Historical excerpt: IDWT_02_Mass.md

Imported 2026-09-08. Historical evidence only; claims are not endorsed by V2. This is a local extract, not a link to or dependency on another repository. Original full-file SHA-256: 79ed92d5d02ba41311cbac4335705b871ac984ce2fa99cb0d316340d323d8962.

Original source lines 633–741

10. Mass Scale Derivation

$m_{\rm scale,3}$ and $m_{\rm scale,4}$ — unified coupling self-consistency condition

The kernel vacuum analysis gives a single fixed-point equation that applies to all sectors uniformly:

In equilibrium, the squared mass of the lightest occupied mode in sector $d$ equals $(g_{dd}/g_{66}) \times m_e^2$.

The lightest occupied mode in sector $d$ has mass $m_{\rm lightest}(d) = m_{\rm scale,d} \times S(n_{\min}(d), d)$, where $n_{\min}(d)$ is the lowest mode index selected by the co-fixed-point. The fixed-point equation is therefore:

$$\bigl(m_{\rm scale,d}\times S(n_{\min}(d),d)\bigr)^2 = \frac{g_{dd}}{g_{66}}\times m_e^2 \;\Longrightarrow\; m_{\rm scale,d} = m_e\times\frac{\sqrt{g_{dd}/g_{66}}}{S(n_{\min}(d),d)}$$

This is one formula for all sectors. The apparent differences between sectors are entirely due to $n_{\min}(d)$ and $S(n_{\min}(d), d)$:

$d=3$ (down quarks): $n_{\min}=1$, $S(1,3)=1$.

$$m_{\rm scale,3} = m_e\times\frac{\sqrt{g_{33}/g_{66}}}{1} = 0.511\times\sqrt{8\sqrt{7}/0.25} = 4.702\ \text{MeV}$$ The lightest occupied mode is the down quark: $m_d = m_{\rm scale,3} \times S(1,3) = 4.702$ MeV. PDG 2024: 4.70 MeV. Error: +0.04%.

$S(1,3) = 1$ makes the normalization factor trivially 1 for $d=3$.

$d=4$ (up-type quarks): $n_{\min}=n_u=3$ (postulate; see Part 2 §5 note), $S(3,4)=15$. The postulate introduces no new number — it reuses the seed $n_u=3$, stating that the up quark occupies the sector floor. It is also data-constrained: $n_{\min}(4)=1$ or $2$ would rescale every up-type mass by $S(3,4)/S(n_{\min},4)$ (a factor $15$ or $3$), excluded by observation. Why the $d=4$ floor sits at the seed index is part of the open index-rationale question (Part 2 §11.7).

$$m_{\rm scale,4} = \frac{m_e\sqrt{g_{44}/g_{66}}}{S(3,4)} = \frac{0.511\times\sqrt{12/(\sqrt{7}\times0.25)}}{15} = 0.1451\ \text{MeV}$$ The lightest occupied mode is the up quark: $m_u = m_{\rm scale,4} \times S(3,4) = m_e \times \sqrt{g_{44}/g_{66}} = 2.177$ MeV (bare count; physical $2.175$ MeV after the §11.9 confinement-binding correction). The fixed-point requires $m_u^2 = (g_{44}/g_{66}) \times m_e^2$ — i.e. the up quark mass satisfies the same fixed-point equation that the down quark satisfies, with $g_{44}$ in place of $g_{33}$. The factor $/S(3,4) = /15$ is the fixed-point condition applied consistently to a sector whose lowest occupied mode is not $n=1$.

Unified check: $m_u/m_d = \sqrt{g_{44}/g_{33}} = \sqrt{(12/\sqrt{7}) \div 8\sqrt{7}} = \sqrt{3/14} = 0.4629$. PDG: 0.462 ± ~0.10. Error: +0.20%.

The down quark is a pure prediction: $m_d = m_{\rm scale,3} \times S(1,3) = m_{\rm scale,3} \times 1 = 4.702$ MeV. PDG 2024: 4.70 MeV. Error: +0.04%.

The ρ Meson — Comb Filter Prediction

The inter-sector comb filter $\mathrm{Im}[\Gamma_{346}(\omega)]$ predicts the $\rho$ meson mass independently of $m_{\rm scale,3}$. Its inputs are the coupling constants $g_{33}=8\sqrt{7}$, $g_{44}=12/\sqrt{7}$, $g_{66}=1/4$ (all derived from seeds and $\mathbb{CP}^3$ complex geometry) and the Jacobi chain delays $\tau_d = 1/(2\sqrt{k_0+d})$ at resonance site $k_0=16$:

$$m_{\rho^*} = \arg\max\,\mathrm{Im}[\Gamma_{346}(\omega)] = 775.794\ \text{MeV} \quad (\text{PDG: }775.260\pm0.250\ \text{MeV},\ {+}0.069\%)$$

No direct mass input is used. The 0.534 MeV residual reflects the accuracy of the comb filter at this order; contributing open items include: (a) isospin breaking absent from the $SU(3)$-symmetric kernel (open item, Part 2 §11), and (b) the leading-order sector phase delay approximation in $\tau_d$ being exact only for $d=10$ (see Part 1 §3b). The agreement is a consistency check of the coupling geometry at the 0.069% level.

Note: $\tau_d = 1/(2\sqrt{k_0+d})$ is a valid description of the inter-sector phase delay at the resonance site $k_0$, where both $d=3$ and $d=6$ modes are evaluated at the same resonance frequency scale set by $k_0=n_s^2=16$. The delay formula does not assume comparable mass scales between sectors — it depends only on the Jacobi chain structure at $k_0$, which is a geometric property of the manifold, not the sector mass scale.

Scale Hierarchy

Unit reference: $m_e$. All sector scales follow from the unified fixed-point $m_{\rm scale,d} = m_e \times \sqrt{g_{dd}/g_{66}} / S(n_{\min}(d), d)$.

The light-quark predictions are parameter-free outputs of the coupling self-consistency derivation. The bare counts sit at d +0.04%, s +0.57%, u +0.77% against PDG 2024; the derived confinement-binding correction (§11.9) brings the colour-carrying strange and up to s +0.49% and u +0.70% — all below PDG measurement precision for light quarks (PDG d: ±10%, s: ±9%).

All sector scales

10b. $g_{22}$ — the kernel back-reaction fixed-point 🔶

Status: 🔶 Structurally motivated (state-counting). $g_{22} = p^2q/2 = 722.5$ is a multiplicity count: the product of available Dirac-eigenspace dimensions across the $d=3$ sector (two kernel legs → $p^2$) and the $d=4$ sector (one leg → q), with the 1/2 from the $\xi\leftrightarrow\xi'$ symmetry of the two-body kernel. It is not a kernel matrix element. Testing whether a genuine $(\xi\cdot\xi')^2$ trace yields $p^2q/2$ (files/idwt.py STEP 2d): the literal Tr[$G_{23}^2 + G_{24}^2$] is additive (∼ $p^2+q$, the wrong structure for a product), and the actual $(\xi\cdot\xi')^2$ overlap returns $\langle r^2\rangle$ magnitudes that scale as the mode index $n$ — O(1), orders of magnitude below the multiplicity product. $p^2q$ appears only as a trace of the identity over rank-(p,p,q) eigenspaces, where the multiplicities are the input. So $g_{22}$ is a state-count (IDOS), on the same footing as the CKM formula: combinatorial, empirically exact, mechanism = counting. The 1/2 and the leg-counting are kernel-motivated; the magnitude is a count of eigenstates, not a dynamical overlap. A count is not a foreign mechanism here — counting is the framework's native currency: the mass law itself is the integrated density of states (Part 8 §5, Theorem S1), and the CKM formula is a relative state count of the Born-ratio form (Part 3 §0.9, Part 1 §2.3). What remains open (🔶) is the operator-level counting theorem — a dynamical selection operator whose spectral-projector trace makes the eigenspace dimensions an output rather than an input.

The $d=3$ self-coupling $g_{33}$ is fixed by the intra-sector confinement condition $g_{\rm eff}(n_s,3) = g_{33}/S(n_s,3) \approx 1$ (Part 2 §8). The $d=2$ sector has no self-confinement — the W is massive but not confined in the quark sense. Its self-coupling $g_{22}$ is instead fixed by the cross-sector back-reaction: the requirement that the $d=2$ vacuum amplitude is consistent with the $d=3$ and $d=4$ quark sector structures at the level $n_s=4$.

The derivation:

Step 1. The positive Dirac eigenvalue $\lambda_{l=3} = 7/2$ on $S^3$ has multiplicity $M_3 = (3+1)(3+2) = 20 = S(n_s,3)$ (Theorem S1, Part 8 §5). Of these 20 eigenstates, $n_u = 3$ are already accounted for by the up-quark sector boundary (Theorem S2, Part 8 §5). The remaining

$$p = M_3^{S^3} - n_u = S(n_s,3) - n_u = 20 - 3 = 17$$ [Notation: p is used here rather than $\alpha$ to avoid collision with the fine structure constant $\alpha = e^2/4\pi \approx 1/137$, which is a separately derived quantity in Part 3 §8.]

eigenstates are available to couple to the $d=2$ sector through $G_{23}$. The kernel is two-body — $(\xi\cdot\xi')^2$ couples two copies of $J^{d=3}$ — so both legs contribute, giving $p^2$.

Step 2. The hockey-stick identity $S(n,d)-S(n,d-1)=S(n-1,d)$ gives the $d{=}4$ eigenstate increment at the up-quark threshold:

$$q = S(n_u,4) - S(n_u,3) = 15-10 = 5 = S(n_u-1,4) = S(2,4)$$

These q eigenstates below the up-quark threshold couple to $d=2$ through a single $G_{24}$ insertion, entering linearly.

Step 3. The kernel $(\xi\cdot\xi')^2 = (\xi'\cdot\xi)^2$ is symmetric under $\xi\leftrightarrow\xi'$, so the vacuum integral double-counts. Divide by 2.

Step 4 (fixed-point). Equate the cross-sector back-reaction to the $d{=}2$ self-coupling:

$$g_{22} = \frac{1}{2}\times p^2\times q = \frac{1}{2}\times17^2\times5 = 722.5 \quad(\text{exact})$$

Comparison with $d=3$: For $d=3$, the intra-sector fixed-point gives $g_{33} \approx S(n_s,3) = 20$ (with a ~5.8% Jacobi correction). For $d=2$, the cross-sector fixed-point gives $g_{22} = (S(n_s,3)-n_u)^2 \times S(n_u-1,4)/2$ — a different structure, reflecting that the EW sector acquires its scale by coupling to both quark sectors, not by self-confinement.

Consequences:

$$m_{\rm scale,2} = m_e\times\sqrt{g_{22}/g_{66}} = m_e\times\sqrt{722.5/0.25} = 27.471\ \text{MeV}$$

IDWT has a sole unit reference $m_e = 0.511$ MeV. All quarks, leptons, bosons, CKM angles, Fermi constant, Weinberg angle, and muon lifetime follow from $m_e$ and seeds $\{n_d=1, n_u=3\}$. □

---