Historical excerpt: IDWT_Appendix_D.md

Imported 2026-09-08. Historical evidence only; claims are not endorsed by V2. This is a local extract, not a link to or dependency on another repository. Original full-file SHA-256: 1365ad5f97f61436ff0977c5c2ca2e5a39749b20b60e1270b719a39e10f9c60f.

Original source lines 685–696

§49b. Mode-selection tests (2026-07-13): three rejected candidates

Earlier mode-selection candidates included a triangular-number projection filter, energy-space avoided-crossing, single-sector Gross–Pitaevskii dynamics, a complete simplex-resonance fixed-point claim, a dynamical-stability functional, a $d{=}12$ cutoff, and a circular $\alpha=1/137$ construction. These candidates did not establish a selection rule. Three further candidates were examined:

Rejected — the OQ35 "$\rho$-meson as a comb-filter spectral peak." An old script located $m_\rho$ as a root $x$ of $\sum_{d\in\{3,4,6\}} g_{dd}\,\tau_d\cos(x\tau_d)=0$ (comb delays $\tau_d=1/(2\sqrt{k_0+d})$), giving $775.79$ MeV ($+0.069\%$). The equation is a sum of cosines and has $\sim$182 roots between $x\in[0,5000]$, spaced every $\sim$14 MeV — dense enough that an arbitrary target mass (tested at 500, 800, 900, 1000, 1100, 1200 MeV, none physically meaningful) lands within $\pm5$ MeV of some root in 5 of 6 cases. The root search itself was bracketed to $[1510,1528]$, a window built around the already-known answer, with no argument for why that root (out of $\sim$182 candidates) is the physical one. Not a prediction; a fit dressed as a resonance condition.

Rejected — an alternate $d{=}6=d{=}4\otimes d{=}2$ Vandermonde decomposition ($g(N,6)=\sum_k g(k,4)g(N{-}k,2)$, verified exact). Checked against every other split of $d=6$ ($1{+}5$, $2{+}4$, $3{+}3$, $5{+}1$): all give the identical result. This is the fully general Chu–Vandermonde identity ($\sum_k\binom{m}{k}\binom{n}{p-k}=\binom{m+n}{p}$ for any $m,n,p$), carrying no information specific to the up-quark/EM sectors. Not distinct from the already-used $d{=}3\otimes d{=}3$ reading.

Rejected — $\wedge^2(\mathbb{R}^4)=6$ as independent corroboration for an untested $g_{4,6}$ coupling. Verified $\dim\mathrm{Sym}^2(\mathbb{R}^n)=\binom{n+1}{2}=\dim\wedge^2(\mathbb{R}^{n+1})$ identically for every $n=1,\ldots,7$ — the same binomial coefficient under two names, not two independent geometric facts that happen to agree at $n=3$. Adds nothing beyond the already-used $\mathrm{Sym}^2(\mathbb{R}^3)=6$.

Restored (low-stakes, both survive the same adversarial check): a second closed form for the top index, $n_{\rm top}=2\,S(2n_s,2)=2\,S(8,2)=72$, with the same one-parameter family giving $n_{\rm charm}=2\,S(n_s,2)=20$ at $k=1$ — logged as a second value identity (files/idwt.py, $n_{\rm top}$ definition; Part 9 T15 note), explicitly not a selection mechanism (a neighbouring-integer check shows 72 is not uniquely reachable this way — 70 and 71 each have more additive representations from known indices); and the structural margins of the sum-free property (files/idwt.py STEP 56; Part 1 §5 note) — $\tau$ dominates the sum of all other $S$-values by $12.5\times$, the $d{=}4$/$d{=}3$ gap past the up quark is $442.75\times$, and the tightest surviving margin anywhere in the spectrum is $S(\nu_1)+S(\nu_2)=13{,}630$ against $S(e)=18{,}564$ (margin $4{,}934$).