A finite normal oscillator projection can imitate the curvature chain's simplex product. Restoring the complete ordinary spatial oscillator changes that frequency. This identifies a concrete requirement for a physical carrier: the finite chain must arise as a controlled isolated band, rather than from an imposed cutoff.

The test does not change the exact surface curvature identity. It tests one possible realization of the chain components as normal modes of the same field. That realization is not derived or adopted here.

Canonical ordinary spatial modes

On one ordinary normal coordinate x, set the oscillator length to one:

a=∂x+x2,a†=−∂x+x2,N=a†a=−∂x2+x2−12.(1) a=\frac{\partial_x+x}{\sqrt2},\qquad a^\dagger=\frac{-\partial_x+x}{\sqrt2},\qquad N=a^\dagger a=\frac{-\partial_x^2+x^2-1}{2}. \tag{1}

The normalized Hermite functions satisfy a phi_n=sqrt(n) phi_(n-1). Their coefficient labels count spatial modes, not physical dimensions. The second ordinary normal direction and the origin of this oscillator profile are not supplied by this test.

Consider the complete normal operator and its Hermitian pair

Tb=b+N+Ga,b>0, G>0,Hb=(0Tb†Tb0).(2) T_b=b+N+Ga,\qquad b>0,\ G>0,\qquad H_b=\begin{pmatrix}0&T_b^\dagger\\T_b&0\end{pmatrix}. \tag{2}

All coefficients in (2) are in a common normal oscillator unit. G is the coefficient of a in this carrier test. Its identification with the earlier chain gap would require this proposed projection to hold. The number operator is second order in the actual coordinate x. Consequently (2) is a spectral carrier test, not an established fundamental first-order causal equation in all spatial directions.

At cutoff q, its upper triangular matrix is

(Tb(q))nn=b+n,(Tb(q))n,n+1=Gn+1,0≤n<q.(3) (T_b^{(q)})_{nn}=b+n,\qquad (T_b^{(q)})_{n,n+1}=G\sqrt{n+1},\qquad 0\le n<q. \tag{3}

Canonical normalization fixes the square roots in (3). At fixed q and large G its smallest singular frequency is

ωq∼(b)qGq−1(q−1)!.(4) \omega_q\sim \frac{(b)_q}{G^{q-1}\sqrt{(q-1)!}}. \tag{4}

For q=d and b=L-1, the numerator is exactly (L-1)L...(L+d-2)=d!D_(L-2,d). The extra factorial in (4) is a common coefficient at a fixed depth. This makes the oscillator a plausible algebraic comparison, but not yet a physical derivation.

The missing component is explicit

Let (x,y)/sqrt(2) be a normalized singular pair of (3). Extend both coefficient vectors by zeros into the full normal space. Although T maps the finite polynomial subspace into itself, T^dagger does not. The complete upper equation contains the additional component

(HbΨ)upper,q=Gq2 yq−1.(5) \boxed{(H_b\Psi)_{{\rm upper},q} =\frac{G\sqrt q}{\sqrt2}\,y_{q-1}.} \tag{5}

In the large-G finite chain, y is concentrated at its last component. The residual (5) is then large while (4) is very small. The finite subspace is not a reducing subspace of the Hermitian spatial operator. Checking the small finite-matrix residual does not check its complete normal field equation.

The simpler displaced annihilator T=b+Ga gives another exact control. In the full space it annihilates the normalized coherent state with coefficients proportional to (-b/G)^n/sqrt(n!). A finite square truncation has a nonzero smallest singular value. That value must converge to zero as the full coherent field is restored.

Solve the complete operator

The Bargmann representation is a unitary mathematical representation of the same one-coordinate oscillator. Its complex argument z does not add a physical spatial dimension. In this representation,

Tbf=(z+G)f′(z)+bf(z).(6) T_b f=(z+G)f'(z)+bf(z). \tag{6}

A zero solution would be proportional to (z+G)^(-b), which is not entire for b>0. The exact inverse on polynomial sources, extended by continuity to the complete Hilbert space, is

(Tb−1h)(z)=∫01tb−1h(tz−G(1−t)) dt.(7) \boxed{(T_b^{-1}h)(z)=\int_0^1 t^{b-1} h(tz-G(1-t))\,dt.} \tag{7}

Differentiating the integrand verifies T_b T_b^(-1)h=h. Composition by tz-G(1-t) has operator norm

∥Ct∥=exp⁡[G2(1−t)2(1+t)].(8) \|C_t\|=\exp\left[\frac{G^2(1-t)}{2(1+t)}\right]. \tag{8}

One can verify (8) by unitary displacement of the contraction f(z) to f(tz); a normalized coherent state attains the norm. For t<1 the contraction is compact. The integral near t=1 tends to zero in operator norm as its interval shrinks, so (7) is compact as well. It is bounded, with norm at most exp(G^2/2)/b. Thus the complete Hermitian pair has a positive smallest singular frequency for b>0, even though it can be extremely small.

The normalized coherent state centered at -G gives the independent lower inverse-norm bound

eG2/2G−2b γ(b,G2)≤∥Tb−1∥≤∫01tb−1exp⁡[G2(1−t)2(1+t)]dt.(9) e^{G^2/2}G^{-2b}\,\gamma(b,G^2) \le\|T_b^{-1}\| \le\int_0^1t^{b-1} \exp\left[\frac{G^2(1-t)}{2(1+t)}\right]dt. \tag{9}

Here gamma(b,x) is the lower incomplete gamma function. For fixed b and G tending to infinity, substitution s=G^2 t shows that both bounds have the same leading value. Consequently

ωfull∼G2bΓ(b)exp⁡(−G2/2).(10) \boxed{\omega_{\rm full}\sim \frac{G^{2b}}{\Gamma(b)}\exp(-G^2/2).} \tag{10}

The degree dependence in (10) differs from (4). Holding a finite cutoff while increasing G is not a uniform approximation to the complete normal tower. In the assigned offsets b=13 and b=35, the large-G ratio instead grows as G^44 Gamma(13)/Gamma(35). The scalar oscillator comparison therefore does not realize the required common simplex frequency family.

Exact inverse coefficients and a remaining-tower bound

The full coefficient inverse of (2) is

(Tb−1)nm=(−G)m−nm!n!Γ(b+n)Γ(b+m+1),m≥n,(11) (T_b^{-1})_{nm}=(-G)^{m-n}\sqrt{\frac{m!}{n!}} \frac{\Gamma(b+n)}{\Gamma(b+m+1)},\quad m\ge n, \tag{11}

and zero otherwise. Its signs separate into a row sign and a column sign. Its absolute entries also separate into r_n c_m on the upper triangle. Logarithmic cumulative sums therefore apply both the inverse and its adjoint in linear time without overflowing their enormous norms. The largest singular value determines the full small frequency.

For a cutoff q beyond max(b,G^2), split the inverse into its retained block A, upper-right block B and remaining diagonal block D. B has rank one. Its column tail is bounded by a geometric series with ratio G sqrt(q+1)/(b+q+1). The numerical range of the remaining normal operator gives

∥D∥≤[b+q−G2(q+q+1)]−1.(12) \|D\|\le\left[b+q- \frac G2\left(\sqrt q+\sqrt{q+1}\right)\right]^{-1}. \tag{12}

The bracket is positive in the reported controls. The norm of the complete inverse is bounded above by the largest singular value of the two-by-two matrix with entries (||A||,||B||;0,||D||). Thus the uncomputed infinite tower has an explicit bound, independently of agreement between two finite cutoffs. Equations (9) give a second complete-space comparison using an ordinary integral.

Verification and interpretation

normal-oscillator-carrier-20260914T015817_286861Z.json passes 69 checks. Independent 90-digit singular values verify the small finite matrices, including b=13 and b=35 at q=6 and G=10000. Actual Hermite functions and their full spatial derivatives verify (1)-(3), and the outside component (5) is retained explicitly. The coherent zero control is canonically normalized in the full space.

The restored-tower controls use up to 4900 normal modes, with a separate extension by 300 modes and bounds (9), (11)-(12). For example, at G=30 the logarithms of the complete frequencies are approximately -381.746751 for b=13 and -301.915689 for b=35. The difference is far from the logarithm of the simplex ratio. At G=60 the corresponding values are -1713.581377 and -1602.320398. Logarithms are necessary because these frequencies are below ordinary floating-point range.

The full singular fields also resolve why the result changes: one normal component lies mainly at small oscillator degree, while its partner lies near degree G^2. At b=13 and G=60 their mean degrees are approximately 0.0473 and 3573.95. These are mode occupations of the specified normal operator, not additional dimensions or assigned particle indices.

This closes the scalar-oscillator carrier test. The finite curvature chain remains a valid specified operator. Its physical realization requires a derived normal band or another same-field construction whose complementary response is controlled. No neutrino mass scale or coupling is identified with (10).