Frequency-domain review

Physical premise (IDWT): Physical spacetime has infinitely many ordinary spatial dimensions plus time. Matter occupies up to ten spatial dimensions; all remaining spatial directions are unbound for every particle. Depth counts a pattern's bound spatial involvement within that space. “Observed” and “unobserved” describe an observer's access, not different kinds of dimension. Older projective and nested-geometry calculations below retain their additional hypotheses; they do not establish a separate physical manifold or compactified direction. See spatial meaning and mathematical scope.

Summary: Corrected frequency-domain conclusions and the remaining hadronic proof target.

RESEARCH NOTES, NOT A PHYSICS MODULE. This is a record of calculations, corrections, and unresolved hypotheses. It does not assert a completed hadronic-mass derivation or a validated replacement for the V2 master dynamics.

1. Where the investigation stands

The most useful progress was not another numerical factor close to a nucleon residual. Several earlier exclusions depended on confusing a static matrix, an observed projection, or an operator coefficient with a frequency generator. Those exclusions were narrowed or withdrawn after exact checks.

Two later results had not yet been recorded in the inspected ledger. First, the point-contact construction and a smooth coherent-density construction have different spectra; the latter does not automatically replace or disprove the graph-supported prescription explicitly chosen in V2. Second, a rank-one source-to-scalar-self-energy response supplies a concrete sufficient mechanism for the common-anchor ratios without measuring or solving for the sources. The unresolved issue is whether that response applies to complete occupied reference frequencies or to coefficients before the simplex multiplier.

The companion contact-kernel note and rank-one-response note contain those previously undocumented calculations. They are live possibilities and exact comparison tests, not a numerical baryon prediction.

The last inspected hadronic ledger ends at Section 76. Its Sections 67--76 contain the earlier saved corrections. The returned execution report confirmed that write and its verification. Later read-only tests did not, by themselves, save the additional findings. This notes package preserves them separately rather than extending an exploratory ledger as though it were a finished module.

2. Observable and notation discipline

The target is a rest-frequency ratio, ultimately

$$z_B=\frac{\omega_B}{\omega_e}, \qquad \omega_e=\frac{m_ec^2}{\hbar}.$$

Use $\varpi_d=\mu_d/\hbar$ when $\mu_d$ denotes the sector energy unit, and distinguish it from an oscillator frequency produced by a constriction model. On the hypothesized ordinary contact branch,

$$\omega_{n,d}=\varpi_d S(n,d), \qquad S(n,d)=\binom{n+d-1}{d}.$$

A specified occupied reference frequency is $\omega_d^\star=\varpi_d S(n_d^\star,d)$, not generally $\varpi_d$. In particular, the depth-four reference has count 15, while the electron reference has count 18564. The pre-prism source-phase proposal $\Omega_{n,d}=S(n,d)-1$ is yet another object; its identification with time-dependent physical dynamics remains conditional in the contact-resonance source, Section 3.

No independently calibrated field amplitude is an input to these notes. Arbitrary normalization of an eigenvector cannot move the eigenvalues of a fixed linear operator. However, replacing an unknown coefficient by an unknown frequency ratio does not determine it, and a nonlinear background dependence cannot be erased by renaming its variables. Genuine coefficients, projected kernels, kinetic metrics, and boundary conditions still need a derivation or a demonstrated cancellation.

3. Corrected static and spectral matrix results

For the adopted test values

$$G=\begin{pmatrix}a&b\\b&c\end{pmatrix}, \qquad a=8\sqrt7,\quad b=4\sqrt6,\quad c=12/\sqrt7,$$

one has $b^2=ac$. The three quantities previously conflated are

$$\underbrace{a-b^2/c}_{\text{actual static Schur complement}}=0, \qquad \underbrace{b^2/c}_{\text{removed relaxation contribution}}=a, \qquad \underbrace{a}_{\text{clamped principal block}}.$$

The frequency-dependent matrix elimination instead uses

$$f(z)=a-z-\frac{b^2}{c-z}, \qquad (c-z)f(z)=\det(G-zI)=z(z-a-c).$$

This is exact away from the excluded complementary denominator. That excluded value must be checked in the original problem. The full matrix eigenvalues are $0$ and $a+c$, not the removed static contribution considered in isolation.

The operator square root also does not commute with projection. Here

$$\sqrt G=\frac{G}{\sqrt{a+c}}, \qquad \frac{\langle3|\sqrt G|3\rangle}{\sqrt a}=\sqrt{14/17}, \qquad \frac{\sqrt{\langle3|G|3\rangle}}{\sqrt a}=1.$$

The reference-band law $(\omega_d^\star/\omega_e)^2=g_{dd}/g_{66}$ alone does not identify the full interacting frequency generator with $\sqrt{G/g_{66}}$. Consequently the earlier universal prohibition on a term linear in a dimensionless coupling was too strong. The particular matrix prescriptions were checked; a different operator map was not disproved. These corrections are already recorded in the hadronic ledger, Section 73.

4. Projection of a signal is not replacement of its generator

For a time-independent linear map $P$ and the same clock,

$$P\bigl(u e^{-i\omega t}\bigr)=(Pu)e^{-i\omega t}.$$

A nonzero projected line retains its frequency. A zero projection removes the line. This is different from physically imposing a new closed dynamics generated by $PHP$ on a subspace that does not reduce $H$.

The exact diagnostic used

$$u=\begin{pmatrix}1/\sqrt6\\\sqrt{5/6}\end{pmatrix}, \qquad H=64uu^\dagger, \qquad P=\operatorname{diag}(1,0).$$

The compressed matrix has its observed entry $32/3$, but the exact projected resolvent is

$$P(z-H)^{-1}P =\left[\frac{1}{6(z-64)}+\frac{5}{6z}\right]P.$$

The factor $1/6$ is the residue at the unchanged frequency 64. The corresponding exact reduced generator is

$$H_{\rm eff}(z)=\frac{32z}{3z-160}.$$

This resolves the earlier color-projection error: the ordered rank-one contact gives $64/6$ under compression, but that is not a universal frequency penalty. If color and orbital factors separate as $I_{\rm color}\otimes A_{\rm mode}$, singlet restriction leaves the allowed orbital eigenfrequencies unchanged. Likewise, annihilation of an identical color-vector product does not exclude a shared orbital mode with a separate antisymmetric color state. These results do not establish that 64 is a physical baryon eigenvalue. See ledger Section 74.

5. A genuine color coefficient, with its sign retained

With fundamental generators normalized by $\operatorname{tr}(T^aT^b)=\delta^{ab}/2$, the directly constructed pair operator is

$$C_{ij}=\sum_aT_i^aT_j^a =\frac12\left(X_{ij}-\frac13I\right).$$

On the normalized determinant singlet $\epsilon\in\Lambda^3\mathbb C^3$, $X_{ij}\epsilon=-\epsilon$. Hence

$$C_{ij}\epsilon=-\frac23\epsilon, \qquad \sum_{i<j}C_{ij}\epsilon=-2\epsilon.$$

All eight generators, their trace normalization, and all three pair operators were checked explicitly. For a derived interaction of the form $\sum_{i<j}C_{ij}\otimes K_{ij}(z)$, the singlet restriction is $-(2/3)\sum_{i<j}K_{ij}(z)$. It becomes $-2K(z)$ only when the remaining pair operators actually coincide.

This is not a frequency prediction. The radial, spin, and normalization content of $K_{ij}$ is still required. A numerical equality with the $2/3$ in $28/9=(2/3)(g_{33}/g_{44})$ does not identify the two constructions. The adopted sector couplings must not silently be equated with the coefficient of a gauge-current pair interaction. The source generator convention appears in the contact-resonance module, Section 14; the diagnostic is recorded in ledger Section 76.

6. What the earlier numerical searches actually tested

These searches used preselected illustrative prescriptions, not a derived composite generator. Their failures therefore do not falsify all V2 hadronic constructions.

For the committed quark contact counts $D_u=15$, $D_d=1$, $D_s=20$, raw constituent products give 225 for $uud$, 15 for $udd$, and 300 for $uds$. This shows why that unqualified product prescription is unsuitable as a common nucleon bulk contribution; it does not establish how physical composite frequencies combine.

A separate carrier-size scan gave:

The number $64\varpi_3$ lay about 2.77 percent below the chosen approximately 309.5 MeV per-constituent residual when the conditional sector scale was used. That proximity does not choose a baryon carrier. Its previous suppression-based exclusion was withdrawn, not replaced by a derivation.

Another scan tried $S(n,6)/S(13,6)$ for $35\le n\le47$. The values bracketing the selected residual ratio near 605.676 were approximately 578.408 at $n=42$ and 661.038 at $n=43$. No operator identification justified borrowing the electron-sector scale for a composite. Scalar $S^5$ harmonic degeneracies were also scanned; their initial values are $1,6,20,50,105,196$. Degeneracy is not automatically a frequency. These are records of unsuccessful exploratory identifications, not universal spectral exclusions or statistical null tests.

The approximately 309--310 MeV quantity itself was a residual allocated per constituent after subtracting chosen quark contributions. It is not, absent an additional decomposition theorem, a separately measured pole that a correct operator must contain. The primary target is the whole baryon rest frequency.

7. The constructive spectral target

Three positions in a common observed three-space give six relative Jacobi coordinates after center-of-mass removal. A fixed Euclidean hyperradius has an $S^5$ angular surface. This is useful effective three-body kinematics, not proof that the baryon is the depth-six electron sector or the depth-five neutrino sector. Native higher-depth coordinates, spinor bundles, and eliminated channels must be accounted for. Ledger Section 75 records the corrected distinction.

For a specified stationary generator $\mathsf H$ in electron-frequency units and an orthogonal model-space projector $P_B$, let $Q_B=I-P_B$. Exact reduction gives

$$\mathsf H_{\rm eff}(z)=P_B\mathsf HP_B+ P_B\mathsf HQ_B(z-Q_B\mathsf HQ_B)^{-1}Q_B\mathsf HP_B,$$

and seeks

$$\ker\left[z_BI_{P_B}-\mathsf H_{\rm eff}(z_B)\right]\ne\{0\}.$$

The complementary inverse has a specified domain; at its excluded values the original problem is checked. This is an exact block identity for a given generator, not a derivation of the missing blocks. A covariant Dirac operator must first be related to a stationary time generator. An off-diagonal map $P_3D_\infty P_4$ has no standalone ordinary frequency spectrum without further structure.

The common-anchor law remains conditional on deriving the projected reference response from the action. The composite blocks and their induced spectral terms are also unresolved; amplitude fits do not supply those derivations.

8. Scope of verification

The packaged regression checks reproduce finite algebra, the illustrative resolvent, color contractions, moving-projector identities, smooth-symbol spectra, and the source-cancellation/count-normalization distinction. They are not Monte Carlo significance tests, a full variational derivation, or a proof of empirical agreement.

The source audit covers the cited passages and reported bounded searches. An unsuccessful search is not a proof that no suitable formula exists. The packaged checks do not include extensive statistical null tests.

9. Source versions

The hashes below identify the source versions present when this notes package was written. They are provenance, not a claim that every statement in those files has been verified. Source links throughout these notes refer to the inspected sections.

Related

• Research status — current proof boundaries and correction guidance.