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

Black-hole thermodynamics, large-$N$TeXN gauge dynamics, and holographic duality relate gravitational and gauge descriptions only after highly specific dynamical structures are supplied [citation]. D-brane and strong-coupling constructions likewise depend on explicit actions, charges, decoupling limits, and observable maps [citation].

The object $\bc$TeX\bc below collects a compact manifold, selected parameters, a lifted map, a horizon family, a gauge family, and a scaling map. Because CVS does not derive a compact solution and MLC does not derive a microstate Hilbert space, the existence of these entries must remain an explicit hypothesis [citation].

The central question is logical sufficiency: assuming that these objects coexist does not establish that they arise from equivalent dynamics. We use counterexamples to identify the missing conditions.

The single-scalar gravitational root [citation], black-hole models [citation], confinement-facing model [citation], and large-$N$TeXN comparison [citation] enter only as conditional inputs. None contains an independently derived duality map for the objects used here.

center 1.16

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

center

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 \left[\partial_{\sigma_a}\partial_{\sigma_b}\VEV(\sigstar)\right]\succ0.

- 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 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 N^2}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).

With this common normalization, a leading-density comparison has the form

\Fgrav(\beta,\mu_i;\Fsel)=\Fg(N,\lambda,\beta;\Fsel)+\Ord(N^{-2}).

proposition: Condition for leading free-energy agreement. Assume expansions

I_E^{\mathrm{lift}}=N^2 i_0(\lambda,\beta)+i_1(\lambda,\beta)+r_{\mathrm{grav}}(N,\lambda,\beta),
\qquad
|r_{\mathrm{grav}}|\le C_{\rm g},

and

-\log Z_{\rm gauge}=N^2j_0(\lambda,\beta)+j_1(\lambda,\beta)+r_{\rm gauge}(N,\lambda,\beta),
\qquad |r_{\rm gauge}|\le C_{\rm q}.

Then

\Fgrav-\Fg=\frac{i_0-j_0}{\beta}
+\frac{i_1-j_1+r_{\rm grav}-r_{\rm gauge}}{\beta N^2}.

In particular, reference holds if and only if $i_0=j_0$TeXi_0=j_0 at leading order.

proof. Substitute both expansions into the normalized definitions of $\Fgrav$TeX\Fgrav and $\Fg$TeX\Fg and subtract. An injective parameter-scaling map does not imply $i_0=j_0$TeXi_0=j_0; that equality must be calculated from the two dynamics or imposed as an additional hypothesis.

theorem: Source-free free energy is insufficient for duality. Equality of partition functions at zero source does not imply equality of correlation functions or an equivalence of theories.

proof. For a real source $J$TeXJ, consider

Z_1[J]=Z_0\exp\!\left(\frac{J^2}{2}\right),
\qquad
Z_2[J]=Z_0\exp(J^2).

They satisfy $Z_1[0]=Z_2[0]=Z_0$TeXZ_1[0]=Z_2[0]=Z_0 and hence have the same source-free free energy. However,

\left.\frac{d^2\log Z_1}{dJ^2}\right|_{J=0}=1,
\qquad
\left.\frac{d^2\log Z_2}{dJ^2}\right|_{J=0}=2.

Their connected two-point functions differ, so the theories cannot be identified by the source-free free energy.

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).

theorem: Index stability does not imply area-entropy matching. The condition

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

does not imply reference, even when $\CHCPhase=0$TeX\CHCPhase=0 and the index is smooth.

proof. Choose a one-dimensional horizon Hilbert space for every $(Q_i,J)$TeX(Q_i,J), so that $\dhor=1$TeX\dhor=1 and $\Shor=0$TeX\Shor=0. Both derivatives in the stability condition vanish. Choose any family with $\Ahor/(4\GNd)$TeX\Ahor/(4\GNd) growing as $N^2$TeXN^2 and set $\CHCPhase=0$TeX\CHCPhase=0. Then $\Sbh=\Ahor/(4\GNd)$TeX\Sbh=\Ahor/(4\GNd) differs from $\Shor$TeX\Shor by order $N^2$TeXN^2, contradicting reference. Entropy matching therefore requires an independent asymptotic calculation of $\dhor$TeX\dhor, not derivative boundedness.

remark. An index may also exhibit cancellations between bosonic and fermionic states, so even a correctly computed index need not equal the total degeneracy. The relation between an index and thermodynamic entropy must be proved for the specified theory and charge sector.

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.

proposition: Single-point scaling non-identifiability. At a single comparison point with $N>0$TeXN>0, $\lambda\ne0$TeX\lambda\ne0, $\muGap>0$TeX\muGap>0, and positive gravitational quantities, the coefficients

\alpha_N=\frac{\Llift^{d-2}}{\GNd N^2},
\qquad
\alpha_\lambda=\frac{\lambda_{\rm coh}}{\lambda},
\qquad
\alpha_T=\frac{\Tstr}{\muGap^2}

make the left-hand side of reference vanish identically. Thus a single-point match cannot identify or test the scaling map.

proof. Substitution cancels each of the three absolute-value terms separately. Predictive content requires the coefficients to be fixed independently and tested at additional points.

definition: Restricted duality on a parameter domain. A restricted duality on a domain $\mathcal U$TeX\mathcal U requires:

- independently defined state spaces or operator algebras on both sides; - an invertible map between physical states or observable algebras, respecting constraints and gauge equivalence; - a source map $J\mapsto J'$TeXJ\mapsto J' such that $Z_{\rm grav}[J]=Z_{\rm gauge}[J']$TeXZ_{\rm grav}[J]=Z_{\rm gauge}[J'] on $\mathcal U$TeX\mathcal U; - matching symmetries, anomalies, conserved charges, and boundary conditions; - controlled finite-$N$TeXN, quantum, and truncation corrections.

theorem: Simultaneous scalar comparisons are insufficient. There exist pairs of model families for which the source-free free energies agree at leading order, the logarithmic state counts differ by only $\Ord(N^0)$TeX\Ord(N^0), and a prescribed parameter-scaling inequality such as reference holds, while no state-space bijection and no equality of generating functionals exists. Therefore simultaneous satisfaction of reference, reference, and reference is insufficient for reference.

proof. Use the generating functionals in reference; their zero-source free energies agree while their two-point functions differ. For state spaces take cardinalities $K_N$TeXK_N and $2K_N$TeX2K_N. Their entropies differ by $\log2=\Ord(N^0)$TeX\log2=\Ord(N^0), but no bijection exists at any finite $N$TeXN. Finally, define the scaling-map coefficients in reference from the displayed positive quantities, making its residual zero. All three scalar comparisons can therefore hold while conditions (ii) and (iii) of reference fail.

corollary: Status of the compact datum. The tuple $\bc$TeX\bc is a container for conditional comparison data. Its definition and the scalar residual bounds do not establish a restricted duality.

proof. The definition of $\bc$TeX\bc assumes the constituent objects, while reference supplies a counterexample to the sufficiency of their scalar comparisons.

remark. If a future construction verifies all conditions of reference, its claim remains restricted to the specified parameter and source domain. The present paper does not construct such a map.

The following are required before a duality claim can be evaluated:

- 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; - the source-dependent generating functionals and their operator dictionaries are not shown to agree.

The present construction does not establish any of these conditions from the CHC root action. In particular, it proves neither AdS/CFT recovery, a black-hole microstate model, nor ultraviolet completion.

A compact scalar target partitions the CHC configuration space into winding sectors and assigns an invariant canonical circumference. These data do not define a horizon index, a gauge-theory state space, or a duality map. Even on a compactified branch, equality of local equations or leading thermodynamic coefficients cannot identify the global spectra or source responses of two theories.

The compatibility-manifold theorem supplies a constructive threshold for future claims. A single $q$TeXq-parameter correspondence map must jointly determine more than $q$TeXq independent spectral, thermodynamic, and response observables, leaving transverse relations in the left null space of its sensitivity matrix. A dictionary with one adjustable comparator per observable has full nuisance rank and is not a correspondence. Thus compact-target structure is admitted as a new global theorem, while the branch-conditioned no-go tests for duality and microscopic horizon counting remain in force.

The closure test for the branch-conditioned duality and horizon indices 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 indices, entropies, spectra, and large-parameter observables. Let $a$TeXa range over the independent constitutive inputs comprising compact branch, horizon index, gauge comparator, and dictionary nuisance.

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 branch map simultaneously preserves index data, observable products, and controlled corrections.

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 observable-wise comparators can fit each quantity while failing to define a common duality. Failure of that condition therefore rejects the proposed microscopic closure before parameter estimation can be counted as evidence for it.

The compact datum is a conditional collection of objects, not a construction of those objects from CHC dynamics. The corrected free-energy proposition shows that leading agreement requires equality of independently calculated coefficients; injectivity of a scaling map cannot supply that equality. The horizon-index stability condition is insufficient for entropy matching, as demonstrated by a constant-index counterexample. Single-point scaling coefficients are non-identifiable because they can be solved directly from the compared quantities.

Even simultaneous agreement of source-free free energy, leading entropy, and large-$N$TeXN scales does not imply a duality. Distinct theories can share those scalar quantities while having unequal state counts and different correlation functions. A future restricted duality claim must construct both theories independently and prove an invertible observable map or equality of source-dependent generating functionals, with symmetries, anomalies, and corrections under control.

The following substitutions illustrate the conditional notation but do not solve field equations or construct a dual pair. 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^{-2}),

\Fgrav(T,\mu_i;\Fsel)
=-a_*T^p+\Ord(\epsdual N^{-2}),

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).

The comparison relations then follow by substitution because they have been imposed in reference--reference. This is a consistency check of the notation, not evidence that a corresponding compact solution or dual pair exists.

Funding and competing interests..

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