Microstate Counting and Planar Large-N Comparator Windows in the CHC Framework

A black hole presents a macroscopic entropy law although the inaccessible states behind the horizon are not directly enumerated. Large-$N$TeXN gauge families display an apparently opposite surprise: as the number of internal channels grows, whole classes of observables simplify into planar and factorized patterns. These two tensions define some of the sharpest benchmark pressure on any proposed gauge--gravity language.[citation]

A benchmark ledger by itself does not yet provide the decisive object. On the gravity side, one wants a count sharp enough to deserve the name of a microstate-counting object rather than a purely thermodynamic proxy. On the gauge side, one wants a comparator built from genuinely planar gauge observables rather than a gravity-side quantity re-expressed twice. The construction below addresses both demands on declared benchmark windows and then asks whether a fixed scaling map can place the two objects in one nontrivial coefficient-level relation.

The benchmark claim remains typed and narrow. No full AdS/CFT theorem is asserted, no string/M equivalence is claimed, and no ultraviolet closure is derived. The result is instead a pair of declared benchmark objects together with a theorem-level coefficient criterion telling us when their leading scaling laws are compatible on a declared family and when that compatibility is structurally absent. Typed benchmark-ledger source summary and cross-sector dictionary synthesis are outside the scope of the present paper.

definition: Declared gravity-side benchmark window. A declared gravity-side benchmark window is a family

\Bg=(\AreaH,\ellstar,\bstate,\epsH)

with the following properties:

- A quasi-stationary accessibility boundary admits a partition into

\QHcell:=\left\lfloor \frac{\AreaH}{\ellstar^2}\right\rfloor

nominal horizon cells. - Each cell carries exactly $\bstate\ge 2$TeX\bstate\ge 2 hidden branch states compatible with the same exterior macro-ledger. - Composition across cells is multiplicative up to a subextensive defect,

\Nmicro = \bstate^{\QHcell}e^{\delta_{\mathrm H}}, \qquad |\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}.

All gravity-side counting claims below are restricted to this declared family.

definition: Declared gauge-side large-$N$TeXN benchmark family. A declared gauge-side benchmark family is a family

\Bq=(N,\lambda,\epsG)

with fixed 't Hooft coupling $\lambda$TeX\lambda and a single-trace observable set $\{\Ord_a\}$TeX\{\Ord_a\} satisfying:

- connected-correlator suppression

\langle \Ord_{a_1}\cdots \Ord_{a_k} \rangle_{\!c}=N^{2-k}C_{a_1\dots a_k}(\lambda)+r_{a_1\dots a_k}(N,\lambda), \qquad |r_{a_1\dots a_k}|\le \epsG N^{1-k};

- free-energy scaling

\FN(\lambda)=N^2 f_0(\lambda)+f_1(\lambda)+o(1);

- spectral-density growth on a declared energy window,

\log \Nplan(E,\lambda)=N^2 s_0(E,\lambda)+o(N^2), \qquad s_0(E,\lambda)>0.

A large-$N$TeXN comparator is genuine only if the planar data and the positive spectral-density coefficient all live on the gauge side of the declared family.

definition: Typed benchmark statuses. For the declared benchmark pressure we use the statuses $\Rec,\Par,\Cmp,\OpenS,\Fail$TeX\Rec,\Par,\Cmp,\OpenS,\Fail with the following meanings:

- $\Rec$TeX\Rec: explicit benchmark map on a declared family. - $\Par$TeX\Par: explicit decisive object on one side together with a declared reduced benchmark relation, but not full benchmark closure. - $\Cmp$TeX\Cmp: necessary benchmark objects are present and admit a declared compatibility test, but no identity theorem is claimed. - $\OpenS$TeX\OpenS: a benchmark-facing interface exists but the decisive relation is still missing. - $\Fail$TeX\Fail: a structurally necessary comparator slot is absent on the declared family.

The two declared families play different roles. The gravity-side family isolates inaccessible branch structure on a quasi-stationary accessibility boundary. The gauge-side family isolates genuine planar simplification and factorization on a fixed-rank observable family. Neither family is allowed to borrow its decisive object from the other side.

definition: Microstate-counting object. On the declared gravity-side benchmark window $\Bg$TeX\Bg, define the decisive microstate-counting object by

\Nmicro(\Bg):=\#\,\mathfrak H_{\mathrm{hid}}(\Bg)=\bstate^{\QHcell}e^{\delta_{\mathrm H}},
\qquad \QHcell=\left\lfloor \frac{\AreaH}{\ellstar^2}\right\rfloor,

where $\mathfrak H_{\mathrm{hid}}(\Bg)$TeX\mathfrak H_{\mathrm{hid}}(\Bg) is the set of hidden branch classes compatible with one fixed exterior macro-ledger on $\Bg$TeX\Bg, and $|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}$TeX|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}. The associated entropy functional is

\Smicro(\Bg):=\log \Nmicro(\Bg).

definition: Decisiveness criterion. The object $\Nmicro$TeX\Nmicro is decisive on $\Bg$TeX\Bg if:

- it is explicitly countable on the declared family; - it distinguishes inequivalent hidden branch classes; - its entropy is stable under coarse graining up to a subextensive defect.

theorem: Decisive Microstate-Counting Object Theorem on the Declared Gravity-Side Window. On the declared gravity-side benchmark window $\Bg$TeX\Bg, the object $\Nmicro$TeX\Nmicro of reference is decisive in the sense of reference. Moreover,

\Smicro(\Bg)=\frac{\log \bstate}{\ellstar^2}\AreaH+\mathcal O\!\left(\sqrt{\frac{\AreaH}{\ellstar^2}}\right)+\mathcal O(1).

In particular, the benchmark-scaling defect relative to an area law,

\Delta_{\BH}(\Bg):=\left|\frac{\Smicro}{\AreaH/(4G)}-1\right|,

obeys

\Delta_{\BH}(\Bg)\le\left|\frac{4G\log \bstate}{\ellstar^2}-1\right|+\mathcal O\!\left(\sqrt{\frac{\ellstar^2}{\AreaH}}\right).

proof. The object is explicitly countable because $\Nmicro$TeX\Nmicro is given by a closed formula on $\Bg$TeX\Bg. Distinct hidden branch assignments differ by at least one horizon cell and therefore correspond to distinct elements of $\mathfrak H_{\mathrm{hid}}(\Bg)$TeX\mathfrak H_{\mathrm{hid}}(\Bg), so the object distinguishes inequivalent hidden classes. Coarse graining changes the count only through the defect term $\delta_{\mathrm H}$TeX\delta_{\mathrm H}, which is subextensive by hypothesis. Taking the logarithm gives

\Smicro = \QHcell\log \bstate + \delta_{\mathrm H}.

Since $\QHcell=\AreaH/\ellstar^2+\mathcal O(1)$TeX\QHcell=\AreaH/\ellstar^2+\mathcal O(1) and $|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}$TeX|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}, the stated entropy expansion follows. Dividing by $\AreaH/(4G)$TeX\AreaH/(4G) yields the bound for $\Delta_{\BH}$TeX\Delta_{\BH}.

remark. The theorem does not claim a universal microscopic model of every black hole. It claims only that on a declared accessibility-boundary family, hidden branch classes define an explicit countable object with area-like scaling. The external benchmark is therefore met at the level of a declared object and a scaling law, not at the level of a universal string microstate construction.

definition: Large-$N$TeXNN gauge-side comparator. On the declared gauge-side family $\Bq$TeX\Bq, define the comparator

\Cn(\Bq):=\bigl(f_0(\lambda),\,s_0(E,\lambda),\,\mathcal P_N\bigr),

where

- $f_0(\lambda)$TeXf_0(\lambda) is the leading planar free-energy density from $\FN=N^2 f_0+f_1+o(1)$TeX\FN=N^2 f_0+f_1+o(1), - $s_0(E,\lambda)$TeXs_0(E,\lambda) is the leading spectral-density coefficient from $\log \Nplan=N^2 s_0+o(N^2)$TeX\log \Nplan=N^2 s_0+o(N^2), - $\mathcal P_N$TeX\mathcal P_N is the planar factorization predicate encoded by the connected-correlator suppression of reference.

proposition: Genuine Gauge-$N$TeXNN Comparator on the Declared Family. If the declared gauge-side family $\Bq$TeX\Bq satisfies the hypotheses of reference, then the object $\Cn(\Bq)$TeX\Cn(\Bq) is a genuine gauge-side large-$N$TeXN comparator. Its leading scaling data are

\frac{\FN(\lambda)}{N^2}=f_0(\lambda)+\mathcal O(N^{-2}),
\qquad
\frac{\log \Nplan(E,\lambda)}{N^2}=s_0(E,\lambda)+o(1),

and the connected correlators satisfy

\langle \Ord_{a_1}\cdots \Ord_{a_k} \rangle_{\!c}=N^{2-k}C_{a_1\dots a_k}(\lambda)+\mathcal O(N^{1-k}).

Therefore $\Cn$TeX\Cn carries genuine planar and spectral large-$N$TeXN data on the gauge side of the declared family.

proof. The three ingredients in reference are explicit gauge-side objects: the leading free-energy density, the leading spectral-density coefficient, and the planar factorization predicate. By hypothesis, each admits a well-defined large-$N$TeXN limit on the declared family. Their collection therefore constitutes a genuine gauge-side comparator rather than a gravity-side rephrasing. The displayed asymptotic relations are the leading parts of the defining formulas in reference.

remark. The proposition constructs a comparator, not a duality. It does not claim string/M-theory equivalence, does not identify any specific conformal dual, and does not derive a complete holographic dictionary. The claim is only that a genuine gauge-side large-$N$TeXN comparator now exists on a declared family. On one declared map, this comparator is the only gauge-side large-$N$TeXN comparator admitted for any typed benchmark ledger or typed synthesis dictionary; no additional comparator object is introduced at those layers.

definition: Declared scaling map and coefficient mismatch. Define the gravity-side and gauge-side scaling variables by

\QHmap:=\frac{\AreaH}{\ellstar^2},
\qquad
\QG:=N^2.

A declared benchmark-coupling family is a family $\Bcpl\subset\Bg\times\Bq$TeX\Bcpl\subset\Bg\times\Bq on which there exists a fixed scaling constant $\gamma>0$TeX\gamma>0 and a remainder $r_\gamma$TeXr_\gamma such that

\Map:\ \QHmap = \gamma\QG + r_\gamma,
\qquad |r_\gamma|\le \epsC\sqrt{\QG},

with one and the same $\gamma$TeX\gamma used across the declared benchmark window. Define the coefficient mismatch by

\kappacpl(E,\lambda;\gamma):=\gamma\log \bstate - s_0(E,\lambda).

proposition: Typed Entropy--Planar Scaling Compatibility on One Declared Map. Assume the hypotheses of reference and a declared benchmark-coupling family $\Bcpl$TeX\Bcpl in the sense of reference. Then the density defect

\Delta_{\mathrm{dens}}(\Bcpl):=\left|\frac{\Smicro}{\QG}-s_0(E,\lambda)\right|

obeys

\Delta_{\mathrm{dens}}(\Bcpl)\le |\kappacpl(E,\lambda;\gamma)| + \mathcal O(\QG^{-1/2}) + o(1).

Equivalently,

\left|\frac{\Smicro}{\QHmap}-\frac{s_0(E,\lambda)}{\gamma}\right|\le \frac{|\kappacpl(E,\lambda;\gamma)|}{\gamma}+\mathcal O(\QG^{-1/2})+o(1).

Hence coefficient-level compatibility on the declared family is governed entirely by the fixed scaling map and the coefficient mismatch $\kappacpl$TeX\kappacpl.

proof. By reference,

\Smicro = \log \bstate\,\QHmap + \mathcal O(\sqrt{\QHmap}),

while reference gives

\log \Nplan(E,\lambda)=s_0(E,\lambda)\QG + o(\QG).

On $\Bcpl$TeX\Bcpl, the declared scaling map yields $\QHmap=\gamma\QG+r_\gamma$TeX\QHmap=\gamma\QG+r_\gamma with $|r_\gamma|\le \epsC\sqrt{\QG}$TeX|r_\gamma|\le \epsC\sqrt{\QG}. Therefore

\Smicro = \gamma\log \bstate\,\QG + \log \bstate\,r_\gamma + \mathcal O(\sqrt{\QG}),

so

\frac{\Smicro}{\QG}=\gamma\log \bstate + \mathcal O(\QG^{-1/2}).

Subtracting $s_0(E,\lambda)$TeXs_0(E,\lambda) yields the first bound. Dividing instead by $\QHmap\sim\gamma\QG$TeX\QHmap\sim\gamma\QG gives the equivalent second bound.

theorem: Declared Benchmark Coupling Theorem on One Fixed Scaling Map. Assume the hypotheses of reference. If there exists a declared benchmark-coupling family $\Bcpl$TeX\Bcpl with one fixed scaling map $\Map$TeX\Map such that

\kappacpl(E,\lambda;\gamma)=0

on the adopted benchmark window, then

\left|\Smicro - \log \Nplan(E,\lambda)\right| = \mathcal O(\sqrt{\QG}) + o(\QG)

and, equivalently,

\lim_{\QG\to\infty}\left(\frac{\Smicro}{\QG}-s_0(E,\lambda)\right)=0,
\qquad
\lim_{\QG\to\infty}\frac{\Smicro}{\log \Nplan(E,\lambda)}=1.

Therefore the gravity-side entropy density and the gauge-side planar spectral density are asymptotically coefficient-matched on the declared family.

proof. The hypothesis $\kappacpl=0$TeX\kappacpl=0 is precisely the coefficient condition $\gamma\log \bstate=s_0(E,\lambda)$TeX\gamma\log \bstate=s_0(E,\lambda). By reference, the density defect then satisfies

\left|\frac{\Smicro}{\QG}-s_0(E,\lambda)\right|=\mathcal O(\QG^{-1/2})+o(1),

which proves the first limit. Multiplying by $\QG$TeX\QG gives

\left|\Smicro - s_0(E,\lambda)\QG\right| = \mathcal O(\sqrt{\QG})+o(\QG).

Since $\log \Nplan(E,\lambda)=s_0(E,\lambda)\QG+o(\QG)$TeX\log \Nplan(E,\lambda)=s_0(E,\lambda)\QG+o(\QG) with $s_0(E,\lambda)>0$TeXs_0(E,\lambda)>0 on the declared spectral window, subtraction yields

\left|\Smicro - \log \Nplan(E,\lambda)\right| = \mathcal O(\sqrt{\QG})+o(\QG).

The ratio statement follows because both numerator and denominator are asymptotic to the same positive leading term $s_0(E,\lambda)\QG$TeXs_0(E,\lambda)\QG.

remark. The theorem is stronger than a mere coexistence statement but weaker than a holographic identity. It asserts coefficient-level and exponent-level compatibility on one declared scaling family. It does not supply a full duality map, a compactification scheme, or a universal gauge--gravity equivalence.

The benchmark consequences of the declared counting and comparator objects are now sharp enough to update the typed ledger.

definition: Updated benchmark statuses on the declared families. Relative to the declared benchmark pressure, the statuses are:

\text{microstate entropy accounting} : \Par,

\text{genuine gauge-side large-}N\text{ comparator} : \Cmp,

\text{coupled entropy--planar benchmark} : \Cmp\text{ on }\Bcpl\text{ satisfying \cref{thm:coupling},}\ \OpenS\text{ otherwise.}

proposition: Typed benchmark-ledger update. On the declared benchmark families, the benchmark status updates as follows.

- The gravity-side microstate benchmark is $\Par$TeX\Par because a decisive counting object and an area-like entropy law are explicitly constructed on $\Bg$TeX\Bg, while no universal microscopic dynamics is claimed. - The gauge-side large-$N$TeXN benchmark is $\Cmp$TeX\Cmp because a genuine comparator exists on $\Bq$TeX\Bq, but no exact gauge--gravity identification is established. - The coupled benchmark is $\Cmp$TeX\Cmp on any declared family $\Bcpl$TeX\Bcpl satisfying reference, and remains $\OpenS$TeX\OpenS on families where no fixed coefficient-matching scaling map can be admitted.

proof. Each status follows directly from reference together with the meanings in reference. The microstate side exceeds mere compatibility because a decisive counting object and a scaling theorem are explicit, but falls short of full recovery because no universal microscopic black-hole model is provided. The gauge side reaches compatibility because genuine large-$N$TeXN data are present, but no exact dual map is proved. The coupled benchmark is compatibility or openness according to whether the declared scaling map and coefficient criterion can be maintained on one adopted window.

The positive benchmark claim fails on the declared families if any of the following occurs.

- The gravity-side count is not truly decisive, for example if the hidden branch alphabet is not fixed on the declared family or if the subextensive defect grows extensively. - The gauge-side comparator is not genuine, for example if planar scaling is imported from gravity-side language rather than built from gauge-side observables. - The benchmark coupling requires a scaling constant $\gamma$TeX\gamma chosen post hoc for each observable instead of one fixed scaling map on the declared family. - The coefficient condition $\gamma\log \bstate=s_0(E,\lambda)$TeX\gamma\log \bstate=s_0(E,\lambda) fails on the admitted window, so coefficient-level compatibility is absent even though both benchmark objects exist. - Entropy matching or planar comparison demands ultraviolet data, compactification structure, or a full holographic dictionary absent from the declared architecture.

The result does not claim:

- full AdS/CFT duality; - string/M-theory equivalence or string/M containment; - ultraviolet completion; - a universal microscopic model of all black-hole entropy; - a complete gauge--gravity unification theorem.

The established result consists only of a decisive gravity-side counting object, a genuine gauge-side large-$N$TeXN comparator, and a declared scaling map under which coefficient-level benchmark compatibility may or may not hold on an adopted benchmark family.

On declared benchmark windows, the construction above fixes a decisive gravity-side counting object, a genuine gauge-side large-$N$TeXN comparator, and a typed compatibility test on one declared scaling map. On a quasi-stationary accessibility-boundary family, inaccessible branch classes define a decisive countable object with area-like entropy scaling. On a declared gauge-side rank family, planar factorization, free-energy scaling, and spectral-density growth with positive leading coefficient define a genuine large-$N$TeXN comparator. A fixed declared scaling map between the gravity-side and gauge-side variables then yields a coefficient-level coupling theorem: when the declared coefficient condition is met, the gravity-side entropy density and the gauge-side planar spectral density are asymptotically compatible on that family. The result is stronger than a merely descriptive benchmark ledger but remains typed, map-conditioned, and narrower than duality. Typed benchmark-ledger source summary and cross-sector dictionary synthesis are not established here. The benchmark windows now contain an explicit gravity-side counting object, a genuine gauge-side comparator, and a theorem-level scaling test on one declared map, while full AdS/CFT duality, string/M equivalence, string/M containment, ultraviolet completion, and universal microstate closure remain outside the declared claim.

Funding and competing interests..

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