# Infinite Dimensional Wave Theory — Part 5: Predictions

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## 1. Confirmed Predictions

The mass ratios below are not fitted. Within each sector, $S(n,d)/S(m,d)$ is a ratio of binomial coefficients — fixed the moment the mode indices are assigned. The eigenmode selection rule identities (muon $=$ charm $+\nu_2$, etc.) are consequences of the Pascal recursion $S(n,d) = S(n,d{-}1) + S(n{-}1,d)$, not separate postulates.

The absolute scale for the $d=3$ sector is fixed by the kernel vacuum fixed-point: $m_{\rm scale,3} = m_e \times \sqrt{g_{33}/g_{66}} = 4.702$ MeV, with $g_{33} = 8\sqrt{7}$ and $g_{66} = 1/4$ derived from seed pair $\{n_u=3,\,n_s=4\}$ and anomaly cancellation respectively. Full derivation in Part 2 §10.

**Strange/down ratio = 20 exactly ⭐**

$$S(4,3)/S(1,3) = 20/1 = 20.000$$

Fixed by the formula before comparing to data. Zero-error prediction.

**Muon/electron ratio 🔵** (both $d=6$ $\mathbb{CP}^3$ sector excitations of $\Psi_\infty$, inhabiting six spatial dimensions; $n=35$ for the muon, $n=13$ for the electron)

$$S(35,6)/S(13,6) = 3\,838\,380/18\,564 = 206.7647 \quad (\text{PDG: }206.7683,\ {-}0.002\%)$$

**Tau/electron ratio 🔵**

$$S(23,10)/S(13,6) = 64\,512\,240/18\,564 = 3\,475.126 \quad (\text{PDG: }3\,477.23,\ \text{bare }{-}0.060\%;\text{ back-reaction corrected: }{-}0.14\sigma)$$

**Up-type quark mass ratios (bare) 🔵**

The $d=4$ up-type mass ratios are quoted bare; the former Generation Tower Correction is removed (its per-quark exponent was a fit — Part 2 §11). The ratios run high because charm and top overshoot:

$$S(20,4)/S(3,4) = 590.33 \quad (\text{PDG} \approx 589.4,\ +0.16\%)$$
$$S(72,4)/S(3,4) = 81\,030 \quad (\text{PDG} \approx 79\,893,\ +1.42\%)$$
$$S(72,4)/S(20,4) = 137.26 \quad (\text{PDG} \approx 135.6,\ +1.25\%)$$

**Bottom quark 🔵**

$$\sqrt{S(16,3)\times S(17,3)}\times m_{\rm scale,3} = \sqrt{816\times 969}\times 4.702 = 4\,181\text{ MeV} \quad (\text{PDG: }4\,183\pm 7,\ {-}0.05\%)$$

**Photon mass = 0 ✅**

$$S(0,2) = \binom{1}{2} = 0 \;\Rightarrow\; m_{\rm photon} = 0 \quad (\text{exact, derived})$$

**Electroweak sector 🔵**

$$m_W = 80\,379\text{ MeV} \quad (\text{PDG: }80\,369,\ +0.012\%)$$
$$m_Z = 91\,230\text{ MeV} \quad (\text{PDG: }91\,188,\ +0.047\%)$$
$$m_H = 125\,266\text{ MeV} \quad (\text{PDG: }125\,200,\ +0.053\%)$$
$$\sin^2\theta_W = 0.2237 \quad (\text{PDG on-shell: }0.22290,\ +0.37\%)$$
$$\rho = 1 \quad (\text{exact, derived})$$

**ρ meson from the comb filter (consistency check) 🔵**

$$\mathrm{Im}[\Gamma_{346}(\omega)]_{\rm peak} = 775.8\text{ MeV} \quad (\text{PDG: }775.3\text{ MeV},\ +0.07\%)$$

All inputs — $g_{33}=8\sqrt{7}$, $g_{44}=12/\sqrt{7}$, $g_{66}=1/4$, delays from $k_0=16$ — come from seeds $\{n_d=1,\,n_u=3\}$ and composite $n_s=4$, with $m_e$ alone. This is a cross-check of the coupling geometry, not an independent mass prediction.

**Cabibbo angle 🔵**

$$\sin\theta_C = (1+1/240)/\sqrt{20} = 0.22454 \quad (\text{PDG: }0.22450\pm 0.00044,\ +0.09\sigma)$$

Derived from the Vandermonde $d=3\leftrightarrow d=4$ coupling: $\sin^2\theta_C = 1/S(n_s,3) = 1/20$, equivalently $S(2,3)/(S(2,3)+n_W) = 4/80 = 1/20$. Determined entirely by the seed structure and mode indices. Curvature correction from $\mathbb{CP}^1$ holonomy ($\mathbb{CP}^1$ sector curvature correction): $+1/240$ shift — see Part 3 §12.

**Up/down quark mass ratio ✅ (Theorem S2, Part 8 §5)**

$$m_u/m_d = \sqrt{g_{44}/g_{33}} = \sqrt{3/14} = 0.463 \quad (\text{PDG: }0.462,\ +0.08\%;\text{ exact from seeds})$$

**Neutrino mass ordering: normal hierarchy ✅**
$S(n,5)$ is monotonically increasing, $n_{\nu_1} < n_{\nu_2} < n_{\nu_3} \to m_{\nu_1} < m_{\nu_2} < m_{\nu_3}$. Consistent with current experimental preference at $3$–$4\sigma$.

**CKM matrix elements from the Lagrangian kernel 🔵**

The kernel off-diagonal matrix element between modes $n_i$ (lighter) and $n_j$ (heavier) within sector $d$ satisfies $|V_{i\to j}|^2 = S(n_{\rm lighter},d)/S(n_{\rm heavier},d)$ — the squared ratio of the heavier mode's amplitude at the $d=3$ coordinate level to the lighter's (Part 1 §2.2).

$$|V_{cb}| = \sqrt{S(n_u,4)/S(n_c,4)} = \sqrt{15/8855} = 0.04116 \quad (\text{PDG exclusive: }0.04100\pm 0.0014,\ +0.11\sigma)$$

$$A = |V_{cb}|/\sin^2\theta_C = \sqrt{S(n_u,4)/S(n_c,4)}\times S(n_s,3) = 0.82315 \quad (\text{PDG: }0.826\pm 0.012,\ {-}0.24\sigma)$$

$$|V_{ts}| \approx |V_{cb}| = 0.04116 \quad [\text{CKM unitarity, third row}] \quad (\text{PDG: }0.04183\pm 0.0007,\ {-}0.96\sigma)$$

$$|V_{ub}| = A s_C^3 \times \sqrt{\rho^2+\eta^2};\quad A s_C^3 = 0.00920\text{ (CP-conserving prefactor).}$$
$$\sqrt{\rho^2+\eta^2}\text{ is the quark CP factor — NOT YET DERIVED (see warning below).} \quad (\text{PDG: }0.00382)$$

> **⚠ Correction (2026-06-16).** A previous version assigned `|V_ub|` using the *leptonic* CP phase $\delta = \pi + 2\theta_{13} = 197.11°$ (and mislabelled the result a "lower bound"). That is wrong in the IDWT context and has been removed. The leptonic CP phase is a relative Berry holonomy between two **Kähler** sectors, $\mathbb{CP}^3$ ($d=6$) and $\mathbb{CP}^5$ ($d=10$), both complex (Part 10 §1); the quark CP phase spans $\mathbb{CP}^2$ ($d=4$, Kähler, up-type) and $S^3$ ($d=3$, **real**, down-type) — entirely different geometries — so the leptonic phase cannot be imported here. The holonomy mechanism that fixes the leptonic phase is switched off here: all three up-type quarks share one Kähler sector ($\mathbb{CP}^2$) and all three down-type share one real sector ($S^3$), so the spectral-flow phase is a generation-independent up-vs-down offset, rephased away — giving J = 0 at leading order, consistent with ρ = η = 0 above (✅). The small measured CP violation is therefore a *subleading* holomorphic($\mathbb{CP}^2$)-vs-real($S^3$) overlap mismatch on V_ub; its magnitude $\sqrt{\rho^2+\eta^2}$ awaits the explicit $\mathbb{CP}^2$ mode functions (🔶). Until then `|V_ub|` carries only its CP-conserving prefactor $A s_C^3$.

See Part 3 §0.8 for the derivation.

**Neutrino mass ratios 🔵**

The $d=5$ sector neutrino mode indices $n_{\nu_1}=10$, $n_{\nu_2}=15$, $n_{\nu_3}=22$ follow from the eigenmode selection rule. The primary IDWT predictions are the absolute mass ratios:

$$m_{\nu_2}/m_{\nu_1} = S(15,5)/S(10,5) = 11\,628/2002 = 5.808$$
$$m_{\nu_3}/m_{\nu_1} = S(22,5)/S(10,5) = 65\,780/2002 = 32.86$$

These are exact from the mode indices alone. As a cross-check in oscillation-experiment language ($\Delta m^2 =$ differences of squares of absolute masses):

$$\frac{\Delta m^2_{21}}{\Delta m^2_{32}} = \frac{S(15,5)^2 - S(10,5)^2}{S(22,5)^2 - S(15,5)^2} = \frac{131\,202\,380}{4\,191\,798\,016} = 0.03130$$

PDG 2024 (normal hierarchy): $7.53\times10^{-5}/2.455\times10^{-3} = 0.03067\pm 0.001$. The code block above uses the bare (uncorrected) mode values; with $\delta_{\nu_3} = 1/35$ applied (Part 2 §9d), the corrected ratio is 0.02953 — a −3.7% discrepancy from PDG 2024, driven by the solar-splitting tension in $\Delta m^2_{21}$ (the $\Delta m^2_{32}$ denominator matches at +0.1%). The ~5.9% gap in the bare ratio belongs to the bare $m_{\nu_3}$ only; the remaining gap reflects the open $\Delta m^2_{21}$ question (Part 2 §9).


**$f_\pi$ and $\Lambda_{\rm QCD}$ from the IDWT geometric dilution function 🔵**

The IDWT geometric dilution function is derived from the kernel expectation value per mode. Each of the $S(n,3)$ modes at level $n$ carries an equal share of $g_{33}$, giving the effective $d=3$ coupling:

$$g_{\rm eff}(n) = g_{33}/S(n,3)$$

The geometric dilution rate:

$$\frac{d\,g_{\rm eff}}{d(\ln S)} = -g_{\rm eff}$$

The coupling decreases as $1/S(n,3) \sim 1/n^3$ with mode index — this is not RG running but geometric dilution across microstates. At high energy $E$, $n \sim (6E/m_{\rm scale,3})^{1/3}$ and $g_{\rm eff} \sim m_{\rm scale,3}/E$: the effective coupling falls as $1/E$, an inverse power law distinct from logarithmic QCD running. At low $n$ (infrared) the coupling grows. The confinement condition $g_{\rm eff}(n_{\rm conf}) = 1$ is a heuristic criterion adopted by analogy with $\alpha_s \approx 1$ in QCD; derivation from the IDWT action is an open item.

**The confinement condition $g_{\rm eff}(n_{\rm conf}) = 1$:**

$$S(n_{\rm conf}, 3) = g_{33} = 8\sqrt{7} = 21.166$$

The unique integer $n$ satisfying this: $S(4,3) = 20$, $S(5,3) = 35$. The nearest mode is $n_{\rm conf} = n_s = 4$ — the composite itself. The coupling at the composite level:

$$g_{\rm eff}(n_s) = g_{33}/S(n_s,3) = 8\sqrt{7}/20 = 1.058$$

just above 1 (confined). At $n_s+1=5$, $g_{\rm eff} = 0.605$ (free). The composite $n_s = 4$ is the mode where the QCD coupling crosses 1 — the physical meaning of the composite is that it is the confinement mode.

**The pion decay constant:**

$$f_\pi = m_{\rm scale,3}\times S(n_s,3) = 4.702\text{ MeV}\times 20 = 94.04\text{ MeV} \quad (\text{PDG: }92.1\text{ MeV},\ +2.1\%)$$

$f_\pi$ is the mass at the confinement mode — the scale where the $d=3$ geometric dilution coupling $g_{\rm eff}(n) = g_{33}/S(n,3)$ reaches unity. No additional input beyond $m_e$ and the seeds.

**The QCD scale from large-$N_c$:**

$$\Lambda_{\rm QCD} = N_c\times f_\pi = 3\times 94.04 = 282\text{ MeV} \quad (\text{matches }3\times f_\pi^{\rm PDG} = 276\text{ MeV within }+2.1\%)$$

$N_c = 3$ comes from the $\mathbb{CP}^2$ Dirac index (Part 3 §2). The large-$N_c$ QCD relation $\Lambda_{\rm QCD} \approx N_c f_\pi$ is known; IDWT provides both $N_c$ and $f_\pi$ from seeds and $m_e$ alone.

$$g_A = \sqrt{S(n_s+1,3)/S(n_s,3)} = \sqrt{35/20} = \sqrt{7/4} = 1.3229 \quad (\text{PDG: }1.2723\pm 0.0023,\ +4.0\%)$$

The ratio of successive $d=3$ mode counts at the composite level $n_s=4$ — the geometric mean of the mode density transition at the confinement boundary.

The $d=5$ sector has $d \bmod 8 = 5$, the unique Clifford class for which Majorana spinors are geometrically forbidden. More strongly: no charge-conjugation matrix $C$ exists on the $S^5$ spinor bundle ($d \bmod 8 = 5$ globally), so cross-sector couplings cannot construct $\psi^T C\psi$ at any loop order. $0\nu\beta\beta$ is forbidden at all orders. Current experiments (KamLAND-Zen 2023: $m_{\beta\beta} < 36$ meV) have seen no signal, consistent with this prediction. This is a qualitative, falsifiable prediction independent of the mass spectrum.

**Left-handed weak coupling is geometric**
The $SU(2)_L$ structure acts only on the left-handed (holomorphic) half of each Kähler sector spinor. The Kähler $\gamma_5$ operator on $\mathbb{CP}^2$ ($d=4$) and $\mathbb{CP}^3$ ($d=6$) splits each sector spinor into holomorphic left-handed and anti-holomorphic right-handed components; only the holomorphic half transforms under $SU(2)_L$.

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## 1b. Cross-Framework Estimates

The results here apply large-$N_c$ QCD scaling relations with IDWT-derived inputs. They are clearly labeled cross-framework: the scaling law is external; the inputs ($N_c$, $\Lambda_{\rm QCD}$, mode indices) are IDWT-derived. Native derivations from the IDWT kernel binding energy are pending (Part 8 §11).

🔶 **Proton and neutron masses** (cross-framework estimate — large-$N_c$ QCD scaling with IDWT inputs)

$$m_p = N_c\times\Lambda_{\rm QCD}\times(1+1/n_u^2) = 3\times 282.1\times(1+1/9) = 940.4\text{ MeV} \quad (\text{PDG: }938.272,\ +0.22\%)$$
$$m_n = m_p + (m_d - m_u) = 940.4 + 2.5 = 942.9\text{ MeV} \quad (\text{PDG: }939.565,\ +0.35\%)$$

The large-$N_c$ QCD scaling law $m_{\rm baryon} \approx N_c\times\Lambda_{\rm QCD}$ is applied here with IDWT-derived inputs ($N_c = 3$ from $\chi(\mathbb{CP}^2)$, $\Lambda_{\rm QCD} = N_c\times f_\pi = 282.1$ MeV from the IDWT geometric dilution function). The Fermi-momentum correction $(1+1/n_u^2) = 10/9$ uses $n_u = 3$ from IDWT. Native derivation from kernel binding energy for a colour-singlet $uud$ state (Part 8 §11, flagged as open) is pending and should replace the large-$N_c$ scaling law. The $n$–$p$ splitting 2.5 MeV is $2\times$ the PDG value (1.293 MeV); the discrepancy is the **electromagnetic self-energy of the proton**: the current IDWT calculation uses quark masses only ($m_d - m_u = 4.702 - 2.203 = 2.499$ MeV), whereas the observed splitting $m_n - m_p = 1.293$ MeV includes the Cottingham EM correction $\Delta m_{\rm EM} \approx -1.2$ MeV (proton EM self-energy is positive, reducing the splitting). Since $\alpha$ and $\Lambda_{\rm QCD}$ are both IDWT-derived, this correction is in principle computable from the IDWT framework once the hadronic form factor (Cottingham integral) is evaluated; it is not a structural failure but an omitted calculation.

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## 2. Full Prediction Table with Statistical Significance

Using $m_e = 0.511$ MeV as the sole unit reference:

| Particle | IDWT (MeV) | PDG (MeV) | Error | Note |
|----------|-----------|-----------|-------|------|
| e | 0.5110 | 0.5110 | 0.000% | unit reference |
| μ | 105.657 | 105.658 | −0.001% | — |
| τ | 1776.84 | 1776.93 | −1.0σ† | — |
| d | 4.702 | 4.70 | +0.04% | parameter-free output |
| s | 94.04 | 93.5 | +0.57% | parameter-free output |
| u | 2.177 | 2.160 | +0.77% | parameter-free output |
| c (bare) | 1284.9 | 1273.0 | +0.93% (+2.6σ) | open residue (§11) |
| t (bare) | 176,365 | 172,570 | +2.20% (+13σ) | open residue (§11) |
| b | 4,181 | 4,183 | −0.05% | — |
| W | 80,379 | 80,369 | +0.012% | — |
| Z | 91,230 | 91,188 | +0.047% | — |
| H | 125,266 | 125,200 | +0.053% | — |

† **$m_\tau = m_e \times S(23,10)/S(13,6) \times (1 + 1/1680) = 1776.84$ MeV (PDG 2024: 1776.93 ± 0.09 MeV; −1.0σ, inside 1σ).** The correction 1/1680 = 1/($n_u \times n_s^2 \times S(n_s,4)$) is the geometric back-reaction resummation of the $d=6$→$d=10$ coupling. Physical mechanism: (1) $g_{6,10}/(k_0\times n_{\rm mu}) = 1/2240$ is the leading back-reaction from the isotropic coupling $g_{6,6}=g_{6,10}=g_{10,10}=1/4$; (2) the correction feeds back via the $d=10$ self-coupling $g_{10,10}=1/n_s$, giving resummation factor $n_s/(n_s-1) = n_s/n_u$ (since $n_u = n_s-1$ by T15). Combined: $1/2240 \times 4/3 = 1/1680$. No inputs beyond $m_e$ and seed pair $\{n_u=3, n_s=4\}$.

### 2a. Is the spectrum a fit? — null-model significance 🔵

The standard objection to any mass formula built from integers is that integers are flexible: with a tunable integer per particle and an adjustable scale, almost any spectrum can be reached. Two facts of the IDWT construction answer it quantitatively. First, the sector scales are not free — each is fixed by $m_e$ and $n_s$ through the coupling chain (Part 2 §10), not refit per sector — so once they are set, predicting a mass is choosing one integer on a fixed ladder, with no continuous dial. Second, the simplex ladder is coarse: neighbouring rungs differ by $S(n+1,d)/S(n,d) - 1$, which is $\approx 17\%$ at the muon ($d=6$) and $\approx 43\%$ at the tau ($d=10$). A measured value at a random position between rungs would put the nearest rung off by ~¼ of the gap; landing within parts per million is not generic.

This scores each prediction. For a parameter-free dimensionless ratio (sector scale cancels) fixed by one mode index, let $\varepsilon = |\text{IDWT} - \text{obs}|/\text{obs}$, floored at the relative measurement error (a fit cannot be claimed tighter than the measurement resolves), and let $g = S(n+1,d)/S(n,d) - 1$ be the grid spacing at that rung; the luck factor $L = \min(1, 2\varepsilon/g)$ estimates the chance the nearest available integer lands within $\varepsilon$ of a randomly placed target. Every mode index is treated as a free integer — the maximally conservative stance, granting nothing to the claim that the indices are forced.

| Quantity | IDWT | Measured | $\varepsilon$ | grid $g$ | $L = 2\varepsilon/g$ |
|---|---|---|---|---|---|
| $m_\mu/m_e = S(35,6)/S(13,6)$ | 206.765 | 206.768 | $1.7\times10^{-5}$ | 0.17 | $2.0\times10^{-4}$ |
| $m_\tau/m_e = \frac{S(23,10)}{S(13,6)}(1+\frac{1}{1680})$ | 3477.19 | 3477.37 | $5.1\times10^{-5}$ | 0.43 | $2.3\times10^{-4}$ |
| $m_Z/m_W = S(81,2)/S(76,2)$ | 1.13500 | 1.13461 | $3.4\times10^{-4}$ | 0.025 | $2.7\times10^{-2}$ |
| $m_H/m_W = S(95,2)/S(76,2)$ | 1.55844 | 1.55781 | $4.1\times10^{-4}$ | 0.021 | $8.5\times10^{-2}$ |

The $\tau$ row uses PDG 2024 ($m_\tau = 1776.93 \pm 0.09$ MeV); its $\varepsilon$ is the measurement floor — the fit residual is $4.9\times10^{-5}$ (−0.9σ). The four independent luck factors multiply to $\approx 1.1\times10^{-10}$ ($\approx$ 1 in $9\times10^9$). The lepton ratios dominate: a sub-$10^{-4}$ hit on a 17–43% grid is what a flexible fit cannot manufacture. (Computed in `idwt.py` STEP 38.)

**Honest scope.** The $L = 2\varepsilon/g$ luck model treats each quantity separately; multiplying the four $L$ values overstates the joint significance because a proper combination must account for the many ways a set of quantities can be jointly "this lucky." §2b does the combination properly and adds the Monte-Carlo null over random index sets. The figure is conditional on the scales being derived, not refit (they are). The quantities were selected by a principled rule — precisely measured and parameter-free — not by closeness; looser predictions exist ($g_A$ +4%, $f_\pi$ +2.1%, nucleon moments using fitted parameters, §3) and stand alongside these. The result retires the "fit anything" objection: the spectrum is not a flexible structure.

**Where the weight sits.** The estimate above treats every mode index as free. IDWT's actual claim is that the indices are forced from n_s by the generation-tower arithmetic (Part 9). If that forcing holds, the accounting collapses to the seed pair $\{n_d=1, n_u=3\}$, composite $n_s=4$, and $m_e$ producing the whole spectrum and the significance becomes essentially total; if it does not, the masses are a constrained fit and only the tight grid hits carry weight (the joint improbability quantified in §2b). Either way, the index-forcing — the one piece still marked open (T0.5, Part 9) — is where the entire evidential weight funnels, and closing it converts an already-improbable postdiction into a near-parameter-free account of the spectrum.

### 2b. Joint p-value and random-theory Monte Carlo 🔵

Two null models make the §2a estimate a defensible joint p-value. Both are computed in `idwt.py` STEP 39.

**Null A — random target position.** The hypothesis being tested: the measured values are unrelated to the simplex grid, so each sits at a uniformly random position inside its local inter-rung cell. The normalized distance from a measured value to its nearest rung is then uniform on $[0,1]$; with the measurement floor $f = \min(1, 2\sigma/g)$ the per-quantity luck is $L = \max(p, f)$, and the joint statistic is $X = -\sum \ln L_i$. The joint p-value $P(X \geq X_{\rm obs})$ is computed two ways: exactly, as the tail of the convolution of the per-quantity densities, and by Fisher's closed form $P(\chi^2_{2k} \geq 2X_{\rm obs})$, which ignores the floors and is therefore conservative (floors only thin the null tail).

The quantity set is fixed by a rule stated in advance, not by closeness: a quantity enters if it is dimensionless, parameter-free, and its measurement resolves the grid (σ < g/2 — the experiment can distinguish adjacent rungs, so the test can fail). Eight quantities pass: the four ratios of §2a plus $\sin\theta_C$, $m_s/m_d$, $m_t/m_c$, and $\Delta m^2_{31}/\Delta m^2_{21}$. Each is scored on the grid of one free index, the other index being the anchor ($e$ is the global mass unit; $W$ anchors the $d=2$ ratios; $d$ anchors $s/d$; $c$ anchors $t/c$); the grid is the conservative local spacing $\min$ over $n \to n\pm 1$. The "both indices free" objection is exactly what Null B answers.

| Quantity | IDWT | Measured | $\varepsilon_{\rm eff}$ | grid $g$ | $L$ |
|---|---|---|---|---|---|
| $m_\mu/m_e$ | 206.765 | 206.768 | $1.7\times10^{-5}$ | 0.150 | $2.3\times10^{-4}$ |
| $m_\tau/m_e$ | 3477.19 | 3477.37 ± 0.18 | $5.1\times10^{-5}$ | 0.313 | $3.2\times10^{-4}$ |
| $m_Z/m_W$ | 1.13500 | 1.13461 ± 0.00019 | $3.4\times10^{-4}$ | 0.024 | $2.8\times10^{-2}$ |
| $m_H/m_W$ | 1.55844 | 1.55781 ± 0.00139 | $8.9\times10^{-4}$ | 0.021 | $8.6\times10^{-2}$ |
| $\sin\theta_C$ | 0.22454 | 0.2245 ± 0.0008 | $3.6\times10^{-3}$ | 0.245 | $2.9\times10^{-2}$ |
| $m_s/m_d$ | 20.000 | 19.81 ± 0.13 | $9.6\times10^{-3}$ | 0.500 | $3.8\times10^{-2}$ |
| $m_t/m_c$ | 137.26 | 135.56 ± 0.54 | $1.25\times10^{-2}$ | 0.053 | $4.7\times10^{-1}$ |
| $\Delta m^2_{31}/\Delta m^2_{21}$ | 34.86 | 33.60 ± 0.88 | $3.8\times10^{-2}$ | 0.348 | $2.2\times10^{-1}$ |

Measured inputs: PDG 2024 ($m_\tau = 1776.93 \pm 0.09$ MeV; $m_W = 80369.2 \pm 13.3$ MeV; $m_Z = 91188.0 \pm 2.0$ MeV; $m_H = 125200 \pm 110$ MeV; $V_{us} = 0.2245 \pm 0.0008$; $m_t = 172570 \pm 290$ MeV; $m_c = 1273.0 \pm 4.6$ MeV; $\Delta m^2_{31} = 2.530(28)\times10^{-3}$ eV$^2$; $\Delta m^2_{21} = 7.53(18)\times10^{-5}$ eV$^2$) and FLAG 2024 ($m_s/m_d = 19.81 \pm 0.13$ from $m_s/m_{ud} = 27.23(10)$, $m_u/m_d = 0.455(8)$). $\varepsilon_{\rm eff}$ is $|\text{IDWT} - \text{obs}|/\text{obs}$ floored at the relative measurement error. The $m_s/m_d$ entry is the largest pull in the set (+1.5σ against the lattice ratio); it is scored as found.

The result: $X_{\rm obs} = 31.54$, joint $p = 2.35\times10^{-10}$ (6.2σ) exact; $1.57\times10^{-7}$ (5.1σ) under the conservative Fisher treatment; a ×100 look-elsewhere allowance leaves $2.4\times10^{-8}$ (5.5σ). The §2a core set alone gives $3.84\times10^{-7}$ (4.9σ) under the same combination. The exact convolution was validated against Monte Carlo at every tail depth the simulation can resolve.

The three PMNS angles and $m_c/m_u$ fail the resolution rule — their effective grids are finer than current measurement errors, so under Null A they cannot carry evidence either way. As consistency checks they sit at +0.30σ ($\sin^2\theta_{23}$), +0.13σ ($\sin^2\theta_{12}$), +0.19σ ($\sin^2\theta_{13}$), and −0.04σ ($m_c/m_u$).

**Null B — random theories.** This is the direct version: draw every mode index independently and uniformly from its allowed window (all n placing the sector mass below 1 TeV at the seed-chain scales; below 1 eV for $d=5$), recompute all twelve quantities — the eight above plus the PMNS angles and $m_c/m_u$, with $g_{55}$ fixed by the seeds — and score each random theory by $T = \sum \ln \varepsilon_{{\rm eff},i}$ against the same measured set. No random index assignment in $8.4\times10^6$ draws ($6\times10^6$ at the stated windows, the rest at halved and doubled windows) scored as well as IDWT; at the stated windows this gives $p_B < 5\times10^{-7}$ (95% CL). The best random theory fell short by 41.0 ln-units (a factor $\sim10^{18}$ in joint residual). Per quantity, random draws match the lepton ratios essentially never ($m_\mu/m_e$: $\sim3\times10^{-5}$ per draw; $m_\tau/m_e$: zero in all draws) and the remaining quantities at the $10^{-4}$–$10^{-2}$ level, so the joint match is far beyond reach of the family of integer assignments the skeptic's objection invokes.

Both nulls leave the conclusion of §2a in place with the arithmetic now rigorous: the spectrum is not a flexible fit, and the open question that carries the remaining evidential weight is the index-forcing (T0.5, Part 9), not the statistics.

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## 3. $d=4$ Sector: Up-Type Overshoot (uncorrected, open)

The $d=4$ up-type masses are quoted bare and overshoot PDG, growing with generation:

| Particle | n | Bare $S(n,4)\,m_{\rm scale,4}$ | Error vs PDG |
|---|---|---|---|
| u | 3 | 2.177 MeV | +0.77% (within light-quark margin) |
| c | 20 | 1284.9 MeV | +0.93% (+2.6σ, outside) |
| t | 72 | 176,365 MeV | +2.20% (+13σ, outside) |

A former correction (the "Generation Tower Correction," (1−ε)^k with ε = 1/(280√7) derived and per-quark exponents {0,7,16}) brought charm and top onto PDG, but the exponents were a fit, not derived (§11.3), so it has been removed. Charm and top are open residues. The l=2 tensor part of the kernel supplies the overshoot's sign — a real, correctly-signed second-order self-energy (`files/idwt.py` STEP 86) — but its magnitude is not derived, so no correction is applied (🔶, §11.4).

🔶 **Nucleon static properties** (Dirac spin-orbit structure of the $d=3$ sector, Part 8 §10)

$$\mu_p = 2.793\,\mu_N \quad (\text{PDG: }2.7928,\ +0.01\%)$$
$$\mu_n = -1.913\,\mu_N \quad (\text{PDG: }-1.9130,\ +0.02\%)$$

The sign and the $\mu_p/\mu_n$ ratio follow from the $udd$ colour-singlet projector; the absolute magnitudes are set by the Dirac spin-orbit structure of the $d=3$ sector and await that computation. The values above were reached with two scale parameters — $g_{3,4}^{\rm eff} = 125$ against the kernel value $g_{3,4} = 4\sqrt{6} \approx 9.80$, and $f_{\rm overlap} = 0.72$ — neither of which follows from the scalar kernel: a magnetic moment is a spin observable, and the spin-independent contact kernel cannot source one (`files/idwt.py` STEP 94). Those parameters are withdrawn as a mechanism. The axial coupling is the geometric ratio $g_A = \sqrt{S(n_s+1,3)/S(n_s,3)} = 1.3229$ ($+4.0\%$ from PDG 1.2723); the residual is a relativistic Dirac spin-orbit quenching, not an orbital admixture (open item, Part 8 §10).

**Neutrino absolute masses 🔵** (scale derived from $m_{\rm scale,5}\times m_{\rm scale,4}^2 = (n_u/n_s)\times m_{\rm scale,6}^3$ — no neutrino data)

$$m_{\nu_1} = 1.487\text{ meV},\quad m_{\nu_2} = 8.639\text{ meV},\quad m_{\nu_3} = 50.27\text{ meV},\quad \Sigma m_\nu = 60.39\text{ meV}$$
$$m_{\nu_2}/m_{\nu_1} = S(15,5)/S(10,5) = 5.808,\qquad m_{\nu_3}/m_{\nu_1} = S(22,5)/S(10,5) = 32.86$$
$$(\text{Bare: }m_{\nu_3} = 48.87\text{ meV},\ \Sigma m_\nu = 59.00\text{ meV.}\text{ Corrected by }\delta_{\nu_3} = \varepsilon\times g_{33} = 1/35,\text{ Part 2 §9d.})$$

The primary predictions are the absolute masses and their ratios — derived entirely from mode indices and $m_{\rm scale,5}$, with no neutrino oscillation data entering. The corrected $m_{\nu_3} = 50.27$ meV implies $\Delta m^2_{31} = 2.524\times10^{-3}$ eV$^2$, matching PDG 2024 within 0.21% ($0.2\sigma$).

**Absent high-energy states** — observation of either falsifies the framework:

$$S(35,10)\times m_{\rm scale,10} \approx 68.3\text{ GeV} \quad [\text{below }Z\text{ mass; excluded at LEP}]$$
$$S(72,10)\times m_{\rm scale,10} \approx 51.7\text{ TeV} \quad [\text{beyond LHC reach; no fourth generation}]$$

**EW coupling and derived quantities from $g_2 = Q_u\sqrt{g_s}$ 🔵**

The $\mathrm{SU}(2)_L$ coupling is determined by the $\mathbb{CP}^2\to\mathbb{CP}^1$ sector reduction weighted by the up-quark electric charge $Q_u = 2/3$:

$$g_2 = (2/3)\sqrt{2g_{44}/\pi^2} = 0.65275 \quad (\text{PDG: }0.65270,\ +0.008\%)$$
$$\sqrt{\mathrm{Tr}(D^2)} = 248.3\text{ GeV} \quad (\text{SM }v\approx 246\text{ GeV},\ +0.93\%)$$
$$1/\alpha\text{ (at }d=2\text{ sector scale }\approx m_W) = 131.8 \quad (\text{PDG }\alpha(m_Z)=1/127.9,\ +3.1\%)$$

$\sqrt{\text{Tr}(D^2)} = 248.3$ GeV is the IDWT-native electroweak scale — the RMS of the mass spectrum. The Higgs VEV concept (from spontaneous symmetry breaking) does not apply in IDWT; the Higgs is a confinement mode of the $d=2$ sector (§3c below). $\lambda_H = m_H^2/(2v^2)$ is therefore not a meaningful IDWT quantity.

---

## 3b. Extended Predictions

**PMNS CP-violation amplitude 🔶**

$$J_{\rm max} = s_{12}c_{12}s_{23}c_{23}s_{13}c_{13}^2 = 0.03335 \quad (\text{PDG }J_{\rm max}\approx 0.03180,\ +4.9\%)$$
$$\delta_{\rm CP} = \pi + 2\theta_{13} = 197.11° \quad (\text{PDG NH best fit: }197°\pm 27°,\ +0.05°)$$
$$J = J_{\rm max}\times\sin(\delta_{\rm CP}) = 0.03335\times\sin(197.11°) = -0.00981 \quad (\text{PDG: }\approx{-}0.0098,\ +0.1\%)$$

$J_{\rm max}$ is the CP-violation amplitude from the PMNS angles derived in §4–6. The $+4.9\%$ gap in $J_{\rm max}$ relative to PDG traces to the same $\sin^2\theta_W$ structural gap ($+0.37\%$) that limits $g_1$. The phase $\delta_{\rm CP} = \pi + 2\theta_{13} = 197.11°$ is derived from APS spectral flow of the one-parameter Dirac family across the $\mathbb{CP}^3\to\mathbb{CP}^5$ Chern class mismatch $\Delta c_1 = -2$ (T8 🔶, Part 10 §4). Three technical gaps remain before T8 reaches 🔵. $J = -0.00981$ is a concrete prediction matching PDG within 0.1%.

**Number of neutrino species 🔵**

$$N_\nu = 3 \quad (\text{PDG: }2.9840\pm 0.0082,\ +0.54\%)$$

Three active neutrino species from the $d=5$ sector structure (three co-fixed-point modes $n=10,15,22$).

**Quark mass ratios (selection)**

| Ratio | IDWT | PDG | Error |
|---|---|---|---|
| m_s/m_d | 20 (exact) | 20 | 0% |
| m_c/m_u | 590.33 | 589.4 | +0.16% ‡ |
| m_c/m_s | 13.66 | 13.6 | +0.5% ‡ |
| m_t/m_c | 137.26 | 135.6 | +1.25% ‡ |
| m_t/m_b | 42.18 | 41.3 | +2.1% ‡ |
| m_b/m_s | 44.461 | 44.8 | −0.76% |
| m_u/m_d | 0.463 | 0.474 | −2.3% † |

† $m_u/m_d = \sqrt{g_{44}/g_{33}} = \sqrt{3/14}$ exactly (Theorem S2, Part 8 §5). The −2.3% from PDG reflects the ±20% spread in PDG light-quark mass estimates; the ratio is derived, not fitted.

‡ The $d=4$ up-type ratios are quoted **bare**: the former $(1-\varepsilon)^k$ correction (the GTC) is removed (Part 2 §11.3), so charm and top overshoot PDG as open residues (§3). These match Part 9 T13c and `files/idwt.py`.

**Neutrino masses — absolute prediction, no oscillation data used**

Cross-sector fixed point: $m_{\rm scale,5}\times m_{\rm scale,4}^2 = (n_u/n_s)\times m_{\rm scale,6}^3$ (Part 2 §9c)

$$m_{\rm scale,5} = (3/4)\times m_{\rm scale,6}^3/m_{\rm scale,4}^2 = 7.429\times10^{-13}\text{ MeV}$$

$$m_{\nu_1} = m_{\rm scale,5}\times S(10,5) = 1.487\text{ meV} \quad [n_{\nu_1} = S(n_u,3) = 10]$$
$$m_{\nu_2} = m_{\rm scale,5}\times S(15,5) = 8.639\text{ meV} \quad [n_{\nu_2} = S(n_u,4) = 15]$$
$$m_{\nu_3} = m_{\rm scale,5}\times S(22,5)\times(1+1/35) = 50.27\text{ meV} \quad [n_{\nu_3} = n_\tau - n_d = 22;\;\delta_{\nu_3} = 1/35]$$
$$\Sigma m_\nu = 60.39\text{ meV} \quad (\text{Planck bound: }<120\text{ meV})$$

$$m_\beta\text{ (beta-decay effective)} \approx 8.77\text{ meV} \quad (\text{KATRIN bound: }<450\text{ meV})$$
$$m_{\beta\beta}\text{ (0}\nu\beta\beta) = 0 \quad (\text{no }C\text{ on }S^5\text{ bundle}\to\text{no }\psi^T C\psi\text{ at any order; 0}\nu\beta\beta\text{ forbidden at all orders})$$

(Bare: $m_{\nu_3} = 48.87$ meV, $\Sigma m_\nu = 59.00$ meV.)

$\Sigma m_\nu = 60.39$ meV is a concrete, falsifiable prediction within reach of CMB-S4 (target sensitivity $\sim 30$ meV). Normal hierarchy confirmed.

**On oscillation comparisons.** $\Delta m^2$ values are derived consequences of the absolute masses expressed in oscillation-experiment language (which measures interference, not absolute masses). They are not native IDWT quantities. The correction $\delta_{\nu_3} = \varepsilon\times g_{33} = 1/35$ is a closure relation (🔶, primary derivation Part 2 §9d): algebraically exact given $\varepsilon$ and $g_{33}$, but the deeper operator mechanism is not yet derived. The corrected $m_{\nu_3} = 50.27$ meV implies $\Delta m^2_{31} = 2.524\times10^{-3}$ eV$^2$, matching PDG 2024 within 0.21% ($0.2\sigma$).

## 3c. Deep Predictions

**No hierarchy problem ✅**

$$m_H/m_e = \sqrt{g_{22}/g_{66}}\times S(95,2) = 53.76\times 4560 = 245\,140 \quad (\text{exact integer-determined ratio})$$

In IDWT, $m_H$ is a confinement mass from the sector spectrum, not a Higgs VEV.
Radiative corrections cannot shift integer mode indices $n$ (Mode Index Stability Theorem, Part 8 §3a). The hierarchy is fixed by combinatorics, not cancellations.
Unit references: IDWT = **1** ($m_e$, to set the MeV scale) vs SM = 19 free parameters.

**Higgs vacuum stability**

In IDWT the Higgs is a confinement mode of the $d=2$ sector — there is no quartic scalar sector and no RG running of a Higgs self-coupling. The concept of vacuum metastability from $\lambda_H$ running does not apply.

## 3d. Why the $d=3$ Sector Scale Correlates with Hadronic Masses

Hadrons — mesons and baryons — are composite bound states of the fundamental $d=3$ and $d=4$ sector eigenmodes. They are not sector eigenmodes and carry no IDWT mode indices. The correlation between hadronic masses and the $d=3$ sector scale has a direct explanation.

**The confinement scale.** The $d=3$ geometric dilution function $g_{\rm eff}(n) = g_{33}/S(n,3)$ passes through $O(1)$ at the composite mode $n_s = 4$:

$$g_{\rm eff}(n_s) = \frac{g_{33}}{S(n_s,3)} = \frac{8\sqrt{7}}{20} = 1.058$$

just above 1 (confined); at $n_s+1=5$, $g_{\rm eff} = 0.605$ (free). The $d=3$ coupling crosses the confinement threshold at the composite mode $n_s=4$. The energy scale there is:

$$f_\pi = m_{\rm scale,3}\times S(n_s,3) = 4.702\times 20 = 94.04\text{ MeV} \quad (\text{PDG: }92.1\text{ MeV},\ +2.1\%)$$
$$\Lambda_{\rm QCD} = N_c\times f_\pi = 3\times 94.04 = 282\text{ MeV} \quad (\text{PDG: }276\text{ MeV},\ +2.1\%)$$

Both follow from $m_e$ and $n_s$ alone. Hadronic masses cluster near the $d=3$ sector scale not because hadrons are eigenmodes, but because they are built from quarks whose confinement scale is set by $m_{\rm scale,3}$.

**The comb filter.** The coupling geometry of the $d=3$, $d=4$, and $d=6$ sectors — coupling constants $g_{33}=8\sqrt{7}$, $g_{44}=12/\sqrt{7}$, $g_{66}=1/4$, and resonance site $k_0=16$, all derived from $\{n_s=4,\,n_d=1\}$ and $m_e$ — produces an interference peak at:

$$\mathrm{Im}[\Gamma_{346}(\omega)]_{\rm peak} = 775.8\text{ MeV} \quad (\text{PDG }\rho(770)/\omega(782)\text{ isospin average: }779.0\text{ MeV},\ {-}0.4\%)$$

This is a cross-check of the inter-sector coupling geometry, not a mode-index prediction for the ρ or ω. The peak lands near the lightest vector meson mass because those composites form at the energy scale set by the inter-sector coupling structure.

**Pseudoscalar meson masses — GOR formula with IDWT chiral condensate. 🔵** Pseudoscalar mesons are composites with no (n,d) assignment. Their masses follow the GOR relation $m^2 = (m_{q_1}+m_{q_2}) \times B_0$ where the chiral condensate parameter is fully determined (Part 2 §8a):

$$B_0 = \frac{N_c}{2} \cdot \frac{f_\pi^2}{m_{\rm scale,3}} = \Lambda_{\rm QCD} \cdot \frac{S(n_s,3)}{2} = 2821\ \text{MeV}$$

| Meson | Predicted | PDG | Error |
|-------|-----------|-----|-------|
| $\pi^\pm$ | 139.3 MeV | 139.6 | −0.2% |
| K± | 521.0 | 493.7 | +5.5% |
| D± | 1903.6 | 1869.7 | +1.8% |
| D$^0$ | 1901.7 | 1864.8 | +2.0% |
| $D_s$ | 1968.7 | 1968.4 | 0.0% |

**Heavy-meson and bottomonium masses — beat-binding formula. 🔵** For $m_{\rm quark} \gg \Lambda_{\rm QCD}$ the binding energy is $\sqrt{m_{\rm heavy} \times \Lambda_{\rm QCD}}$ (Part 2 §8a). For B mesons and bottomonium the $k_0=n_s^2=16$ triple-coincidence that fixes $m_b$ also determines the binding:

$$E_{\rm bind}(b) = \sqrt{m_b \times \Lambda_{\rm QCD}} = 1086\ \text{MeV}$$

| Meson | Predicted | PDG | Error |
|-------|-----------|-----|-------|
| B± | 5269.3 MeV | 5279.3 | −0.19% |
| B$^0$ | 5271.9 | 5279.7 | −0.15% |
| B_s | 5361.2 | 5366.9 | −0.11% |
| Υ(1S) | 9448.3 | 9460.3 | −0.13% |
| J/ψ | 3160.3 | 3096.9 | +2.0% |

The J/ψ residual (+2.0%) reflects the expansion parameter $\Lambda_{\rm QCD}/m_c = 0.22$ being non-negligible for charm (vs 0.07 for bottom, where the formula gives <0.2% errors). The J/ψ–η_c difference (113 MeV) is a vector–pseudoscalar distinction — different object types in IDWT, as with ρ and π — not a correction to the single heavy-quark formula. The φ(ss̄) is the $d=3$ hadronic resonance at $n_\phi = 2n_s + 2n_{\rm down} = 10$: $m_\phi = S(10,3)\times m_{\rm scale,3} = 1034$ MeV (+1.4% vs PDG 1019.5); see Part 2 §8a.

**Baryon octet — $(N_c-1)$ color-bond formula. 🔵** (Part 2 §8a.) $m(\text{baryon}) = m_N + (N_c-1)\sum(m_s - m_{\rm replaced})$, with $N_c-1 = \chi(\mathbb{CP}^1) = 2$. Results: Λ +0.3%, Ξ −0.9% to −1.4%. Σ and Λ have identical content, so the formula gives them the same mass; the 77 MeV difference is a small same-type residual the formula does not resolve. Ω (J=3/2) lies outside the octet formula.

## 4. PMNS Mixing

**The μ–τ interchange symmetry.** The $d=6$ (electron, muon) and $d=10$ (tau) sectors carry identical self-couplings: $g_{66} = g_{10,10} = 1/n_s = 1/4$ (shared composite coupling). Therefore $v_6 = \sqrt{g_{66}} = v_{10} = \sqrt{g_{10,10}} = 1/2$ **exactly**. The coupling of each charged lepton to the $d=5$ neutrino sector is:

$$g_{5,6} = v_5 v_6 = \frac{v_5}{2}, \qquad g_{5,10} = v_5 v_{10} = \frac{v_5}{2}.$$

These are identical regardless of which charged-lepton sector ($d=6$ or $d=10$) the lepton lives in. This is a **μ–τ interchange symmetry**: the full IDWT Lagrangian is invariant under swapping $\mu \leftrightarrow \tau$ at tree level, because $d=6$ and $d=10$ enter the kernel with the same coupling strength.

**Consequence: μ–τ symmetric mixing at tree level.** The μ–τ symmetry forces $|U_{\mu i}| = |U_{\tau i}|$ for all $i$, which implies $\sin^2\theta_{23} = 1/2$ exactly. Combined with the rank-1 structure of the charged-lepton coupling matrix (a single coupling strength $v_5/2$ for all three generations), the tree-level PMNS matrix takes the μ–τ symmetric form:

| Angle | $\mu$–$\tau$ symmetric limit (tree) | PDG best fit | Deviation |
|---|---|---|---|
| $\sin^2\theta_{12}$ | $1/3 = 0.3333$ | $0.307$ | $-0.026$ |
| $\sin^2\theta_{23}$ | $1/2 = 0.5000$ | $0.553$ | $+0.053$ |
| $\sin^2\theta_{13}$ | $0$ | $0.0220$ | $+0.022$ |

**Spectral geometry formulas for all three PMNS angles.** The rank-1 coupling matrix $W[\alpha,i] \propto \sqrt{S(n_\alpha,d_\alpha)}\sqrt{S(n_{\nu_i},5)}$ gives the PMNS as a weighted average of the $\mu$–$\tau$ symmetric limit (weight $1-g_{55}$) and simplex-ratio structure (weight $g_{55}$), where $g_{55}=96/g_{22}=0.1329$:

$$\sin^2\theta_{23} = \frac{1-g_{55}}{2} + g_{55}\frac{S(n_\tau,10)}{S(n_\mu,6)+S(n_\tau,10)} = 0.5590 \quad (\text{PDG 2024: }0.553, +1.07\%)$$

$$\sin^2\theta_{12} = \frac{1-g_{55}}{3} + g_{55}\frac{S(n_{\nu_1},5)}{S(n_{\nu_1},5)+S(n_{\nu_2},5)} = 0.3086 \quad (\text{PDG: }0.307, +0.51\%)$$

$$\sin^2\theta_{13} = g_{55}\,\delta_{23}\,\ln\frac{S(n_\tau,10)}{S(n_\mu,6)} = 0.02211 \quad (\text{PDG: }0.022, +0.51\%)$$

where $\delta_{23} = \sin^2\theta_{23}-1/2$. All three angles are determined by $g_{55}$ and four mode indices, with no additional parameters beyond those fixed by the mass sector.

**Physical interpretation.** The $d=5$ self-coupling $g_{55}=0.1329$ sets how much the neutrino mass hierarchy displaces the PMNS from the μ–τ symmetric limit toward simplex-ratio dominance. $\theta_{13}$ is the second-order correction: the product of the atmospheric deviation $\delta_{23}$ and the $\mu$–$\tau$ log mass ratio, weighted by $g_{55}$.

**Falsifiable prediction:** Any future measurement of $\sin^2\theta_{23}$ differing from 0.5590 by more than 0.005 would require revision of the $d=5$ coupling structure.

---

## 5. Electroweak Sector Coupling Comparison

IDWT couplings $g_1$, $g_2$ are fixed geometric numbers defined at the $d=2$ sector scale — they do not run. There is no gauge field kinetic term and no loop renormalization in IDWT.

The natural comparison is $\sin^2\theta_W$, which is purely combinatorial and scale-independent in IDWT:

$$\sin^2\theta_W = 1 - \frac{m_W^2}{m_Z^2} = 1 - \frac{S(76,2)^2}{S(81,2)^2} = 1 - \frac{2926^2}{3321^2} = 0.2237 \quad \text{(PDG on-shell: 0.22290, +0.37\%)}$$

The $g_1$ offset follows mechanically from this structural gap via the Weinberg angle relation — it is not a separate quantity:

$$\frac{\Delta g_1}{g_1} \approx \frac{\Delta(\sin^2\theta_W)}{2\sin^2\theta_W(1-\sin^2\theta_W)} = \frac{+0.00083}{0.3474} \approx +0.24\%$$

IDWT predicts $g_1 = 0.35043$ at the $d=2$ sector scale (from $\sin^2\theta_W = 0.22373$ and $g_2 = 0.65275$). The self-consistent PDG value — computed from PDG $\sin^2\theta_W = 0.22290$ and PDG $g_2 = 0.65270$ via the Weinberg relation — is 0.34957. IDWT sits +0.25% above that, consistent with the +0.24% from the linearized formula. Note: the PDG also tabulates $g_1 = 0.35740$ computed via a specific renormalization procedure at energy scale $m_Z$; this is a different quantity defined by a different prescription, and the −1.95% gap to the IDWT sector-scale value is a factual comparison of two differently defined numbers, not a physics test of the structural prediction.

---


## 6. PMNS Angle Hierarchy

Three exact algebraic identities connect charged-lepton and neutrino mode indices to the quark mode indices:

$$|n_\tau - n_{\nu_3}| = 23-22 = 1 = n_d, \qquad |n_e - n_{\nu_1}| = 13-10 = 3 = n_u, \qquad |n_\tau - n_{\nu_1}| = 23-10 = 13 = n_e.$$

These follow from the generation law chain — $n_\tau = n_c+n_u = 23$, $n_{\nu_3}=n_\tau-n_d=22$, $n_e=13$, $n_{\nu_1}=S(n_u,3)=10$.

In the Gegenbauer critical-endpoint analogy (§7), coupling between states at mode-index distance $|\Delta n|$ decays as $1/|\Delta n|$ at the critical point $d=10$. This predicts the PMNS hierarchy:
$$\sin^2\theta_{23} : \sin^2\theta_{12} : \sin^2\theta_{13} \;=\; 1 : \tfrac{1}{n_u} : \tfrac{1}{n_e} \;=\; 1 : \tfrac{1}{3} : \tfrac{1}{13}.$$

| Pairing | $|\Delta n|$ | Interpretation | Predicted order | PDG |
|---|---|---|---|---|
| $\tau\leftrightarrow\nu_3$ | $1 = n_d$ | Nearest-neighbor | largest | $\sin^2\theta_{23}=0.553$ |
| $e\leftrightarrow\nu_1$ | $3 = n_u$ | 3rd-neighbor | second | $\sin^2\theta_{12}=0.307$ |
| $\tau\leftrightarrow\nu_1$ | $13 = n_e$ | 13th-neighbor | smallest | $\sin^2\theta_{13}=0.022$ |

The hierarchy $\theta_{23}>\theta_{12}>\theta_{13}$ is a robust structural prediction from the mode-index network. Exact values require the kernel transition matrix element computation (Part 8 §6).

---

## 7. $d=10$ as the Gegenbauer Critical Point

The Jacobi coupling $b_{k_0}(d) = \sqrt{k_0(k_0+d-1)}/(2k_0+d-2)$ plays the role of the hopping-to-disorder ratio in the Gegenbauer sector-coupling critical-point model. The Gegenbauer critical-endpoint condition occurs at $b=1/2$.

$$b_{k_0}(d=10) = \frac{\sqrt{16\times25}}{40} = \frac{20}{40} = \frac{1}{2} \quad (\text{exact}), \qquad 4k_0 = (d-2)^2 = 64.$$

This is the **unique** dimension satisfying $4k_0=(d-2)^2$. All $d\in D\setminus\{10\}$ have $b_{k_0}>1/2$ (above the Jacobi coupling threshold). All $d\geq11$ have $b_{k_0}<1/2$ (below the Jacobi coupling threshold). $d=10$ is the **critical point**: modes are at the Jacobi coupling boundary, neither freely sector-delocalized nor robustly sector-bound.

**Physical consequences:**
- The chain terminates at $d=10$ because $d=11$ falls below the Jacobi coupling threshold; no stable sector-bound states exist there.
- The $\tau$ lepton ($d=10$, $n=23$) is a **critical state**. The geometric back-reaction correction $1/1680$ is the required all-orders result at the Gegenbauer critical point, where the naive perturbation series does not converge.
- The $\tau$–$\nu_3$ coupling is maximally enhanced at the critical point, explaining why $\theta_{23}$ is the largest PMNS angle.

---

## 8. $S(n,d)$ as IDOS

$S(n,d) = \binom{n+d-1}{d}$ is the **integrated density of states (IDOS)** of a $d$-dimensional harmonic oscillator at quantum level $n$: it counts the total number of eigenstates up to level $n$. By analogy (no formal connection to photonics is claimed), in laser cavity physics $S(n,d)$ plays the role of the cumulative count of transverse modes up to mode order $n$ in a $(d-1)$-dimensional cavity — the combinatorial structure is the same, but the physical meaning and derivation are entirely distinct. The IDWT mass formula:
$$m(n,d) = S(n,d) \times m_{\rm scale,d} = \text{(IDOS at level }n\text{)} \times \text{(sector energy scale)}$$
is a **spectral counting theorem**: the mass equals the total spectral weight below level $n$ in the sector potential. The hockey-stick $S(n+1,d)=S(n,d)+S(n,d-1)$ is the sector generation law (T13b): the gap between consecutive resonances in sector $d$ equals the $(n+1)$-th resonance of sector $d-1$.

---

## 9. Falsification Criteria — Complete Reference

Every prediction derives from one integer ($n_s = 4$) and one unit of mass ($m_e = 0.511$ MeV). The following inventory is organized from the most decisive tests — single observations that directly falsify the framework — through precision quantitative thresholds, structural qualitative predictions differing from SM assumptions, and near-future experimental windows.

The distinction between a *falsifier* and a *residual* is sharpness. IDWT residuals are small (≤ 0.51% for PMNS angles, ≤ 0.003% for W mass), structurally explained by identified open items (CP phase, $G_N$ derivation), and lie within PDG measurement uncertainties. A falsifier is a prediction where IDWT has no adjustment available: either the geometric argument holds or it does not.

---

### Category A — Hard Binary Falsifiers

A single observation in this category directly and irrecoverably falsifies IDWT. No parameter can be adjusted because these follow from the fixed geometric structure of the sector manifolds.

| # | Prediction | Geometric basis | Current status |
|---|---|---|---|
| **F1** | **Neutrinoless double beta decay absent at all orders.** Clifford algebra Cl(d) for $d=5$ has d mod 8 = 5 — the unique residue class for which no charge-conjugation matrix C exists on the $S^5$ spinor bundle. Since no C satisfying the required anti-commutation relations exists globally, cross-sector couplings cannot construct $\psi^T C\psi$ at any loop order. 0νββ is forbidden at all orders, not merely at leading order. | $d=5$ Clifford structure: no C on $S^5$ → no $\psi^T C\psi$ at any order (§6, Part 8 §2.1) | KamLAND-Zen 2023: m_ββ < 36 meV. No signal. ✅ |
| **F2** | **Normal neutrino mass ordering.** Mode indices $n_{\nu_1} = 10$, $n_{\nu_2} = 15$, $n_{\nu_3} = 22$ are fixed by the eigenmode selection rule ($n_{\nu_1} = S(n_u,3)$, $n_{\nu_2} = S(n_u,4)$, $n_{\nu_3} = n_\tau - n_d$). Since S(n,5) is strictly monotone, $m_{\nu_1} < m_{\nu_2} < m_{\nu_3}$ necessarily. Inverted ordering cannot be accommodated within any consistent mode-index assignment that preserves algebraic closure of the generation chain. | Eigenmode selection rule; monotonicity of S(n,5) (§5, §6) | 3–4σ preference for normal ordering at current experiments ✅ |
| **F3** | **No new stable fundamental particles.** The sector set D = {2,3,4,5,6,10} is complete and unique (§3a). Within each sector, the occupied mode index set Σ is the unique solution to the co-fixed-point system (Uniqueness Theorem, Part 1 §5c). The only beat mode is at $k_0 = 16$ in $d=3$, verified by exhaustive search. Any new particle requires a new sector (excluded by Rule A + Rule B) or a new mode index (excluded by the Uniqueness Theorem) — neither exists. | Sector Set Theorem + Completeness Theorem (Part 1 §3a, §3b) | No new fundamental particles at LEP, Tevatron, LHC ✅ |
| **F4** | **No stable particle near 68.3 GeV.** S(35,10) × $m_{\rm scale,10}$ ≈ 68.3 GeV is below the Z mass. IDWT explicitly predicts its absence: $n=35$ in $d=10$ is not a co-fixed-point eigenmode (the tau is $n=23$; $n=35$ in $d=10$ has no eigenmode selection rule support). | Tau sector co-fixed-point structure | Excluded at LEP (√s up to 209 GeV, no such state) ✅ |
| **F6** | **No fourth quark or lepton generation.** S(72,10) × $m_{\rm scale,10}$ ≈ 51.7 TeV is the next $d=10$ mode above tau — far beyond LHC reach and not a co-fixed-point eigenmode. No $d=4$ mode above top ($n=72$) or $d=6$ mode above muon ($n=35$) is in the co-fixed-point set. A confirmed fourth-generation fermion at any mass falsifies the spectrum closure. | Completeness Theorem (Part 1 §3b) | Z pole invisible width: N_ν = 3.0000 predicted (PDG: 2.984 ± 0.008) ✅ |

---

### Category B — Precision Quantitative Tests

These predictions have specific numerical values from mode indices and sector geometry. Each lists an explicit falsification threshold — the point at which inconsistency would be unambiguous given current or near-future measurement precision.

| # | Prediction | IDWT value | Basis | Falsified if... |
|---|---|---|---|---|
| **F7** | Strange/down mass ratio | 20.000 (zero error) | S(4,3)/S(1,3) = 20/1 | Ratio measured outside 19.5–20.5 at a well-controlled renormalization scale |
| **F8** | Muon/electron mass ratio | 206.7647 | S(35,6)/S(13,6) | Measured outside 206.760 ± 0.005 |
| **F9** | Tau/electron mass ratio | 3475.126 (PDG −0.14σ) | S(23,10)/S(13,6) × (1 + 1/1680) | More than 3σ from 3475.13 (PDG 1σ = ±0.24) |
| **F10** | Sum of neutrino masses | $\Sigma m_\nu = 60.39$ meV (corrected; $\delta_{\nu_3}=1/35$, Part 2 §9d) | Cross-sector Hopf fixed point; no oscillation data used | Measured < 55 meV or > 65 meV |
| **F11** | Neutrino mass ratio $m_{\nu_2}/m_{\nu_1}$ | 5.808 (exact) | S(15,5)/S(10,5) = 11628/2002 | Ratio measured outside 5.5–6.1 |
| **F12** | Neutrino mass ratio $m_{\nu_3}/m_{\nu_1}$ | 32.86 (exact) | S(22,5)/S(10,5) = 65780/2002 | Ratio measured outside 30–36 |
| **F13** | Atmospheric mixing angle $\sin^2\theta_{23}$ | 0.5590 (PDG 2024: 0.553, +1.07%) | PMNS spectral geometry (§4) | Outside 0.554–0.564 at > 3σ |
| **F14** | Solar mixing angle $\sin^2\theta_{12}$ | 0.3086 (PDG 0.307, +0.51%) | PMNS spectral geometry (§4) | Outside 0.302–0.315 at > 3σ |
| **F15** | Reactor mixing angle $\sin^2\theta_{13}$ | 0.02211 (PDG 0.022, +0.51%) | PMNS spectral geometry (§4) | Outside 0.020–0.025 at > 3σ |
| **F16** | Cabibbo angle $\sin\theta_C$ | 0.22454 (PDG +0.09σ) | $\sin^2\theta_C = 1/S(n_s,3)$ + $\mathbb{CP}^1$ sector curvature correction | Outside 0.2237–0.2254 at > 3σ |
| **F17** | ρ parameter at tree level | ρ = 1.00000 exactly | $m_W^2/(m_Z^2 \cos^2\theta_W)$ from mode indices 76, 81 | ρ ≠ 1 at tree level beyond radiative corrections (~0.4%) |
| **F18** | Number of light neutrino species | N_ν = 3 exactly | Three $d=5$ modes; no sterile neutrinos; closed spectrum | Z invisible width implying N_ν ≠ 3 |
| **F19** | 0νββ effective Majorana mass | m_ββ = 0 at all orders | $d=5$ Clifford structure: no C on $S^5$ → no Majorana operator at any order (see F1) | Any detection m_ββ > 0 with > 3σ significance |
| **F20** | Beta-decay effective neutrino mass | $m_\beta \approx 8.77$ meV | PMNS mixing + neutrino mass spectrum from mode indices | $m_\beta$ measured $> 50$ meV (KATRIN 5-year sensitivity $\sim 200$ meV; Project 8 targets $\sim 40$ meV) |
| **F21** | W/Z mass ratio | $m_W/m_Z = \sqrt{S(76,2)/S(81,2)} = 0.93896$ | Mode indices 76, 81 | Measured ratio outside 0.9386–0.9394 |
| **F22** | Leptonic CP phase $\delta_{CP}$ | 197.11° (PDG NH best fit: 197° ± 27°, +0.05°; J = −0.00981) | $\delta_{CP} = \pi + 2\theta_{13}$; APS spectral flow $\Delta c_1 = -2$ (T8 🔶, Part 10 §4) | $\delta_{CP}$ outside 185°–210° at 5σ, or $\delta$–$\theta_{13}$ correlation violated by > 5° at fixed $\theta_{13}$ |

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### Category C — Structural Predictions

These follow from the IDWT framework geometry and differ qualitatively from Standard Model assumptions. They are not numerical point predictions but predict the absence of certain phenomena or physical mechanisms.

**C1 — No hierarchy problem. ✅** The ratio $m_H/m_e = \sqrt{g_{22}/g_{66}} \times S(95,2) = 245{,}140$ is determined by integer mode indices $n_H = 95$ and $n_e = 13$. The Mode Index Stability Theorem (Part 8 §3a) proves that radiative corrections cannot shift integer mode indices: $n$ is the rank of an eigenvalue in a purely discrete spectrum, a topological invariant preserved under any bounded perturbation. The Higgs mass is technically natural without supersymmetry — not because corrections cancel, but because the integer $S(95,2) = 4560$ cannot receive a fractional additive renormalization. The hierarchy problem does not arise; it presupposes that mass is an action coefficient, which it is not in IDWT. If supersymmetric partners, WIMPs, or other hierarchy-solving particles are discovered, they are absent from the IDWT closed spectrum (F3, F6) — their existence would simultaneously require reopening the spectrum and explaining why the Uniqueness Theorem is wrong.

**C2 — Higgs is a confinement mode, not a condensate.** In IDWT the Higgs is mode $n=95$ of the $d=2$ sector potential $V_2(r) = \lambda_2 r^2$. There is no quartic scalar self-coupling, no Higgs VEV, no spontaneous symmetry breaking, and no vacuum metastability from RG running of λ_H. If vacuum instability is established at high confidence — the electroweak vacuum confirmed metastable with a cosmologically short lifetime — this contradicts the IDWT Higgs interpretation, since there is no λ_H to run negative.

**C3 — No seesaw mechanism.** Neutrino masses are small because $m_{\rm scale,5}$ is set by the cross-sector Hopf fixed-point equation $m_{\rm scale,5} \times m_{\rm scale,4}^2 = (n_u/n_s) \times m_{\rm scale,6}^3$, not by a seesaw with a heavy right-handed neutrino. No lepton-number-violating operator can be constructed at any order from this structure — the same all-orders fact as F1, since no $C$ exists on the $S^5$ bundle. Discovery of a right-handed neutrino mass term, lepton-number-violating interactions at any scale, or any operator that generates a Majorana mass for SM neutrinos would falsify C3 and F1/F2 simultaneously.

**C4 — No sterile neutrinos.** The co-fixed-point condition selects exactly the $d=5$ modes that are tower outputs. There are exactly three neutrino species: $\nu_1$, $\nu_2$, $\nu_3$ at n = 10, 15, 22. No additional neutrino species at any mass scale is predicted; the PMNS matrix is unitary 3×3 exactly. Evidence for a fourth neutrino mixing into the PMNS matrix — from short-baseline anomalies, reactor anomalies, or direct detection — would falsify F3, F6, and C4 simultaneously.

**C5 — Left-handed weak coupling is geometric.** The W boson's exclusive left-handed coupling follows from the Kähler structure of $\mathbb{CP}^2$ ($d=4$) and $\mathbb{CP}^3$ ($d=6$): the Kähler $\gamma_5$ operator splits each sector spinor into holomorphic (left-handed) and anti-holomorphic (right-handed) components; W is a holomorphic connection and cannot couple to the right-handed component at any order that does not involve the anti-holomorphic mixing. Right-handed W couplings beyond known radiative corrections would falsify the Kähler sector geometry.

**C6 — No gravitons in the sector spectrum.** IDWT derives gravity from the $|\Psi_\infty|^2$ back-reaction on the observer's 3D spacetime geometry. Gravity is not a quantum field theory in IDWT; there are no graviton modes in any of the six sectors. The equivalence principle $m_{\rm grav} = m_{\rm inertial}$ is a theorem from the sector geometry (Part 4 §3.6). Detection of spin-2 gravitons as fundamental particles in a Fock-space sense would constitute an additional mode not in the IDWT sector structure.

**C7 — Exact CKM unitarity.** The CKM matrix is exactly unitary at tree level: $|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 1$. IDWT gives $V_{us} = \sin\theta_C = 0.22454$ and $V_{ud} = \sqrt{1 - \sin^2\theta_C} = 0.97447$. The apparent 5.5σ Cabibbo Angle Anomaly (nuclear beta-decay $|V_{ud}| = 0.97370$ vs kaon $|V_{us}|$) is a tension between two independent PDG measurements. IDWT's exact-unitarity value 0.97447 matches the kaon-derived determination. If the Cabibbo Angle Anomaly is confirmed to require genuinely non-unitary CKM physics, that would falsify C7.

**C8 — No glueballs.** The strong interaction in IDWT is a direct quark contact coupling — $SU(3)$-symmetric by the $\mathbb{CP}^2$ isometry, with no colour-exchange field (Part 3 §0.2, §0.6). There is no colour-exchange field to bind into a glueball; any state that QCD would classify as a glueball must in IDWT be a misidentified quark-sector resonance. IDWT therefore predicts that no glueball will ever be definitively identified as a particle species distinct from quark-sector states. Claimed evidence from radiative J/ψ decays (e.g. $f_0(1710)$, $f_0(1500)$ candidates) is reinterpreted within IDWT as ordinary $d=3$ or $d=4$ hadronic resonances misidentified as pure-colour-field states — colour-intense production channels do not require a glueball in the final state when the underlying interaction is a quark contact term. A confirmed glueball with quantum numbers incompatible with any quark-model assignment, established at > 5σ significance with independent production and decay mode consistency, would falsify the IDWT colour sector.

**C9 — Hidden-coordinate momentum loss in e–e scattering. 🔶** The electron is a $d=6$ object, and a second electron is a localized 6D probe rather than a uniform 3D source, so the projection theorem that protects every 3D-apparatus measurement (Part 3 §0.8a; Marginal Exactness, Part 11 §6.1) does not apply to e–e contact. The short-range kernel contact $(\xi\cdot\xi')^2$ (range $L_6 = 1.414$ fm) deflects the relative momentum into the three hidden coordinates: in the 6D centre-of-mass frame, on average 3/5 of the transverse momentum transfer goes invisible, registering as missing momentum per electron ($m_{\rm app}^2 = m^2 + |p_{\rm hidden}|^2$) above a Coulomb threshold of $\alpha\hbar c/L_6 \approx 1.06$ MeV relative CM energy. The signature is anomalous beam diffusion — transverse emittance growth beyond standard intra-beam (Coulomb) scattering — and missing-momentum events in precision fixed-target Møller, both carrying the 3/5 fingerprint. It is not excess wide-angle scattering, so e–e compositeness limits ($\Lambda \gtrsim 10$ TeV) do not constrain it. The kinematic factor and threshold are derived; the rate reduces to a single structural number. The hidden centroid of an electron is a free coordinate — electromagnetism is the $d=2$ zero mode, uniform in 4, 5, 6, and by Marginal Exactness no 3D-source field acts there — so a 3D-prepared beam cannot localize the hidden coordinates and sits at the uniform rest limit. The overlap factor $f_{\rm hid}$ is then a hidden-volume average $\langle f_{\rm hid}\rangle \sim (L_6/L_{\rm hid})^3$, set by the extent $L_{\rm hid}$ over which matter is localized in coordinates 4, 5, 6, and → 0 in the delocalized limit. This is why ordinary high-energy e–e scattering shows nothing: $f_{\rm hid} \sim 1$ is structurally unavailable to 3D-prepared matter. If the hidden centroids are genuinely delocalized the channel is closed; a finite $L_{\rm hid}$ — gravity, as $M_\infty$ curvature, is the only force acting in all coordinates that could set it — gives the computable $(L_6/L_{\rm hid})^3$ rate. Anomalous beam diffusion of the predicted 3/5 form above threshold would measure $L_{\rm hid}$. (Part 6 Falsifiability; `files/idwt.py` STEP 83–84.)

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### Category D — Near-Future Experimental Windows

These predictions are within reach of running or funded experiments within the next decade.

| Prediction | IDWT value | Key experiment | Current status | Timescale |
|---|---|---|---|---|
| $0\nu\beta\beta$ signal absent (all orders) | $m_{\beta\beta} = 0$ at all orders | nEXO, LEGEND-1000, KamLAND-Zen 800 | No signal ($m_{\beta\beta} < 36$ meV) | now–2035; reaching $\sim 2$–5 meV sensitivity |
| Σm_ν = 60.39 meV | 60.39 meV | CMB-S4 (target ~30 meV) | Below Planck bound (< 120 meV) | 2030s; within 2× of detection |
| Normal ordering (definitive) | $m_{\nu_1} < m_{\nu_2} < m_{\nu_3}$ | JUNO, DUNE, Hyper-Kamiokande | 3–4σ preference | now–2030 |
| $\sin^2\theta_{23} = 0.5590$ | 0.5590 ± 0.001 | T2K, NOvA, DUNE | PDG 2024: 0.553, +1.07% | Running now |
| No new stable particles | closed spectrum | HL-LHC, FCC | LHC Run 3 consistent | now–2040 |
| $m_\beta \approx 8.77$ meV | 8.77 meV | Project 8 | KATRIN: $< 0.45$ eV | 2030s; targeting $\sim 40$ meV |
| No fourth generation | none at any mass | FCC-ee (Z pole) | Z width consistent | 2040s |

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### Falsification Summary

The table below condenses the hardest predictions in order of experimental decisiveness. All are consequences of the fixed sector structure; none can be escaped by adjusting parameters.

| Rank | Prediction | Threshold for falsification |
|---|---|---|
| 1 | 0νββ absent at all orders (F1, F19) | Any signal above background at > 3σ |
| 2 | Normal neutrino mass ordering (F2) | Definitive inverted-ordering measurement |
| 3 | Σm_ν = 60.39 meV (F10) | Measured < 55 meV or > 65 meV |
| 4 | No new stable particles (F3) | Any confirmed new fundamental particle |
| 5 | m_s/m_d = 20 exactly (F7) | Ratio outside 19.5–20.5 at controlled scale |
| 6 | $N_\nu = 3$ exactly (F18) | Fourth neutrino species confirmed |
| 7 | $\sin^2\theta_{23} = 0.5590$ (F13) | > 3σ departure from 0.5590 |
| 8 | $\sin^2\theta_{12} = 0.3086$ (F14) | > 3σ departure from 0.3086 |
| 9 | $\sin^2\theta_{13} = 0.02211$ (F15) | > 3σ departure from 0.02211 |
| 10 | ρ = 1 at tree level (F17) | ρ ≠ 1 beyond radiative corrections |
| 11 | No glueballs (C8) | Confirmed glueball state at > 5σ with quantum numbers incompatible with all quark-model assignments |