A Causal Determinant Law and Exact Normal-Curvature Defects

This is a new candidate constitutive action, not a derivation of the former curvature-squared law. It retains its target metric and canonical normal connection. It does not retain its Hodge gate, independently varied Gauss constraint or flux guides. The calculation uses field-value coordinates on the already declared normal-orientation target; these are not additional physical spatial dimensions.

Definition and Physical Domain

Let the ordinary sector spacetime have signature $(+,-,\ldots,-)$ and spatial depth $d$. Let $\phi^A$ be local coordinates on the positive Riemannian orientation target, with metric $G_{AB}$. For the earlier orthogonal complex structure $J$, choose $G_{AB}d\phi^Ad\phi^B=\|dJ\|_F^2$. On the complete rank-three pure-ray target this is sixteen times the unit-vector Fubini–Study metric. The target dimension is $q(q-1)$, where the declared ten-parent prescription assigns $q=4,3,3,2,2,0$ at depths $2,3,4,5,6,10$. This prescription and its odd-depth pullbacks remain conditional.

For positive constants $T$ and $\lambda$, define

$$\gamma_{\mu\nu}=\eta_{\mu\nu}-\lambda G_{AB}\partial_\mu\phi^A\partial_\nu\phi^B, \qquad \Delta=\frac{\det\gamma}{\det\eta},\qquad \mathcal L_J=T(1-\sqrt\Delta).$$

The chosen branch requires the spatial slices to be spacelike for $\gamma$ and a strictly positive lapse. A positive determinant alone is not a general signature test. Here the spatial block is automatically negative definite, so its Schur complement supplies the precise domain below. If the earlier quadratic coefficient is $f^2/2$, set $T\lambda=f^2$. The independent neutral comparison field may retain $\mathcal L_u=(\partial u)^2/2-\Omega^2u^2/2$; this adds a unit-speed principal block and does not provide the old dielectric coupling.

At a point choose orthonormal target coordinates. Write $v=\partial_t\phi$ and let $Y$ have columns $\partial_i\phi$. Put

$$A=I_d+\lambda Y^TY,\qquad H=I_r+\lambda YY^T, \qquad\delta=1-\lambda v^TH^{-1}v.$$

Then $\Delta=\det A\,\delta$ and the physical domain is exactly $\delta>0$. The target response matrix is

$$Q=I_r+\lambda M\gamma^{-1}M^T =(H-\lambda vv^T)^{-1} =H^{-1}+\frac\lambda\delta H^{-1}vv^TH^{-1}>0,$$

where $M=(v,Y)$. This follows from the Schur complement and the rank-one inverse formula. Arbitrarily large spatial gradients are permitted. Coordinate $\|v\|$ can exceed $1/\sqrt\lambda$ when an aligned spatial gradient is present; the invariant domain condition is $\delta>0$, not the homogeneous speed bound applied indiscriminately.

Equation and Complete Principal Symbol

The Euler equation is

$$\partial_\mu\bigl(\sqrt\Delta\,\gamma^{\mu\nu}G_{AB}\partial_\nu\phi^B\bigr) -\frac12\sqrt\Delta\,\gamma^{\mu\nu}(\partial_A G_{BC}) \partial_\mu\phi^B\partial_\nu\phi^C=0.$$

Constant $T\lambda$ cancels. Freeze a first jet in target normal coordinates and use a rank-one gradient variation $\delta M=h\xi^T$. For $p=\xi^T\gamma^{-1}\xi$, $m=M^Th$, $a=h^Th$, $b=\xi^T\gamma^{-1}m$ and $c=m^T\gamma^{-1}m$, the determinant lemma gives exactly

$$\frac{\Delta(M+s h\xi^T)}{\Delta(M)} =1-2\lambda bs+ \bigl[\lambda^2(b^2-pc)-\lambda ap\bigr]s^2.$$

Differentiating its square root proves the full contracted Hessian,

$$\frac{d^2}{ds^2}\mathcal L_J(M+s h\xi^T)\bigg|_{s=0} =T\lambda\sqrt\Delta\,(h^TQh)\,(\gamma^{\mu\nu}\xi_\mu\xi_\nu).$$

Polarization therefore gives the principal matrix $T\lambda\sqrt\Delta\,Q_{AB}\gamma^{\mu\nu}\xi_\mu\xi_\nu$. No squared first variation has replaced the complete Hessian. Antisymmetric spacetime-index contributions to the unrestricted Hessian cancel under the characteristic contraction. Every target polarization has the same cone. Since $Q>0$ and $\gamma^{00}=1/\delta>0$, the time Hessian is positive and the frozen system has a wave-equation principal reduction. On smooth coefficients bounded away from $\delta=0$, this supplies the usual local strongly hyperbolic structure. It is not a global smooth-existence theorem.

Exact Speeds and Causality

Set $b=\lambda A^{-1}Y^Tv$. The inverse metric satisfies

$$\gamma^{-1}(\xi,\xi)= \frac{(\xi_0-b\cdot k)^2}{\delta}-k^TA^{-1}k.$$

For a unit spatial covector $n$, the signed characteristic roots are

$$c_\pm=b\cdot n\pm\sqrt{\delta\,n^TA^{-1}n}.$$

These are covector roots; their sign convention can reverse with the Fourier convention, without affecting the absolute speed bound. To prove $|c_\pm|\le1$, put $w=\sqrt\lambda H^{-1/2}v$ and $z=\sqrt\lambda H^{-1/2}Yn$. Then $b\cdot n=w\cdot z$, $\delta=1-\|w\|^2$ and $n^TA^{-1}n=1-\|z\|^2$. Cauchy–Schwarz twice gives

$$|w\cdot z|+\sqrt{(1-\|w\|^2)(1-\|z\|^2)}\le1.$$

The ray-cone statement is equally direct. For a $\gamma$-null tangent vector $V$, $\eta(V,V)=\lambda\|d\phi(V)\|_G^2\ge0$. Thus its spatial group velocity also lies within the ordinary light cone. No physical target-space dimensions are being added by this proof.

A homogeneous affine target geodesic is an exact source-free solution with $\delta=1-\lambda\|v\|^2$. All its polarizations have squared principal speed $\delta$, which stays positive on the timelike branch. At its boundary speeds tend to zero and the energy diverges. The earlier curvature-squared action instead has an admissible smooth moving background with negative transverse spatial stiffness; this determinant law changes that large-gradient constitutive response rather than proving the old background was unphysical in the old action.

Positive Energy and Its Boundary

The momentum and energy density in target normal coordinates are

$$\Pi=T\lambda\frac{\sqrt{\det A}}{\sqrt\delta}H^{-1}v, \qquad e_J=T\left(\frac{\sqrt{\det A}}{\sqrt\delta}-1\right)\ge0.$$

If the independent neutral field is included, its nonnegative energy is added. For a fixed finite spatial jet, $e_J$ diverges as $\delta\downarrow0$. A pointwise energy bound $e_J\le E_*$ implies $\delta\ge[T/(T+E_*)]^2$. A finite total integral alone does not imply a uniform pointwise bound, so the calculation does not exclude caustics, concentration or approach to the domain boundary during evolution.

What Changes at Fourth Order

Let $B^\mu{}_\nu=\eta^{\mu\rho}G_{AB}\partial_\rho\phi^A\partial_\nu\phi^B$. The small-gradient expansion is

$$\mathcal L_J= \frac{T\lambda}2\operatorname{tr}B +\frac{T\lambda^2}8\bigl[2\operatorname{tr}(B^2)-(\operatorname{tr}B)^2\bigr] +O((\partial\phi)^6).$$

The first term retains the full declared ray metric. The fourth-order term differs from the previous $-\epsilon\|\widehat F\|^2/4$. Along a single target geodesic all canonical commutators vanish, yet the determinant has a nonzero fourth-order self-kinetic term. At rank three, orthogonal target pairs with equal metric Gram matrices can also have different commutator norms. Thus metric-only determinant dynamics cannot inherit the old curvature energy or its Hodge activation by a relabeling of coefficients.

The canonical connection $\widehat A=-JdJ/2$ and curvature $\widehat F=-[dJ,dJ]/4$ remain available geometric observables. Their previously derived holonomy still belongs to the same target geometry. This action by itself does not identify that holonomy with an observed force or explain higher-sector force absence.

A Static Localization Test Before Further Modeling

On a flat ordinary $\mathbb R^d$ sector, consider a smooth finite-energy static orientation map for which the dilation variation is admissible. Let $J_R(x)=J(x/R)$. At a fixed reference point let $x_j\ge0$ be the $d$ spatial strain eigenvalues after the $R^{-2}$ scaling and put $P=\prod_j(1+x_j)$. The pointwise dilation derivative of $R^d(\sqrt P-1)$, divided by $R^{d-1}$, is

$$d(\sqrt P-1)-\sqrt P\sum_j\frac{x_j}{1+x_j} =\sqrt P\left(\sum_j y_j-d\sqrt{\prod_jy_j}\right), \qquad y_j=\frac1{1+x_j}.$$

For $d\ge2$, AM–GM and $0<\prod_jy_j\le1$ imply

$$\sum_j y_j\ge d(\prod_jy_j)^{1/d} \ge d\sqrt{\prod_jy_j}.$$

For $d>2$, the second inequality is strict wherever a strain eigenvalue is nonzero. Hence the determinant orientation energy alone has no nonconstant finite-energy static critical point at the occupied depths three, four, five or six under these flat-domain assumptions. Depth ten has no orientation target in the present prescription. At depth two equality requires equal strain eigenvalues; conformal calibrated maps can remain scale-degenerate rather than gaining a selected radius. This does not rule out rotating patterns, fields with a potential, supported curved spatial geometry or complete same-field actions. It does establish that repairing principal causality alone does not supply localized quarks or the mass law.

Exact Moving Geometric Flux Defects

The static test has a constructive boundary case. A two-dimensional conformal orientation map solves the full determinant equation exactly. For any $q\ge2$, use one real four-direction block carrying quaternionic orthogonal complex structures $I_1,I_2,I_3$, with $I_aI_b=-\delta_{ab}+\epsilon_{abc}I_c$ and $\operatorname{tr}(I_a^TI_b)=4\delta_{ab}$. The remaining $2q-4$ normal directions have constant complex structure. This block is a totally geodesic target two-sphere: its tangent Lie triple products remain within the same block.

For two ordinary spatial coordinates $x,y$ and any width $a>0$, put

$$n=\frac{(2ax,2ay,a^2-x^2-y^2)}{a^2+x^2+y^2}, \qquad J=n_a I_a\oplus J_{\rm rest}.$$

Its two spatial strains are equal,

$$\langle\partial_iJ,\partial_jJ\rangle_F=s\delta_{ij}, \qquad s=\frac{16a^2}{(a^2+x^2+y^2)^2}.$$

Consequently $\sqrt\Delta=1+\lambda s$ and the complete static equation reduces to the harmonic-map equation. Direct substitution gives $\Delta n+|\nabla n|^2n=0$. The energy and transverse stress are exactly

$$e_J=T\lambda s=f^2s,\qquad \int_{\mathbb R^2}e_J\,dx\,dy=16\pi f^2,\qquad \Theta^i{}_j=0\quad(i,j=x,y).$$

The last identity follows from the complete canonical stress,

$$\Theta^\mu{}_\nu=T\sqrt\Delta\,\gamma^{\mu\rho}\eta_{\rho\nu}-T\delta^\mu{}_{\nu}.$$

The width remains free. Relative to the earlier normal frame, the canonical curvature is also exact:

$$\widehat F_{xy}=-f_aJ_{\rm block},\qquad f_a=\frac{2a^2}{(a^2+x^2+y^2)^2},\qquad \int\operatorname{tr}_{\mathbb C}\widehat F=-4\pi i.$$

It has determinant Chern number two in the convention $c_1=(i/2\pi)\operatorname{tr}F$. The vacuum spin line has Chern number minus one, consistent with $F_L=-\operatorname{tr}F_W/2$. Its traceless normal curvature has squared real Frobenius norm

$$\|\widehat F^0_{xy}\|_F^2=4f_a^2\left(1-\frac2q\right).$$

Thus $q=2$ retains only central curvature, $q=3$ has a nonzero traceless fraction $1/3$, and $q=4$ has fraction $1/2$. This supplies an actual finite-width configuration carrying the earlier determinant/normal curvature, instead of identifying the group name with a force. The determinant action admits it at all the occupied sectors with $q\ge2$, so it does not explain higher-sector force absence.

At spatial depth two the object has finite total energy. At higher depths it is a codimension-two defect with finite energy per unit volume of its remaining $d-2$ directions. In particular it is a line at depth three and a sheet at four; it is not a localized particle or a four-dimensional flux tube. Its longitudinal stress is minus its energy in contravariant spatial components, giving tension $16\pi f^2$. A transverse ordinary Lorentz boost is an exact moving solution, with energy per remaining volume $16\pi f^2/\sqrt{1-v^2}$ and transverse momentum $16\pi f^2v/\sqrt{1-v^2}$. This is a complete inertial response of the constructed classical defect, not the observed particle mass law.

The width variation is not a normalizable collective mode on the infinite transverse plane:

$$\|\partial_aJ\|_F^2=\frac{16r^2}{(a^2+r^2)^2},\qquad \int_{r<L}\|\partial_aJ\|_F^2d^2x =16\pi\left[\log\left(1+\frac{L^2}{a^2}\right)+\frac{a^2}{a^2+L^2}-1\right].$$

It diverges logarithmically as $L\to\infty$. A finite containing geometry or a changed tail can alter this conclusion, but must be calculated rather than treating the free static width as a finite-inertia oscillator. No electric source confinement or quantized particle identification follows from this magnetic geometric flux.

External Mathematical Context and Scope

The determinant is the standard scalar/multifield Dirac–Born–Infeld mathematical structure, used here as a candidate orientation constitutive law without adopting its original particle or extra-dimensional interpretation. Gibbons and Herdeiro, Born–Infeld Theory and Stringy Causality (2001), https://arxiv.org/abs/hep-th/0008052, provide related causal-cone mathematics. Langlois, Renaux-Petel, Steer and Tanaka, Primordial fluctuations and non-Gaussianities in multi-field DBI inflation (2008), https://arxiv.org/abs/0804.3139, establish common propagation speeds in their multifield setting. The mixed-jet symbol, energy, cone bound and static dilation argument above are explicitly derived for this candidate and must be checked against the stated orientation geometry.

The positive result is a common causal local principal structure on the full timelike branch, reusing the established metric at every occupied depth. The open tasks are the same-field origin, dynamical compatibility of sector projections, physical source response, sustaining motion, frequency selection and color-force identification. The result supplies a controlled alternative action, not a completed physical theory or a new preference for three/four.

Reproduction

The source-coupled continuation derives the constrained curvature force, complete reciprocal stress and current-driven stability controls for independent retained coefficients. The single-ray phase completion separately checks the amplitude and phase omitted by the orientation norm. Its fixed-amplitude lift has infinite transverse phase energy, while a smooth amplitude-zero lift has finite complete gradient energy and a nonzero unrestricted dynamical residual. Neither continuation identifies the root defect with an observed particle.

The single checker and recorded report preserve 1,829 exact conditions across 60 mixed-jet cases. They compare direct perturbed determinants with the complete Hessian, exact positive normal factors, characteristic roots and complete Legendre energy at all occupied nonempty normal carriers. The depth-ten empty carrier and independent neutral scalar are explicit controls. The same script checks the static dilation derivative, an actual quaternionic orientation map and canonical curvature, determinant Chern number, transverse stress, boost invariants and the logarithmic width norm. The general statements rely on the displayed proofs, rather than interpreting finite checks as all-background evidence. No particle simulation is used. Running the script creates a dated report alongside the executed source.