# Infinite Dimensional Wave Theory — Part 6: Open Questions

What follows is an honest list of what the framework has not yet derived, with precise statements of what is needed.

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## Reframing: What Is Actually Open

A systematic review of the labelled "open items" against PDG experimental uncertainties shows:

**Now consistent with experiment after revised derivation:**
- **Light-quark mass residuals ($d$ +0.04%, $s$ +0.57%, $u$ +0.77% vs PDG 2024):** All within PDG $1\sigma$ light-quark uncertainties, as is the bottom quark (−0.05%). These are parameter-free predictions that agree with experiment, not scheme-conversion residuals. The $d=4$ up-type quarks are open residues at bare values: charm +0.93% (+$2.6\sigma$) and top +2.20% (+$13\sigma$); the former $(1-\varepsilon)^k$ correction has been removed as a fitted parameter — see Part 2 §11.
- **$\Lambda_{\rm QCD}$ "−9%":** IDWT defines $\Lambda_{\rm IDWT} = 3f_\pi = 282$ MeV. The PDG "300–340 MeV" is a hadronic-scheme empirical range. IDWT's value matches $3f_\pi$(PDG) $= 276$ MeV within +2.2% and $m_\rho/2.74 = 283$ MeV within −0.3%. The comparison label "−9%" was to an ill-defined target.

**Reclassified as data-dependent, not definitively wrong:**
- **$m_{\nu_3}$ 2.5–2.9% shortfall (current data):** $\Delta m^2$ is not a native IDWT quantity — it is oscillation-experiment language for a technique that cannot access absolute masses. Expressed in IDWT's own terms, $m_{\nu_3} = 48.87$ meV is 2.5–2.9% below current oscillation inference (PDG 2023: $\Delta m^2_{31} = 2.523\times10^{-3}$ eV$^2$; PDG 2024 Normal Order: $\Delta m^2_{31} = 2.530\times10^{-3}$ eV$^2$, from $\Delta m^2_{32} = 2.455 + \Delta m^2_{21} = 0.0753$). The correction $\delta_{\nu_3} = 1/35$ gives $m_{\nu_3}^{\rm corr} = 50.267$ meV, implying $\Delta m^2_{31} = 2.5246\times10^{-3}$ eV$^2$ — within 0.05% of PDG 2023 and within 0.2σ of PDG 2024 (uncertainty $\pm0.028\times10^{-3}$). Structural source: $n_{\nu_3} = n_{\nu_1} + n_{\nu_2} - n_u$ is the only neutrino mode derived by inclusion-exclusion (not a primary Pascal evaluation); $S(n_s,4) = 35$ is the leading seed-level $d=4$ denominator, the same factor appearing in the $\tau$ geometric back-reaction correction ($1/1680 = 1/(n_u \times n_s^2 \times S(n_s,4))$). Closure relation 🔶 (primary derivation Part 2 §9d): $\delta_{\nu_3} = \varepsilon \times g_{33} = 1/35$, where $\varepsilon = 1/(280\sqrt{7})$ is the $\ell=2$ self-energy scale (§11) and $g_{33} = 8\sqrt{7}$ is the $d=3$ coupling (T9); the $\sqrt{7}$ factors cancel algebraically ($g_{\rm coeff} \times g_{33} = n_s^2 = k_0$, so $\varepsilon \times g_{33} = k_0/(k_0 \times n_{\rm mu}) = 1/35$). The algebraic identity is exact given independently-derived $\varepsilon$ and $g_{33}$; the deeper operator mechanism — why the $l=2$ T2 cross-term acts with exactly this product — is not yet derived. Corrected: $\Sigma m_\nu = 60.393$ meV; uncorrected: 59.00 meV. Both are within current cosmological bounds; CMB-S4 will distinguish them.

**Genuine small residuals:**
- **$\sin^2\theta_W$ +0.37%:** The mode-index prediction is exact; it sits +0.37% ($3.6\sigma$ at PDG precision) from the PDG on-shell value. IDWT produces one structural value, so this is a genuine small residual.
- **$g_1$ +0.25% (vs self-consistent PDG):** IDWT $g_1 = 0.35043$, derived from $\sin^2\theta_W = 0.22373$ and $g_2 = 0.65275$ via the Weinberg relation. The $g_1$ derived self-consistently from PDG on-shell $\sin^2\theta_W = 0.22290$ and PDG $g_2 = 0.65270$ is 0.34957; IDWT sits +0.25% above it — entirely the $\sin^2\theta_W$ structural gap (+0.37%) propagated via the Weinberg angle relation. Not a separate quantity. The PDG tabulated value 0.35740 is in MS-bar at $m_Z$, a different scheme and scale; the −1.95% gap to that figure is the scheme/scale offset, not a physics discrepancy.

**Genuinely open (computation not done):**
- PMNS mixing angles — spectral geometry formulas derived:
  $\sin^2\theta_{23}=0.5590$ (PDG 2024: 0.553, +1.07%), $\sin^2\theta_{12}=0.3086$ (PDG 0.307, +0.51%),
  $\sin^2\theta_{13}=0.02211$ (PDG 0.022, +0.51%). From $g_{55}=96/g_{22}$ and mode indices.
  CP phase $\delta_{CP} = \pi + 2\theta_{13} = 197.11°$ derived via APS spectral flow (T8 🔶, Part 10); $J = -0.00981$.
  Three technical gaps remain before T8 reaches 🔵 (spectral flow coefficient rigour, sign from T6 matrix, equivalence with coupling-space holonomy integral).
- $g_1$ comparison offset — IDWT $g_1 = 0.35043$ is +0.25% above the self-consistent PDG value (derived from PDG $\sin^2\theta_W$ and $g_2$ via the Weinberg relation), entirely the $\sin^2\theta_W$ structural gap (+0.37%) propagated. The −1.95% gap to the PDG tabulated value 0.35740 is a scheme/scale offset (MS-bar at $m_Z$ vs $d=2$ sector scale); no loop order or scale translation changes a structural prediction.

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## Open

**$f_\pi$ and $\Lambda_{\rm QCD}$**
The IDWT geometric dilution function gives $g_{\rm eff}(n) = g_{33}/S(n,3)$, with confinement at $g_{\rm eff} = 1$ (heuristic criterion by analogy with $\alpha_s \approx 1$; native derivation from the IDWT action is open) → $S(n_{\rm conf},3) = g_{33} = 8\sqrt{7} \approx 21.17$. The nearest integer solution is $n_{\rm conf} = n_s = 4$ (the composite itself). The confinement mass scale is:

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

The pion decay constant is the mass at the seed level — the seed is the confinement mode. $\Lambda_{\rm QCD} = N_c \times f_\pi = 3 \times 94.04 = 282$ MeV, matching $3\times f_\pi$(PDG) $= 276$ MeV within +2.1% — this relation uses $N_c = 3$ from $\chi(\mathbb{CP}^2)$. **🔶 Proton mass phenomenological scaling ansatz** (temporary estimate — native derivation from kernel binding energy pending — Part 8 §11): $m_p = N_c \times \Lambda_{\rm QCD} \times (1 + 1/n_{\rm up}^2) = 3 \times 282.1 \times (10/9) = 940.4$ MeV (PDG: 938.3, +0.22%). This is a large-N_c QCD scaling law with IDWT-derived inputs; it is not a prediction from the IDWT kernel and should be replaced when the kernel binding energy calculation is complete.

**Composite hadron masses — two-formula picture (🔵 established; φ and baryon octet open).** Mesons are not IDWT eigenmodes and carry no (n,d) assignment. The composite mass programme uses two standard hadronic forms — the chiral GOR relation and the heavy-quark binding form — evaluated with IDWT-derived inputs ($f_\pi$, $\Lambda_{\rm QCD}$, and $N_c$ from $\chi(\mathbb{CP}^2)$). They are labeled cross-framework comparisons, not kernel derivations; the regime boundary $m_{\rm quark} \sim \Lambda_{\rm QCD} = 282$ MeV that separates them is QCD's organizing principle, not a feature derived from the IDWT kernel (see Part 2 §8a and Part 5 §3d for predictions):

*Chiral regime ($m_q \ll \Lambda_{\rm QCD}$):* the Gell-Mann–Oakes–Renner relation $m^2 = (m_{q1}+m_{q2}) \times B_0$ where $B_0 = (N_c/2)\cdot f_\pi^2/m_{\rm scale,3} = \Lambda_{\rm QCD}\cdot S(n_s,3)/2 = 2821$ MeV. Verified: $\pi^\pm$ (−0.2%), $K$ (5–6%), $D$ (1–2%), $D_s$ (0.0%).

*Heavy-quark regime ($m_q \gg \Lambda_{\rm QCD}$):* $m = m_{q1}+m_{q2}+\sqrt{m_{\rm heavy}\times\Lambda_{\rm QCD}}$. Verified: $B^\pm$ (−0.19%), $B^0$ (−0.15%), $B_s$ (−0.11%), $\Upsilon(1S)$ (−0.13%), $J/\psi$ (+2.0%). The binding energy $\sqrt{m_b\times\Lambda_{\rm QCD}} = 1086$ MeV is determined by $n_s$ alone through the $k_0=n_s^2=16$ triple-coincidence.

Three open items: (a) $\phi(s\bar{s})$: the strange quark sits at the regime boundary — neither formula works at leading order; (b) the vector–pseudoscalar differences ($J/\psi - \eta_c$, etc.) are object-type distinctions — vector mesons are sector resonances, pseudoscalars are composites, as with $\rho$ and $\pi$ — not corrections to a single formula; the small same-type $\Sigma$–$\Lambda$ residual (77 MeV) is likewise not resolved by the composite formula; (c) baryon octet $\Lambda$, $\Sigma$, $\Xi$, $\Omega$: the proton formula works but extending to strange-containing baryons shows ${\approx}15\%$ systematic underestimate, suggesting an additive mass contribution per strange quark not yet derived.

**Full CKM matrix and CP violation**
The Cabibbo angle is derived including curvature correction: $\sin\theta_C = (1 + \chi(\mathbb{CP}^1)/(24\cdot S(n_s,3)))/\sqrt{S(n_s,3)}$ $= (1+1/240)/\sqrt{20} = 0.22454$ (Part 3 §12), matching PDG $|V_{us}| = 0.22450 \pm 0.00044$ at +0.09σ. $|V_{cb}| = \sqrt{S(n_u,4)/S(n_c,4)} = \sqrt{15/8855} = 0.04116$ (Part 3 §0.8; PDG exclusive: 0.04100 ± 0.0014, +0.11σ). CP-violating phase $\delta_{CP} = \pi + 2\theta_{13} = 197.11°$ now derived (T8 🔶, Part 10); $|V_{ub}|$ lower bound: $A s_C^3 = 0.00920$ (CP phase = 90°). Full $|V_{ub}|$ prediction awaits T8 reaching 🔵. Note: $V_{ud} = \sqrt{1-\sin^2\theta_C} = 0.97447$ is not a separate prediction but the unitarity complement of $\sin\theta_C$. The apparent 5.5σ gap against the nuclear beta-decay value (0.97370 ± 0.00014) is the Cabibbo Angle Anomaly — a pre-existing ~3σ unitarity deficit in the Standard Model itself. IDWT enforces exact CKM unitarity by construction; $\sqrt{1-|V_{us}|^2_{\rm PDG} - |V_{ub}|^2_{\rm PDG}} = 0.97447$ matches IDWT exactly. IDWT predicts $\sin\theta_C = 0.22454$; the PDG value is 0.22450 ± 0.00044 (+0.09σ agreement). The unitarity gap is between two independently measured PDG quantities, independent of IDWT's predictions.

**Neutrino mass scale**
The $d=5$ mass scale follows from the cross-sector fixed point $m_{\rm scale,5} \times m_{\rm scale,4}^2 = (n_u/n_s) \times m_{\rm scale,6}^3$, giving $m_{\rm scale,5} = 7.429 \times 10^{-13}$ MeV. This is the $d=5$ analog of the $g_{22}$ back-reaction equation. See Part 2 §9c. Absolute masses: $m_{\nu_1} = 1.487$ meV, $m_{\nu_2} = 8.639$ meV, $m_{\nu_3} = 50.27$ meV (corrected; bare 48.87 meV), $\Sigma m_\nu = 60.39$ meV (bare 59.00 meV; $\delta_{\nu_3} = 1/35$, Part 2 §9d).

**Lorentz completion 🔶**
Scalar covariance, spin-½ from the sector Dirac operator, chirality from Kähler $\gamma_5$ on $\mathbb{CP}^2$ and $\mathbb{CP}^3$, and particle/antiparticle from the complex spinor are all established. Remaining: full Spin(9) decomposition for $d=10$. The main open item is the explicit $D_\Xi$ spectrum matching $m_{\rm scale,d} \times f(S(n,d))$.

**🎯 Sector set D as spectral gap structure of the IDWT ground state 🔶**

The sector set D = {2,3,4,5,6,10} is currently derived from $M_\infty$'s topology and geometry: Gegenbauer criticality ($4k_0=(d-2)^2$, $k_0=16$), Hopf fibration pairing ($d=3$,4,5,6), Euler characteristic constraints, and the Gegenbauer fixed-point closure condition (Part 9 T3). These are properties of $M_\infty$ independent of any specific field configuration. Two open theorems follow from this structure.

**Open Theorem A.** Let $\Psi_\infty$ be the IDWT ground state and define the directional covariance operator Ĉ = ∫ |∂$\Psi_\infty$⟩⟨∂$\Psi_\infty$| dv. Does Ĉ have eigenvalue spectral gaps at exactly d = 2, 3, 4, 5, 6, 10 — and no others? If yes, the sector set D is recoverable as the set of stable spectral plateau indices of Ĉ, providing an operator-theoretic characterization that is independent of and complementary to the topological derivation via T3. This would constitute a self-consistency theorem: the sectors selected by $M_\infty$'s geometry are exactly the ones at which the ground-state field concentrates its directional activity.

**Open Theorem B (does not hold in the stated form).** The conjecture that the sector mass scales satisfy $m_{\rm scale,d} \propto \sqrt{\lambda_d}$, with $\lambda_d$ the d-th eigenvalue of $\hat{C}$, fails. For a sector ground Gaussian the covariance eigenvalue scales as $\sqrt{\lambda_d} \propto g_{dd}^{1/3}$ (with $\lambda_d = (g_{dd}/2)^{2/3}$, Part 4 §3.10.4), while the derived mass scales (Part 2 §10) are not powers of $g_{dd}$: $m_{\rm scale,6}$ is mode-anchored at $m_e/S(13,6)$ and $m_{\rm scale,5} = (n_u/n_s)\cdot m_{\rm scale,6}^3/m_{\rm scale,4}^2$ is a cubic combination of the chain. The ratio $m_{\rm scale,d}/\sqrt{\lambda_d}$ spans about twelve orders of magnitude across D, so the mass-scale hierarchy is not the covariance spectrum. The $m_{\rm scale,d}$ stay anchored to $m_e$ and the coupling chain; the ground-state covariance does not supply them.

Note: the covariance operator construction is not a new foundation for IDWT — $d$ is not defined by $\hat{C}$, because $\hat{C}$ requires knowing $\Psi_\infty$ which presupposes the sector structure. Proving Theorem A would show the geometric and operator derivations of $D$ are consistent. It bears on sector selection only: it does not address the top mode index $n_{\rm top}$ or the even-level mode selection within a sector, which are separate questions.

**Rule A coupling-exhaustiveness 🔵.** Part 9 T3 Rule A excludes d ∈ {7,8,9}; all three cases are now closed. $d=8$: the Euler-characteristic gap ($\chi(\mathbb{CP}^4)=N_c+2=5$, seed-unmatched; no kernel fixed point for $g_{88}$; T15b). $d=9$: an active $S^9$ would carry, on the $S^1$-invariant block of $L^2(S^9)=\bigoplus_m L^2(\mathbb{CP}^4,O(m))$, a $\mathbb{CP}^4$-symmetric kernel self-consistency with coupling proportional to $g_{99}$; by T15b that equation form has no seed-anchored solution, so no value of $g_{99}$ makes $S^9$ self-consistent — an obstruction independent of how the activation is attempted. $d=7$: closed by the deposit level-count (🔵; `files/idwt.py` STEP 100). The MC-2 deposit bijection (STEP 74e, ✅) places physical modes at sites j=α+β+2 for α∈{0,1,2} (U(2) rank) and β∈{0,1,2,3} (U(3) rank); the maximum total degree is p=2+3=5, giving exactly six deposit levels j=2,...,7. The six elements of D={2,3,4,5,6,10} saturate all six levels; a seventh sector would require p=6, needing α≥3 or β≥4 — impossible under U(2)×U(3). Separately, $d=7$ cannot replace $d=10$ because the Gegenbauer endpoint is fixed at $d=10$ by Rule B independently. The two constraints close $d=7$.

**G_N from infinite-dimensional gravity — $\Lambda$ shown irreducible 🔵**
Gravity is a phenomenon of $M_\infty$ — it has no sector boundary and couples to every particle through that particle's dimensional complexity (encoded in mass). $G_N = G_\infty/(4\pi)$: a 3D observer integrates a source over its hidden coordinates, leaving the sector-independent ordinary Newtonian coupling (the $4\pi$ is the 3D Green's-function constant). The vacuum-sector contribution ($d > 10$) does not enter: curvature from $d ≤ 10$ does propagate into $d > 10$ (no hard wall), but T5 establishes that $d > 10$ modes are scattering states — any disturbance disperses rather than localizing — and $d > 10$ is Ricci-flat in vacuum ($R_{ab}=0$), so it sources no curvature. **The absolute scale $G_\infty$ (equivalently $G_N$) is a second irreducible dimensional input** alongside $m_e$: it is not derived from the sector combinatorics, and the two scales are different kinds of quantity with no hierarchy to explain between them. What remains genuinely open is whether an independent geometric determination of $G_\infty$ — for instance from beyond-product cross-sector metric mixing ($G_{\mu a}\neq 0$) — can reduce the count to one input.

**Gravity bound within, gradient-free without — the dark-matter corollary.**
A mass sources a curvature gradient only in the dimensions it is localized in; in the directions where the source is uniform (free) the sourced geometry is translation-invariant and carries no gradient (Part 4 §3.8 — the gravitational counterpart of the STEP 64 no-bath argument). A falsifiable corollary follows: dark matter cannot be higher-dimensional gravity leaking into our three. ✅ A $d=3$-bound body — ordinary baryonic matter and the instruments built from it — is uniform in $d>3$, so by the same theorem it feels zero gradient from anything gravitating in those hidden dimensions, however strong; there is no handle for hidden-sector self-gravity to act on it. A budget check agrees: charge neutrality gives roughly one electron per proton and $m_e/m_p\approx1/1836$, so the total electron rest-mass is $\approx0.05\%$ of the baryonic mass, far short of the dark-matter budget. If IDWT has dark matter it is therefore not a new gravity but candidate hidden resonances — EM-dark, mass-carrying modes gravitating on us through ordinary shared-$d=3$ nesting with their full scalar mass — whose abundance, stability, and clustering are not derived from the equations of motion 🔶.

**The gravitational scale $G_\infty$ and the cosmological constant 🔶**
The absolute gravitational scale $G_\infty$ (equivalently $G_N = G_\infty/(4\pi)$) is a second dimensional input, not derived from the sector combinatorics; IDWT fixes the *structure* of gravity (curvature of $M_\infty$, sector-independent Newtonian law) but not its magnitude. The cosmological constant is a separate open item — $\Lambda_{\rm eff}$ from the unoccupied-mode vacuum energy (Part 4 §4, Part 8 §13), with the suppression mechanism not yet derived. Neither bears on the particle-physics observables: the dimensionless quantities ($\sin^2\theta_W$, mixing angles) are purely combinatorial and the masses follow from $m_e$, all independent of the gravitational scale. (The small $\sin^2\theta_W$ residual, $+0.37\%$, and the $g_1$ offset propagated from it via the Weinberg relation, are structural and unrelated to gravity.)

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## Structural Findings on Blocked Computations

**PMNS angles — now derived from spectral geometry (T6, Part 9).** The charged-current coupling matrix between $d=5$ and $d=6$/10 is rank-1, giving $\mu$–$\tau$ symmetric mixing at the coupling-symmetry level. The physical PMNS is a weighted average of the μ–τ symmetric limit (weight $1-g_{55}$) and the simplex-ratio structure (weight $g_{55}=96/g_{22}=0.1329$), with results: $\sin^2\theta_{23}=0.5590$ (PDG 2024: 0.553, +1.07%), $\sin^2\theta_{12}=0.3086$ (PDG 0.307, +0.51%), $\sin^2\theta_{13}=0.02211$ (PDG 0.022, +0.51%). The CP-violating phase δ is derived in Part 10 ($\delta_{CP} = \pi + 2\theta_{13} = 197.11°$); see the entry below for status.

**$\delta_{\rm CP}$ — derived structurally, proof incomplete (T8 🔶).** The leptonic CP phase $\delta_{CP} = \pi + 2\theta_{13} = 197.11°$ is derived via APS spectral flow from the Chern-class step $\Delta c_1 = c_1(\mathbb{CP}^5) - c_1(\mathbb{CP}^3) = 2$ between the two charged-lepton sectors (Part 10 §1–4, T8 🔶). Three technical gaps remain before T8 reaches 🔵; see Part 10 §4 for the gap inventory. *Framework hygiene: the spectral-flow construction must remain anchored to the IDWT Dirac operator $D$ on $M_\infty$ — it is the index theory of IDWT's own operator on the sector (Hopf/$\mathbb{CP}$) geometry, not a gauge-theory Chern–Simons or instanton calculation. Every step must be expressible in sector-geometry terms.*

**CKM and PMNS have different structural origins.** The CKM arises from intra-sector mixing within the $d=3$/4 quark sectors. The PMNS arises from the spectral geometry of the cross-sector coupling between $d=5$ (neutrinos) and $d=6$/10 (charged leptons) — specifically from the ratio of $d=5$ self-coupling to the EW self-coupling $g_{22}$, and from the mode-index hierarchy within each sector.

**$g_2$, $\alpha$, $G_F$ — derived.** The $SU(2)_L$ coupling is determined by the $\mathbb{CP}^2\to\mathbb{CP}^1$ sector reduction: $g_2 = Q_u\sqrt{g_s} = (2/3)\sqrt{2g_{44}/\pi^2} = 0.65275$ (PDG: 0.65270, +0.008%). The Fermi constant follows from the standard electroweak relation $G_F = g_2^2/(4\sqrt{2}\,m_W^2)$ (SM formula with IDWT inputs) $= 1.1658\times10^{-5}$ GeV$^{-2}$ (PDG: $1.1664\times10^{-5}$, −0.05%). Both $g_2$ and $m_W$ are independently derived from IDWT — no Higgs VEV enters. The $d=2$-sector-scale α gives $1/\alpha = 131.8$. W total width $\Gamma_W = 2044$ MeV (−2%), Z total width $\Gamma_Z = 2444$ MeV (−2.0%), muon lifetime $\tau_\mu = 2.190\times10^{-6}$ s (−0.3%). See Part 3 §0.7. $g_1 = 0.35043$ at the $d=2$ sector scale (from $\sin^2\theta_W = 0.22373$ and $g_2 = 0.65275$ via the Weinberg relation). The self-consistent PDG value (from PDG $\sin^2\theta_W = 0.22290$ and PDG $g_2 = 0.65270$) is 0.34957; IDWT sits +0.25% above it — exactly the $\sin^2\theta_W$ structural gap propagated. The PDG tabulated $g_1 = 0.35740$ is in MS-bar at $m_Z$; the −1.95% gap to that figure is a scheme/scale offset, not a physics discrepancy. IDWT couplings do not run.

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## Foundational Open Items

The following items are open derivations or missing foundational elements within the IDWT framework. Each is recorded here to give referees a complete picture of what has and has not been established.

**Mode functions $\chi_{n,d}(\xi)$ — explicit (resolved).** The modes are the eigenfunctions of the harmonic self-binding operator $H_d = -\Delta_d + \lambda_d r^2$ (Part 4 §3.10): $\chi_{n_r,l,m} = N_{n_r,l} r^l L_{n_r}^{(l+d/2-1)}(\sqrt{\lambda_d}\,r^2) e^{-\sqrt{\lambda_d}\,r^2/2} Y_{lm}(\Omega)$, with closed-form normalization, level degeneracy $C(N+d-1,d-1)$, and cumulative count (IDOS) $= S(n,d)$ verified exactly in every sector. "Mode n" is the space of degree-$(n-1)$ homogeneous polynomials in $d+1$ sector coordinates ($\dim\,\mathrm{Sym}^{n-1} = S(n,d)$). These are normalizable and ready for kernel overlap integrals; the remaining work is to *use* them (see "CKM mixing formula not derived from kernel" below, and the $a_4$-gauge term in Part 4 §3.12.4).

**$V_d(r)$ potential form — saturation dropped, pure harmonic adopted (narrowed).** The earlier saturating ansatz $V_d = \lambda_d r^2/(1+r^2)$ is dropped: it has no localized bound state in $d=5$,6,10, so it cannot host the spectrum. The adopted potential is pure harmonic $V_d = \lambda_d r^2$ (Part 4 §3.10.2): confining, purely discrete spectrum, IDOS = S(n,d) exact, and the $\lambda_d$ self-consistency (§3.10.3) becomes exact rather than circular. The kernel self-energy derives the $r^2$ coefficient near a localized source; the narrow residual is that extending the pure-harmonic form to all r is adopted on those grounds rather than derived term-by-term from the action.

**Co-fixed-point stability — EOM-level necessary condition established; sufficient selection open (🔶, MC-4).** Non-co-fixed-point modes are excluded by two mechanisms (Part 7 §1.2): odd-level modes by an exact $l$-parity zero (⭐), even-level modes by dephasing in the infinite-dimensional coupled tower (🔵). The dephasing coupling, timescale $\tau \sim 1/({\rm ME}\times m_{\rm scale,3})$, and monotone initial amplitude loss are now derived from the linearised Dirac–Bogoliubov–de Gennes operator $L_{n,d}$ of the fundamental nonlinear Dirac EOM (MC-4.2, MC-4.3; `files/idwt.py` STEP 35–36) — EOM consequences, not kernel-level assertions. The even-level exclusion cannot be sharpened to an exact cut (the seed-coupling overlap has no node at the kernel weight $s=3/2$; `files/idwt.py` STEP 34). All 15 $\Sigma_{\rm pairs}$ modes are shown to have real $\sigma(L_{n,d})$ via the full-tower BdG check: worst-case ratio $|\Delta|/|\epsilon| = 0.148$ at the seed ($n=1$, $d=3$), well below 1 for all 15; every dangerous (nearest-gap) channel is exactly decoupled by $l$-parity; $d=5$ is exactly stable ($B=0$, no $C$ on $S^5$) (`files/idwt.py` STEP 36). This is a **necessary** stability condition derived from the EOM.

What remains open: **(a)** the strict no-recurrence proof for the non-$\Sigma$ dephasing — irreversibility is motivated by cross-sector incommensurate $m_{\rm scale,d}$ ratios (no common revival period), but a rigorous proof requires a continuous spectral measure for the reduced amplitude in the infinite-dimensional limit; **(b)** the **sufficient** selection — why the composite indices ($n_{\rm top}=72$, $n_\mu=35$, $n_\tau=23$, the inclusion-exclusion edges) land in $\Sigma_{\rm pairs}$ rather than other BdG-stable even-level modes. MC-4.4 (`files/idwt.py` STEP 37) shows this question is closed as a negative: no EOM-energetic condition — Hartree self-consistency, cross-sector energy resonance, kernel eigenvector, or conserved-charge level set — selects the DAG. The IE edge indices are pure index (count) relations with no on-shell energy closure at either candidate energy (mass ratio 0.235, E_HO ratio 0.757/1.054). The kernel supplies the moves (parity-conserving level steps $\Delta N\in\{0,\pm 2\}$; additive products at leading level $n_1+n_2-1$) and a kernel reading of the IE edge form (two-density product, ground-state offset gives $n_1+n_2$, 🔶), but its action set is strictly larger than the DAG (reach/DAG $\approx 15$ in $d=3$). The selection is therefore a **directed index derivation** — an acyclic generator closure whose generators are motivated (hockey-stick ⭐, $\chi$-seeding ✅, Hopf routing ✅) but whose firing-as-a-set is a postulated selection rule, not an EOM consequence. MC-4.5 tested five combinatorial/topological angles — finite minimality, IE-offset topology, sector coverage, DAG symmetry, and Schur-positivity/branching completeness — and found clean negatives for all five: NS is not the unique minimal closed set (proper subsets exist; generic-schema lift yields 561 false positives); the IE additive offset is non-uniform (no universal topological $k=0$ rule); coverage forces only 6 modes, not 15; Aut(DAG) is trivial (rigidity, not forcing); and NS is no complete/Schur-positive collection. MC-4.6 tested the characteristic-class route: all characteristic numbers of $\mathbb{CP}^2\times\mathbb{CP}^3\times\mathbb{CP}^5$ were computed (Euler characteristic, Todd genus, signature, $\hat{A}$-genus, all Pontryagin numbers, sample Chern numbers). All Pontryagin numbers vanish by a structural obstruction (the $\mathbb{CP}^3$ factor forces an odd $h_2$-power in the fundamental class that no Pontryagin monomial can match); signature $= 0$; $\hat{A}$-genus $= 0$; Todd genus $= 1$; $c_1^{10} \approx 1.1\times10^{10}$. The Euler characteristic $\chi = 72$ is the unique nonzero, non-trivial, purely topological characteristic number, giving $n_{\rm top} = c_{\rm top}(\mathbb{CP}^2\times\mathbb{CP}^3\times\mathbb{CP}^5)$ the status of a characteristic-class theorem (T15b, upgraded). However, starting from the product geometry alone — without T15 — the value 72 cannot be derived as $n_{\rm top}$. The characteristic-class route is now exhausted. The co-fixed-point condition T0.5 remains 🔶; no further avenue is currently identified.

**Time-dependent dynamics — stationary modes characterised; native decay-rate formalism established (🔶).** The fundamental EOM is the first-order nonlinear Dirac equation $(i\gamma^\mu\partial_\mu + \Sigma_d D_d)\Psi_\infty = V_{\rm kernel}[\Psi_\infty]\cdot\Psi_\infty$ (Part 1 P1; MC-4.1). Static sector modes $\chi_{n,d}$ are mean-field self-consistent solutions: the $\ell=0$ kernel acting on the mode's own density gives a c-number eigenvalue shift (the $\lambda_d$ self-consistency of Part 4 §3.10), while the $\ell=2$ off-diagonal part drives amplitude leakage into the mode tower (MC-4.2). The linearised Dirac–BdG operator $L_{n,d}$ governs that leakage; the dephasing coupling and timescale are EOM consequences (MC-4.3, `files/idwt.py` STEP 35–36). The frequency-domain redistribution of a sector excitation into other modes — the IDWT content of what is observed as decay — now has a native formalism (🔶; `files/idwt.py` STEP 97, 103–110). The $\ell=0$ kernel transition matrix element $\langle\chi_{n_f,d_f}|K|\chi_{n_i,d_i}\rangle$ is evaluated (STEP 97), and the native transition width is the classical Wigner–Weisskopf rate of the mode amplitude into the observable-radiation continuum, $\Gamma = 2\pi|g|^2\rho_{\rm rad}$ — verified by direct classical simulation, with no second quantization and no S-matrix (STEP 105). Two coupling structures appear: the scalar kernel contact $(\xi\cdot\xi')^2$ (level-local; governs persistence and dephasing) and the vector $\mathbb{CP}^2$ holonomy connection (the chiral electroweak channel, STEP 107). The muon weak lifetime is reproduced end-to-end from IDWT inputs — $\Gamma_\mu = G_F^2 m_\mu^5/192\pi^3$ with $G_F$ derived (Part 3), the V–A vertex the holonomy connection (STEP 107), and the $192\pi^3$ the native 3-body Dalitz integral (STEP 110). What remains open (🔶): the absolute normalization constant of the scalar-kernel (radiation) channel, tied to the vacuum-density seeding $\rho_{\rm vac}$ and the coupling assignment $\lambda = \mathrm{ME}\times m_{\mathrm{scale},d}$ (STEP 105).

**Sector transition rates — native formalism established (🔶).** Decay in IDWT is the single wave's sector excitation redistributing into other sector modes; there are no asymptotic in/out states and no scattering amplitudes — the rate is a frequency-domain quantity set by the kernel coupling between modes. This is now realised as the classical Wigner–Weisskopf rate from the kernel matrix elements $\langle\chi_{n_f,d_f}|K|\chi_{n_i,d_i}\rangle$ (`files/idwt.py` STEP 97, 103): the discrete mode amplitude decays into the observable-radiation continuum at $\Gamma = 2\pi|g|^2\rho_{\rm rad}$, a classical (not quantized) result confirmed by direct simulation (STEP 105). The native phase space coincides with the relativistic one — the per-leg classical resonance factor reproduces the $1/2\omega$ normalization (STEP 106) — so the low-energy correspondence with the SM Fermi EFT is exact in structure rather than assumed, and the muon lifetime is native end-to-end (STEP 110). Two-channel stability follows: the scalar kernel $\Delta N=2$ link lands on a member for none of the 15 modes, so member decays run entirely through the chiral electroweak channel, and the absolutely stable set is exactly $\{\gamma, u, e, \nu_1, \nu_2, \nu_3\}$ (STEP 108). Open (🔶): the absolute normalization of the scalar-kernel radiation channel ($\rho_{\rm vac}$ / coupling assignment, STEP 105).

**Geometric dilution as a testable mode-index prediction.** IDWT predicts $g_{\rm eff}(n,d) = g_{dd}/S(n,d)$ — a power-law decrease of the effective coupling with mode index $n$ (this is the IDWT dilution mechanism, not RG running; IDWT couplings do not run). Open: identify an observable that resolves individual modes $n$, so the $S(n,d)$ dilution law can be tested directly.

**CKM mixing formula — state-counting, not a kernel overlap (resolved 🔶).** Tested with the explicit mode functions: the T2 density-density kernel overlap between two harmonic modes is $O_{ij} = \langle r^2\rangle_i\langle r^2\rangle_j/d$ (the $(\xi\cdot\xi')^2$ factorizes under isotropy), which scales as the mode index $n^1$. The CKM formula $|V_{ij}|^2 = S(n_{\rm lighter},d)/S(n_{\rm heavier},d)$ scales as $n^d$ ($n^4$ in $d=4$) — the two differ by ×40–×4400 across the up-type quarks. So the formula is **not** the kernel matrix element $\langle\chi|K|\chi\rangle$; it is a mode-count (IDOS) ratio — legitimately 🔶 combinatorial rather than a dynamical kernel overlap. The formula remains empirically good (Part 3 §0.8: $|V_{cb}|$, $|V_{us}|$ at ${\approx}0.1\sigma$); its mechanism is counting, not the T2 overlap.

**Stability of the 15-particle spectrum — BdG necessary condition shown; sufficient selection not in energetics (🔶).** The Uniqueness Theorem (Part 1 §3b) shows the spectrum is the unique solution to the co-fixed-point system. The $L_{n,d}$ Dirac–BdG stability check (MC-4.3, `files/idwt.py` STEP 36) shows all 15 $\Sigma_{\rm pairs}$ modes have real spectrum — a **necessary** condition from the EOM. MC-4.4 (`files/idwt.py` STEP 37) establishes that the sufficient selection is not in the EOM energetics: no energy resonance at the additive DAG edges, no Hartree lock, and the kernel action set is strictly larger than the DAG. What remains open: **(a)** the sufficient selection of the generator closure — MC-4.4 (EOM-energetics), MC-4.5 (combinatorial/topological completeness), and MC-4.6 (characteristic-class invariant of $\mathbb{CP}^2\times\mathbb{CP}^3\times\mathbb{CP}^5$) have all returned negative results; no further avenue is currently identified; **(b)** sector transition rates (what prevents or governs decay between the 15 modes; IDWT has no S-matrix). The second question is part of the open time-dependent dynamics problem above.

**Bottom quark beat stability.** The bottom quark is identified as a beat mode at $k_0=16$ in $d=3$. Its stability as a beat — why the $|A_{16}|=|A_{17}|$ amplitude ratio is preserved — is asserted but not dynamically derived. Open: derive the stability of the $k_0=16$ beat from the IDWT equation of motion.

**Born rule — not derived.** The density $\rho(r,t) = \int|\Psi_\infty|^2 d\xi$ is stated (Part 1 §2.3) without derivation from the IDWT action. With $\Psi_\infty$ a single physical field, its squared modulus is the wave's own intensity; that this intensity governs the observed frequency of detections is asserted, not derived. Open: either derive $\rho = \int|\Psi_\infty|^2$ as the detection-frequency law from the IDWT action, or add it to the postulate list explicitly (P5: Born rule).

**Effective discreteness — not derived.** There is no account of how the continuous single field $\Psi_\infty$ yields the effective discreteness of what is registered. Open: provide an IDWT-native account in which sector localization of the one wave produces that effective discreteness, or add it as an explicit postulate — without importing measurement-collapse machinery, which the single-wave ontology does not contain.

**$m_e$ as unit reference — conventional choice (resolved).** Any of the 15 particle masses could serve as the dimensionful unit reference; the genuine IDWT predictions are the mass ratios. The electron is a $d=6$ $\mathbb{CP}^3$ sector excitation of $\Psi_\infty$ at mode index $n=13$ — a 6D object whose observable mass is its $d=6$ sector eigenvalue — but it holds no geometrically privileged role as the reference. $m_e$ is chosen because it is measured with high accuracy (relative uncertainty $\sim3\times10^{-10}$), fixing the overall scale with the least input uncertainty. The unit choice is conventional, justified by measurement precision, not by sector geometry.

**Spin-1 bosons (W, Z, photon) from spinor framework — not worked out.** W and Z are spin-1 — how does this emerge from the Dirac spinor $\Psi_\infty$? The $\chi(\mathbb{CP}^1)=2$ helicity count for the photon is noted, but the spin-1 polarization tensor structure for W and Z is not derived. Open: derive the spin-1 polarization structure of bosons from the $d=2$ sector Dirac operator spectrum.

**$g_{22}$ — multiplicity count, not a kernel trace (resolved 🔶).** Tested whether a genuine $(\xi\cdot\xi')^2$ kernel trace yields $g_{22} = (1/2)p^2 q$ (`files/idwt.py` STEP 2d): it does not. The literal $\mathrm{Tr}[G^2_{23} + G^2_{24}]$ is additive ($\sim p^2+q$), the wrong structure for a product; the actual T2 overlap ($O \sim \langle r^2\rangle\langle r^2\rangle/d$) is $O(1)$, orders of magnitude below the target; and $p^2 q$ appears only as a trace of the identity over rank-$(p,p,q)$ eigenspaces, where the multiplicities are the input. So $g_{22} = p^2 q/2$ is a state-count (IDOS multiplicity product), like the CKM formula — combinatorial 🔶, empirically exact, mechanism = counting. The 1/2 ($\xi\leftrightarrow\xi'$ symmetry) is the one genuinely kernel-derived piece. An operator-level upgrade would require a dynamical selection operator (not the density-density kernel) whose spectral-projector trace makes the eigenspace dimensions an output — the same EOM-level gap as co-fixed-point stability (MC-4) and the CKM mechanism.

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## Falsifiability

The complete falsification inventory is in Part 5 §9 — the canonical location for IDWT's testability, organized into hard binary falsifiers (Category A), precision quantitative thresholds (Category B), structural predictions without SM equivalent (Category C), and near-future experimental windows. Part 1 §8 gives a brief summary with a pointer to Part 5 §9.

### The colour sector and $e^+e^-$ → 3-jet events 🔶

IDWT forbids a propagating colour field. Colour is the $d=4$ kernel contact term, and the action carries no Yang–Mills kinetic term (Part 3 §0.2, §0.6), so there is no colour-octet vector mode and nothing for a quark to radiate. This is a structural commitment of the framework, and it meets the $e^+e^-$ annihilation data at a point that is currently unresolved.

Two-jet production sits comfortably inside IDWT: the $q\bar q$ angular distribution $1+\cos^2\theta$ follows from spin-$\tfrac12$ quark pair production, which the framework has. The open challenge is the three-jet final state. Four features of the 3-jet data are, in the Standard Model, the signature of a radiated spin-1 colour-octet gluon, and each constrains a no-gluon account:

- **Soft/collinear scaling.** The 3-jet rate grows as $\alpha_s\ln^2(1/y_{\mathrm{cut}})$ as the jet-resolution parameter $y_{\mathrm{cut}}\to 0$ — the behaviour of bremsstrahlung of a massless propagating vector. A point-like contact vertex carries no soft or collinear enhancement and gives a finite rate with no such logarithm.
- **3D kinematic balance.** Three-jet events are coplanar and balance 3-momentum in the centre-of-mass frame, with the third jet carrying up to $\approx\tfrac13\sqrt{s}$ of visible energy. Momentum directed into coordinates an observer does not share registers as missing 3-momentum, not as a balanced visible jet (Part 1 §3h–§3i) — so the energy of the third jet is demonstrably in the observable slice.
- **Flavour universality.** Third jets accompany down-type primaries ($b,s,d$ — all $d=3$, $S^3$, with no coordinate beyond the observable three) at the same rate as up-type; $b\bar b g$ is well measured. Any mechanism that sources the third jet from the up-type $d=4$ extra coordinate predicts a near-zero rate for the down-type majority of accessible flavours.
- **Jet substructure.** Gluon and quark jets differ measurably, the multiplicity ratio trending toward $C_A/C_F = 9/4$. This difference is carried by an octet colour charge, for which IDWT has no object; the framework as written predicts no such systematic difference.

The contact terms $g_{dd'}$ and the sector geometry are the only structure available, and they are excluded as a source of the third jet *upstream of any amplitude calculation*: the contact vertex fails the soft/collinear scaling, a hidden-coordinate recoil fails kinematic balance, and the $d=3$ down-type quarks that also produce third jets have no extra coordinate to recoil into. This is a cross-framework comparison, not a derivation: the QCD reading is named to locate the gap, not imported.

The honest statement of the tension: IDWT predicts no propagating colour-octet degree of freedom; the $e^+e^-$ → 3-jet soft/collinear scaling, flavour universality, and quark/gluon substructure differences are the strongest existing evidence for one. Reconciling them — either by deriving these four features from the contact-term and sector structure without a propagating octet mode, or by identifying what in the data the framework genuinely cannot accommodate — is open. If the three-jet phenomenology is shown to *require* a propagating colour-octet vector, that contradicts the IDWT colour sector as currently formulated.

### A six-dimensional probe: anomalous beam diffusion in e–e scattering 🔶

Most tests of the hidden sector dimensions fail for a structural reason: a 3D apparatus is uniform in the hidden coordinates, so by the projection theorem (Part 3 §0.8a) its potentials act only on the observable slice and the hidden coordinates separate off as spectators. This is Marginal Exactness (Part 11 §6.1), and it is why no chemistry, spectroscopy, or tunnelling measurement made with 3D apparatus can deviate from the standard result. The way around it is to probe a 6D object with another 6D object. Two electrons, each localized in all six of its coordinates, are not uniform in the hidden directions, so the projection theorem does not apply to their mutual interaction — the one place its scope condition genuinely fails.

The consequence is a definite kinematic prediction. Long-range e–e repulsion is the $d=2$ zero mode on the shared observable coordinates and reproduces ordinary Møller scattering exactly. The short-range kernel contact $(\xi\cdot\xi')^2$, of range $L_6 = 1.414$ fm (Part 1 §3e; `files/idwt.py` STEP 59–60), acts on all six shared coordinates and can deflect the relative momentum into the hidden directions 4, 5, 6. In the six-dimensional centre-of-mass frame the perpendicular scattering space is five-dimensional — two observable, three hidden — and the kernel is SO(6)-invariant, so on average three-fifths of the transverse momentum transfer goes into coordinates the detector cannot resolve. The two electrons recoil with equal and opposite hidden momenta, conserving the total observable momentum while each registers missing momentum, $m_{\rm app}^2 = m^2 + |\mathbf{p}_{\rm hidden}|^2$. The channel is Coulomb-blocked below a centre-of-mass relative energy $\alpha\hbar c/L_6 \approx 1.06$ MeV and opens above it; being a point contact, its cross-section is flat in energy ($\lesssim \pi L_6^2 \approx 63$ mb) where the Møller cross-section falls as $1/E^2$.

The signature is missing momentum, not excess wide-angle scattering, so existing e–e compositeness limits ($\Lambda \gtrsim 10$ TeV) do not constrain it — they bound an additional observable hard-scattering amplitude, not momentum leaving into invisible coordinates. The effect registers instead as anomalous beam diffusion, transverse emittance growth beyond standard intra-beam (Coulomb) scattering, above the ∼1 MeV relative-energy threshold, and as missing-momentum events in precision fixed-target Møller — both carrying the 3/5 fingerprint. The rate carries a hidden-coordinate overlap factor: two electrons whose centroids are separated by $\Delta X$ in coordinates 4, 5, 6 overlap as $f_{\rm hid} = \exp(-\Delta X_{456}^2/4L_6^2)$, the Gaussian overlap of two profiles of width $L_6$. That separation is set structurally by where matter sits in the hidden coordinates. Nothing in ordinary laboratory physics localizes an electron there: electron–electron and electron–nucleus electromagnetism is the $d=2$ zero mode, supported on the observable coordinates and uniform in 4, 5, 6, and by Marginal Exactness every 3D-source field is uniform in the hidden coordinates. The hidden centroid is therefore a free translational coordinate, and a free direction at rest is uniform (Part 1 §3i). A beam prepared by a 3D source cannot localize the hidden coordinates, so it sits at this uniform limit, and $f_{\rm hid}$ is an average over delocalized centroids rather than a fixed-separation Gaussian — a hidden-volume overlap $\langle f_{\rm hid}\rangle = (4\pi L_6^2)^{3/2}/L_{\rm hid}^3 \sim (L_6/L_{\rm hid})^3$, where $L_{\rm hid}$ is the extent over which matter is localized in coordinates 4, 5, 6. This is why ordinary high-energy e–e scattering shows nothing: $f_{\rm hid}\sim 1$ is not available to 3D-prepared matter, and the channel is volume-suppressed for structural reasons rather than tuned away. The open quantity reduces to the single number $L_{\rm hid}$. If the hidden centroids are genuinely delocalized the channel is closed and the framework predicts no anomalous diffusion from it; if some interaction localizes matter in coordinates 4, 5, 6 over a finite $L_{\rm hid}$ — gravity, as M$_\infty$ curvature, is the only force acting in all coordinates that could — the rate carries the computable $(L_6/L_{\rm hid})^3$ suppression. The kinematic factor and threshold are derived; the rate reduces to $L_{\rm hid}$, and anomalous beam diffusion of the predicted 3/5 form above threshold would measure it. The full derivation is in `files/idwt.py` STEP 83–84.