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

Black-hole thermodynamics supplies a macroscopic area entropy, while large-$N$TeXN gauge theories exhibit planar organization and factorization [citation]. A microscopic account must identify actual quantum states and a Hamiltonian or path integral; a duality must additionally relate observables on both sides.

The proposed construction assigns $b_*$TeXb_* labels to each of $Q$TeXQ nominal horizon cells and separately assumes a gauge theory with $N^2$TeXN^2 spectral growth. We distinguish what is proved by this combinatorics from what would require gravitational dynamics. We then test whether equality of the two leading growth coefficients has nontrivial content.

The result is primarily negative. The cell model proves an area-scaled count for an explicitly chosen product set, but accessibility does not select that set. Planar scaling is assumed rather than derived from CHC, and leading-coefficient equality does not determine a duality.

definition: Cell-counting family. 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 is assigned a finite alphabet $\mathcal B$TeX\mathcal B with $|\mathcal B|=\bstate\ge2$TeX|\mathcal B|=\bstate\ge2. - The proposed hidden-state set $\mathfrak H_Q$TeX\mathfrak H_Q has an integer cardinality $N_Q\ge1$TeXN_Q\ge1 whose logarithmic defect

\delta_{\mathrm H}:=\log N_Q-\QHcell\log\bstate

satisfies

\qquad |\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}.

The integer sequence $N_Q$TeXN_Q is part of the hypothesis; it is not determined by exterior accessibility.

definition: Assumed gauge-side large-$N$TeXN 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.

The two families play different roles. The gravity-side family postulates an inaccessible-state count, whereas the gauge-side family postulates planar asymptotics. Neither set of assumptions derives the other.

definition: Nominal configuration count. Let $\mathcal B=\{1,\ldots,\bstate\}$TeX\mathcal B=\{1,\ldots,\bstate\} and define the nominal cell-configuration set and the hypothesized hidden-state count by

\mathcal C_Q:=\mathcal B^{\QHcell},
\qquad
\Nmicro(\Bg):=N_Q=\bstate^{\QHcell}e^{\delta_{\mathrm H}},
\qquad \QHcell=\left\lfloor \frac{\AreaH}{\ellstar^2}\right\rfloor,

where $N_Q\in\mathbb N$TeXN_Q\in\mathbb N and $|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}$TeX|\delta_{\mathrm H}|\le \epsH\sqrt{\QHcell}. The associated logarithmic count is

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

definition: Physical microstate criterion. The integer $N_Q$TeXN_Q represents a physical black-hole microstate count only if:

- a Hilbert space or phase space is derived from a specified gravitational theory; - $N_Q$TeXN_Q counts distinct physical states satisfying common macroscopic charges and constraints; - gauge redundancies and constraints are quotiented rather than counted; - the state dynamics or partition function yields the proposed degeneracy.

theorem: Conditional area scaling of the cell count. Under reference,

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

In particular, the relative discrepancy from the Bekenstein--Hawking coefficient obeys

\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. By definition of the integer sequence and its defect,

\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 expansion follows. Dividing by $\AreaH/(4G)$TeX\AreaH/(4G) and applying the triangle inequality yields the bound for $\Delta_{\BH}$TeX\Delta_{\BH}. The proof uses the assumed growth law for $N_Q$TeXN_Q; it does not establish the physical microstate criterion.

theorem: Accessibility does not determine degeneracy. Fix any exterior macrostate $m$TeXm and any positive integer $K$TeXK. There exists a state space $X_K$TeXX_K and an accessibility map $\pi_K:X_K\to\{m\}$TeX\pi_K:X_K\to\{m\} whose hidden fiber has cardinality $K$TeXK. Therefore exterior indistinguishability alone places no nontrivial constraint on $N_Q$TeXN_Q.

proof. Take $X_K=\{1,\ldots,K\}$TeXX_K=\{1,\ldots,K\} and define $\pi_K(j)=m$TeX\pi_K(j)=m for every $j$TeXj. All states have the same exterior image, while $|\pi_K^{-1}(m)|=K$TeX|\pi_K^{-1}(m)|=K. Since $K$TeXK is arbitrary, neither $K=\bstate^Q$TeXK=\bstate^Q nor any other degeneracy follows from accessibility alone.

corollary: Status of the product-set construction. The exact identity $|\mathcal C_Q|=\bstate^Q$TeX|\mathcal C_Q|=\bstate^Q proves the combinatorial area law $\log|\mathcal C_Q|=Q\log\bstate$TeX\log|\mathcal C_Q|=Q\log\bstate. It becomes a black-hole entropy result only after a bijection between $\mathcal C_Q$TeX\mathcal C_Q and physical constrained states is independently derived.

proof. The product rule gives $|\mathcal B^Q|=|\mathcal B|^Q=\bstate^Q$TeX|\mathcal B^Q|=|\mathcal B|^Q=\bstate^Q. The final statement follows from reference.

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: Consequences of the assumed large-$N$TeXNN scalings. If the gauge-side family $\Bq$TeX\Bq satisfies the hypotheses of reference, then

\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}).

These conclusions are conditional restatements of the assumed gauge-side asymptotics; they are not derived from the CHC scalar action.

proof. Divide the free energy and logarithmic state count by $N^2$TeXN^2 and use the remainders in reference. The correlator statement is the first hypothesis with its remainder written in big-$O$TeXO notation.

remark. Planar scaling occurs in many gauge theories and does not identify a particular gravitational dual. A duality requires substantially more information than the three asymptotic coefficients collected in $\Cn$TeX\Cn.

definition: 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 comparison 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: Conditional entropy--planar scaling bound. 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: Conditional leading-coefficient identity. 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.

Thus the two assumed counting sequences have the same leading logarithmic density on the comparison 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.

theorem: Post hoc scaling degeneracy. At any fixed $(E,\lambda)$TeX(E,\lambda) with $s_0(E,\lambda)>0$TeXs_0(E,\lambda)>0 and any $\bstate>1$TeX\bstate>1, there is a unique positive number

\gamma_\star(E,\lambda)=\frac{s_0(E,\lambda)}{\log\bstate}

for which $\kappacpl(E,\lambda;\gamma_\star)=0$TeX\kappacpl(E,\lambda;\gamma_\star)=0. Hence single-point coefficient matching has no evidential force if $\gamma$TeX\gamma is chosen from the matched coefficient.

proof. The equation $\kappacpl=0$TeX\kappacpl=0 is linear in $\gamma$TeX\gamma and $\log\bstate>0$TeX\log\bstate>0. Solving it gives reference; positivity and uniqueness follow immediately.

corollary: Constraint from a fixed map on a window. Let $\bstate$TeX\bstate and $\gamma$TeX\gamma be constant on a connected energy--coupling domain $\Omega$TeX\Omega. Exact coefficient matching at every point of $\Omega$TeX\Omega holds if and only if

s_0(E,\lambda)=\gamma\log\bstate

is constant on $\Omega$TeX\Omega.

proof. If matching holds, the displayed equality makes $s_0$TeXs_0 equal to a constant. Conversely, if $s_0$TeXs_0 equals that constant, the mismatch vanishes throughout $\Omega$TeX\Omega.

theorem: Leading entropy does not determine a state correspondence. For every $s>0$TeXs>0 there exist two sequences of finite state sets $(X_N)$TeX(X_N) and $(Y_N)$TeX(Y_N) satisfying

\lim_{N\to\infty}\frac{\log|X_N|}{N^2}
=\lim_{N\to\infty}\frac{\log|Y_N|}{N^2}=s,

while $|X_N|\ne|Y_N|$TeX|X_N|\ne|Y_N| for every $N$TeXN. Therefore equality of leading entropy densities does not imply a bijection of states, equality of partition functions, or a duality.

proof. Take $|X_N|=\lceil e^{sN^2}\rceil$TeX|X_N|=\lceil e^{sN^2}\rceil and $|Y_N|=2\lceil e^{sN^2}\rceil$TeX|Y_N|=2\lceil e^{sN^2}\rceil. The logarithms differ by exactly $\log2$TeX\log2, which vanishes after division by $N^2$TeXN^2, while the finite cardinalities are unequal. Unequal finite cardinalities preclude a bijection. Since no dynamics or observable algebra has been specified, equality of partition functions or a duality cannot follow.

remark. The conditional identity in reference is an asymptotic comparison only. reference show why it cannot be promoted to holographic evidence without an independently fixed map and a correspondence of observables and dynamics.

The conditional formulas above become a physical microstate or duality result only if additional structures are constructed:

- a gravitational Hilbert space or phase space with constraints and gauge equivalences; - an integer state count derived from that space rather than postulated through its asymptotic growth; - a specified gauge theory whose planar and spectral coefficients are calculated rather than assumed; - an independently fixed relation between gravitational and gauge parameters; - an invertible map of observables or equality of generating functionals, including subleading corrections.

Without these ingredients, the cell count is a combinatorial model and the large-$N$TeXN comparison is an asymptotic analogy. It establishes neither AdS/CFT duality, string/M-theory equivalence, ultraviolet completion, nor a microscopic derivation of black-hole entropy.

Integer winding labels connected components of the compact scalar configuration space, but the existence of these labels does not produce the exponentially large, energy-resolved microscopic degeneracy required for black-hole entropy. The winding-energy bound constrains representatives within each sector; it is not a count of horizon microstates. Likewise, a shared leading area or $N^2$TeXN^2 coefficient remains non-identifying when subleading spectra and source responses are unconstrained.

A stronger comparator must use one frozen map to predict several independent quantities. When its parameter dimension is lower than the combined observable dimension, the left-null space of the sensitivity matrix supplies relations that distinct candidate theories can fail. Observable-wise coefficient matching raises the rank and destroys precisely these tests. Compact topology therefore enriches the field theory but leaves the microstate and large-$N$TeXN no-go conclusions intact.

The closure test for the microstate and planar comparator windows 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 entropy, degeneracies, planar coefficients, and finite-parameter corrections. Let $a$TeXa range over the independent constitutive inputs comprising state-counting measure, ensemble, large-parameter scaling, and observable dictionary.

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 microscopic ensemble and one shared dictionary fix both counting and comparator coefficients.

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 winding labels are superselection data and are not microscopic degeneracies without a derived state map. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The exact product-set construction has area-scaled logarithmic cardinality, but the assignment of those configurations to physical black-hole states is not derived. Accessibility alone allows an arbitrary hidden degeneracy. The large-$N$TeXN asymptotics are likewise hypotheses about an independently specified gauge theory, not consequences of the CHC root action.

Leading-coefficient matching is non-identifying: it can always be enforced at one point by choosing $\gamma$TeX\gamma, and a fixed $\gamma$TeX\gamma across a window imposes the strong condition that $s_0(E,\lambda)$TeXs_0(E,\lambda) be constant there. Even exact agreement of the leading entropy density does not furnish a state bijection or an observable dictionary. The strongest valid result is therefore a set of conditional asymptotic identities and no-go theorems specifying what a future microscopic construction must add.

Funding and competing interests..

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