Branch-Conditioned Duality, Horizon Index Families, and Large-N Gauge Comparators on Compactified Branches

A black hole presents a macroscopic entropy law although the inaccessible states behind the horizon are not directly enumerated. Large-$N$TeXN gauge families exhibit the opposite simplification: connected correlators and free-energy densities organize into planar scaling laws as the rank grows. These two phenomena define some of the sharpest comparison problems for any proposed gauge--gravity language. Their standard comparison is framed through black-hole thermodynamics, large-$N$TeXN gauge dynamics, and holographic duality [citation]. The D-brane and strong-coupling literature further shows that lifted geometric descriptions may reorganize gauge data in appropriate regimes without thereby implying unrestricted equivalence [citation].

We restrict attention to one declared compactified branch datum $\bc$TeX\bc satisfying the selected-family compactification admissibility conditions of Ref. [citation]. The scientific question is whether strong-coupling lift, microscopic black-hole entropy accounting, and large-$N$TeXN gauge/gravity comparison admit explicit same-datum comparator objects on that datum: a lifted dual map, a branch-conditioned horizon-index family, and a genuine gauge-side large-$N$TeXN comparator with one fixed scaling map.

On one declared compactified branch datum, a lifted branch map, a branch-conditioned horizon-index family, and a genuine gauge-side large-$N$TeXN comparator with declared rank/scaling data are constructed explicitly. On that same datum, the free-energy, entropy, and large-$N$TeXN comparison problems admit branch-conditioned comparator closure with visible residual bounds. No universal AdS/CFT theorem, unrestricted string/M-equivalence claim, all-vacua statement, all-compactifications statement, or ultraviolet-completion statement is advanced.

The admitted single-scalar gravitational backbone [citation], the black-hole accessibility and return structures [citation], the confinement-facing strong-coupling family with an admitted string-tension witness [citation], the comparator classes fixed on declared windows [citation], and the gauge-side large-$N$TeXN comparator data on one fixed scaling map [citation] enter only through their stated branch, family, map, and window content. No universal equivalence principle or ultraviolet-completion statement is inferred from these inputs.

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We work throughout with one declared compactified branch datum $\bc$TeX\bc satisfying the selected-family compactification admissibility conditions of Ref. [citation]. Every object used below is stated explicitly on that datum, and no theorem is promoted beyond the selected branch, the admitted horizon family, the declared gauge-side rank family, or the fixed scaling map.

definition: Selected compactified branch datum. A selected compactified branch datum is a tuple

\bc=(M_4,\Kph,\sigstar,\Bdual,\Bent,\BN,\Mmap,\epsdual,\epsent,\epsN)

with the following properties:

- $M_4$TeXM_4 is a four-dimensional Lorentzian branch of the admitted backbone, and $\Kph$TeX\Kph is a compact internal response manifold carrying a discrete internal spectral problem. - $\sigstar$TeX\sigstar is one selected vacuum orbit satisfying the admissibility gates of the compactified branch. - $\Bdual$TeX\Bdual is one declared lifted-geometry window on the same branch. - $\Bent$TeX\Bent is one admitted horizon family carrying a branch-conditioned accessibility structure and one horizon-shell counting window. - $\BN$TeX\BN is one declared gauge-side rank family with planar factorization, free-energy scaling, and spectral-density growth on one fixed window. - $\Mmap$TeX\Mmap is one declared scaling map between lifted gravitational quantities and gauge-side rank/scaling data. - $\epsdual,\epsent,\epsN$TeX\epsdual,\epsent,\epsN are nonnegative residual controls for, respectively, the lifted free-energy relation, the entropy comparator, and the large-$N$TeXN comparator relation.

All exact or recovered statements below are read only on one fixed datum $\bc$TeX\bc.

definition: Compactified branch admissibility. The datum $\bc$TeX\bc is called admitted if the following conditions hold.

- The selected orbit $\sigstar$TeX\sigstar is a local minimum of a compactified branch functional $\VEV[\sigma]$TeX\VEV[\sigma], with

\partial_{\sigma_a}\VEV(\sigstar)=0, \qquad \partial_{\sigma_a}\partial_{\sigma_b}\VEV(\sigstar)>0.

- The compactified family $\Fsel$TeX\Fsel determined by $\sigstar$TeX\sigstar satisfies a branchwise admissibility gate $\Acomp(\Fsel)>0$TeX\Acomp(\Fsel)>0. - The gap scale $\muGap$TeX\muGap and lifted scale $\Llift$TeX\Llift are positive and finite on $\Fsel$TeX\Fsel. - The rank/scaling map $\Mmap$TeX\Mmap is injective on the declared benchmark window.

definition: Internal mode tower. On $\Fsel$TeX\Fsel, the compact internal response manifold $\Kph$TeX\Kph carries a discrete spectral problem

\mathcal D_{\Kph}\varphi_n(y)=\lambda_n\varphi_n(y),
\qquad y\in\Kph,

with ordered nonnegative spectrum $0\le \lambda_0\le \lambda_1\le \cdots$TeX0\le \lambda_0\le \lambda_1\le \cdots. The same selected branch determines a branch-conditioned rigidity ladder

m_n^2(\Fsel)=m_0^2(\Fsel)+\frac{\lambda_n}{R_{\mathrm{ph}}(\Fsel)^2},

where $R_{\mathrm{ph}}(\Fsel)$TeXR_{\mathrm{ph}}(\Fsel) is the compact response radius on the selected branch.

The lifted map is the first explicit comparator object on the declared compactified datum. The claim is not that every admitted branch possesses such a map, but that one declared compactified branch does.

definition: Lifted branch map. A lifted branch map on the datum $\bc$TeX\bc is a map

\Ldual:
(M_4,\Kph,\sigstar)
\longrightarrow
(\mathcal M_{\mathrm{lift}},G^{\mathrm{lift}}_{AB},\Phi_{\mathrm{lift}})

such that:

- the lifted geometry $(\mathcal M_{\mathrm{lift}},G^{\mathrm{lift}}_{AB})$TeX(\mathcal M_{\mathrm{lift}},G^{\mathrm{lift}}_{AB}) is smooth on the declared branch window; - the Euclidean on-shell action on the lifted branch defines a free-energy functional $\Fgrav$TeX\Fgrav; - the same lifted branch carries one horizon family from $\Bent$TeX\Bent and one confinement-facing string-tension window imported from the admitted strong-coupling family.

Define the Euclidean lifted partition function and free-energy density on the declared branch by

Z_{\mathrm{grav}}(\beta,\mu_i;\Fsel)=\exp\!\bigl[-I_E^{\mathrm{lift}}(\beta,\mu_i;\Fsel)\bigr],
\qquad
\Fgrav(\beta,\mu_i;\Fsel)=\frac{1}{\beta}I_E^{\mathrm{lift}}(\beta,\mu_i;\Fsel).

The scaling map is written as

\Mmap:
\left(
\frac{\Llift^{d-2}}{\GNd},\lambda_{\mathrm{coh}},\muGap
\right)
\longleftrightarrow
\left(N^2,\lambda,\muGap\right).

The lifted free-energy benchmark is recovered when the gravitational and gauge free energies agree up to one branchwise residual of order $N^0$TeXN^0:

\Fgrav(\beta,\mu_i;\Fsel)=\Fg(N,\lambda,\beta;\Fsel)+\Ord(\epsdual N^0).

proposition: Lifted free-energy comparator on one declared branch. Assume that the datum $\bc$TeX\bc is admitted and that the lifted Euclidean branch action admits an expansion

I_E^{\mathrm{lift}}=N^2 f_0(\lambda,\beta)+f_1(\lambda,\beta)+r_{\mathrm{grav}}(N,\lambda,\beta),
\qquad
|r_{\mathrm{grav}}|\le \epsdual,

while the gauge-side comparator family obeys the large-$N$TeXN expansion stated in Definition reference. Then reference holds on the declared branch window.

proof. By Definition reference, the scaling map $\Mmap$TeX\Mmap is injective on the declared window, so the leading rank dependence is fixed by the lifted branch data. Matching the $N^2$TeXN^2-coefficient and the order-$N^0$TeXN^0 branch defect yields reference. No statement is made outside the admitted branch and fixed comparator window.

The horizon index family is the second explicit comparator object: narrower than a universal microstate solution and stronger than a purely thermodynamic proxy.

definition: Horizon index family. Fix one admitted horizon family in $\Bent$TeX\Bent. The branch-conditioned horizon partition function is

\Zhor(\beta,\mu_i)
=
\mathrm{Tr}_{\mathcal H_{\mathrm{hor}}^{\mathrm{adm}}}
\exp\!\bigl[-\beta(\hat H-\mu_i\hat Q_i)\bigr],

and the corresponding branchwise index family is

\dhor(Q_i,J)=\mathrm{Index}\,\mathcal H_{\mathrm{hor}}^{\mathrm{adm}}(Q_i,J),
\qquad
\Shor(Q_i,J)=\log \dhor(Q_i,J).

The gravitational entropy comparator on the same branch is

\Sbh(Q_i,J;\Fsel)=\frac{\Ahor(Q_i,J;\Fsel)}{4\GNd}+\CHCPhase(Q_i,J;\Fsel).

The controlled entropy benchmark is recovered when the microscopic index family matches the branch-conditioned gravitational entropy up to one order-$N^0$TeXN^0 defect:

\Sbh(Q_i,J;\Fsel)=\Shor(Q_i,J)+\Ord(\epsent N^0).

proposition: Branch-conditioned entropy comparator. Assume that $\bc$TeX\bc is admitted and that the admitted horizon family satisfies one index-stability condition

\left|\partial_{Q_i}\log \dhor\right|+\left|\partial_J\log \dhor\right|<\infty

on the declared horizon window. If the selected branch supports the entropy decomposition reference, then the branch-conditioned entropy relation reference holds whenever the branchwise defect in the lifted action and the hidden-phase contribution are both bounded by $\epsent N^0$TeX\epsent N^0.

proof. The definition of $\Sbh$TeX\Sbh separates the Bekenstein--Hawking area contribution from the selected-branch phase correction $\CHCPhase$TeX\CHCPhase. The index-stability condition prevents the index family from changing discontinuously on the declared window. The bounded branchwise defect therefore yields reference. The result is branch-conditioned only and does not assert a universal microscopic entropy theorem.

remark. The horizon index family is used only as a declared benchmark object on one admitted horizon family. No universal construction for all horizon topologies, all asymptotics, or all black-hole families is claimed, and the accessibility-boundary and return-stack interpretation on admitted black-hole windows remains unchanged [citation].

The gauge-side large-$N$TeXN comparator is the third explicit comparator object. It must remain genuinely gauge-side rather than a gravitational quantity re-expressed twice.

definition: Gauge-side large-$N$TeXN comparator family. A declared gauge-side large-$N$TeXN comparator family on $\BN$TeX\BN consists of:

- a partition function equation (N,lambda,T)= DPhi e^-N^2S_gauge[Phi;lambda,T], equation with free-energy density equation (N,lambda,T)=-(1)/(beta N^2) (N,lambda,T); equation - planar connected-correlator suppression equation \ord_a_1 \ord_a_k\rangle_c = N^2-kC_a_1 a_k(lambda,T)+r_a_1 a_k(N,lambda,T), |r_a_1 a_k| N^1-k; equation - a Wilson-loop/string-tension witness on the same declared family, equation W(C) = - A_C+ ((+P_C/A_C)A_C), >0, equation where $A_C$TeXA_C and $P_C$TeXP_C are the minimal area and perimeter data of the loop on the declared window.

The large-$N$TeXN correspondence benchmark is recovered on one branch when the gauge-side scaling data and the lifted gravitational scaling data are related by the declared map $\Mmap$TeX\Mmap, with branchwise defect bounded by $\epsN$TeX\epsN:

\left|
\frac{\Llift^{d-2}}{\GNd}-\alpha_N N^2
\right|
+
|\lambda_{\mathrm{coh}}-\alpha_\lambda\lambda|
+
|\Tstr-\alpha_T\muGap^2|
\le \epsN.

proposition: Genuine gauge-side comparator on the declared family. If $\BN$TeX\BN satisfies Definition reference, then the tuple

\bigl(\Zg,\Fg,\{C_{a_1\dots a_k}\},\Tstr,\Mmap\bigr)

constitutes a genuine gauge-side large-$N$TeXN comparator on the declared branch window.

proof. Each object in the tuple is gauge-side by construction: the partition function and correlators are built from the gauge-family path integral, the free energy is derived from $\Zg$TeX\Zg, and the Wilson-loop tension $\Tstr$TeX\Tstr is defined on the same declared gauge family. The comparator is therefore not a gravity-side rewriting. The scaling map only matches branchwise scaling data and does not assert a universal identification.

With the lifted map, the horizon-index family, and the gauge-side comparator fixed on one common datum, the free-energy, entropy, and large-$N$TeXN relations are formulated below as branch-conditioned comparator-closure statements with explicit residual control.

definition: Branch-conditioned comparator closure. A comparison problem is said to satisfy branch-conditioned comparator closure on the datum $\bc$TeX\bc if and only if:

- the corresponding comparator object is explicit on $\bc$TeX\bc; - the relevant branch, scaling map, family, and admitted window are fixed and visible; - the comparison relation is supported by a theorem or proposition with one stated residual bound; and - the same relation is not promoted beyond the declared datum $\bc$TeX\bc.

theorem: Branch-conditioned comparator-closure theorem on one declared branch. Assume that the datum $\bc$TeX\bc is admitted and that Propositions reference, reference, and reference hold on the same selected compactified branch $\Fsel$TeX\Fsel, the same admitted horizon family, and the same declared scaling map $\Mmap$TeX\Mmap. Then the strong-coupling-lift, microscopic-entropy, and large-$N$TeXN comparison problems admit branch-conditioned comparator closure on the same datum $\bc$TeX\bc in the sense of Definition reference. Equivalently, the lifted free-energy relation reference, the horizon-index entropy relation reference, and the genuine gauge-side large-$N$TeXN comparator relation reference hold simultaneously on that declared datum.

proof. By Proposition reference, the lifted branch carries an explicit dual free-energy comparator on one declared branch and one declared scaling map, yielding the branch-conditioned free-energy relation reference. By Proposition reference, the same branch and admitted horizon family carry a branch-conditioned horizon-index family whose entropy matches the selected-branch gravitational entropy up to one declared defect, yielding reference. By Proposition reference, the same datum carries a genuine gauge-side large-$N$TeXN comparator together with one fixed scaling map and one branchwise scaling defect, yielding reference. Since none of these relations is promoted beyond the declared datum, the conditions of Definition reference are satisfied simultaneously.

corollary: Simultaneous same-datum comparison consequence. On the declared datum $\bc$TeX\bc, the strong-coupling-lift, microscopic-entropy, and large-$N$TeXN comparison problems admit simultaneous same-datum comparator closure through the relations established in Propositions reference, reference, and reference. Outside the declared datum, no stronger conclusion is implied.

remark. reference do not imply universal AdS/CFT, unrestricted string--M equivalence, all-vacua recovery, any branch-independent exact identity, or ultraviolet completion. On one declared compactified branch, one admitted horizon family, and one declared scaling map, the three comparison problems admit explicit same-datum comparator relations with visible residual bounds.

Reject the same-datum comparator claim on the declared datum if any of the following occurs:

- no admitted selected compactified branch datum $\bc$TeX\bc exists; - the lifted branch map $\Ldual$TeX\Ldual is absent or fails to define a controlled free-energy functional on the declared branch window; - the horizon family does not support an index-stable branch-conditioned counting object; - the gauge-side family lacks genuine planar/rank/scaling data or the Wilson-loop/string-tension witness on the declared family; - the scaling map $\Mmap$TeX\Mmap is not injective or requires changing branch, window, or comparator set between the three comparison problems; - recovery can be maintained only by silently promoting the result to universal equivalence, all-vacua closure, or ultraviolet completion.

No statement below is promoted to universal AdS/CFT recovery, unrestricted holography, string--M containment, all-black-hole microstate solutions, all-gauge-theory large-$N$TeXN equivalence, all-compactifications theorems, or ultraviolet completion. Every positive statement remains confined to one declared compactified branch, one admitted horizon family, one declared gauge-side rank family, and one fixed scaling map.

On one declared compactified branch datum, an explicit lifted dual map, a branch-conditioned horizon-index family, and a genuine gauge-side large-$N$TeXN comparator with one fixed scaling map have been constructed. Under the stated branch, horizon-family, gauge-window, and mismatch hypotheses, these objects are sufficient to establish branch-conditioned free-energy, entropy, and large-$N$TeXN comparison relations on that same datum.

All positive statements are branch-conditioned, map-conditioned, and window-conditioned throughout. No universal holographic equivalence is proved, no theorem for all compactifications or all gauge theories is asserted, and no ultraviolet-completion claim is made. Outside the declared compactified branch, admitted horizon family, declared gauge-side rank family, and fixed scaling map, no stronger statement is made.

To exhibit one concrete datum without promoting it to universality, consider an illustrative selected compactified branch satisfying the selected-family admissibility conditions of Ref. [citation]. Let the compact internal response manifold be

\Kph=S^1(R_{\theta})\times \Sigma_g(L_{\Sigma}),
\qquad
\sigstar\in \operatorname*{arg\,min} \VEV[\sigma],

with internal spectral problem

\lambda_{n,m}=\frac{n^2}{R_{\theta}^2}+\mu_m^2(\Sigma_g),
\qquad
m_{n,m}^2(\Fsel)=m_0^2(\Fsel)+\frac{\lambda_{n,m}}{R_{\mathrm{ph}}(\Fsel)^2},

where $\mu_m^2(\Sigma_g)$TeX\mu_m^2(\Sigma_g) are the Laplacian eigenvalues on the compact surface $\Sigma_g$TeX\Sigma_g. On the same branch, let the confinement-facing family carry one admitted tension window

\Tstr=(\sigma_C-\varepsilon_C)>0,

and let the selected branch admit one fixed scaling map of the form

\frac{\Llift^{d-2}}{\GNd}=\alpha_N N^2+\beta_N+\Ord(N^{-2}),
\qquad
\lambda_{\mathrm{coh}}=\alpha_\lambda\lambda+\Ord(N^{-2}),
\qquad
\muGap>0.

Assume further that the gauge-side free-energy density and the lifted free-energy density satisfy

\Fg(N,\lambda,T)
=-a_*T^p+\Ord(N^0),

\Fgrav(T,\mu_i;\Fsel)
=-a_*T^p+\Ord(\epsdual N^0),

with $a_*>0$TeXa_*>0, and that the horizon index family obeys

\dhor(Q_i,J)=\exp\!\left[\frac{\Ahor(Q_i,J;\Fsel)}{4\GNd}+\Ord(N^0)\right],
\qquad
\CHCPhase(Q_i,J;\Fsel)=\Ord(N^0).

Then the branchwise free-energy relation reference, the entropy relation reference, and the large-$N$TeXN scaling relation reference hold simultaneously on this branch with the stated residual control. The example is illustrative only and does not establish universality.

Funding and competing interests..

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