A Typed Benchmark Ledger for Duality, Microstate Entropy, and Large-N Correspondence in the CHC Framework

Strong-coupling lift/duality, black-hole microstate entropy accounting, and large-$N$TeXN gauge/gravity correspondence provide three external benchmark classes for any would-be gauge--gravity unification program.[citation] We test those classes against an adopted comparator set using a typed benchmark ledger.

The declared comparator set fixes a nontrivial gravitational backbone: an admitted single-scalar branch with controlled recovery to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}}, explicit dark-sector branches, and black-hole accessibility/return constructions. It also contains one selected compactified family carrying a compact internal response manifold, a selected vacuum orbit, and an admitted ultraviolet closure family on one declared completion family.[citation] In addition, on one admitted family it contains a benchmark-facing matter-coupled gauge sheet, an anomaly-closed gauge--chiral sheet, a realized electroweak-facing completion family, confinement-facing closure, a fixed observable fit-window basket, and a bound-state vibrational microscopic object set.[citation] What the declared typed benchmark ledger does not contain is an explicit strong-coupling lift map, a microscopic entropy-counting theorem, or the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation] and not imported here. The compactified family removes any claim that the comparator set lacks compactification or family-conditioned ultraviolet closure altogether, but it does not by itself furnish a theorem of equivalence to string/M-theory, a decisive benchmark comparator closure, or a universal unified field theory.

The comparison remains benchmark-specific and does not assert containment or equivalence. We do not claim that reproduces string duality, derives D-brane microstate counting, or already contains a gauge/gravity dual pair. We claim only that the comparator set adopted here admits a typed benchmark ledger in which each of the three fixed benchmark classes receives a status---recovered, partially recovered, benchmark-compatible, open, or failed---with explicit reasons and non-claims. These labels classify benchmark classes only on the declared comparator set; they are not universal theory judgments. The substantive result established here is a typed benchmark ledger on that comparator set: benchmark pressure is converted into an explicit recovered / partially recovered / benchmark-compatible / open / failed classification rather than into unsupported benchmark inflation.

definition: Fixed benchmark classes. The benchmark classes of this paper are exactly the following three:

\Bdual : \text{strong-coupling lift / duality benchmark},

\Bent : \text{black-hole microstate entropy-accounting benchmark},

\BlargeN : \text{large-}N\text{ gauge/gravity correspondence benchmark}.

A compactification / selected-vacuum / ultraviolet-closure note may be discussed only as secondary benchmark context and is not itself an additional benchmark class.

definition: Typed ledger statuses. For any benchmark class $\mathsf{B}$TeX\mathsf{B}, the benchmark ledger assigns exactly one status from the set

\{\Recovered,\ \Partial,\ \Compatible,\ \OpenS,\ \Failed\}.

The meanings are fixed as follows.

- Recovered: the declared CHC comparator set contains an explicit comparator object, a declared domain or family, and a theorem/proposition-level map that reproduces the benchmark target on that domain. - Partially recovered: the declared CHC comparator set contains an explicit comparator object and a theorem/proposition-level map, but only on a proper subwindow or reduced family that omits at least one benchmark ingredient. - Benchmark-compatible: the declared CHC comparator set contains explicit comparator objects satisfying a necessary structural criterion for the benchmark, but no theorem-level benchmark map is yet present. - Open: a benchmark comparator is partially present, but the decisive lift/ counting/ correspondence object is not yet built, and the declared comparator set does not prove that such a completion is impossible. - Failed: the declared comparator set lacks, or explicitly excludes, a structurally necessary comparator slot in a way that blocks the benchmark window on the declared comparator set.

Statuses are assigned in the order recovered, partially recovered, benchmark-compatible, open, and failed, except that overrides weaker statuses whenever a structurally necessary comparator slot is explicitly absent from the declared comparator set. These labels classify benchmark classes only on the declared comparator set and do not by themselves assert equivalence, containment, or completion.

proposition: Typed Benchmark Ledger Proposition. Let $\Ledger$TeX\Ledger be the benchmark ledger defined by reference and evaluated on the declared CHC comparator set. Then

\Ledger(\Bdual)=\Compatible,
\qquad
\Ledger(\Bent)=\OpenS,
\qquad
\Ledger(\BlargeN)=\Failed.

The justification of each status is given in reference.

definition: Present CHC comparator objects. The benchmark comparison uses the following CHC-side comparator objects and no others.

- The gravitational backbone

\Ggrav:=(g_{\mu\nu},\HH,M_{\mathrm{eff}}^2(\HH),U_{\mathrm{eff}}(\HH),\Xigrad),

meaning the admitted single-scalar scalar--tensor branch with controlled recovery to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}} on a small-gradient window.[citation] - The selected compactified completion family

\Gcomp:=(\Kph,\sigstar,\Guv),

meaning one declared compact phase fiber, one selected vacuum orbit, and one admitted ultraviolet closure family on one selected compactified CHC family.[citation] - The black-hole comparator stack

\GBH:=(\Dobs,\Iglob,\Aacc,\Xith,\Kret),

meaning the finite observation-domain accessibility split, the horizon-threshold diagnostic, the exact return identity on a fixed retarded-time split, and the stationary delayed-return kernel on the admitted reduced family.[citation] - The benchmark-facing gauge-sheet / anomaly / electroweak-facing / confinement / fixed-family observable / vibrational stack

\Gbench:=(\mathcal S_{\mathrm{gauge}},\mathcal A_{\mathrm{anom}},\mathcal E_{\mathrm{EW}},\mathcal C_{\mathrm{conf}},\mathcal W_{\mathrm{fit}},\mathcal V_{\mathrm{vib}}),

meaning the matter-coupled gauge sheet and benchmark representation cell, generator-resolved anomaly closure on that sheet, a realized electroweak-facing completion family, confinement-facing closure, one fixed observable basket on one admitted family, and a bound-state vibrational microscopic object set on one declared family.[citation] - The explicitly missing decisive benchmark-closure slots

\Gmiss:=(\cO_{\mathrm{lift}},\cO_{\mathrm{micro}},\cO_{N}),

meaning the explicit lifted duality map, the microscopic entropy-counting theorem, and the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation] and not imported into the declared typed benchmark ledger.

All downstream benchmark claims are read only relative to these declared comparator objects on the declared CHC comparator set.

The first comparator records the strongest exact external correspondence available in the declared comparator set: an admitted single-scalar, luminality-safe scalar--tensor branch with controlled recovery to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}}. The second records the strongest family-conditioned compactification result: one selected compactified family carrying a compact phase fiber, a selected vacuum orbit, and an admitted ultraviolet closure family. The third records the strongest black-hole-side analytical closure: horizons interpreted as accessibility boundaries together with exact and reduced return laws. The fourth records the benchmark-facing gauge-sheet / gauge--chiral / electroweak-facing / confinement / fixed-family observable / vibrational objects realized on one admitted family. The fifth records the decisive benchmark-closure slots that remain absent from the declared typed benchmark ledger. In particular, the large-$N$TeXN failure does not arise from the total absence of gauge-facing structure, but from the absence, on the declared typed benchmark ledger, of the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation] and not imported here.

The Witten benchmark asks for more than an extra scalar or a strong-coupling parameter. In its benchmark form, the target is an explicit strong-coupling lift and duality structure: one description must pass to another while preserving an admitted object set, with a concrete map to a lifted regime such as the type-IIA to eleven-dimensional relation.[citation]

For this benchmark we use the following criterion.

definition: Duality/lift compatibility criterion. A benchmark-compatible duality/lift window exists if the candidate theory contains:

- an admitted branch whose departures from the Einstein limit are controlled by an explicit hierarchy parameter; - one selected compactified family carrying an admitted ultraviolet closure family on a declared domain; and - no internal statement forbidding the introduction of a stronger lifted description,

while still lacking an explicit lifted geometry, duality group, or strong-coupling map.

proposition: Duality/Lift Benchmark Proposition. Relative to the strong-coupling lift / duality benchmark of [citation], the declared CHC comparator set is but neither nor .

Reason..

The declared CHC comparator set satisfies reference: it contains an admitted scalar--tensor branch with controlled recovery, an explicit hierarchy variable $\Xigrad$TeX\Xigrad, and one selected compactified family with an admitted ultraviolet closure family.[citation] What is missing is the lifted benchmark content itself: no explicit lifted geometry is constructed, no duality group is defined, and no theorem-level strong-coupling lift map exists on the declared benchmark ledger. Therefore the benchmark is not recovered or partially recovered. It is benchmark-compatible because the declared comparator set is structurally non-obstructive to an unconstructed strong-coupling lift sector, but it remains only compatible.

remark. This compatibility status is deliberately narrow. It does not license any statement of the form `` contains M-theory'' or `` has derived string duality.'' The declared comparator set does not contradict the duality/lift benchmark, while still lacking the benchmark map itself.

The Strominger--Vafa benchmark is sharper than a horizon-language resemblance. It asks for an admitted family on which microscopic degeneracy accounting reproduces the black-hole entropy law,

S_{\mathrm{micro}}=\log \Omega(Q)=\frac{\Area}{4G}+\cdots,

with a concrete charge family $Q$TeXQ and an explicit count or protected index.[citation]

The declared CHC comparator set contains nontrivial black-hole-side comparator objects: an accessibility-boundary interpretation of the horizon together with an exact return identity on a fixed retarded-time split, a stationary delayed-return kernel, and a sharp-limit control law.[citation] It also contains one selected compactified family with an admitted ultraviolet closure family.[citation] These are genuine black-hole-side and completion-side structures, but they are not yet microscopic entropy accounting.

proposition: Entropy-Accounting Benchmark Proposition. Relative to the microstate entropy benchmark of [citation], the declared CHC comparator set is .

Reason..

The benchmark is not because the declared CHC comparator set contains no CHC microstate family, no index or degeneracy map $\Omega(Q)$TeX\Omega(Q), and no theorem equating a CHC counting formula to $\Area/4G$TeX\Area/4G on an admitted black-hole family. It is not because the declared comparator set already provides nontrivial comparator objects---accessibility partition, threshold diagnostic, exact return identity, stationary return kernel, and a selected compactified family with admitted ultraviolet closure---and nothing in the declared comparator set proves that a microscopic accounting map is impossible. The correct status is therefore : the benchmark has a genuine CHC-side interface, but the decisive counting object is not yet built on the declared typed benchmark ledger.

remark. The declared status is intentionally stronger than a vague analogy and intentionally weaker than entropy recovery. The declared CHC comparator set can discuss horizon accessibility, delayed return, and one selected compactified family, but it cannot yet claim microscopic entropy accounting.

The large-$N$TeXN benchmark is taken here in the supergravity/gauge window opened by Maldacena's AdS/CFT proposal, sharpened by the Gubser--Klebanov--Polyakov and Witten holographic dictionaries, and codified for the present comparison by Itzhaki, Maldacena, Sonnenschein, and Yankielowicz. Its minimal requirement is a non-Abelian gauge-side comparator with a rank parameter $N$TeXN, a coupling parameter, and a declared regime in which a gravitational description is matched to a large-$N$TeXN planar family.[citation]

proposition: Large-$N$TeXN Correspondence Benchmark Proposition. Relative to the large-$N$TeXN gauge/gravity benchmark of [citation], the declared CHC comparator set is on the declared comparator set.

Reason..

The benchmark is not in the weak sense of a missing detail; it fails on the declared comparator set because a structurally necessary comparator slot is absent. The declared CHC comparator set contains gauge-facing objects, and it also contains a selected compactified family, but the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation], together with its declared rank/scaling data, is not imported into the declared typed benchmark ledger. Since reference declares failure whenever a necessary comparator slot is explicitly absent, the correct status is . This is a failure of the declared comparator set, not a proof of permanent impossibility for future CHC extensions.

Compactification, vacuum selection, and admitted ultraviolet closure are not one of the three fixed benchmark classes, but they provide necessary secondary context. The declared CHC comparator set contains one selected compactified family carrying a compact phase fiber, a selected vacuum orbit, and an admitted ultraviolet closure family on one declared completion family.[citation] That family-conditioned completion result is part of the comparator background used here. It does not by itself furnish a lifted duality map, a microscopic entropy-counting theorem, or a genuine gauge-side large-$N$TeXN comparator on the declared typed benchmark ledger. The compactified-family note is therefore secondary and non-recovering with respect to the three fixed benchmark classes.

A separate comment is required for the Higgs-adjacent language present on the CHC comparator set. admits scalar-background and order-parameter analogies through the universal field $\HH$TeX\HH, the admitted scalar--tensor branch, and the common-potential constraints of the dark-energy and dark-matter sides. The declared comparator set also contains a realized electroweak-facing completion family with a complex doublet and matrix-valued Yukawa maps on one admitted family.[citation] But these objects remain family-conditioned and non-identical to the backbone order-parameter object; they do not collapse the declared typed benchmark ledger into a recovered Standard-Model or UV-complete interface. The compactified-family note therefore remains secondary and non-recovering.

The substantive result established here is the typed benchmark ledger on the declared CHC comparator set: the three benchmark classes are not compressed into one slogan, but separated into explicit recovered / partially recovered / benchmark-compatible / open / failed windows.

center 1.18

Figure or table content is omitted from the web reader; use the canonical manuscript for the exact object.

center

The open problems are correspondingly narrow and typed. For the duality/lift benchmark, the missing object is an explicit lifted branch or duality map. For the entropy benchmark, the missing object is a microscopic state-counting or index map on an admitted black-hole family. For the large-$N$TeXN benchmark, the missing object on the declared typed benchmark ledger is the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation], together with the declared rank/scaling map that is not imported on the benchmark windows used here. These are not cosmetic gaps; they are exactly the objects any later extension would need to supply before a stronger benchmark verdict could be attempted.

We have tested exactly three benchmark classes: strong-coupling lift / duality, black-hole microstate entropy accounting, and large-$N$TeXN gauge/gravity correspondence. On the declared CHC comparator set, their typed statuses are

\Ledger(\Bdual)=\Compatible,
\qquad
\Ledger(\Bent)=\OpenS,
\qquad
\Ledger(\BlargeN)=\Failed.

What has not been established is equally important: there is no theorem of equivalence between and string/M-theory, no lifted duality map on the declared typed benchmark ledger, no microscopic entropy-counting theorem, and no imported gauge-side large-$N$TeXN comparator construction on the declared comparator set. The failure frontier is explicit on the declared typed benchmark ledger: without a lifted duality map, a microscopic entropy-counting theorem, and the separate gauge-side large-$N$TeXN comparator construction reserved for Ref. [citation] and not imported here, no stronger benchmark claim is licensed on the declared typed benchmark ledger.

The typed benchmark ledger fixes only the declared status labels together with the corresponding open and failure regions. The selected compactified family furnished in Ref. [citation] provides family-conditioned compactification, vacuum selection, and admitted ultraviolet closure on one declared family, but it is treated here only as secondary benchmark context. It does not by itself recover any of the three benchmark classes. The ledger therefore does not license any claim that is equivalent to, contains, or recovers string/M-theory.

Funding and competing interests..

No external funding was received for this work. The author declares no competing interests.