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 formally specified benchmark ledger.

The declared comparator set fixes a single-scalar gravitational branch, explicit phenomenological dark-sector branches, and black-hole response constructions. It also contains a compact response family with a finite spectral weighting.[citation] The latter does not constitute ultraviolet completion because its propagator retains the ordinary $1/p^2$TeX1/p^2 asymptotic tail. Gauge, anomaly, electroweak, confinement, fit-window, and vibrational models are also available on stipulated families.[citation] These constructions do not contain an explicit strong-coupling lift, a microscopic entropy-counting map, or a state--operator correspondence.

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 formally specified 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 formally specified 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: Formally specified 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: Formally specified 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)=\OpenS,
\qquad
\Ledger(\Bent)=\OpenS,
\qquad
\Ledger(\BlargeN)=\Failed.

The justification of each status is given in reference.

proof. The duality claim lacks an explicit state--operator map and source-dependent generating-functional identity, so it satisfies neither the recovered nor partially recovered conditions. The entropy claim lacks a microscopic family and degeneracy map. The large-$N$TeXN claim lacks a correspondence object on the comparator set specified in reference. Applying the mutually exclusive cases of reference gives the displayed assignments. These assignments concern evidential sufficiency on the stated comparator set and do not imply impossibility of later extensions.

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 compact response family

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

meaning one declared compact phase fiber, one selected vacuum orbit, and one finite weighted propagator on a stipulated 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 formally specified benchmark ledger.

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

The first comparator is a single-scalar scalar--tensor branch with controlled recovery to GR+$\Lambda_{\mathrm{eff}}$TeX\Lambda_{\mathrm{eff}}. The second is a compact response construction with a finite weighted propagator, not a proof of ultraviolet finiteness. The third consists of conditional black-hole response laws. The fourth collects stipulated gauge-facing models. The fifth lists the decisive correspondence objects that remain absent.

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 compact response 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: Structural data do not imply a duality. Relative to the strong-coupling lift / duality benchmark of [citation], the declared CHC comparator set is and neither nor . A scalar--tensor branch and a compact response space alone do not imply a duality.

proof. A duality must at minimum identify state spaces and operator algebras and preserve source-dependent correlation functions. None of these maps is contained in $\Ggrav$TeX\Ggrav, $\Gcomp$TeX\Gcomp, or $\GBH$TeX\GBH. Non-implication is constructive: attach to the same displayed background data two quantum sectors with generating functionals $Z_1[J]=Z_0\exp(aJ^2)$TeXZ_1[J]=Z_0\exp(aJ^2) and $Z_2[J]=Z_0\exp(bJ^2)$TeXZ_2[J]=Z_0\exp(bJ^2) for $a\ne b$TeXa\ne b. They agree at $J=0$TeXJ=0 and therefore on every source-free scalar comparator listed here, but their connected two-point functions differ. Thus the listed structural data cannot entail a duality. The absence of a no-go statement is not positive compatibility evidence, so the scientifically warranted assignment is open.

Reason..

The declared CHC comparator set contains an admitted scalar--tensor branch, an explicit hierarchy variable $\Xigrad$TeX\Xigrad, and one selected compact response family.[citation] The finite spectral weighting in the latter is a regulator ansatz rather than an ultraviolet completion: its propagator retains a $1/p^2$TeX1/p^2 tail and its representative one-loop integral is quadratically divergent. No explicit lifted geometry, duality group, state map, operator map, or source-dependent equality is constructed. The benchmark therefore remains open.

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 an accessibility-boundary interpretation, a return identity on a fixed retarded-time split, and a stipulated delayed-return kernel.[citation] It also contains one compact response family.[citation] These are macroscopic or phenomenological structures and do not perform microscopic entropy accounting.

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

proof. Fix a macroscopic horizon area $\Area$TeX\Area. For every positive integer $K$TeXK, a microscopic candidate can be formed with $K$TeXK states while retaining the same macroscopic area and the same accessibility/return observables. Its entropy is $\log K$TeX\log K. Choosing distinct $K$TeXK therefore leaves all declared macroscopic comparators unchanged while changing the microscopic entropy. Hence those comparators do not determine $\Omega(Q)$TeX\Omega(Q) or imply $\log\Omega(Q)=\Area/(4G)+\cdots$TeX\log\Omega(Q)=\Area/(4G)+\cdots. Since no independent impossibility theorem is available, the result is open rather than failed.

Reason..

The benchmark is not because the declared CHC comparator set contains no microscopic state family, index, or degeneracy map $\Omega(Q)$TeX\Omega(Q) and no theorem equating a count to $\Area/4G$TeX\Area/4G. No impossibility theorem is established either; the claim therefore remains open.

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.

proof. On the comparator set defined in reference, the rank-dependent gauge/gravity map is absent by construction, so the stated failure follows from item (v) of reference. Even after importing a leading $N^2$TeXN^2 comparator, coefficient agreement would remain insufficient: two families may have free energies $F_1(N)=aN^2+O(1)$TeXF_1(N)=aN^2+O(1) and $F_2(N)=aN^2+O(1)$TeXF_2(N)=aN^2+O(1) while their state multiplicities, operator spectra, and source responses differ. Therefore a correspondence requires an explicit map and source-dependent tests, not only planar scaling.

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 formally specified 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 and vacuum selection are not among the three target claims, but provide secondary context. The declared comparator set contains a compact phase fiber, a selected vacuum orbit, and a finite spectral weighting.[citation] The associated loop integral remains quadratically divergent, so this construction is not an ultraviolet completion. Nor does it furnish a lifted duality map, a microscopic entropy theorem, or a gauge/gravity correspondence.

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 formally specified 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 formally specified 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.

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The open problems are correspondingly narrow and formally specified. 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 formally specified 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.

The compact root scalar supplies a global target circumference, winding sectors, and a sharp gradient-energy bound. None of these objects is an invertible state--operator map, a source-dependent generating-functional identity, or a protected microscopic degeneracy. The compact-target theorem therefore adds genuine global structure without satisfying any necessary condition for duality or horizon microstate counting in this analysis.

A future correspondence proposal must also be overidentified. If a $q$TeXq-parameter benchmark map controls $m>q$TeXm>q spectra, correlators, source responses, and entropy data, its image has codimension at least $m-q$TeXm-q; left-null response combinations are the local tests. Adding a separate matching coefficient to each benchmark can reproduce leading quantities while making the map locally onto. Such coefficient matching remains a comparator, not evidence for a duality.

The closure test for the duality benchmark ledger is applied to a dimensionless observable vector $y\in\mathbb R^m$TeXy\in\mathbb R^m formed from fixed reference scales and the declared basket of spectra, correlators, entropy counts, and large-parameter residuals. Let $a$TeXa range over the independent constitutive inputs comprising the two theories, operator dictionary, state map, large-parameter limit, and error measure.

proposition: Functional saturation, finite closure, and sector admissibility. Suppose the unrestricted prediction map $F:a\mapsto y$TeXF:a\mapsto y is continuously differentiable on a Banach space of constitutive inputs. If $D_aF$TeXD_aF is surjective and has a bounded right inverse at the calibration point, the unrestricted family is locally open in observable space and supplies no nonzero local equality restriction on $y$TeXy. Suppose instead that a single microscopic closure replaces $a$TeXa by finite parameters $\theta\in\mathbb R^p$TeX\theta\in\mathbb R^p, with profiled nuisance coordinates $\eta\in\mathbb R^q$TeX\eta\in\mathbb R^q. If

J=D_\eta F_{\rm cl}D_\theta F_{\rm cl},
 \qquad \operatorname{rank}J=r<m,

then there are $m-r$TeXm-r independent first-order restrictions

w^{\mathsf T}\delta y=0,
 \qquad w\in\ker J^{\mathsf T}.

If the rank is constant locally, these restrictions are tangent to a compatibility manifold of codimension $m-r$TeXm-r. For this sector, the finite closure is admissible only if one invertible dictionary with matched partition data determines every comparator before tests are evaluated.

proof. Split surjectivity gives a bounded right inverse $R$TeXR with $D_aF\,R=I_m$TeXD_aF\,R=I_m. The Banach-space submersion theorem then makes $F$TeXF locally onto a neighborhood of the calibrated observable vector. Any smooth equality holding throughout that image must therefore vanish on an open set and contributes no model-specific local restriction. Under finite closure, the attainable first-order variations are exactly the column space of $J$TeXJ. Its orthogonal complement is $\ker J^{\mathsf T}$TeX\ker J^{\mathsf T}, whose dimension is $m-r$TeXm-r by rank--nullity, which proves reference. The constant-rank theorem supplies the stated local manifold. The sector condition is necessary because a collection of separately normalized analogies does not establish a duality. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

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 formally specified statuses are

\Ledger(\Bdual)=\OpenS,
\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 formally specified 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 formally specified 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 formally specified benchmark ledger.

The finite compact response construction of Ref. [citation] does not recover any of the three benchmark classes and does not establish ultraviolet completion. The available evidence therefore does not support a 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.