Dynamics — root dossier (source of truth v1.7, current 2026-07-12)

Honest status header. This root dossier states the parent action, the boundary laws, and the reduction map for the Dynamics root, together with the later Dynamics refinements. The scoped constructions it records are:

the corrected SG-5 electroweak-breaking Higgs Actor — a minimal \(SU(2)\)-equivariant \(\mathcal O(1)\) scalar bundle with primitive hypercharge \(q_H=3\), giving \(SU(2)_L\times U(1)_Y\to U(1)_{\rm em}\), one massless photon, and the tree relation \(\rho=1\) at \(d\le4\); the legacy Wilson-line/Hosotani winding Higgs and its winding-number protection are RETIRED;
the SG-6 Wilsonian zero-mode stabilization — a CONSTRUCTION-ANCHOR positive Wilsonian potential; the target radii and scalar masses are matching parameters, not derived numbers; the full-13D compactification branch is CLOSED-NEGATIVE / NOT-STABLE-AS-WRITTEN;
the Gap-05 vacuum-energy SEQUESTERING mechanism — a construction-anchor that cancels field-independent offsets; no value of \(\Lambda_{\rm obs}\) is generated;
the Gap-08 \(R^2\)-scalaron primordial mechanism — a construction-anchor in which the \(R^2\) coefficient is calibrated (the parent coefficient \(\gamma\) is a matching parameter; the scalaron mass is calibrated by \(A_s\)).

Key OPEN items: a UV-derived 13D potential; full compactification / KK stability; a unique reheating; and the full UV completion.

The per-gate dossiers at /gates/ remain the ultimate source of truth; this root dossier is reconciled to the corrected 2026-07-12 state.

0. How to read this dossier

Dynamics is the fourth of four roots. The framework’s canonical object is an ordered quadruple — Shape, Scale, Granularity, and Dynamics — and no physical claim is closed until it is expressed in all four. This file is the Dynamics root: it specifies the parent action, the field domains, the orbifold boundary contract, and the reduction map that carries thirteen-dimensional physics down to a four-dimensional effective theory.

Three reading disciplines are enforced throughout, and every reviewer should hold this dossier to them:

Every load-bearing field is an Actor in the action or the boundary data. A structure is not physical merely because its bundle name has been written down; it must occur in \(\mathfrak D\) and its observable must be produced by the reduction map.
Every dimensionful quantity carries exactly one honesty label. A number is either measured calibration, a matching parameter, a dynamically generated quantity, a derived ratio, a held-out prediction, or a consistency check. This dossier never lets a matching parameter masquerade as a derivation.
A construction anchor stays a construction anchor. Several of the mechanisms below (the Higgs Actor, the stabilization potential, the sequester, the primordial scalaron) are declared structures whose consistency is certified but whose existence the Stage does not force. They are labelled as such and never quietly promoted.

The document has three movements. Part IV (sections 12–17) is the locked Dynamics skeleton itself: the typed definition, the parent-action, the geometric connection ansatz, the hypercharge boundary contract, the reduction status, and the vacuum condition. The later Dynamics refinements (SG-5, SG-6, Gap-05, Gap-08) fold in the addenda that post-date the skeleton, each scoped and each labelled. A consolidated status ledger closes the file. Where the deep source and the corrected per-gate dossiers ever disagreed, the corrected 2026-07-12 version governs, and those corrections are itemized at the end.

The four roots, in one paragraph each

The canonical object is

\[ \boxed{\mathfrak T=(\mathfrak S,\mathfrak L,\mathfrak G,\mathfrak D)}, \]

whose four entries answer different questions. Shape \(\mathfrak S\) asks what structures exist and how they are related — the scale-free typed architecture, including the thirteen-dimensional Stage \(X_{13}=\mathcal M_{3,1}\times K_6\times S^2\times I_\chi\) with \(K_6=SU(3)/T^2\) and \(I_\chi=S^1_\chi/\mathbb Z_2\). Scale \(\mathfrak L\) asks which dimensionless ratios, moduli, flows, and calibrations apply; the internal radii \(R_6,R_2,R_\chi\) and the two shape coordinates are moduli, and \(M_{\rm Pl}\) is a measured calibration. Granularity \(\mathfrak G\) asks which physical records are distinguishable at finite resources; it supplies finite audit windows and tolerances but supplies no universal spacetime cell and no derivation of \(\hbar\). Dynamics \(\mathfrak D\) — this file — asks what action, boundary laws, evolution, and reduction map govern them.

Part IV — Dynamics

12. Typed definition

Dynamics is the ordered quadruple

\[ \boxed{ \mathfrak D=(S_{13},\mathcal H,\mathrm{BC},\mathcal R_4) } \]

with:

\(S_{13}\): the parent action;
\(\mathcal H\): the field/operator domains and the state space;
\(\mathrm{BC}\): the orbifold and boundary conditions;
\(\mathcal R_4\): the reduction / effective-field-theory map.

A physical claim is not closed unless every load-bearing Actor occurs in the action or the boundary data and the relevant observable is produced by the reduction map. This is the operational content of “Dynamics governs.” It is what separates a genuine mechanism from a structure that has merely been named. The four items above are not decorative; each is a place where a claimed derivation can be audited. If a claim uses a field that never appears in \(S_{13}\) or in \(\mathrm{BC}\), or uses a boundary condition that was never declared, or requires an observable that \(\mathcal R_4\) does not actually produce, the claim is not closed — regardless of how elegant its algebra looks.

13. Locked parent-action skeleton

The minimum locked action is

\[ \boxed{ \begin{aligned} S_{13}={}&\int_{X_{13}}d^{13}x\sqrt{|G|}\Bigg[ \frac{M_*^{11}}{2}\bigl(R[G]-2\Lambda_{13}\bigr) -\frac{1}{4g_{13,Y}^{2}}F^Y_{MN}F_Y^{MN}\\ &\qquad\qquad +\bar\Psi\,i\Gamma^M\bigl(\nabla_M-iq_YB_M\bigr)\Psi +\mathcal L_{\rm declared\ actors} \Bigg] +S_{\rm fixed\ sets}+S_{\rm required\ ct}. \end{aligned} } \]

The first term is thirteen-dimensional Einstein–Hilbert gravity with a bulk cosmological constant \(\Lambda_{13}\) and the higher-dimensional Planck mass \(M_*\). The second is the field strength of the one independent gauge connection in the theory. The third is the minimally coupled fermionic Actor. The remaining pieces — localized matter, boundary terms, and the required counterterms — are placeholders that must be enumerated before any full-theory claim; they are written schematically here precisely so that no downstream gate can quietly assume content that was never declared.

Important consequences follow directly from the structure of this action:

\(SU(3)_c\) and \(SU(2)_L\) vectors arise from the metric sector, as geometric Kaluza–Klein connections of the internal Killing algebras; they are not duplicated in \(\mathcal L_{\rm declared\ actors}\). This non-duplication is essential: an independent higher-dimensional \(SU(3)\) or \(SU(2)\) Yang–Mills connection would produce a second massless color or weak vector, which is forbidden.
\(B_M\) is the one independent principal-bundle connection, carrying \(U(1)_Y\) hypercharge. Color and weak isospin come from geometry; hypercharge does not.
Every localized term and every counterterm must be enumerated; the placeholders \(\mathcal L_{\rm declared\ actors}\), \(S_{\rm fixed\ sets}\), and \(S_{\rm required\ ct}\) are not yet a complete action.
Higgs, flavor, proton-safety, and anomaly terms are gate-owned additions. They may not be assumed merely because their bundle names appear elsewhere. Each is added — and audited — by the gate that owns it.

The gauge-origin table is worth stating explicitly because it is the load-bearing architectural choice of the whole programme:

4D algebra	Locked origin	Carrier / Actor
\(\mathfrak{su}(3)_c\)	geometric KK / Ehresmann connection	Killing algebra of \(K_6=SU(3)/T^2\)
\(\mathfrak{su}(2)_L\)	geometric KK / Ehresmann connection	Killing algebra of the round \(S^2\)
\(\mathfrak u(1)_Y\)	independent principal connection	\(U(1)_Y\) bundle \(P_Y\to X_{13}\)

14. Geometric connection ansatz

The precise sense in which an isometry generates a gauge connection is a metric expansion, not a naming convention. For Killing vectors \(K_A^m\) on \(K_6\times S^2\), the metric is expanded as

\[ \boxed{ ds^2=g_{\mu\nu}dx^\mu dx^\nu +g_{mn}(y) \bigl(dy^m+K_A^mA_\mu^A dx^\mu\bigr) \bigl(dy^n+K_B^nA_\nu^B dx^\nu\bigr) +R_\chi^2d\chi^2+\cdots . } \]

Under an \(x\)-dependent internal isometry, the fields \(A_\mu^A\) transform as gauge connections whose structure constants are exactly the structure constants of the Killing algebra. This is the operational content of “an isometry generates a gauge connection”: the off-diagonal metric modes along the exact internal Killing fields become four-dimensional connections after reduction. Merely writing down an isometry group is not enough — the metric ansatz and the reduction map are what actually produce the gauge field. A centralizer computation, or a symmetry statement without the reduction, does not substitute for this.

The four-dimensional gauge kinetic matrix that this ansatz produces has the normalization-explicit form

\[ (g^{-2}_{\rm geom})_{AB} =M_*^{11}\int_{Y_9}d^9y\sqrt{g_9}\, g_{mn}K_A^mK_B^n, \]

with the normalization of the Killing basis fixed as part of the boundary/parity data. For the independent hypercharge connection, the analogous relation is \(g_1^{-2}=g_{13,Y}^{-2}\,\operatorname{Vol}(Y_9)\) before threshold and wave-function corrections. These are matching relations, not numerical predictions, until the moduli, the Killing normalization, the matter spectrum, and the threshold calculation are all frozen.

15. Hypercharge boundary contract

On the chirality interval \(I_\chi=S^1_\chi/\mathbb Z_2\), hypercharge uses gauge-field parity, not an interval isometry — the interval has no connected \(U(1)\) rotation. The contract is

\[ B_\mu(x,-\chi)=+B_\mu(x,\chi), \qquad B_\chi(x,-\chi)=-B_\chi(x,\chi), \]

with an even gauge parameter. Therefore \(B_\mu\) has exactly one constant four-dimensional zero mode — the physical hypercharge photon component — while \(B_\chi\) has no even scalar zero mode in the minimal sector. This is what supplies a single low-energy hypercharge gauge field without an extra scalar.

Fermion parities are separate, representation-valued matrices \(P_0,P_\pi\) assigned at the two fixed points, satisfying

\[ P_0^2=P_\pi^2=1. \]

Even interval components carry a constant four-dimensional zero mode; odd components do not. This is the mechanism by which the interval routes chirality: it selects one four-dimensional Weyl zero mode and removes its vectorlike mirror. The detailed zero-mode table — which representation is even at which fixed point — belongs to the chirality and representation gates, not to this root file; the root file only fixes the form of the contract.

16. Reduction status

The locked reduction map keeps the full Kaluza–Klein theory conceptually and defines a low-energy effective theory by integrating out the massive modes:

\[ \mathcal R_4:(G,B,\Psi,\ldots)_{13D} \longrightarrow (g_{\mu\nu},A_\mu^{A(0)},B_\mu^{(0)},\psi^{(0)},\ldots)_{4D} +\text{higher-dimensional operators}. \]

An exact nonlinear truncation to a finite set of massless fields is not assumed. This is a deliberate and consequential restraint. Linear reduction certifies the massless gauge algebra and its low-energy gauge symmetry; it does not certify that every uplift of the finite four-dimensional theory solves the full thirteen-dimensional equations of motion. If a later claim requires such an uplift, it must supply a separate consistent-truncation theorem — a much stronger statement than linear reduction. The gate that certifies the massless gauge algebra (SG-2) certifies only that; it does not silently import a consistent-truncation result it has not proven.

17. Vacuum condition

A proposed vacuum is physical only if both a stationarity and a positivity condition hold on the physical modulus subspace, after the gauge and constraint directions are removed:

\[ \frac{\partial V_{\rm eff}}{\partial m_i}=0, \qquad \operatorname{Hess}(V_{\rm eff})\succeq0. \]

A negative physical mode is a live falsification test: it is a genuine adverse result about the vacuum, and it cannot be reclassified as a Granularity artifact, a bookkeeping convention, or a non-gating residual. This condition is the yardstick against which the SG-6 stabilization results below are measured, and it is exactly why the full-13D compactification branch is reported honestly as closed-negative rather than dissolved: a real negative eigenvalue of the physical Hessian is a real instability, and no taxonomy rule may convert it into a resolved positive vacuum.

Later Dynamics refinements

The four subsections that follow fold in Dynamics addenda that post-date the v1.7 Part IV skeleton. Each is a scoped construction-anchor and does not upgrade any terminal in Part IV. Provenance is stated inline, and each subsection ends with an explicit list of what is not claimed. Where any of these refinements once carried a stronger legacy grade, the corrected 2026-07-12 status governs and the retired claim is named.

SG-5 electroweak-breaking Higgs Actor scoped realized

Source: v1.7 root authority, Part IX (v1.3 SG-5 addition), retained through v1.7, reconciled to the SG-5 governing correction of 2026-07-12. Corrected status: CLOSED-SCOPED / EWSB-REALIZED-GIVEN-MINIMAL-EQUIVARIANT-HIGGS-DOUBLET-AND-NEUTRAL-VEV / RENORMALIZABLE-ZERO-MODE-MASS-MATRIX-CERTIFIED / RHO-TREE-DIMENSION4-EQUALS-ONE / V-EW-MEASURED-CALIBRATION / CONSTRUCTION-ANCHOR.

Why the legacy gauge-Higgs picture was retired

The legacy SG-5 treated the Standard-Model Higgs as a Wilson-line / Hosotani mode — an internal component of a higher-dimensional gauge connection. Such a scalar is adjoint-valued in the parent gauge algebra before symmetry breaking. But the locked electroweak gauge algebra is only

\[ \mathfrak{su}(2)_L\oplus\mathfrak u(1)_Y, \]

whose adjoint content is

\[ (\mathbf3,0)\oplus(\mathbf1,0), \]

and contains no \((\mathbf2,+\tfrac12)\) scalar. A genuine gauge-Higgs doublet would require a larger parent electroweak group and an additional boundary-breaking contract, neither of which exists in the locked action. This representation mismatch is decisive. The legacy Wilson-line / Hosotani doublet, its winding number \(n_H=1\), and its claimed topological mass protection are therefore RETIRED. The interval \(I_\chi\) is likewise not a hypercharge circle and supplies no hypercharge Wilson line: hypercharge is the independent connection \(B_M\), not momentum around an interval. The corrected SG-5 replaces the winding Higgs with a minimal equivariant scalar bundle.

Shape: the minimal equivariant Higgs Actor

The canonical parent Higgs Actor is one complex scalar

\[ \boxed{ \Phi_H\in\Gamma(\mathcal E_H), \qquad \mathcal E_H =\pi_{S^2}^{*}\mathcal O(1)\otimes L_Y^{3}\otimes\mathcal L_{\chi,+}. } \]

Its properties are:

\(\mathcal O(1)\to S^2\simeq\mathbb{CP}^1\) is the minimal positive \(SU(2)\)-equivariant line bundle;
\(H^0(\mathbb{CP}^1,\mathcal O(1))\cong\mathbb C^2\), transforming as one irreducible \(SU(2)\) fundamental;
\(L_Y^3\) implements primitive integer hypercharge \(q_H=3\), hence \(Y_H=q_H/6=1/2\);
\(\mathcal L_{\chi,+}\) supplies even interval parity;
\(\Phi_H\) is trivial over \(K_6\), a color singlet, and a family singlet.

The fixed Chern connection on \(\mathcal O(1)\) is internal equivariant bundle data; it does not add another four-dimensional \(U(1)\) gauge boson. That the lowest multiplet is exactly a doublet is a clean cohomological fact: for the standard equivariant line bundles on \(\mathbb{CP}^1\),

\[ H^0\!\left(\mathbb{CP}^1,\mathcal O(n)\right) \cong\operatorname{Sym}^{n}(\mathbb C^2)^*, \qquad \dim_{\mathbb C}H^0\!\left(\mathbb{CP}^1,\mathcal O(n)\right)=n+1 \]

for \(n\ge0\). Hence \(H^0(\mathbb{CP}^1,\mathcal O(1))\cong\mathbb C^2\) is one \(SU(2)\) fundamental. Within the declared shelf of positive equivariant line bundles, \(n=1\) is the unique minimal choice whose lowest multiplet has dimension two. This Actor is a construction anchor: the Stage does not independently force its existence. The minimality statement is shelf-relative — it says \(n=1\) is minimal within a declared shelf — not that nature must use this Actor.

Dynamics: parent scalar action and reduction

The canonical action becomes \(S_{13}^{(v1.3)}=S_{13}^{(v1.2)}+S_H\), with

\[ \boxed{ \begin{aligned} S_H=-\int_{X_{13}}d^{13}x\sqrt{|G|}\Big[ &G^{MN}(D_M\Phi_H)^\dagger D_N\Phi_H +m_{13,H}^2\Phi_H^\dagger\Phi_H\\ &+\xi_HR[G]\Phi_H^\dagger\Phi_H +\lambda_{13,H}(\Phi_H^\dagger\Phi_H)^2 \Big]+S_{H,\rm fixed}. \end{aligned} } \]

The covariant derivative uses the Levi-Civita connection, the fixed \(\mathcal O(1)\) Chern connection, the geometric \(SU(2)_L\) Killing action, and the independent \(U(1)_Y\) connection with \(q_H=3\). No independent weak Yang–Mills connection is added. For an orthonormal basis \(s_\alpha(y)\), \(\alpha=1,2\), of the selected lowest \(\mathcal O(1)\) multiplet, and the normalized even interval profile \(f_0(\chi)\),

\[ \Phi_H(x,y,\chi)=f_0(\chi)\sum_{\alpha=1}^{2}H_\alpha(x)s_\alpha(y)+\Phi_H^{\rm heavy}, \qquad H(x)=\begin{pmatrix}H_1(x)\\H_2(x)\end{pmatrix}. \]

After integrating over the normalized internal profiles and canonically normalizing \(H\),

\[ \boxed{ D_\mu H= \left(\partial_\mu-i g_2W_\mu^a\frac{\sigma^a}{2} -i g_YB_\mu\frac12\right)H. } \]

The Pauli matrices here are not inserted as a substitute for the geometry; they are the matrix representation of the geometric \(SU(2)\) action on the two-dimensional lowest-mode space just constructed. The selected mode is not automatically massless: its effective quadratic coefficient contains the parent mass \(m_{13,H}^2\), the internal covariant-Laplacian eigenvalue, the curvature coupling \(\xi_H R[G]\), the fixed-set terms, and threshold corrections. Their combined result is labelled an effective matching parameter.

The honest input and the certified output

The renormalizable zero-mode potential is

\[ V_4(H)=-\mu_H^2H^\dagger H+\lambda_H(H^\dagger H)^2, \qquad v^2=\frac{\mu_H^2}{\lambda_H}, \qquad \langle H\rangle=\frac1{\sqrt2}\begin{pmatrix}0\\v\end{pmatrix}. \]

SG-5 certifies the branch conditioned on \(\mu_H^2>0\), \(\lambda_H>0\), and this neutral vacuum. The sign and magnitude of \(\mu_H^2\), the value of \(\lambda_H\), and stability in every heavy or modulus direction are not derived by SG-5. These inequalities define the scoped vacuum being certified; they are not outputs. On that vacuum the unbroken generator is \(Q=T_3+Y\): on the lower component \(T_3=-\tfrac12\) and \(Y_H=+\tfrac12\), so \((T_3+Y_H)\langle H\rangle=0\), leaving one connected \(U(1)\) unbroken and breaking the other three generators.

From \((D_\mu H)^\dagger(D^\mu H)\), in the real-vector basis \((W^1,W^2,W^3,B)\), the renormalizable mass matrix is

\[ M^2=\frac{v^2}{4} \begin{pmatrix} g_2^2&0&0&0\\ 0&g_2^2&0&0\\ 0&0&g_2^2&-g_2g_Y\\ 0&0&-g_2g_Y&g_Y^2 \end{pmatrix}, \]

whose neutral \((W^3,B)\) block has determinant zero and rank one for nonzero \(g_2,g_Y,v\) — the single massless combination is the photon. With \(\tan\theta_W=g_Y/g_2\) the eigenvalues are

\[ M_W^2=\frac{g_2^2v^2}{4}, \qquad M_Z^2=\frac{(g_2^2+g_Y^2)v^2}{4}, \qquad M_\gamma^2=0, \]

and therefore

\[ \boxed{ \rho_{\rm tree}^{(d\le4)} =\frac{M_W^2}{M_Z^2\cos^2\theta_W}=1. } \]

The scope of this identity is stated carefully. It is a tree-level, dimension-four statement, not an exact all-orders or full-Kaluza–Klein result. The full reduced theory contains loops and operators of dimension six and higher; in particular a custodial-violating operator of the schematic form \((c_{HD}/\Lambda^2)\,|H^\dagger D_\mu H|^2\) can shift the mass relation. The physical comparison must therefore retain

\[ \rho_{\rm phys}=1+\Delta\rho_{\rm loops}+\Delta\rho_{\rm KK/EFT}+\Delta\rho_{\rm extra\ VEVs}. \]

An earlier claim that the internal overlap calculation turns the standard one-doublet identity into a new independent geometric prediction is not correct and is not made here: once one canonically normalized doublet with \(Y_H=1/2\) acquires a neutral vacuum, \(\rho_{\rm tree}=1\) follows from the representation alone; the overlap integrals fix the common wave-function normalization and the matching of \(v\), nothing more.

Scale roles for SG-5

Quantity	Honesty label
\(v=(\sqrt2\,G_F)^{-1/2}\)	MEASURED-CALIBRATION
\(g_2,\ g_Y\)	matching quantities (fixed by low-energy gauge data) unless independently predicted
\(\mu_H^2,\ \lambda_H\)	effective matching parameters
\(M_W^2/M_Z^2,\ \rho_{\rm tree}^{(d\le4)}\)	derived ratios in the scoped EFT
Higgs mass \(m_h^2=2\lambda_H v^2\)	relation, not a prediction without \(\lambda_H\)
\(v/M_{\rm KK},\ v/M_{\rm Pl}\)	open hierarchy mechanism

What SG-5 does not claim

that the 13D Stage forces the Higgs Actor to be present;
that the sign or magnitude of the Higgs quadratic term is derived;
that the electroweak hierarchy is solved;
that a Wilson-line / Hosotani Higgs exists in the locked gauge architecture (it does not — retired);
that all heavy Kaluza–Klein, modulus, or additional scalar directions are stable (that is SG-6);
that \(\rho=1\) remains exact after loops and higher-dimensional operators;
that the physical Higgs mass is predicted;
that the single-doublet vacuum is the unique vacuum of the complete 13D theory.

SG-6 Wilsonian zero-mode stabilization construction-anchor full-13D closed-negative

Source: v1.7 root authority, Part X (v1.4 SG-6 addition) and Part XI (v1.5 full-stability adjudication), reconciled to the SG-6 governing correction of 2026-07-12. SG-6 is a two-branch gate: a scoped construction-anchor that is CLOSED-SCOPED, and a full-compactification branch that is CLOSED-NEGATIVE / NOT-STABLE-AS-WRITTEN.

Why the prior “resolved” closure does not follow

The legacy SG-6 rolled a symmetry-forced critical point, two independently negative curvature layers, and an unspecified fermionic one-loop sign up into a single resolved grade. That roll-up does not follow from those calculations, and the corrected dossier states so plainly. Several named defects force the correction:

Permutation symmetry does not force the full gradient to vanish. At the symmetric point \(u_1=u_2=u_3\), \(S_3\) symmetry forces only the gradient in the two traceless shape directions to vanish. It says nothing about the common breathing direction, the \(S^2\) radius, the interval radion, or the Higgs radial direction. An \(S_3\)-invariant gradient may be proportional to \((1,1,1)\); symmetry removes only the doublet component.
The old metric coordinates double-counted one degree of freedom. Using both an overall \(K_6\) radius \(R_6\) and three unrestricted squashing variables \((u_1,u_2,u_3)\) lets a common rescaling be absorbed into \(R_6\). The corrected metric imposes \(u_1u_2u_3=1\), leaving two shape coordinates plus the breathing radius, for five independent homogeneous metric moduli \((R_6,R_2,R_\chi,s_1,s_2)\).
A negative shape mode is a failure of the unstabilized branch, not a resolved gate. The pure-curvature symmetric-\(K_6\) sector carries a certified negative Hessian eigenvalue; no taxonomy rule may convert a physical negative mode into a resolved positive vacuum.
An unknown one-loop sign remains unknown. A fermionic graded-Casimir sign that affects the physical Hessian cannot be waved through as “intrinsically unpinnable” while retaining a resolved gate; a complete theory must specify enough operator, bundle, boundary, regulator, and renormalization data to produce one physical Hessian. Moreover the old one-loop spectrum is nonportable, because SG-3 replaced the family-count construction and SG-5 replaced the Wilson-line Higgs with an explicit \(\mathcal O(1)\) scalar — all loop and Casimir calculations involving the superseded content must be recomputed.
Admissibility restrictions do not dynamically stabilize a modulus. Rejecting points outside a chosen chamber restricts the candidate space; it is not a potential barrier, a restoring force, or a positive mass. Stabilization must appear in Dynamics through the effective action.

Branch 1 — the scoped Wilsonian construction anchor (CLOSED-SCOPED)

The canonical homogeneous metric coordinates are \(q_m=(\beta_6,\beta_2,\beta_\chi,s_1,s_2)\) with \(\beta_i=\ln(R_i/R_i^\star)\) and

\[ \begin{aligned} u_1&=\exp\!\left(\tfrac{s_1}{\sqrt2}+\tfrac{s_2}{\sqrt6}\right),\\ u_2&=\exp\!\left(-\tfrac{s_1}{\sqrt2}+\tfrac{s_2}{\sqrt6}\right),\\ u_3&=\exp\!\left(-\tfrac{2s_2}{\sqrt6}\right), \end{aligned} \qquad u_1u_2u_3=1. \]

In unitary gauge \(H=\tfrac1{\sqrt2}(0,\,v+\varphi)^{\mathsf T}\); the three electroweak Goldstone directions are gauge and are removed. The physical zero-mode scalar vector therefore also contains the Higgs radial fluctuation \(\varphi\):

\[ q=(\beta_6,\beta_2,\beta_\chi,s_1,s_2,\varphi). \]

The Einstein-frame radion kinetic matrix, for product-factor dimensions \((d_6,d_2,d_\chi)=(6,2,1)\), is

\[ G_\beta=\begin{pmatrix}24&6&3\\6&4&1\\3&1&3/2\end{pmatrix}, \]

with leading principal minors \((24,60,66)\) — positive definite by Sylvester’s criterion. The two shape coordinates are locally orthonormalized at the reference point, and \(\varphi\) has the usual canonical kinetic term, so the full physical scalar kinetic metric is positive definite.

Pure higher-dimensional Einstein curvature does not stabilize the locked symmetric \(K_6\) branch; a modulus-potential source is required. SG-6 adopts the minimal honest completion — an explicit renormalized four-dimensional potential for the retained zero modes — and does not pretend it was derived from the locked 13D terms. Below the first Kaluza–Klein threshold,

\[ \begin{aligned} V_{\rm eff}={}&\Lambda_4 +\frac{M_{\rm Pl}^2}{2}\left[ m_6^2\beta_6^2+m_2^2\beta_2^2+m_\chi^2\beta_\chi^2 +m_s^2(s_1^2+s_2^2) \right]\\ &+\frac{\lambda_H}{4}\left[(v+\varphi)^2-v^2\right]^2 +O(q^3/\Lambda_{\rm EFT}), \end{aligned} \]

with \(m_6^2,m_2^2,m_\chi^2,m_s^2>0\), \(\lambda_H>0\), \(v>0\). This is a declared Wilsonian construction anchor, not a derivation from the previously locked 13D terms. The coefficients are the total renormalized coefficients after every contribution included at the chosen matching scale; the diagonal form is the minimal audit representative because every positive-definite real quadratic form can be diagonalized by a nonsingular local field redefinition.

At the reference point \(q_\star=(0,0,0,0,0,0)\), the gradient vanishes exactly — now from an explicit Dynamics object in every physical scalar direction, not from \(S_3\) symmetry alone — and the Hessian is

\[ H_\star=\operatorname{diag} \left( M_{\rm Pl}^2m_6^2,\ M_{\rm Pl}^2m_2^2,\ M_{\rm Pl}^2m_\chi^2,\ M_{\rm Pl}^2m_s^2,\ M_{\rm Pl}^2m_s^2,\ 2\lambda_Hv^2 \right)\succ0. \]

All six leading principal minors are positive. Because the physical scalar kinetic metric \(K_\star\) is also positive definite, every generalized physical mass-squared eigenvalue is positive: \(H_\star x=m^2 K_\star x\) gives \(m^2=(x^{\mathsf T}H_\star x)/(x^{\mathsf T}K_\star x)>0\) for every nonzero physical scalar \(x\). The Higgs radial mass-squared is exactly \(2\lambda_Hv^2\). The additive constant \(\Lambda_4\) does not enter this Hessian and is not predicted by SG-6.

The target radii \(R_6^\star,R_2^\star,R_\chi^\star\) are matching parameters; the renormalized modulus masses are matching coefficients; \(v\) remains measured calibration. No numerical radius, unification scale, or hierarchy is derived. The scoped branch produces no blind output; it is a consistency realization, not a parameter-free prediction. It reopens if the declared kinetic metric develops a nonpositive direction, if any renormalized quadratic coefficient crosses zero, if a retained scalar zero mode was omitted, if a supposed gauge direction is actually physical, or if matching/loop corrections make the total Hessian indefinite.

The v1.4 canonical status is recorded exactly as:

Canonical status:
  CLOSED-SCOPED / LOCAL-ZERO-MODE-STABILITY-REALIZED-GIVEN-
  EXPLICIT-POSITIVE-WILSONIAN-POTENTIAL /
  FULL-13D-AND-COMPLETE-KK-STABILITY-OPEN /
  LOCKED v1.4 / CHANGE-CONTROLLED.

Legacy negative result, retained correctly

The legacy pure-curvature computation indicates a negative \(K_6\) shape direction at the symmetric point. The corrected ledger records it exactly as a negative result, not as a resolved gate:

PURE-CURVATURE SYMMETRIC-K6 BRANCH:
  CLOSED-NEGATIVE / LOCAL SHAPE SADDLE
  (certified -1/3 I2 shape-doublet mode).

The two determinant-one homogeneous shape directions of \(K_6\) are physical transverse metric deformations. In logarithmic coordinates the scalar-curvature Hessian restricted to that orthonormal doublet is \(\operatorname{Hess}_{\rm shape}(\mathrm{Scal}_{K_6})=\tfrac13 I_2\); the curvature contribution to the compactification potential has the opposite sign, \(V_{\rm EH,shape}\propto-\mathrm{Scal}_{K_6}\), so

\[ \operatorname{Hess}_{\rm shape}V_{\rm EH}=-\tfrac13 I_2 \]

in the frozen unit-radius convention. This is consistent with the classification of invariant Einstein metrics on generalized Wallach spaces: the standard metric on the full flag manifold \(SU(3)/T^2\) sits in the adverse-sign case for a potential proportional to minus scalar curvature.

Branch 2 — the full 13D / complete-KK / global / tunnelling attack (CLOSED-NEGATIVE)

The stronger question that Branch 1 explicitly left open is attacked directly, and the honest verdict is sharper than “open”: the present 13D branch is not a fully stable compactification as written. This does not weaken the exact scoped statement of Branch 1; it is a separate, adverse result about the stronger claim.

SG-6F — FULL COMPACTIFICATION STABILITY

Current locked 13D branch, as written:
  CLOSED-NEGATIVE / NOT A FULLY STABLE COMPACTIFICATION.

Reason:
  (1) the Einstein-curvature contribution contains the certified
      negative homogeneous K6 shape-doublet block;
  (2) the positive zero-mode Wilsonian potential has no specified
      13D uplift;
  (3) the parent action still contains unresolved actor, fixed-set,
      and counterterm placeholders, so no unique full fluctuation
      operator exists;
  (4) complete KK, global, and bounce questions are therefore not
      computable from the locked data.

Four independent obstructions are worth stating in full, because each is a place a naive stability claim fails:

The v1.4 potential does not uplift automatically. Many inequivalent 13D actions reduce to the same displayed quadratic zero-mode potential while producing different nonzero KK masses, boundary spectra, higher-order interactions, global minima, and Euclidean bounce actions. The uplift must be written explicitly before its full-theory consequences exist. The parent action still carries placeholders rather than an enumerated action with all coefficients, topological terms, boundary tensions, regulator prescription, and counterterms; the second variation depends on precisely those missing terms.
A zero-mode certificate cannot imply KK stability. A projected-out zero mode may coexist with a tachyonic first surviving mode: with \(m_n^2=n^2/R^2+\mu^2\), \(n\ge1\), odd parity removes \(n=0\), yet \(\mu^2=-2/R^2\) gives \(m_1^2=-1/R^2<0\). A positive retained-zero-mode Hessian contains no theorem about the omitted tower; a finite numerical scan is diagnostic only unless paired with an analytic lower bound for every representation family.
A local Hessian does not determine the global potential. The two one-field potentials \(V_+(x)=\tfrac12x^2+x^4\) and \(V_-(x)=\tfrac12x^2-2x^4+\tfrac12x^6\) share \(V'(0)=0\) and \(V''(0)=1\), yet \(V_+\) has a unique global minimum at zero while \(V_-\) has a lower remote minimum (at \(x^2=(4+\sqrt{13})/3\), with value \(-13\sqrt{13}/27-46/27<0\)). The \(O(q^3/\Lambda_{\rm EFT})\) remainder is therefore load-bearing for global stability and cannot remain unspecified.
Tunnelling is not computable from a local mass matrix. The semiclassical decay exponent is fixed by a Euclidean solution of the full field equations including gravity; a local mass matrix cannot construct or exclude a Coleman–De Luccia or Hawking–Moss solution. The channel census includes decompactification, transitions among flux sectors, brane/defect nucleation, collapse of compact cycles, and bubble-of-nothing geometries; flux stabilization alone does not automatically remove bubble-of-nothing decays, and the interval formulation needs a dedicated fixed-set and cobordism analysis rather than importing the circle result. Current verdict: not computable from the locked data — not because tunnelling is inaccessible, but because the action, global potential, charged defects, and Euclidean boundary-value problem have not been specified.

Dimension adjudication

Two dimension changes were considered and disposed of honestly:

14D ONE-FACTOR EXTENSION
  REJECTED-DOMINATED / NOT A STABILITY REPAIR.

Adding one independent factor multiplies the old \(K_6\) shape Hessian by a positive volume/Weyl factor and therefore preserves its sign, while adding a radion and a KK tower. It does not repair the negative shape mode. Fourteen dimensions may be reconsidered only if a new topology is proved essential to a stabilizing quantized structure that cannot be represented in 13D; no such necessity is presently established.

11D M4 x CP2 x S2 x I_chi
  RESEARCH-FORK / NOT CANONICAL /
  RE-RUN SG-1 THROUGH SG-6 BEFORE PROMOTION.

The finite constraint search selects \(X_{11}^{\rm fork}=\mathcal M_{3,1}\times\mathbb{CP}^2\times S^2\times I_\chi\) as the lowest-liability geometry research fork, because it preserves the required gauge Lie algebras, has three even cohomology classes, has no homogeneous flag-manifold shape doublet, and uses symmetric factors with more tractable spectra. This is not a canonical change and inherits no gate closure automatically.

What SG-6 closes and does not close

SG-6 closes only the local stability of the declared renormalized zero-mode EFT. It does not close the UV or 13D origin of the stabilization potential, stability of the complete KK tower, absence of ghosts or boundary instabilities, global or tunnelling stability, naturalness or the electroweak hierarchy, or the compactification radii or the vacuum-energy value. The stronger full-theory question — perturbative and nonperturbative, UV-derived — is OPEN / NOT CLAIMED, and the current 13D branch as written is closed-negative.

Gap-05 vacuum-energy sequestering construction-anchor

Source: Shape–Scale–Granularity–Dynamics v2.1 Addendum — Gap-05 Stability (Dynamics addendum post-dating v1.7 Part IV).

New Rulebook principle. The field-independent zero of the complete finite four-dimensional Einstein-frame effective theory is treated as redundant. This is a construction anchor justified by an invisible-spectator argument; it is not retrospectively relabelled as a consequence of the old Shape. The point of the mechanism is to cancel constant, field-independent contributions to the vacuum energy without generating a value.

Shape. The metric Stage remains \(\mathcal M_4\times K_6\times S^2\times I_\chi\). Shape additionally records the unique relative orientation class

\[ [\Omega_9]\in H^9(X_9,\partial X_9), \qquad \int_{X_9}\Omega_9=1. \]

This class is topological / boundary data and carries no metric dimension.

Actors. Two compact twelve-form potentials \(A_{12},\widehat A_{12}\) are added, whose relative zero modes are the nonpropagating four-dimensional three-form potentials used by the sequester. Two rigid variables \(\Lambda\) and \(\vartheta\) are added, with no kinetic terms.

Scale. \(M_{\rm Pl}\) is a measured anchor; \(\mu\) is a declared EFT scale at or above the retained matter cutoff; \(\Delta\Lambda\) is a measured boundary/flux anchor. No value of \(\Lambda_{\rm obs}\) is generated by the construction. This is the crux of the honesty statement: the sequester removes offsets; it does not compute the observed cosmological constant.

Granularity. Granularity supplies the finite inventory of matter, KK, modulus, and graviton modes included in \(\mathcal L_{\rm EFT}\). The cancellation is exact for the field-independent contributions from that inventory. Granularity does not derive the residual value and does not promote the result above the cutoff.

Dynamics. The controlling action is the local four-dimensional Einstein-frame sequestering action with a Gauss–Bonnet global constraint. The naïve full-measure 13D cosmological term is explicitly rejected by a breathing-volume thought experiment: a term that scales with the internal volume would respond to a spectator rescaling of the compact geometry, which the physical vacuum energy must not do.

Boundary. The twelve-form zero modes use relative boundary conditions on the orbifold fixed sets; the global four-form fluxes are boundary data and must be included in the branch manifest.

Cross-gate rule. A four-dimensional operator coupling the strong-CP axion to a sequestering four-form is excluded unless it descends from an explicit local 13D gauge-invariant parent. Discovery of such a parent reopens both the sequestering gate and the strong-CP gate.

Gap-08 \(R^2\)-scalaron primordial mechanism construction-anchor

Source: Shape–Scale–Granularity–Dynamics v2.2 Addendum — Gap-08 (retains the v2.1 Gap-05 sequestering).

Supersession. This addendum retains v2.1 Gap-05 sequestering and adds a scoped primordial Dynamics Actor. It supersedes the earlier physical-completion use of a root-dissolution disposition for Gap-08. The old negative result against the modulus claim \(\lambda^2=1/6\) remains valid and archived; the addendum does not revive it.

Shape. The 13D Stage is unchanged: \(\mathcal M_{3,1}\times SU(3)/T^2\times S^2\times I_\chi\). The gravitational Actor is extended from Einstein–Hilbert to a scalar-curvature \(f(R)\) sector containing

\[ \mathcal R_{13}+\gamma\,\mathcal R_{13}^2/M_{13}^2. \]

No new metric dimension is introduced. The scalaron is an Actor generated by the higher-curvature Dynamics — the propagating scalar degree of freedom that an \(f(R)\) theory carries in addition to the graviton.

Scale.

Cosmological Einstein equations use the reduced Planck mass \(M_P=(8\pi G)^{-1/2}\).
The ordinary Planck mass is retained only as a separately labelled laboratory calibration.
The scalar amplitude \(A_s\) calibrates the scalaron mass \(M\).
The parent coefficient \(\gamma\) is a matching parameter, not a geometric prediction.
\(N_*\), the number of e-folds, is constrained by reheating and is frozen to a declared interval before computing the prediction band.

Granularity. The primordial certificate is restricted to physical frequencies below the operational cutoff. No trans-cutoff state, infinite KK tower, or UV-complete vacuum claim is made.

Dynamics. The scoped four-dimensional branch contains:

\(R+R^2/(6M^2)\) gravity;
its equivalent canonical scalaron;
the adiabatic Mukhanov–Sasaki initial-value problem;
slow-roll exit at \(\epsilon=1\);
scalaron reheating through stress-trace / anomaly couplings;
heavy-modulus decoupling.

Cross-gate conditions.

SG-6 / UQF-10 must provide \(m_I^2\ge100\,H_*^2\) for every retained orthogonal zero-mode scalar in the adopted manifest, so that no light spectator spoils the single-field trajectory.
Full all-KK stability remains open and cannot be imported from this gate.
Gap-05 sequestering removes only field-independent offsets; discovery of an order-one sequestering distortion of the scalaron trajectory reopens both gates.
Gap-10 baryogenesis must use the reheating output of this mechanism rather than an unrelated free thermal history.
The 11D \(\mathbb{CP}^2\) research fork inherits no scalaron compactification result without its own reduction.

Consolidated status ledger

The following table is the honest one-line disposition of every Dynamics leg recorded on this page, reconciled to the corrected 2026-07-12 state. It never strengthens a terminal: a construction-anchor stays a construction-anchor, a measured quantity stays measured, an open item stays open, and a closed-negative result stays closed-negative.

Leg	Corrected status	Honesty note
Parent-action skeleton (Part IV)	Locked / change-controlled	Placeholders \(\mathcal L_{\rm declared\ actors}\), \(S_{\rm fixed\ sets}\), \(S_{\rm required\ ct}\) not yet enumerated
Geometric connection ansatz	Locked	\(SU(3)_c\), \(SU(2)_L\) geometric; \(U(1)_Y\) independent; linear reduction only
Hypercharge boundary contract	Locked	Parity-based; one \(B_\mu\) zero mode; fermion parities gate-owned
Reduction map \(\mathcal R_4\)	Locked (linear)	Exact nonlinear consistent truncation NOT assumed; separate theorem required
SG-5 electroweak breaking	CLOSED-SCOPED / construction-anchor	\(\rho_{\rm tree}^{(d\le4)}=1\); \(v\) measured; Wilson-line Higgs RETIRED
SG-6 local zero-mode stability	CLOSED-SCOPED / construction-anchor	Target radii, modulus masses = matching parameters; \(\Lambda_4\) not predicted
SG-6 pure-curvature symmetric \(K_6\)	CLOSED-NEGATIVE	Local shape saddle, \(-\tfrac13 I_2\) doublet mode
SG-6F full 13D / KK / global / tunnelling	CLOSED-NEGATIVE / NOT-STABLE-AS-WRITTEN	14D one-factor REJECTED-DOMINATED; 11D fork = candidate only
Gap-05 vacuum-energy sequestering	Construction-anchor	Cancels field-independent offsets; no \(\Lambda_{\rm obs}\) generated
Gap-08 \(R^2\)-scalaron primordial	Construction-anchor	\(\gamma\) matching; scalaron mass calibrated by \(A_s\); \(N_*\) frozen before comparison

Key open items

a UV-derived 13D potential that produces the positive Wilsonian zero-mode coefficients rather than declaring them;
full compactification / complete-KK stability, including a spectral lower bound over every representation family, global-minimum comparison, and a tunnelling / bubble-of-nothing analysis on the interval;
a unique reheating history feeding baryogenesis (Gap-10), rather than a free thermal input;
the full UV completion of the parent action, with all localized Actors, fixed-set terms, regulator, and counterterms enumerated;
a target-blind geometric origin for the Higgs Actor, the electroweak scale, and the flavor spurions — each currently a matching or measured input.

Provenance. Source of truth = Shape, Scale, Granularity, and Dynamics — Source of Truth, canonical version 1.7 (locked 2026-07-11), Part IV — Dynamics (typed definition, locked parent-action skeleton, geometric connection ansatz, hypercharge boundary contract, reduction status, vacuum condition), plus Part IX (SG-5 electroweak breaking) and Parts X–XI (SG-6 local stability and full-stability adjudication). The SG-5 and SG-6 sections are reconciled to the governing corrections of 2026-07-12 in the per-gate dossiers; where the deep source and the corrected dossiers ever differed, the corrected version governs. Later Dynamics refinements folded in: v2.1 Addendum — Gap-05 Stability (sequestering) and v2.2 Addendum — Gap-08 (\(R^2\)-scalaron). Page current 2026-07-12. The per-gate dossiers at /gates/ remain the ultimate source of truth; this root dossier is reconciled to the corrected 2026-07-12 state.