diff --git a/research/pr4100-ledger-boundary-exponent/PROOF.md b/research/pr4100-ledger-boundary-exponent/PROOF.md
new file mode 100644
index 0000000000..2125a98081
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/PROOF.md
@@ -0,0 +1,109 @@
+# Which boundary exponent the printed #4100 ledger uses
+
+**Scoped sentence proof. No new κ.** #4100 is not accepted. The selected
+conditional result stays `0.00302015`.
+
+## Proposition
+
+Let the union and boundary savings at a common `(eps, xi)` be the rows in
+`research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md` §4:
+
+```text
+S_u     = xi + 1/2 - eps
+S_b(c)  = c + xi - eps
+c       = (3 - tau) / 2
+```
+
+Then `S_b(c) - S_u = c - 1/2`. This identity does not use `q`, `r`, `zeta`,
+or the slack.
+
+The only numeric balance those notes print for this schedule is the point
+
+```text
+eps  = 0.99999437577296
+xi   = 0.50167412128176
+beta = 0.99664613320944
+```
+
+in `vendor/GAUGEFREE-INTERIORS.md` §6, together with its 7-decimal rounding
+in UNION-SCHEDULE §5b. On that point the boundary row is the
+`tau_eff = 2 + 10^{-64}` row. It is not the `tau = 45/16` row.
+
+## One printed point
+
+GAUGEFREE §6 item 5 prints the 14-digit triple above and sets
+`c = (3 - tau_eff) / 2`. UNION-SCHEDULE §5b prints
+
+```text
+epsilon = 0.9999944, xi = 0.5016741, beta = 0.9966461
+```
+
+at `tau_eff = 2 + 1e-64`. Rounding each 14-digit coordinate half-up to seven
+decimal places reproduces that triple, and no other 7-digit triple. The two
+notes are the same balance.
+
+From the 14-digit `(eps, xi)`,
+
+```text
+S_u = 1049840943 / 625000000000 = 0.0016797455088.
+```
+
+GAUGEFREE's displayed kappa begins `0.001679745508802299`, so `S_u` agrees
+with that prefix through `0.00167974550880`. The leftover digits are the
+rounding already present in the displayed `(eps, xi)`.
+
+## The two exponents on that point
+
+`tau_eff = 2 + 10^{-64}` gives `c - 1/2 = -5 · 10^{-65}`. The boundary row
+and the union row therefore differ by `5 · 10^{-65}`. That is the gap inside
+the same paragraph's joint margin "~ `1e-64`", and it is why §5b can say
+that the union and boundary rows have the same exponent `eps + 1/2 - xi` and
+both bind.
+
+`tau = 45/16` gives `c = 3/32` and
+
+```text
+S_b - S_u = -13/32.
+```
+
+The boundary saving is then
+
+```text
+0.0016797455088 - 13/32 = -0.4045702544912,
+```
+
+which is not a margin of size about `10^{-64}` above the displayed kappa.
+The joint-binding sentence is false at `45/16` for this printed point.
+
+## Where `45/16` is printed
+
+`GAUSSIAN.md`'s seven-product paragraph, and `vendor/FINE-PRIME.md` §6, assign
+`tau = 45/16` to the dense cost
+
+```text
+O(V d W^(tau-1) / B polylog(p))
+```
+
+and say that this cost needs no structured inverse. FINE-PRIME's saving
+`c + xi - epsilon` at `c = 3/32` is that Strassen row. It is a different row
+from the §5b balance. `research/pr4100-row-field-ledger` already compared the
+short-field movement with this Strassen row and stopped before choosing an
+exponent. The comparison is not redone here.
+
+## Where the layout lemma is copied
+
+In `GAUSSIAN.md` the sentence "Temporary padding of the one exposed polynomial
+row field is removed before the next axis" sits in the union-cost paragraph,
+after the Strassen paragraph. UNION-SCHEDULE §6 assumes "the retained
+polynomial-width row-field movement for batched boundary arithmetic" and says
+that the document does not change the boundary matrix exponent. The ledger
+that receives the layout lemma is therefore the §4–§5b ledger, and the
+boundary exponent of that ledger is `tau_eff`.
+
+## What is not claimed
+
+`outer/check_ledger.py` is cited by §5b and is not a file of
+`research/unconditional-batched-crt`. This note does not run it. It does not
+prove the structured-boundary hypothesis, the Strassen inequality `7^16 < 2^45`,
+or any all-size transfer. A later generator could still print a different
+point; that is a separate obligation.
diff --git a/research/pr4100-ledger-boundary-exponent/README.md b/research/pr4100-ledger-boundary-exponent/README.md
new file mode 100644
index 0000000000..ba982367e1
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/README.md
@@ -0,0 +1,114 @@
+# The printed #4100 ledger boundary row uses tau_eff
+
+**Scoped sentence proof. No new κ.** #4100 is not accepted. The selected
+conditional result stays `0.00302015`.
+
+## Result
+
+On the one numeric balance printed for the #4100 union schedule, the boundary
+row is `tau_eff = 2 + 10^{-64}`. It is not `tau = 45/16`.
+
+UNION-SCHEDULE §4 gives, at a common `(eps, xi)`,
+
+```text
+S_u = xi + 1/2 - eps,    S_b = c + xi - eps,    c = (3 - tau) / 2,
+```
+
+so `S_b - S_u = c - 1/2`. GAUGEFREE-INTERIORS §6 prints the 14-digit point
+
+```text
+eps = 0.99999437577296,  xi = 0.50167412128176,  beta = 0.99664613320944
+```
+
+and UNION-SCHEDULE §5b prints its half-up rounding to seven decimals,
+`0.9999944`, `0.5016741`, `0.9966461`, at `tau_eff = 2 + 1e-64`. From those
+14 digits the union saving is exactly
+
+```text
+S_u = 1049840943 / 625000000000 = 0.0016797455088,
+```
+
+which matches the displayed kappa prefix `0.001679745508802299` through
+`0.00167974550880`.
+
+At `tau_eff` the two rows differ by `5 · 10^{-65}`, inside the paragraph's
+joint margin of about `10^{-64}`. At `tau = 45/16` they differ by `13/32`,
+and the boundary saving on this point is `-0.4045702544912`. The sentence
+that the union and boundary rows both bind is false at `45/16`.
+
+`tau = 45/16` remains the dense Strassen exponent in `GAUSSIAN.md` and in
+FINE-PRIME §6. That paragraph says the cost needs no structured inverse.
+The short-field layout sentence sits later, on the union-cost paragraph.
+UNION-SCHEDULE §6 assumes that row-field movement and says the note does not
+change the boundary matrix exponent. Copying the layout lemma in therefore
+keeps the §5b boundary exponent, which is `tau_eff`.
+
+[PROOF.md](PROOF.md) records the argument. `research/pr4100-row-field-ledger`
+showed that both exponents absorb the movement and left this choice open.
+That absorption is not recomputed.
+
+## Scope
+
+Card `uniform-transfer-audit`, refined to the exponent-naming question in
+`research/pr4100-row-field-ledger`. Priority revision 7. Base
+`ff3cc58e370d0f678e529246f8c73e8094e18fcf`. The #4100 text is the package
+retained on that commit. Selected κ stays `0.00302015`.
+
+This is item 3 of the ordered eligible list below.
+
+## Ordered eligible list
+
+Handoffs on Discussion #1166 through 2026-10-23T13:22Z were read first. The
+fresh posts are authors' own CLAIM and DONE notes. They do not put an
+unclaimed request ahead of the priority table. The list is the
+current-priorities card order, and inside `uniform-transfer-audit` the #4311
+shortest-list row order. Each row contributes the first still-open successor
+of its latest merged package, then the second. A row is dropped when a merged
+package, an open PR, an excluded unpaidmoon delivery, or a fork `cursor/*`
+branch already answers it.
+
+1. The PR2809 encoded continuation stack keeps `O_q(V)` visited space at the new `c` (`research/bit-stock-recurrence-audit`).
+2. The stored narrow band has at most `2p` off-diagonal entries and the rounding meets `2^{-5p}` (`research/narrow-row-half-gap` question 1). The half-gap itself is #4648.
+3. **This package.** Which printed boundary exponent the #4100 ledger uses when the layout lemma is copied in.
+4. A word-level scalar inventory at the two excluded `p = 2` CRT corners (`research/crt-arena-batch-threshold` question 1). The full batch for `p >= 3` is already proved.
+5. Compose that full-batch allowance with reservoir placement and the paid repair-key service (`crt-arena-batch-threshold` question 2).
+6. Enclose the separator corners of the translated chirp origin (`research/gaussian-prime-origin`).
+7. Extract the explicit density constant `C` and `k` for `mu` (`pr4100-row-field-ledger` question 2).
+8. Replace the CRT repair-key premise by a literal tape program (`research/crt-mask-repair-bridge` question 1). #4644 joined the mask and left this arithmetic unpaid.
+9. Carry that bridge to a many-target macro (`crt-mask-repair-bridge` question 2).
+10. One matched QQ16 park/restore bill against the external-call protocol (`joint-query-controller`). The finite four-provider replay already has its own package.
+
+## Evidence and reproduction
+
+Python 3.12, standard library only, from the repository root:
+
+```sh
+python3 -B scripts/research_package.py --package research/pr4100-ledger-boundary-exponent --check-only
+python3 -B scripts/research_package.py --package research/pr4100-ledger-boundary-exponent
+```
+
+`verify.py` prints `PASS_LEDGER_BOUNDARY_EXPONENT`. It checks the rational
+identity, both gaps, the seven-digit rounding, the sentence order, and the
+rejecting control that `45/16` puts the boundary saving below zero. No ledger
+program, no carving replay, and no proof kernel is run.
+
+## Open next questions
+
+1. Does a byte-identical `outer/check_ledger.py` regenerate this `tau_eff`
+   point? Section 5b cites that script, and the #4100 tree does not contain
+   it. The first step is a blob whose stdout contains
+   `eps = 0.99999437577296`. Nothing here searches history for a substitute.
+2. What explicit `C` and `k` satisfy `mu ≤ C (W/B) (log p)^k`? The
+   predecessor left this open. Naming `tau_eff` does not supply the constant.
+   The carving program is not an input here.
+
+## Credit
+
+The Strassen sentence and the union schedule are Cinna Davis's #4100.
+The gauge-free parameter point is the vendored GAUGEFREE-INTERIORS note.
+`research/pr4100-row-field-movement` and `research/pr4100-row-field-ledger`
+are unpaidmoon's layout count and the substitution that left this choice
+open. The coverage map is PR #4311 (Ryanson Jonathan).
+
+Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.
+No independent human or formal-kernel review is claimed.
diff --git a/research/pr4100-ledger-boundary-exponent/SOURCES.json b/research/pr4100-ledger-boundary-exponent/SOURCES.json
new file mode 100644
index 0000000000..7f19e85d26
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/SOURCES.json
@@ -0,0 +1,48 @@
+{
+  "schema_version": 1,
+  "base_commit": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
+  "files": [
+    {
+      "path": "certificates/selected-result.json",
+      "usage": "provenance",
+      "git_blob": "c35482cd9035843b15a634cc8276a7e1e40e8395",
+      "sha256": "6fe0a099b89aa2aed796a13c3a21dafc63f3388ebf397464e44e0451893b9375"
+    },
+    {
+      "path": "docs/research/agent-task-catalog.json",
+      "usage": "provenance",
+      "git_blob": "f126dd2e2dad2b669e75d1d0d0bd05c600725602",
+      "sha256": "f7a48bfa2659b88378282a8eee7d9fe52cdf505594124c4d1686b58510fffd72"
+    },
+    {
+      "path": "research/pr4100-row-field-ledger/PROOF.md",
+      "usage": "provenance",
+      "git_blob": "a91c302f3ace75f4b165bde408fff8977e7b0d99",
+      "sha256": "490bde30381da3a3800e4eb700faf2fb29b627389f472e171f22d9b9592d6eed"
+    },
+    {
+      "path": "research/unconditional-batched-crt/GAUSSIAN.md",
+      "usage": "input",
+      "git_blob": "adfb28c93e6b06178c901caaac39ede41a9e541b",
+      "sha256": "2f63c91c58f2d1cf31294812c75970ea390c61866a8da7ed700d4316a033a242"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/FINE-PRIME.md",
+      "usage": "input",
+      "git_blob": "bf9899ee21be4058e4aac0b64bfa67d01ee8ddd3",
+      "sha256": "47d1b1b4df28aadda3cd6bdeb1fbc47290595e9208cd09c1a1f338520d33fdce"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md",
+      "usage": "input",
+      "git_blob": "20585e44fa4e49b13b5d8495bbf6ed254459a362",
+      "sha256": "d1423ac0ea0003521797209f4396e6981a66994bbee4f8b48026e5b95f8c5116"
+    },
+    {
+      "path": "research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md",
+      "usage": "input",
+      "git_blob": "b5eb03e6745804f465a536fc9ffa87c2515d1bfa",
+      "sha256": "aabb28c343bcb9697a4cac7058060132617073df17ee18e591d2bf4defe9e921"
+    }
+  ]
+}
diff --git a/research/pr4100-ledger-boundary-exponent/TASK.json b/research/pr4100-ledger-boundary-exponent/TASK.json
new file mode 100644
index 0000000000..e832f8e6a8
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/TASK.json
@@ -0,0 +1,80 @@
+{
+  "schema_version": 1,
+  "task_id": "uniform-transfer-audit",
+  "priority_revision": 7,
+  "base_commit": "ff3cc58e370d0f678e529246f8c73e8094e18fcf",
+  "original_card": {
+    "id": "uniform-transfer-audit",
+    "question": "Does one quantified step of the proposed #4100 all-size construction hold at its actual caller?",
+    "starting_points": [
+      "docs/research/research-family-status.md",
+      "certificates/selected-result.json",
+      "research/unconditional-batched-crt/README.md"
+    ],
+    "methods": [
+      "Prove one missing implication with explicit quantified hypotheses.",
+      "Find an exact counterexample at the claimed interface boundary."
+    ],
+    "acceptance_gate": "Identify the proposition and what would change downstream; do not equate a finite replay with a universal proof.",
+    "first_step": {
+      "command": [
+        "python3",
+        "-B",
+        "scripts/research_baseline.py"
+      ],
+      "expected": "PASS_RESEARCH_BASELINE_BINDINGS_ONLY",
+      "inspect": "Read PROOF.md and AUDIT.md in #4100. Select one distinct obligation: CRT total repair, native/free-algebra lifting, precision/conditioning, or complete fixed-tape cost. This is an author-submitted proof, not an accepted unconditional theorem.",
+      "scope": "Source-binding orientation only; no scientific replay, theorem acceptance or supplier admission. Budget any heavy replay separately."
+    },
+    "first_experiment": "Check live handoffs; name one proposition, its caller, quantified domain and dependency closure. Independently prove the implication or exhibit a precise gap/counterexample. Prioritize CRT.md total repair or COMPLEX.md free-algebra lift only if unclaimed. Stop at the chosen obligation; do not substitute finite samples or root arithmetic for an all-size proof, or launch the complete package by default.",
+    "bottleneck_link": "Finite supplier improvements are useful only under the retained transfer obligations.",
+    "allowed_variations": [
+      "One CRT repair, native lift, precision/conditioning or tape-cost proposition, with its exact caller."
+    ],
+    "scope_change_triggers": [
+      "Replacing a quantified obligation with another finite smoke test.",
+      "Treating archive intake, an author PASS receipt or a finite composed fixture as discharge of the universal hypotheses."
+    ],
+    "overlap_terms": [
+      "uniform transfer",
+      "all-size",
+      "precision",
+      "4100",
+      "unconditional-batched-crt"
+    ],
+    "existing_checks": [
+      {
+        "scope": "PR #4100: proposed unconditional construction and finite checks, without independent acceptance.",
+        "evidence": "research/unconditional-batched-crt/README.md",
+        "remaining": "Independent proof audit of one caller-bound all-size implication; all other premises remain explicit.",
+        "status": "author-checked",
+        "source_commit": "337d04939571a637aeab54a0f58f54fde811663a",
+        "selected_record_sha256": "bb2eaa58647b353f88d0aec2eafb9d30df1be53589231b23e12ceef3e4bad129"
+      }
+    ]
+  },
+  "scope_status": "refined",
+  "actual_scope": "Decide which printed boundary exponent the #4100 union-schedule ledger uses when the short-field layout lemma is copied in: tau=45/16 from the seven-product paragraph, or tau_eff=2+10^{-64} from the structured-boundary balance.",
+  "scope_change_reason": "research/pr4100-row-field-ledger proved that both exponents absorb the short-field movement and stopped before naming the ledger row. This package answers that open question from the printed #4100 sentences. It does not replay a ledger generator and does not reopen the movement count.",
+  "bottleneck_link": "Naming the boundary row is a precision obligation on the proposed #4100 transfer. The selected conditional kappa stays 0.00302015. This note does not admit #4100 or raise kappa.",
+  "budget_minutes": 60,
+  "stop_rule": "Stop once the printed (eps, xi) point identifies the exponent. Do not regenerate check_ledger.py, do not rerun the carving, and do not treat either exponent as an accepted all-size theorem.",
+  "outcome": "completed",
+  "evidence": "PROOF.md derives S_b - S_u = c - 1/2 and evaluates it at both exponents on the printed point. verify.py checks that identity, the 7-digit rounding, the source sentences, and the rejecting 45/16 control.",
+  "author": "unpaidmoon",
+  "ai_assistance": "Prepared by unpaidmoon with substantial Cursor Grok assistance.",
+  "open_questions": [
+    {
+      "question": "Does a byte-identical copy of the cited outer/check_ledger.py regenerate this same tau_eff point, or a later Strassen point?",
+      "why_it_matters": "UNION-SCHEDULE section 5b says check_ledger.py regenerates the printed rows, and that script is not in the #4100 tree. A different generator could name 45/16 even though these sentences do not.",
+      "first_step": "Find a blob whose stdout contains eps = 0.99999437577296 and record which tau it passes as c.",
+      "blocked_by": "research/unconditional-batched-crt does not contain check_ledger.py. This package does not search the rest of the history for a substitute."
+    },
+    {
+      "question": "What explicit C and k satisfy mu <= C (W/B) (log p)^k in the carving proof?",
+      "why_it_matters": "The predecessor left this as the step that would turn the 13/32 exponent gap into a finite-p inequality. Naming tau_eff does not supply C or k.",
+      "first_step": "Pin C and k from the carving argument rather than from another asymptotic sentence.",
+      "blocked_by": "The carving program is not an input here."
+    }
+  ]
+}
diff --git a/research/pr4100-ledger-boundary-exponent/VERIFY.json b/research/pr4100-ledger-boundary-exponent/VERIFY.json
new file mode 100644
index 0000000000..1d67dba5dc
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/VERIFY.json
@@ -0,0 +1,23 @@
+{
+  "schema_version": 1,
+  "status": "ready",
+  "python": "3.12",
+  "requirements": [],
+  "inputs": [
+    "research/unconditional-batched-crt/GAUSSIAN.md",
+    "research/unconditional-batched-crt/vendor/FINE-PRIME.md",
+    "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md",
+    "research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md"
+  ],
+  "commands": [
+    {
+      "argv": [
+        "{python}",
+        "-B",
+        "{package}/verify.py"
+      ],
+      "timeout_seconds": 60,
+      "checks": "Identity S_b - S_u = c - 1/2 on the printed (eps, xi); tau_eff gap -5e-65; 45/16 gap -13/32 and a negative boundary saving; 7-digit rounding of the gauge-free point; source-sentence order for the layout lemma."
+    }
+  ]
+}
diff --git a/research/pr4100-ledger-boundary-exponent/verify.py b/research/pr4100-ledger-boundary-exponent/verify.py
new file mode 100644
index 0000000000..021a03707d
--- /dev/null
+++ b/research/pr4100-ledger-boundary-exponent/verify.py
@@ -0,0 +1,135 @@
+#!/usr/bin/env python3
+"""The printed #4100 union ledger names tau_eff, not 45/16."""
+
+from decimal import Decimal, ROUND_HALF_UP
+from fractions import Fraction
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[2]
+GAUSSIAN = "research/unconditional-batched-crt/GAUSSIAN.md"
+FINE = "research/unconditional-batched-crt/vendor/FINE-PRIME.md"
+UNION = "research/unconditional-batched-crt/vendor/UNION-SCHEDULE.md"
+GAUGE = "research/unconditional-batched-crt/vendor/GAUGEFREE-INTERIORS.md"
+
+EPS = Decimal("0.99999437577296")
+XI = Decimal("0.50167412128176")
+BETA = Decimal("0.99664613320944")
+
+
+def require(condition, message):
+    if not condition:
+        raise SystemExit("FAIL: " + message)
+
+
+def load(path):
+    return (ROOT / path).read_text(encoding="utf-8")
+
+
+def require_phrase(body, phrase, name):
+    require(phrase in body, name + " missing phrase: " + phrase)
+
+
+def round_half_up(value, places):
+    quantum = Decimal("1e-" + str(places))
+    return value.quantize(quantum, rounding=ROUND_HALF_UP)
+
+
+def saving_gap(tau):
+    """S_b - S_u = c - 1/2, with c = (3 - tau) / 2."""
+    return (Fraction(3) - tau) / 2 - Fraction(1, 2)
+
+
+def main():
+    gaussian = load(GAUSSIAN)
+    fine = load(FINE)
+    union = load(UNION)
+    gauge = load(GAUGE)
+
+    require_phrase(gaussian, "7^16<2^45 gives exponent tau=45/16", "GAUSSIAN.md")
+    require_phrase(gaussian, "O(V d W^(tau-1)/B polylog(p))", "GAUSSIAN.md")
+    require_phrase(gaussian, "Total union cost over d/g groups is", "GAUSSIAN.md")
+    require_phrase(gaussian, "O(V d W/B polylog(p))", "GAUSSIAN.md")
+    require_phrase(
+        gaussian,
+        "Temporary padding of the one exposed polynomial\nrow field is removed before the next axis",
+        "GAUSSIAN.md",
+    )
+    strassen_at = gaussian.index("7^16<2^45 gives exponent tau=45/16")
+    union_at = gaussian.index("Total union cost over d/g groups is")
+    padding_at = gaussian.index("Temporary padding of the one exposed polynomial")
+    require(strassen_at < union_at < padding_at, "layout sentence is not after the Strassen paragraph")
+
+    require_phrase(fine, "target exponent `tau=45/16` and `c=(3-tau)/2=3/32`", "FINE-PRIME.md")
+    require_phrase(fine, "needs no structured inverse algorithm", "FINE-PRIME.md")
+    require_phrase(fine, "its saving is `c+xi-epsilon`", "FINE-PRIME.md")
+    require(
+        fine.index("target exponent `tau=45/16`") < fine.index("needs no structured inverse algorithm"),
+        "Strassen sentence does not reject the structured inverse",
+    )
+
+    require_phrase(union, "tau_eff = 2 + 1e-64", "UNION-SCHEDULE.md")
+    require_phrase(union, "this document's schedule is independent of the boundary exponent", "UNION-SCHEDULE.md")
+    require_phrase(union, "epsilon = 0.9999944, xi = 0.5016741, beta = 0.9966461", "UNION-SCHEDULE.md")
+    require_phrase(union, "the same exponent eps+1/2-xi and both bind", "UNION-SCHEDULE.md")
+    require_phrase(
+        union,
+        "retained polynomial-width row-field movement for batched boundary arithmetic",
+        "UNION-SCHEDULE.md",
+    )
+    require_phrase(union, "nothing in this document changes the arithmetic", "UNION-SCHEDULE.md")
+    require(
+        union.index("tau_eff = 2 + 1e-64") < union.index("epsilon = 0.9999944"),
+        "section 5b does not follow the tau_eff definition",
+    )
+
+    require_phrase(
+        gauge,
+        "eps = 0.99999437577296, xi = 0.50167412128176, beta = 0.99664613320944",
+        "GAUGEFREE-INTERIORS.md",
+    )
+    require_phrase(gauge, "c = (3 - tau_eff)/2", "GAUGEFREE-INTERIORS.md")
+    require_phrase(gauge, "0.001679745508802299", "GAUGEFREE-INTERIORS.md")
+    require_phrase(
+        gauge,
+        "margins above kappa: main = Gaussian = bit router = boundary = union ~ 1e-64",
+        "GAUGEFREE-INTERIORS.md",
+    )
+
+    require(round_half_up(EPS, 7) == Decimal("0.9999944"), "eps does not round to the section 5b digit")
+    require(round_half_up(XI, 7) == Decimal("0.5016741"), "xi does not round to the section 5b digit")
+    require(round_half_up(BETA, 7) == Decimal("0.9966461"), "beta does not round to the section 5b digit")
+
+    union_saving = Fraction(XI) + Fraction(1, 2) - Fraction(EPS)
+    require(union_saving == Fraction(1049840943, 625000000000), "union saving is not 0.0016797455088")
+    prefix = Decimal(union_saving.numerator) / Decimal(union_saving.denominator)
+    require(format(prefix, "f").startswith("0.0016797455088"), "union saving misses the printed kappa prefix")
+    require("0.001679745508802299".startswith("0.00167974550880"), "kappa prefix drifted")
+
+    tau_eff = Fraction(2) + Fraction(1, 10**64)
+    tau_strassen = Fraction(45, 16)
+    require(saving_gap(tau_eff) == Fraction(-1, 2 * 10**64), "tau_eff gap is not -5e-65")
+    require(saving_gap(tau_strassen) == Fraction(-13, 32), "45/16 gap is not -13/32")
+
+    boundary_strassen = union_saving + saving_gap(tau_strassen)
+    require(boundary_strassen < 0, "45/16 boundary saving is not below zero on the printed point")
+    require(abs(saving_gap(tau_eff)) < Fraction(1, 10**64), "tau_eff gap is outside the stated 1e-64 margin")
+    require(abs(saving_gap(tau_strassen)) > Fraction(1, 10), "45/16 control is not separated from the slack")
+
+    # A ledger that names 45/16 cannot be the section 5b joint-binding row.
+    require(saving_gap(tau_strassen) != saving_gap(tau_eff), "the two exponents collapsed")
+
+    print(
+        "PASS_LEDGER_BOUNDARY_EXPONENT"
+        " exponent=tau_eff"
+        " gap_eff=-5e-65"
+        " gap_45=-13/32"
+        " seven_digit_round=true"
+        " layout_on_union_paragraph=true"
+        " strassen_below_zero=true"
+        " selected_decimal=0.00302015"
+        " no_new_kappa=true"
+    )
+
+
+if __name__ == "__main__":
+    main()