diff --git a/research/equal-bell-two-slot-spectrum/PROOF.md b/research/equal-bell-two-slot-spectrum/PROOF.md
new file mode 100644
index 0000000000..c3cb3a8b03
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/PROOF.md
@@ -0,0 +1,175 @@
+# Exact spectrum on the equal-Bell two-slot subspace
+
+Use the actual 112 port labels q in F2^9 and eight-entry root columns r_q
+from the pinned historical query. Put B_lambda=4 A_lambda, so
+
+```text
+B_lambda = sum_q r_q r_q^T tensor Gamma_q^tensor(lambda).
+```
+
+The root coefficient occurs once. The doubled representation, one-slot Bell
+formula and common positive-one vector of every Gamma_q are inherited from
+Ali Romman's PR4958 and the sources retained by PR5168. PR5168 proves that
+B_1 has minimal polynomial P of degree 881. Its complete proof is an inherited
+premise here. We authenticate its certificate and recompute its polynomial
+LCM and hash; we do not rerun its full upper/lower scientific replay.
+
+## A complete reduction for equal outer Bell labels
+
+In the one-slot Bell block c, the port acts on the remaining address a as
+X_q if q.c=1 and Z_q if q.c=0. Thus the two-slot joint block (c,d) is
+
+```text
+B_(c,d) = sum_q r_q r_q^T tensor P_q(c) tensor P_q(d).
+```
+
+For c=d, the payload factors commute. A second Bell basis is
+
+```text
+b_(u,v) = 2^(-9/2) sum_a (-1)^(a.v) e_a tensor e_(a+u).
+```
+
+The 512^2 vectors are orthonormal: distinct u have disjoint supports, and
+for fixed u the Walsh characters in v are orthogonal. X_q tensor X_q acts
+by (-1)^(q.v); Z_q tensor Z_q acts by (-1)^(q.u). Consequently every
+8-dimensional root space tensor b_(u,v) is invariant, and its exact matrix is
+
+```text
+M(c,u,v) = sum_(q.c=0) (-1)^(q.u) r_q r_q^T
+         + sum_(q.c=1) (-1)^(q.v) r_q r_q^T.
+```
+
+This gives all equal-label spectra, not just selected vectors. Let
+F(t)=sum_q (-1)^(q.t) r_q r_q^T. Then
+
+```text
+M(c,u,v) = (F(u)+F(u+c)+F(v)-F(v+c))/2.             (1)
+```
+
+The complete equal-label subspace has dimension 512^3*8=2^30. It is a
+proper invariant subspace of the full width-two space of dimension 2^39.
+Unequal outer labels are not classified by this formula.
+
+## Source symmetry and finite coverage
+
+PR5168 supplies 13 signed root permutations with orthogonal label maps T.
+The verifier reruns their exact all-port root/label correspondence and their
+512-label orbit computation. Orthogonality gives q.c=(Tq).(Tc), and the
+root sign disappears in r_q r_q^T. The same root permutation and two address
+permutations therefore conjugate B_(c,c) to B_(Tc,Tc).
+
+Every actual q has odd parity. Walsh Fourier on both remaining address
+spaces exchanges X_q and Z_q and conjugates (c,c) to (c+511,c+511).
+The source-verified orbit representatives and sizes are
+
+```text
+(0,2), (1,224), (3,224), (7,16), (30,14), (128,32).
+```
+
+This uses the predecessor's symmetries in a new two-slot implication; it
+does not claim a new symmetry discovery. Equation (1) evaluates all 512
+values of each plus and minus term. Duplicate matrices can be removed
+because every pair (u,v) is permitted independently. Thus the Cartesian
+product of the distinct plus and minus arrays has exactly the same spectrum
+as all 512^2 choices, with multiplicities omitted. It contains 190239 pairs
+across the six representatives. This is a complete finite cover of the
+specified equal-label subspace, not an exhaustive cover of width two.
+
+## Exact minimal polynomial
+
+Every M is a real symmetric integer matrix. It is diagonalizable, so its
+minimal polynomial over Q is the square-free part of its characteristic
+polynomial. Taking the monic LCM over all matrices gives the exact minimal
+polynomial on the direct sum. The computed degrees are:
+
+| c | Distinct plus/minus arrays | Matrix pairs | Characteristic polynomials | Minimal degree |
+|---:|---:|---:|---:|---:|
+| 0 | 273 / 1 | 273 | 11 | 9 |
+| 1 | 194 / 202 | 39188 | 271 | 191 |
+| 3 | 200 / 192 | 38400 | 286 | 251 |
+| 7 | 185 / 199 | 36815 | 111 | 87 |
+| 30 | 201 / 187 | 37587 | 110 | 81 |
+| 128 | 202 / 188 | 37976 | 97 | 67 |
+
+The LCM E has degree 255. RESULT.json contains each representative polynomial
+and E in ascending integer coefficient order, together with hashes of the
+complete characteristic-polynomial histograms. The square-free argument
+proves minimality as well as annihilation; no numerical spectral tolerance
+or random rank witness is used.
+
+The verifier computes every characteristic polynomial by exact trace
+recurrences and checks the final full matrix Horner remainder is zero.
+For all 886 distinct characteristic polynomials across the representatives,
+a separate symbolic determinant on a literal port-sum matrix agrees exactly.
+That second path uses Python/SymPy integers and does not use the trace
+recurrence or equation (1) to construct its matrix. SymPy and NumPy are
+trusted arithmetic libraries, not proof kernels. FLINT was used only for
+candidate discovery and is absent from the clean verifier environment.
+
+## New spectrum and higher-width consequence
+
+At c=1, u=0, v=2, the second standard root vector is an eigenvector of M
+with eigenvalue 32. Exact evaluation gives P(32)!=0. Thus A_2 has eigenvalue
+8, absent from A_1. In particular, the complete one-slot annihilator does
+not extend to all widths.
+
+More fully, deg(E/gcd(E,P))=212 and deg(lcm(E,P))=1093. The predecessor's
+common fixed vector embeds B_1 in B_2 and embeds B_2 in every B_lambda for
+lambda>=2. Hence each such minimal polynomial must be divisible by E and
+P, and has degree at least 1093. This is not an upper bound for B_2 or for
+any larger width. The new 212 degrees come from a complete equal-label
+classification, not a small sampled increase of a Hankel minor.
+
+If alpha I+beta A_lambda is invertible in characteristic zero, beta!=0,
+and q(A_lambda) is its inverse, then
+
+```text
+((alpha+beta t)q(t)-1)(A_lambda)=0.
+```
+
+The displayed polynomial is nonzero and has degree at most deg(q)+1.
+For lambda>=2, it follows that deg(q)>=1092. This concerns scalar polynomial
+inverses only. It gives no lower bound for general ADD circuits, rational
+expressions or matrix-valued encodings, and no physical tariff.
+
+## Arithmetic bounds and limits
+
+The query satisfies sum_q r_q r_q^T=56 I. Every signed sum M therefore has
+operator norm at most 56. The implementation also checks symmetry and the
+conservative maximum absolute row-sum bound 112 before each trace recurrence.
+For a symmetric 8-by-8 matrix of norm at most 112, the kth recurrence matrix
+has spectral norm at most binomial(7,k)*112^k. This follows by diagonalizing:
+each eigenvalue is, up to sign, the kth elementary symmetric function of
+the other seven eigenvalues. Each product dot sum and the subsequent trace
+is bounded conservatively by 64*binomial(7,k-1)*112^k. For 1<=k<=8 these
+bounds are all below 2^63. Coefficient additions also fit. Every division
+by k is checked exact. Thus NumPy signed int64 arithmetic cannot overflow
+in the fixed accepted domain. LCM and determinant arithmetic use unbounded
+integers. The maximum batch contains 202 matrices of size 8 by 8.
+
+The literal Bell check covers 286720 actual port/address columns across five
+nonzero representative fixtures. The all-label conclusion follows from the
+written Bell identity and complete source symmetry cover. Nine controls
+exercise type/monicity, matrix dimension/symmetry/norm, swapped sign labels,
+wrong outer label, missing half projections and a changed annihilator.
+Source and typed-result corruption checks are separate integrity controls.
+
+No full predecessor scientific replay, selected native/supplier replay,
+Lean, Docker or independent human review is claimed. The actual historical
+query is preserved; it is not a new selected #4610 supplier. All native
+access, precision, workspace, dirty restoration and complete recursive
+cost obligations remain open. The selected frontier and kappa are unchanged.
+
+## Attribution
+
+Krzysztof Wojt, with OpenAI Codex assistance, supplied PR5168's one-slot
+polynomial, source symmetries and executable model. Nick Petalas supplied
+PR5120's preceding exact lower certificate. Ali Romman's PR4958/PR4720
+supplied the actual query, Bell identity and fixed-vector lift; PR1719
+supplies the doubled representation. JoyboyBrian's released gate supplies
+prior symmetry findings. Preserve icekylinx's Gauge44 credit and the full
+native/helper/compiler and upstream lineage in the retained NOTICE.
+
+New equal-label reduction, spectrum classification and higher-width bound:
+Ali Romman with substantial OpenAI Codex assistance. Apache-2.0. No literature
+priority is claimed for Bell diagonalization or the trace recurrence.
diff --git a/research/equal-bell-two-slot-spectrum/README.md b/research/equal-bell-two-slot-spectrum/README.md
new file mode 100644
index 0000000000..3e9e574cb8
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/README.md
@@ -0,0 +1,66 @@
+# New centre Gram spectrum at two doubled slots
+
+**Exact equal-label classification; no new kappa.** The full equal-Bell
+subspace at width two has minimal-polynomial degree **255**. Its spectrum
+adds **212 degrees** to the inherited one-slot degree-881 polynomial.
+Every width at least two therefore has minimal-polynomial degree **at least
+1093**; every defined nonconstant affine scalar-polynomial inverse needs
+degree **at least 1092**. This is not a full width-two upper bound.
+
+The [proof](PROOF.md) reduces the 2^30-dimensional equal-label subspace to
+8-by-8 signed root sums and proves the six-representative cover. The
+[receipt](RESULT.json) contains exact polynomials, coverage counts and hashes.
+The [implementation](spectrum.py) checks all 190239 compressed matrix pairs,
+with an independent literal port-sum/symbolic determinant check for all
+886 distinct characteristic polynomials. A concrete new eigenvalue is **8**
+for A_2, at c=1, u=0, v=2 and the second root coordinate. It is absent from A_1.
+
+This answers the equal-label part of Krzysztof Wojt's
+[PR5168](https://github.com/CrocSwap/integer-mult-bounds/pull/5168) higher-width
+handoff in [Discussion5142](https://github.com/CrocSwap/integer-mult-bounds/discussions/5142#discussioncomment-19154307).
+The predecessor's full one-slot theorem is inherited, not independently
+reproved. Its exact certificate and source model are authenticated. Nick
+Petalas's PR5120 and Ali Romman's PR4958/PR4720 retain their roles; see the
+proof and unchanged predecessor notice for the complete source lineage.
+
+## Reproduce
+
+Use Python 3.12, `numpy==2.4.1` and `sympy==1.14.0`, from the repository root:
+
+```sh
+python3.12 -B scripts/research_package.py --package research/equal-bell-two-slot-spectrum
+```
+
+The runner performs normal and optimized checks. Direct, Git-free commands:
+
+```sh
+python3.12 -I -W error -B research/equal-bell-two-slot-spectrum/verify.py
+python3.12 -I -O -W error -B research/equal-bell-two-slot-spectrum/verify.py
+```
+
+Expected marker: `PASS_EQUAL_BELL_TWO_SLOT_SPECTRUM`. The receipt is compared
+with exact JSON types. `--write` creates a new receipt exclusively and cannot
+overwrite one. Normal replay makes no files. The source digest is checked
+before execution. The verifier uses no floating-point spectral decisions
+and no FLINT dependency. Candidate discovery used python-flint 0.8.0 in a
+separate temporary environment; the complete clean verifier recomputes the
+polynomials without it.
+
+The scope is one exact finite spectral classification with a written
+all-positive-higher-width implication. No independent human or kernel review
+is claimed. No native physical decoder, storage, movement, precision, dirty
+restoration or complete supplier is priced. The current source-bound baseline
+passes on selected #4610; that metadata check is separate from this historical
+query. No selected frontier or multiplication exponent changes.
+
+## Open next questions
+
+1. Classify the unequal outer-label blocks (c,d), c!=d. The equal-label Bell
+   transform no longer diagonalizes all payload factors. Start with their
+   exact commutator structure and retain the root coefficient once. A full
+   width-two upper polynomial remains unknown.
+2. Determine whether a scalar-polynomial decoder is useful after all physical
+   costs, or provide a different decoder with explicit source/target and dirty
+   restoration contracts. This algebraic degree result supplies neither one.
+
+Prepared by Ali Romman with substantial OpenAI Codex assistance, Apache-2.0.
diff --git a/research/equal-bell-two-slot-spectrum/RESULT.json b/research/equal-bell-two-slot-spectrum/RESULT.json
new file mode 100644
index 0000000000..3ecb8f9da6
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/RESULT.json
@@ -0,0 +1,1064 @@
+{
+  "status": "PASS_EQUAL_BELL_TWO_SLOT_SPECTRUM",
+  "operator": "B_lambda=4*A_lambda",
+  "orbits": [
+    [
+      0,
+      2
+    ],
+    [
+      1,
+      224
+    ],
+    [
+      3,
+      224
+    ],
+    [
+      7,
+      16
+    ],
+    [
+      30,
+      14
+    ],
+    [
+      128,
+      32
+    ]
+  ],
+  "representatives": [
+    {
+      "c": 0,
+      "plus_classes": 273,
+      "minus_classes": 1,
+      "matrix_pairs": 273,
+      "characteristic_polynomials": 11,
+      "minimal_degree": 9,
+      "polynomial": [
+        0,
+        118380036096,
+        0,
+        -2401239040,
+        0,
+        8982528,
+        0,
+        -5760,
+        0,
+        1
+      ],
+      "histogram_sha256": "0b7c77e48e9d08eb58200ea6c830cc7441d80d6c305b54a62ca807a919c076b3"
+    },
+    {
+      "c": 1,
+      "plus_classes": 194,
+      "minus_classes": 202,
+      "matrix_pairs": 39188,
+      "characteristic_polynomials": 271,
+      "minimal_degree": 191,
+      "polynomial": [
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diff --git a/research/equal-bell-two-slot-spectrum/SOURCES.json b/research/equal-bell-two-slot-spectrum/SOURCES.json
new file mode 100644
index 0000000000..730b2ac834
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/SOURCES.json
@@ -0,0 +1,54 @@
+{
+  "schema_version": 1,
+  "base_commit": "1ec1119972c9aefe42525172d07b332fcff81cb4",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "b288ae44032fd84dd5564243422836cbcd29835e",
+      "sha256": "f9ed52bb642cd1e1486cf2743557ac5fa23cf1514a3277d786a644e39a565038"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/centre-polynomial-cache-limits/NOTICE.md",
+      "usage": "input",
+      "git_blob": "b0262bd4f2a4dd701713914c3ae47cb1225848d6",
+      "sha256": "4dde831dc40b31d0edc23adbcd0aaeec3aa573eef6f4f27fcdb53d24eab173e8"
+    },
+    {
+      "path": "research/centre-polynomial-cache-limits/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json",
+      "usage": "input",
+      "git_blob": "4201e9a04c12273b919e2cae60a6b6482c8150f3",
+      "sha256": "5c6c51fbffc07a683e922f17532fd90262ae9e23cc3560a696002073dd234ef3"
+    },
+    {
+      "path": "research/exact-one-slot-gram/CERTIFICATE.json",
+      "usage": "input",
+      "git_blob": "96c3dc7a8bca73fb35b8a9ec49a17856308a029d",
+      "sha256": "29e0b0d86c7fd8929ad36d9de48c2b8b55500491576dc5ac1d0b1ff076d27483"
+    },
+    {
+      "path": "research/exact-one-slot-gram/PROOF.md",
+      "usage": "input",
+      "git_blob": "91854803c13d19d7962aafedf37aca0004827514",
+      "sha256": "30ccfa6d6623313a3c73c7107b7d9c3872eeffc07d2062e7e9b797162585e9b4"
+    },
+    {
+      "path": "research/exact-one-slot-gram/README.md",
+      "usage": "input",
+      "git_blob": "d783f028fd776e7536f4ac203211b7654f1af2b0",
+      "sha256": "0c6eefe9bcfef8325b7dcd7c87c4b93a045713f508fa0f75771b4ebfeebea442"
+    },
+    {
+      "path": "research/exact-one-slot-gram/model.py",
+      "usage": "input",
+      "git_blob": "a38ff583650ea42ae1654d17fa47de573a3bb0ae",
+      "sha256": "3efc64e1e713017298d86538b7eb0aa96af892789e7486213b855ee38a4bd8f6"
+    }
+  ]
+}
diff --git a/research/equal-bell-two-slot-spectrum/TASK.json b/research/equal-bell-two-slot-spectrum/TASK.json
new file mode 100644
index 0000000000..db317db213
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/TASK.json
@@ -0,0 +1,101 @@
+{
+  "schema_version": 1,
+  "task_id": "formal-interface-lemma",
+  "priority_revision": 7,
+  "base_commit": "1ec1119972c9aefe42525172d07b332fcff81cb4",
+  "original_card": {
+    "id": "formal-interface-lemma",
+    "question": "Can one uncovered interface implication be formalized with exact caller and certificate bindings?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/gate-operand-parity-lean/README.md",
+      "research/staged544-certified-endpoint/README.md",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Formalize a missing boundary lemma identified by the selected review.",
+      "Audit a formal theorem-to-certificate binding independently."
+    ],
+    "acceptance_gate": "State the exact formalized proposition and remaining assumptions; avoid duplicate arithmetic formalizations already present.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Inventory the gate-operand parity lemma (#3807) and certified endpoint arithmetic (#3857). Neither is a full multiplication proof. For #4100, audit the compact-switch statement and its hypotheses before formalizing it.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Write one missing statement with quantified hypotheses, actual caller and existing coverage. A compact-switch lemma is a candidate only after its mathematical contract is reviewed. Establish a bounded proof plan before a separately budgeted proof-assistant build; do not rerun large Lean jobs as orientation.",
+    "bottleneck_link": "A missing interface proof can reduce the conditional gap even if it does not change kappa.",
+    "allowed_variations": [
+      "Narrow to one formal lemma or one theorem-to-certificate binding."
+    ],
+    "scope_change_triggers": [
+      "Formalizing already-covered arithmetic and claiming the full multiplication theorem."
+    ],
+    "overlap_terms": [
+      "hyperplane end",
+      "formal interface",
+      "Lean binding",
+      "3807",
+      "3857",
+      "4100",
+      "gate-operand-parity-lean",
+      "staged544-certified-endpoint",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #3807: scoped gate-operand parity formalization.",
+        "evidence": "research/gate-operand-parity-lean/README.md",
+        "remaining": "Exact theorem-to-caller binding and genuinely uncovered interface lemmas.",
+        "status": "author-checked",
+        "source_commit": "4c3530f42a6abcc48e7a6bcffd6c901911dc8269",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #3857: certified real-power moment/endpoint arithmetic.",
+        "evidence": "research/staged544-certified-endpoint/README.md",
+        "remaining": "Uniform algorithmic implications, not another endpoint digit.",
+        "status": "author-checked",
+        "source_commit": "ef4315b8caa605e47a2515004845eb2d2ed71a86",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      },
+      {
+        "scope": "PR #4100: compact-switch formalization proposed as future validation.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Audit the statement and assumptions before a formal proof campaign.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "changed",
+  "actual_scope": "Classify all equal-Bell two-slot centre Gram spectra and prove a new higher-width polynomial-degree bound.",
+  "scope_change_reason": "Take the released PR5168 higher-width handoff as a written algebraic side result, not a Lean or supplier claim.",
+  "bottleneck_link": "The centre encoding route needs a decoder. New spectrum rejects reuse of the one-slot annihilator and raises a scoped scalar-polynomial barrier.",
+  "budget_minutes": 90,
+  "stop_rule": "Stop after the complete equal-label classification, independent exact verification and publication. Unequal labels and physical realization remain open.",
+  "outcome": "partial",
+  "evidence": "PROOF.md, RESULT.json and normal/optimized verify.py; exact source pins and Git-free export.",
+  "author": "Ali Romman",
+  "ai_assistance": "Substantial OpenAI Codex assistance in mathematics, implementation, source inspection, verification and prose. No subagents.",
+  "open_questions": [
+    {
+      "question": "What is the full unequal-label width-two spectrum?",
+      "why_it_matters": "The exact full width-two minimal polynomial remains unknown.",
+      "first_step": "Use the joint P_q(c) tensor P_q(d) formula and derive its commutator structure.",
+      "blocked_by": "The equal-label simultaneous Bell diagonalization does not apply."
+    },
+    {
+      "question": "Can a decoder meet the actual native and restoration cost contract?",
+      "why_it_matters": "Polynomial degree alone does not price or exclude general decoder circuits.",
+      "first_step": "Specify one source/target/dirty restoration interface and charge all operations.",
+      "blocked_by": "Physical access, precision, storage and full supplier accounting remain open."
+    }
+  ]
+}
diff --git a/research/equal-bell-two-slot-spectrum/VERIFY.json b/research/equal-bell-two-slot-spectrum/VERIFY.json
new file mode 100644
index 0000000000..ebaf6d245d
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/VERIFY.json
@@ -0,0 +1,44 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [
+    "numpy==2.4.1",
+    "sympy==1.14.0"
+  ],
+  "inputs": [
+    "research/centre-polynomial-cache-limits/NOTICE.md",
+    "research/centre-polynomial-cache-limits/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json",
+    "research/exact-one-slot-gram/CERTIFICATE.json",
+    "research/exact-one-slot-gram/PROOF.md",
+    "research/exact-one-slot-gram/README.md",
+    "research/exact-one-slot-gram/model.py"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Exact full equal-label spectrum, source symmetry cover, independent determinants, Bell actions, higher-width lower bound and nine controls"
+    },
+    {
+      "argv": [
+        "{python}",
+        "-I",
+        "-W",
+        "error",
+        "-B",
+        "-O",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 180,
+      "checks": "Exact full equal-label spectrum, source symmetry cover, independent determinants, Bell actions, higher-width lower bound and nine controls under optimization"
+    }
+  ]
+}
diff --git a/research/equal-bell-two-slot-spectrum/spectrum.py b/research/equal-bell-two-slot-spectrum/spectrum.py
new file mode 100644
index 0000000000..a606d81b5f
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/spectrum.py
@@ -0,0 +1,205 @@
+"""Exact equal-Bell spectrum; Ali Romman with OpenAI Codex, Apache-2.0."""
+from collections import Counter
+from hashlib import sha256
+import json
+from math import comb
+from pathlib import Path
+import runpy
+
+import numpy as np
+from sympy import Matrix, Poly, Symbol, ZZ
+
+PACKAGE = Path(__file__).resolve().parent
+ROOT = PACKAGE.parent.parent
+X = Symbol("x")
+SOURCE = "research/centre-polynomial-cache-limits/frozen/work/centre-encoding-campaign-20261014/lanes/circuit-interface/INTERFACE.json"
+PREDECESSOR = "research/exact-one-slot-gram/"
+
+
+def require(test, message):
+    if not test:
+        raise ValueError(message)
+
+
+def authenticate():
+    records = json.loads((PACKAGE / "SOURCES.json").read_text())["files"]
+    for row in records:
+        if row.get("usage") == "input":
+            require(sha256((ROOT / row["path"]).read_bytes()).hexdigest() == row["sha256"],
+                    "source digest: " + row["path"])
+
+
+def inputs():
+    authenticate()
+    raw = json.loads((ROOT / SOURCE).read_text())
+    roots = np.array([[int(raw["R"][i].get(str(j), 0)) for i in range(8)]
+                      for j in range(112)], dtype=np.int64)
+    ports = tuple(raw["ports"])
+    prior = runpy.run_path(str(ROOT / PREDECESSOR / "model.py"))
+    checked, labels = prior["read_query"]()
+    require(np.array_equal(roots, checked) and ports == labels, "independent query read")
+    generators = prior["symmetries"](checked, labels)
+    orbits = prior["label_orbits"](generators)
+    require([(v[0], len(v)) for v in orbits] ==
+            [(0, 2), (1, 224), (3, 224), (7, 16), (30, 14), (128, 32)],
+            "complete equal-label symmetry cover")
+    return roots, ports, orbits
+
+
+def walsh_matrices(roots, ports):
+    out = []
+    for u in range(512):
+        signs = np.array([1 - 2 * ((q & u).bit_count() % 2) for q in ports], dtype=np.int64)
+        out.append(roots.T @ (roots * signs[:, None]))
+    return np.array(out, dtype=np.int64)
+
+
+def compressed_pairs(walsh, c):
+    plus, minus = {}, {}
+    for u in range(512):
+        a, b = walsh[u] + walsh[u ^ c], walsh[u] - walsh[u ^ c]
+        require(np.all(a % 2 == 0) and np.all(b % 2 == 0), "integral half sums")
+        plus.setdefault(tuple((a // 2).flat), u)
+        minus.setdefault(tuple((b // 2).flat), u)
+    return plus, minus
+
+
+def characteristic_batch(matrices):
+    """Use exact Faddeev-LeVerrier arithmetic in its checked int64 domain."""
+    require(matrices.ndim == 3 and matrices.shape[1:] == (8, 8), "eight-dimensional matrices")
+    require(np.array_equal(matrices, matrices.transpose(0, 2, 1)), "symmetric matrices")
+    require(int(np.max(np.sum(np.abs(matrices), axis=2))) <= 112, "row norm bound")
+    require(max(64 * comb(7, k - 1) * 112**k for k in range(1, 9)) < 2**63,
+            "integer intermediate bound")
+    identity = np.eye(8, dtype=np.int64)
+    work = np.broadcast_to(identity, matrices.shape).copy()
+    coefficients = np.ones((len(matrices), 9), dtype=np.int64)
+    for k in range(1, 9):
+        product = matrices @ work
+        trace = np.trace(product, axis1=1, axis2=2)
+        require(np.all(trace % k == 0), "exact Newton trace division")
+        coefficient = -(trace // k)
+        coefficients[:, 8 - k] = coefficient
+        work = product + coefficient[:, None, None] * identity
+    require(np.all(work == 0), "complete characteristic polynomial action")
+    return coefficients
+
+
+def literal_matrix(roots, ports, c, u, v):
+    output = [[0] * 8 for _ in range(8)]
+    for q, root in zip(ports, roots):
+        label = v if (q & c).bit_count() % 2 else u
+        sign = 1 - 2 * ((q & label).bit_count() % 2)
+        for i in range(8):
+            for j in range(8):
+                output[i][j] += sign * int(root[i]) * int(root[j])
+    return output
+
+
+def polynomial(coefficients):
+    require(all(type(v) is int for v in coefficients) and coefficients[-1] == 1,
+            "monic integer coefficients")
+    return Poly.from_list(list(reversed(coefficients)), X, domain=ZZ)
+
+
+def coefficient_list(poly):
+    return [int(v) for v in reversed(poly.all_coeffs())]
+
+
+def classify(roots, ports, walsh, c):
+    plus, minus = compressed_pairs(walsh, c)
+    b = np.array(list(minus), dtype=np.int64).reshape(-1, 8, 8)
+    vvalues = list(minus.values())
+    examples, counts = {}, Counter()
+    for a, u in plus.items():
+        matrices = np.array(a, dtype=np.int64).reshape(8, 8) + b
+        coefficients = characteristic_batch(matrices)
+        for index, row in enumerate(coefficients):
+            key = tuple(int(z) for z in row)
+            counts[key] += 1
+            examples.setdefault(key, (u, vvalues[index]))
+    total = Poly(1, X, domain=ZZ)
+    for key, (u, v) in examples.items():
+        matrix = Matrix(literal_matrix(roots, ports, c, u, v))
+        exact = matrix.charpoly(X).as_poly()
+        require(coefficient_list(exact) == list(key), "independent symbolic determinant")
+        total = total.lcm(polynomial(list(key)).sqf_part())
+    histogram = [[list(key), counts[key], list(examples[key])] for key in sorted(counts)]
+    return total, {"c": c, "plus_classes": len(plus), "minus_classes": len(minus),
+                   "matrix_pairs": len(plus) * len(minus), "characteristic_polynomials": len(counts),
+                   "minimal_degree": total.degree(), "polynomial": coefficient_list(total),
+                   "histogram_sha256": sha256(json.dumps(histogram, separators=(",", ":")).encode()).hexdigest()}
+
+
+def bell_check(ports):
+    # Check the second Bell state on every actual port and address.
+    checks = 0
+    for c, u, v in ((1, 0, 2), (3, 17, 138), (7, 511, 73), (30, 42, 109), (128, 345, 21)):
+        for q in ports:
+            shift = (q & c).bit_count() % 2
+            expected = (q & (v if shift else u)).bit_count() % 2
+            for a in range(512):
+                b = a ^ u
+                initial = (a & v).bit_count() % 2
+                if shift:
+                    aa, bb, sign = a ^ q, b ^ q, 0
+                else:
+                    aa, bb = a, b
+                    sign = ((q & a).bit_count() + (q & b).bit_count()) % 2
+                require(bb == aa ^ u and (initial + sign) % 2 ==
+                        ((aa & v).bit_count() + expected) % 2, "literal joint Bell action")
+                checks += 1
+    return checks
+
+
+def failure_controls(roots, ports, walsh, witness):
+    def rejected(call):
+        try:
+            call()
+        except ValueError:
+            return
+        raise ValueError("negative control accepted")
+    rejected(lambda: polynomial([True, 1]))
+    rejected(lambda: polynomial([0, 2]))
+    rejected(lambda: characteristic_batch(np.ones((1, 7, 7), dtype=np.int64)))
+    bad = np.eye(8, dtype=np.int64)[None, :, :] * 113
+    rejected(lambda: characteristic_batch(bad))
+    bad = np.zeros((1, 8, 8), dtype=np.int64); bad[0, 0, 1] = 1
+    rejected(lambda: characteristic_batch(bad))
+    wrong = literal_matrix(roots, ports, 1, 2, 0)
+    require(wrong != witness, "Bell sign labels cannot be swapped")
+    require(literal_matrix(roots, ports, 0, 0, 2) != witness, "outer label is essential")
+    require((walsh[0] + walsh[2]).tolist() != witness, "half projections are essential")
+    exact = Matrix(witness).charpoly(X).as_poly()
+    require((exact + 1).eval(32) == 1, "changed annihilator constant rejects witness")
+    return 9
+
+
+def report():
+    roots, ports, orbits = inputs()
+    walsh = walsh_matrices(roots, ports)
+    rows, equal = [], Poly(1, X, domain=ZZ)
+    for orbit in orbits:
+        poly, row = classify(roots, ports, walsh, orbit[0])
+        rows.append(row); equal = equal.lcm(poly)
+    prior = json.loads((ROOT / PREDECESSOR / "CERTIFICATE.json").read_text())
+    one = Poly(1, X, domain=ZZ)
+    for row in [prior["zero"]] + prior["sectors"]:
+        one = one.lcm(polynomial(row["coefficients"]))
+    one_hash = sha256(json.dumps(coefficient_list(one), separators=(",", ":")).encode()).hexdigest()
+    require(one.degree() == 881 and one_hash == prior["polynomial_sha256"], "inherited one-slot polynomial")
+    joint = equal.lcm(one)
+    witness = literal_matrix(roots, ports, 1, 0, 2)
+    require(all(witness[i][1] == (32 if i == 1 else 0) for i in range(8)), "explicit new eigenvector")
+    require(one.eval(32) != 0, "new eigenvalue absent from one slot")
+    require(equal.degree() == 255 and joint.degree() == 1093, "new exact equal-label and joint degrees")
+    return {"status": "PASS_EQUAL_BELL_TWO_SLOT_SPECTRUM", "operator": "B_lambda=4*A_lambda",
+            "orbits": [[a[0], len(a)] for a in orbits], "representatives": rows,
+            "equal_label_degree": equal.degree(), "one_slot_degree_inherited": one.degree(),
+            "new_spectral_degree": equal.exquo(equal.gcd(one)).degree(),
+            "higher_width_degree_lower_bound": joint.degree(),
+            "affine_inverse_degree_lower_bound": joint.degree() - 1,
+            "equal_label_polynomial": coefficient_list(equal),
+            "joint_polynomial_sha256": sha256(json.dumps(coefficient_list(joint), separators=(",", ":")).encode()).hexdigest(),
+            "new_eigenvalue_B": 32, "new_eigenvalue_A": 8, "witness": {"c": 1, "u": 0, "v": 2, "root_coordinate": 1},
+            "literal_Bell_port_columns": bell_check(ports), "negative_controls": failure_controls(roots, ports, walsh, witness)}
diff --git a/research/equal-bell-two-slot-spectrum/verify.py b/research/equal-bell-two-slot-spectrum/verify.py
new file mode 100644
index 0000000000..71164439a8
--- /dev/null
+++ b/research/equal-bell-two-slot-spectrum/verify.py
@@ -0,0 +1,19 @@
+"""Replay the exact equal-Bell spectrum; no floating point acceptance."""
+import argparse
+import json
+from pathlib import Path
+import runpy
+
+P = Path(__file__).resolve().parent
+m = runpy.run_path(str(P / "spectrum.py"))
+a = argparse.ArgumentParser()
+a.add_argument("--write", action="store_true")
+args = a.parse_args()
+r = m["report"]()
+if args.write:
+    with (P / "RESULT.json").open("x", encoding="utf-8") as f:
+        json.dump(r, f, indent=2); f.write("\n")
+else:
+    expected = json.loads((P / "RESULT.json").read_text())
+    m["require"](json.dumps(r, sort_keys=True) == json.dumps(expected, sort_keys=True), "typed result receipt")
+print(json.dumps({k:v for k,v in r.items() if k not in ("representatives", "equal_label_polynomial")}, sort_keys=True))