An Exact Scalar-Curvature Constraint on Pure Count Carriers

The minimum-carrier comparison can be sharpened without a parameter scan. At the common ground preparation c=3/5, the proper-current count action has a growing spatial disturbance at depths six and ten for every real choice of its local scalar-ratio curvature. The result concerns the one-column carrier, rather than borrowing the earlier full-matrix rigidity theorem.

This limits one derivative-free Lorentz-scalar interaction class and preparation. It does not reject the assigned excited particles, other physical geometries, additional same-field degrees of freedom, derivative interactions, or finite-lived states. The known preferred-time persistence and actual count/rotation identities remain valid in their own actions.

1. What Is Allowed to Vary

Write V=n f(s/n) in radius-one units, n=sqrt(j_mu j^mu), x=s/n. Preserve f and f' at the stationary value x=4/5, so the same complete background, count clock and scalar source survive. Let its second derivative be completely arbitrary and define

$$\gamma=\frac{c^2 f''(x)}{\lambda},\qquad c=3/5,\quad \lambda=\frac{d}{2c}. \tag{1}$$

The affine envelope has gamma=0. The integer-minimum action in the previous comparison has gamma=1. These are controls inside the complete real gamma range; the following result is not a failure of only those two endpoints.

The full linearized source acquires, in addition to the proper-current term already retained, the scalar contribution

$$2f''(x)(\beta-xI)\Phi_0\, \operatorname{Re}\operatorname{tr} [(\beta-xI)\Phi_0]^\dagger\delta\Phi, \tag{2}$$

with rho_0=1 in this normalization. Thus the action is being changed explicitly; the original between-shell pair dynamics is not presumed unchanged.

2. Exact Closure of a Small Spatial Family

Use the global sphere operators and seed C defined in the preceding note. For the fixed Clifford vector label A=0 take

$$X_0C,\quad X_0\Gamma_XC,\quad \Gamma_0C,\quad\Gamma_X\Gamma_0C, \tag{3}$$

and their imaginary multiples. At depths two, three, four, six and ten these eight real fields close exactly under the full disturbance generator for every real gamma. At depth ten the label is the axis of the explicit seed's three-form. This smaller axial family is closed even though the larger all-label candidate in the numerical comparison leaks.

The proof reduces every polynomial coefficient modulo X dot X=1. It uses X_0 squared=1-sum_(A>0) X_A squared and compares all coefficients of the actual field and source. Exact matrix equality holds, without a finite-point closure assumption. At depth five this fixed-label family does not close: 64 residual coefficients remain. Its larger numerically closed family requires the complex-structure partner and is not assigned the following eight-dimensional polynomial.

Let L_gamma be the exact real time generator on (3). Its polynomial is

$$\det(\sigma I-L_\gamma)= \sigma^2[\sigma^6+a\sigma^4+b\sigma^2+c_0].$$

The nonzero squared frequencies z=-sigma squared are roots of

$$\boxed{q_\gamma(z)=z^3-a z^2+b z-c_0.} \tag{4}$$

All roots must be real and nonnegative to avoid exponential growth. For a real monic cubic this is equivalent to a,b,c_0>=0 and nonnegative discriminant. Indeed real roots follow from the discriminant condition; for any t>0, q(-t)=-(t cubed+a t squared+b t+c_0)<0, so none can be negative. Repeated roots may still have secular dynamics; this test does not by itself establish bounded linear response.

3. Depth Six: The Necessary Sum Bound Is Already Enough

The exact cubic is

$$q_\gamma(z)=z^3+ \frac{6002500\gamma-6957906}{60025}z^2 +\frac{39659577-33680060\gamma}{60025}z +\frac{89536720\gamma-106427020}{60025}. \tag{5}$$

Nonnegative roots require their sum to be nonnegative, giving

$$a=\frac{2(3478953-3001250\gamma)}{60025}\ge0, \qquad \gamma\le\frac{3478953}{3001250}. \tag{6}$$

The discriminant is a rational quartic in gamma. Its exact polynomial is retained in the report. Sturm counting establishes no discriminant root in the entire half-line in (6), and its value at zero is negative. Since zero belongs to that half-line, the discriminant is strictly negative everywhere in the necessary domain. Thus that domain has a complex squared-frequency pair. Outside it, the sum condition fails. No real gamma can make all three squared frequencies nonnegative real.

The real structure and spinor orbit argument in the minimum-carrier note relates every pure balanced depth-six seed by a sphere isometry. Consequently this same ground-preparation constraint is not evaded by rotating to another pure balanced seed. It remains a ground result at the specified c, not an all-radius or assigned electron theorem.

4. Depth Ten: A Closed Axial Family with Negative Discriminant

For the explicit one-column depth-ten seed, the corresponding cubic is

$$q_\gamma(z)=z^3+ \frac{3404947562500\gamma-3706068422794}{12257811225}z^2 +\frac{29349873517409-26276440323500\gamma}{12257811225}z +\frac{130649796706000\gamma-144970394453500}{12257811225}. \tag{7}$$

Its discriminant quartic has no real root at all and is negative at gamma=0. Therefore it is negative for every real gamma. The axial family has a nonreal squared-frequency pair throughout the real curvature range. This is a result for that explicit depth-ten seed, not a classification of every allowed pure depth-ten carrier or a tau decay calculation.

5. Occupied-Depth Controls and Verification

The same exact coefficient calculation tests all occupied depths. At two, three and four it gives intervals allowing nonnegative real squared frequencies in this low family: gamma is at or below approximately -0.9222, 0.2791 and -0.5762, respectively. The exact algebraic endpoints and cubics are in the report. These intervals are not all-harmonic or nonlinear stability windows. Depth five has failed fixed-label closure and requires its larger coupled family. Depth six and the axial depth-ten family have empty allowed sets, as proved above.

minimal-carrier-exact-curvature.py and its report retain the exact real matrices, coefficient identities, cubic discriminants, complete real coefficient inequalities and Sturm counts. Independent direct NumPy field sources at five new positions and three curvatures per closed depth agree with the exact matrices to below 1.3e-15 relatively, including nonzero curvatures. This checks the operator as well as its polynomial certificate.

An initial implementation used a real-assumption polynomial variable that SymPy's characteristic-polynomial routine replaced by a distinct symbol of the same name. The resulting degree-zero interpretation was corrected by explicitly substituting the routine's returned generator. Its exact source and failure record remain preserved. The repaired calculation and the independently verified final calculation agree.

The useful conclusion is a constraint on a simplification of the count-supporting action. Changing its scalar curvature alone cannot repair these six-/ten-dimensional ground controls. A successful covariant continuation must change a premise that this calculation actually holds fixed, or establish viable assigned excited states despite these controls.