Confinement-Facing Color-Flux Conditions and Hadronization Grammar on a Declared Strong-Coupling Family of the CHC Gauge Sheet

No isolated color carrier is ever seen as an asymptotic output; in benchmark QCD language this is familiarly phrased as the absence of isolated quarks. Hard colored activity reappears instead as jets that neutralize into hadrons. The physical image is familiar even outside specialist QCD language: one can pull color hard enough to make a jet, but never hard enough to extract a permanently free colored endpoint. Confinement and hadronization are the standard benchmark words attached to that fact. They appear in lattice strong-coupling arguments, in area-law and static-potential criteria, and in fragmentation pictures that turn energetic colored activity into hadronic output rather than free color [citation].

The admitted matter-coupled gauge sheet and the realized anomaly-admissible gauge--chiral branch establish a benchmark-facing gauge architecture and a consistency filter, but they do not yet close the color sector dynamically. The missing question is therefore not whether an admitted gauge sheet resembles QCD in a loose way. The question is whether the admitted sheet supports a declared confinement law that excludes isolated color externalization and forces color-neutral hadronization closure on a stated strong-coupling family.

The -native gain sought here is narrow and explicit. Confinement is read as branch-completion of color transport, not merely as a slogan that a potential becomes large. Hadronization is read as completion of color-flux bookkeeping into color-neutral exits, not as a black-box fragmentation fit. The analysis therefore remains theorem-driven. It does not attempt a full review of QCD, full lattice-QCD equivalence, or complete hadron spectroscopy. It asks only whether a declared strong-coupling family on the admitted gauge sheet supports a mathematically coherent color-flux closure law and a color-neutral terminal-state map.

The front phenomena are fixed throughout:

- no isolated color track despite hard scattering, and - jet neutralization / hadronization with increasing color-separation cost.

These two phenomena are treated as consequences of one declared closure grammar on an admitted family.

We work on a declared strong-coupling family

\Fconf=(\Gsheet,\Wsc,\Bcolor,\Hexit),

where $\Gsheet$TeX\Gsheet denotes the admitted matter-coupled gauge sheet, $\Wsc$TeX\Wsc the declared strong-coupling window, $\Bcolor$TeX\Bcolor the benchmark family of color-bearing sheet states, and $\Hexit$TeX\Hexit the family of admissible terminal exits. The imported gauge-side data are only the matter-coupled sheet, the realized anomaly-admissible closure, and the generator/representation content fixed on their declared domains [citation]. No further Standard-Model or ultraviolet structure is silently imported here.

definition: Declared strong-coupling family. The family reference is admitted if the following hold simultaneously:

- the gauge sheet $\Gsheet$TeX\Gsheet carries the declared color representation data and local transport law on the imported admitted domain; - the strong-coupling window $\Wsc$TeX\Wsc supports benchmark Wilson-loop and static-source observables; - the benchmark state family $\Bcolor$TeX\Bcolor contains the color-bearing excitations whose asymptotic fate is to be tested; and - the terminal exit family splits as equation = H_M H_B H_rem, equation with mesonic, baryonic, and neutral-remnant sectors only.

All claims below are restricted to this declared family.

definition: Open-color externalization predicate. For $b\in\Bcolor$TeXb\in\Bcolor and separation parameter $L$TeXL in the declared strong-coupling window, define

\Extcol(b;L)=1

if there exists an asymptotic branch carrying nonzero open color on $b$TeXb at separation $L$TeXL whose total branch energy remains bounded as $L\to\infty$TeXL\to\infty. Otherwise set $\Extcol(b;L)=0$TeX\Extcol(b;L)=0.

remark. The predicate $\Extcol$TeX\Extcol concerns only asymptotic branch output. It does not deny the existence of short-lived color-bearing substructure inside an interaction region on finite scales.

The closure object is a color-flux functional defined on spanning sheets of benchmark loops.

definition: Color-flux closure functional. For a loop $C$TeXC and a spanning sheet $\Sigma$TeX\Sigma with $\partial\Sigma=C$TeX\partial\Sigma=C, define

\PhiC[\Sigma,C]=\sigmaC\,\Area(\Sigma)+\muC\,\Per(C)+\delta_C[\Sigma,C],

with

\sigmaC>0,
\qquad
\muC\ge 0,
\qquad
|\delta_C[\Sigma,C]|\le \epsC\,\Area(\Sigma),
\qquad 0\le \epsC<\sigmaC.

The admitted Wilson-loop sector satisfies the area-law criterion if

-\log\bigl|\langle W(C)\rangle\bigr|=
\inf_{\Sigma:\,\partial\Sigma=C}\PhiC[\Sigma,C].

proposition: Wilson-Loop Area-Law Proposition. Let $C_{R,T}$TeXC_{R,T} be a rectangular loop of spatial size $R$TeXR and Euclidean time extent $T$TeXT in the declared strong-coupling window. If reference and reference hold on the admitted family, then the corresponding static separation energy

\VC(R):=\lim_{T\to\infty}-\frac{1}{T}\log\bigl|\langle W(C_{R,T})\rangle\bigr|

obeys

\VC(R)\ge (\sigmaC-\epsC)R+2\muC.

In particular, the admitted family carries a positive-tension linear lower envelope for the static separation energy.

proof. For a rectangular loop, $\Area(C_{R,T})=RT$TeX\Area(C_{R,T})=RT and $\Per(C_{R,T})=2(R+T)$TeX\Per(C_{R,T})=2(R+T). By reference and reference,

-\log\bigl|\langle W(C_{R,T})\rangle\bigr|\ge (\sigmaC-\epsC)RT+2\muC(R+T).

Dividing by $T$TeXT and taking the $T\to\infty$TeXT\to\infty limit yields reference.

corollary: Positive-Tension Lower-Envelope Consequence. On the same declared family,

\liminf_{R\to\infty}\frac{\VC(R)}{R}\ge \sigmaC-\epsC>0.

In particular, the static separation energy is unbounded as $R\to\infty$TeXR\to\infty on the admitted strong-coupling family.

proof. Immediate from reference.

A positive-tension flux grammar is enough to exclude stable isolated color externalization on the declared family.

theorem: Color-Flux Closure Theorem. Assume that the declared strong-coupling family reference satisfies the Wilson-loop area-law criterion of reference, that every finite-energy asymptotic branch on the family is represented in the terminal exit family $\Hexit$TeX\Hexit, and that open-color externalization would require a branch with bounded asymptotic separation energy as $L\to\infty$TeXL\to\infty. Then

\Extcol(b;L)=0

for every admissible $b\in\Bcolor$TeXb\in\Bcolor in the asymptotic separation limit. Consequently, only color-neutral terminal exits survive on that family.

proof. Suppose, for contradiction, that some admissible $b\in\Bcolor$TeXb\in\Bcolor supports an asymptotic branch with nonzero open color and bounded separation energy as $L\to\infty$TeXL\to\infty. By reference,

\VC(L)\ge (\sigmaC-\epsC)L+2\muC,

with $\sigmaC-\epsC>0$TeX\sigmaC-\epsC>0. Hence $\VC(L)\to\infty$TeX\VC(L)\to\infty as $L\to\infty$TeXL\to\infty, contradicting the assumed bounded-energy asymptotic branch. Therefore no stable asymptotic open-color branch exists. Since every finite-energy asymptotic branch is represented in $\Hexit$TeX\Hexit, the surviving terminal exits are necessarily color neutral.

corollary: Color Externalization No-Go. On the admitted strong-coupling family, no stable isolated color carrier appears as an asymptotic output despite hard scattering.

proof. Immediate from reference.

remark. The theorem is exact only on the declared strong-coupling family. It is not a universal confinement statement outside that family, and it does not by itself provide full QCD closure.

Confinement closure fixes what cannot survive asymptotically. The next task is to organize which neutral exits do survive.

definition: Hadronization branch kernels. Let

\EC(L):=\VC(L)-\VC(L_0)

denote the excess flux energy above a fixed microscopic completion scale $L_0$TeXL_0 on the declared family. For nonnegative kernels $\chi_M$TeX\chi_M, $\chi_B$TeX\chi_B, and $\chi_{\mathrm{rem}}$TeX\chi_{\mathrm{rem}} with support on $[0,\infty)$TeX[0,\infty), define

\GM(L):=\sum_f A_f\,\chi_M\!\left(\EC(L)-2m^{\mathrm{eff}}_f\right),

\GB(L):=\sum_{f_1,f_2,f_3} B_{f_1f_2f_3}\,\chi_B\!\left(\EC(L)-m^{\mathrm{eff}}_{f_1}-m^{\mathrm{eff}}_{f_2}-m^{\mathrm{eff}}_{f_3}\right),

\GRm(L):=R_0\,\chi_{\mathrm{rem}}\!\left(\EC(L)-E^{\mathrm{eff}}_{\mathrm{rem}}\right),

where the coefficients $A_f$TeXA_f, $B_{f_1f_2f_3}$TeXB_{f_1f_2f_3}, and $R_0$TeXR_0 are nonnegative on the declared family. The normalized branch fractions are

\PM=\frac{\GM}{\GT},\qquad
\PB=\frac{\GB}{\GT},\qquad
\PRm=\frac{\GRm}{\GT},\qquad
\GT:=\GM+\GB+\GRm.

proposition: Hadronization Branch Normalization Proposition. On the declared admitted strong-coupling family,

\PM+\PB+\PRm=1.

Moreover, every branch selected by reference terminates in the color-neutral family

\Hexit=\mathfrak H_{M}\sqcup\mathfrak H_{B}\sqcup\mathfrak H_{\mathrm{rem}}.

proof. Equation reference follows directly from the definition of $\GT$TeX\GT. By construction, $\GM$TeX\GM counts mesonic pair-completion channels, $\GB$TeX\GB baryonic completion channels, and $\GRm$TeX\GRm neutral-remnant channels. Each channel in the declared family is color neutral at asymptotic exit, so every terminal branch selected by reference lies in $\Hexit$TeX\Hexit.

corollary: Color-neutral exit theorem. Every admissible asymptotic exit on the declared strong-coupling family carries no open color.

proof. Combine reference with reference.

remark. The hadronization branch normalization is intentionally weaker than a full hadron-spectrum model. It fixes terminal channel classes and their normalization, not complete resonance content or differential fragmentation shapes.

The declared family allows one to separate three levels of statement: the exclusion of asymptotic open color, the first opening of neutral terminal exits, and the observable weighting of those exits once the neutralization threshold has been crossed.

definition: First neutralization threshold. For the kernels in reference, write $\GM(E)$TeX\GM(E), $\GB(E)$TeX\GB(E), and $\GRm(E)$TeX\GRm(E) for the same expressions viewed as functions of an excess-flux energy argument $E\ge 0$TeXE\ge 0. Define

E_{\mathrm{th}}:=\inf\{E\ge 0:\ \GM(E)+\GB(E)+\GRm(E)>0\}.

If the excess-flux energy $\EC(L)$TeX\EC(L) is evaluated on the declared family, define the corresponding first neutralization scale by

R_{\mathrm{br}}:=\inf\{L\ge L_0:\ \GT(L)>0\}.

All threshold statements in what follows are restricted to the declared strong-coupling family.

proposition: String-Breaking Scale Proposition. Assume that $\EC(L)$TeX\EC(L) is continuous and nondecreasing for $L\ge L_0$TeXL\ge L_0 on the declared strong-coupling family, with $\EC(L_0)=0$TeX\EC(L_0)=0. Then:

- if $\EC(L)<E_{\mathrm{th}}$TeX\EC(L)<E_{\mathrm{th}}, then no terminal neutralization channel is open and $\GT(L)=0$TeX\GT(L)=0; - if $E_{\mathrm{th}}<\infty$TeXE_{\mathrm{th}}<\infty and $\sup_{L\ge L_0}\EC(L)>E_{\mathrm{th}}$TeX\sup_{L\ge L_0}\EC(L)>E_{\mathrm{th}}, then there exists a first neutralization scale $R_{\mathrm{br}}$TeXR_{\mathrm{br}} such that equation (R_br)=E_th equation and $\GT(L)>0$TeX\GT(L)>0 for $L>R_{\mathrm{br}}$TeXL>R_{\mathrm{br}} in a right neighborhood.

proof. By definition of $E_{\mathrm{th}}$TeXE_{\mathrm{th}}, the inequality $\EC(L)<E_{\mathrm{th}}$TeX\EC(L)<E_{\mathrm{th}} forbids any open terminal channel, hence $\GT(L)=0$TeX\GT(L)=0. If $E_{\mathrm{th}}<\infty$TeXE_{\mathrm{th}}<\infty and $\EC$TeX\EC eventually exceeds $E_{\mathrm{th}}$TeXE_{\mathrm{th}}, continuity and monotonicity give a first $R_{\mathrm{br}}$TeXR_{\mathrm{br}} at which $\EC$TeX\EC reaches the threshold value. The positivity of at least one channel kernel above threshold then yields $\GT(L)>0$TeX\GT(L)>0 to the right of $R_{\mathrm{br}}$TeXR_{\mathrm{br}}.

proposition: Terminal Exit-Ratio Identity. On any declared strong-coupling window for which $\GT(L)>0$TeX\GT(L)>0,

\PM(L):\PB(L):\PRm(L)=\GM(L):\GB(L):\GRm(L),

and, whenever the denominators are nonzero,

\frac{\PM(L)}{\PB(L)}=\frac{\GM(L)}{\GB(L)},
\qquad
\frac{\PM(L)}{\PRm(L)}=\frac{\GM(L)}{\GRm(L)},
\qquad
\frac{\PB(L)}{\PRm(L)}=\frac{\GB(L)}{\GRm(L)}.

Thus the relative mesonic, baryonic, and neutral-remnant weights are fixed by the same declared branch kernels that define the hadronization law.

proof. Each identity follows directly from reference after dividing by $\GT(L)>0$TeX\GT(L)>0.

proposition: Branch-Average Observable Window Proposition. Let $J$TeXJ be any declared asymptotic terminal observable determined only by fixed channel values $J_M$TeXJ_M, $J_B$TeXJ_B, and $J_{\mathrm{rem}}$TeXJ_{\mathrm{rem}} on $\mathfrak H_M$TeX\mathfrak H_M, $\mathfrak H_B$TeX\mathfrak H_B, and $\mathfrak H_{\mathrm{rem}}$TeX\mathfrak H_{\mathrm{rem}}, respectively. On any declared strong-coupling window with $\GT(L)>0$TeX\GT(L)>0,

\langle J\rangle_L=\PM(L)J_M+\PB(L)J_B+\PRm(L)J_{\mathrm{rem}}.

In particular, the meson fraction, baryon fraction, neutral-remnant fraction, and any fixed linear jet-to-hadron summary observable are exact branch averages on the declared family.

proof. The channel law reference defines a normalized probability measure on the terminal exit family. Equation reference is the corresponding weighted average for any observable constant on each terminal sector.

remark. The threshold law reference, the exit-ratio identity reference, and the observable window reference belong to the declared strong-coupling family only. They do not claim full event-generator closure, complete hadron spectroscopy, or all-sector QCD phenomenology.

Jet neutralization window..

When hard scattering injects large color-bearing activity, reference excludes asymptotic open-color tracks. The visible consequence is a jet that resolves into color-neutral hadronic output. In this window, the branch law is a closure statement for the outgoing hadronized spray rather than a full Monte Carlo shower model [citation]. Under the declared continuity and monotonicity hypothesis, the threshold at which this neutralized output first opens is given by reference, and the corresponding asymptotic summary observables are branch averages in the sense of reference.

Mesonic window..

If

\GM\gg \GB,\,\GRm,

then the admitted family is meson-dominated and pair completion provides the leading neutral-output channel. In that regime the exact exit-ratio identity reference yields mesonic dominance directly at the level of branch weights. This is the natural strong-coupling image of flux-tube breaking into color-neutral mesonic pieces.

Baryonic window..

If

\GB\gg \GM,\,\GRm,

then the admitted family is baryon-dominated and terminal exits are predominantly color-neutral baryonic composites. The same ratio identity fixes the baryonic branch share relative to the mesonic and neutral-remnant channels on that window. No exact baryon spectrum is claimed.

Neutral-remnant window..

If

\PRm\not\ll 1,

then the admitted family contains neutral-remnant channels, such as soft color-singlet clusters or glue-rich neutral exits. Their exact weight is again part of the same exit-ratio identity and not an independent post-hoc supplement. These channels are allowed because color-neutrality is a branch condition rather than an after-the-fact naming convention.

Benchmark confinement scale..

The lower bound

\VC(R)\ge (\sigmaC-\epsC)R+2\muC

provides the benchmark confinement scale on the declared family. The quantity $\sigmaC-\epsC$TeX\sigmaC-\epsC is therefore the admitted string-tension parameter for the closure window and not a universal QCD claim outside that family. Together with the declared threshold $E_{\mathrm{th}}$TeXE_{\mathrm{th}}, it yields the first neutralization scale through reference under the declared continuity and monotonicity hypothesis for the excess-flux energy.

Reject the confinement claim on the declared family if any of the following occurs:

- no mathematically coherent color-flux functional of the form reference exists on the declared family; - no area-law or equivalent positive-tension lower-envelope criterion can be shown with positive effective tension $\sigmaC-\epsC$TeX\sigmaC-\epsC; - an asymptotic open-color branch survives on the same family; - hadronization requires terminal exits not carried by $\Hexit$TeX\Hexit; - color-neutrality appears only after ad hoc phenomenological patching rather than through the declared branch law; or - the result depends on silently importing full QCD structures absent from the admitted gauge sheet.

The declared closure law does not provide full lattice-QCD equivalence, complete hadron spectroscopy, exact chiral-symmetry-breaking closure, finite-temperature or finite-density QCD, heavy-ion phenomenology, full collider event-generator phenomenology, or universal confinement outside the declared family.

No isolated color carrier is observed as an asymptotic output, and energetic colored activity returns instead as color-neutral hadronic branches. On the declared admitted strong-coupling family of the matter-coupled gauge sheet, that visible fact is encoded by a single closure grammar. The Wilson-loop sector supplies a positive-tension area-law criterion, the associated static separation energy carries a positive-tension linear lower envelope and is therefore unbounded as the color separation increases on the admitted family, and the resulting Color-Flux Closure Theorem excludes stable isolated color externalization on that family. The same closure grammar then resolves terminal exits through a normalized hadronization branch law into mesonic, baryonic, and neutral-remnant channels, all of them color neutral.

The result is not a replacement for full QCD. It is a declared confinement and hadronization closure law on an admitted family. Its -native gain is to read confinement as branch-completion of color transport and hadronization as completion of color-flux bookkeeping into neutral exits. On that same family, the first neutralization threshold is obtained under the declared continuity and monotonicity hypothesis of the String-Breaking Scale Proposition, the mesonic/baryonic/neutral-remnant shares obey an exact terminal exit-ratio identity, and declared jet-to-hadron observables are branch averages rather than external supplements. The claim fails exactly where the declared color-flux functional, area-law criterion, or color-neutral exit family fail. Within the admitted strong-coupling family, however, the bridge from the declared gauge-sheet architecture to color-neutral terminal-output bookkeeping is mathematically explicit.

Funding and competing interests..

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