: move(reg, frame)
+            assert all(cur[reg] == norm(frame) for reg in regs)
+        A = lambda s: ('a', phys[s])
+        leaf = S.srcop
+        def operation(i):
+            op = ops[i]
+            if op[0] == 'add':
+                _, p, o, n = op; frame = ('n', n)
+                gate([('y', self.target_of[p]) if p in elim else A(p), A(o)], frame)
+            else:
+                for g in sorted(op[2], key=lambda s: dim(S.start_key(s))):
+                    frame = S.start_key(g)
+                    gate([A(op[1]), ('y', self.target_of[g]) if g in elim else A(g)], frame)
+        for s in range(self.R):
+            if s not in self.level and s not in elim:
+                assert cur[A(s)] == ('0',) and s == phys[s], 'zero-prelude read of a first occupant at frame zero'
+        for s in self.early_v: gate([A(s), ('x', leaf[s])], ('v', s))
+        for i in self.ph1: operation(i)
+        centres = 0
+        for s, c in S.ret.items():
+            move(A(s), ('ret', c))
+            assert all(cur['y', t] == ('0',) for t, T in enumerate(S.trip) if c in T)
+            centres += 1
+        for kind, s in self.events():
+            if kind == 'read':
+                f = ('sigma', s)
+                if s in self.donor_of: move(A(s), f)
+                assert cur[A(s)] == f, ('read frame', s)
+                for t, c in self.adj[s].items():
+                    assert c > 0
+                    gate([('y', t)], f)
+            elif kind == 'v': gate([A(s), ('x', leaf[s])], ('v', s))
+            elif kind == 'vred': gate([('x', leaf[s]), ('y', self.target_of[s])], ('v', s))
+            else: operation(s)
+        for s, (c, T) in S.out.items():
+            if s not in elim: gate([A(s), ('y', S.tid[T])], ('out', 0, T))
+        for t, T in enumerate(S.trip): move(('y', t), ('out', 0, T))
+        for r in set(phys.values()): move(('a', r), ('F',))
+        for n, _ in S.xs: move(('x', n), ('F',))
+        return dict(moves=moves, seq=seq, hist=hist, centres=centres, phys=phys)
+
+    # ---------------------------------------------------------------------------------------- inherited nestings
+    def inherited(self):
+        """Adjacent frame pairs whose nesting the pinned PR97 audits already certify (check_lifted.py D and
+        check_frames.py X/Q, relied on by #144): consecutive keys of every original slot chain, of every X chain in
+        its original V order, and of every y chain in readout order with positive Z support; plus sigma <= start and
+        sigma <= t_T^perp.  Chains are nested sequences, so any ordered pair taken from one chain is nested too."""
+        S = self.S
+        def norm(k): return ('out', 0, tuple(k[2])) if k[0] == 'out' else k
+        keep = S.sel; S.sel = set(self.orig_sel)
+        chains = []
+        for s in range(S.R): chains.append([norm(k) for k in S.chain_keys(s)])
+        S.sel = keep
+        for n, us in S.xs: chains.append([('n', n)] + [('v', s) for s in us] + [('F',)])
+        byt = defaultdict(list)
+        for s in self.readout:
+            for t, c in self.adj[s].items():
+                if c > 0: byt[t].append(('sigma', s))
+        for t, T in enumerate(S.trip): chains.append([('0',)] + byt[t] + [('out', 0, T), ('F',)])
+        return chains
+
+    def row(self, led, record):
+        """#144's shared-core bit row from the actual ledger moves."""
+        S = self.S; h, v = S.h, S.v
+        aux, source, target = led['hist']['a'], led['hist']['x'], led['hist']['y']
+        assert +source == Counter(r for path in S.xdata() for r in path if r), 'source chains unchanged'
+        assert sum(r * n for r, n in target.items()) == v * (h - 1) and led['centres'] == h
+        gauges = Counter(S.f[s] for s in self.retained)
+        children = Counter()
+        for local in (aux, source, target, Counter({h - 1: h})):
+            for r, n in local.items():
+                if r and n: children[r] += 3 * n
+        for r, n in gauges.items(): children[3 * r] += n
+        children[2] += 2 * v
+        Rn = len(set(led['phys'].values()))
+        assert Rn == S.R - len(self.elim) - len(self.donor_of)
+        enc = lambda H: {str(r): n for r, n in sorted(H.items()) if n}
+        row = dict(record)
+        row.update(R=Rn, W_per_vertex=2 * v + Rn, selected_roles=sum(gauges.values()), selected_rank_histogram=enc(gauges),
+                   auxiliary_histogram=enc(aux), source_data_histogram=enc(source), target_data_histogram=enc(target),
+                   child_histogram=enc(children), rank_per_vertex=sum(r * n for r, n in children.items()),
+                   eliminated_roles=len(self.elim), recycled_roles=len(self.donor_of),
+                   omitted_roles=len(self.orig_sel) - len(self.elim) - len(self.level))
+        row.pop('selection_file', None); row.pop('selection_sha256', None)
+        assert row['W_per_vertex'] * row['m'] - row['rank_per_vertex'] == row['deficit_per_vertex'] == 2024
+        return row
+
+    def new_moves(self, led):
+        """Distinct moves of the ledger that are not an ordered pair of one inherited certified chain."""
+        where = defaultdict(dict)
+        for ci, chain in enumerate(self.inherited()):
+            for i, k in enumerate(chain): where[k].setdefault(ci, i)
+        def certified(a, b):
+            if a == ('0',) or b == ('F',): return True
+            wa, wb = where.get(a), where.get(b)
+            if not wa or not wb: return False
+            small, big = (wa, wb) if len(wa) <= len(wb) else (wb, wa)
+            return any(ci in big and wa[ci] <= wb[ci] for ci in small)
+        return sorted((m for m in led['moves'] if not certified(m[1], m[2])), key=str)