q3 — unrestricted dual and exact 41-point hunt

Replay:

sh compute/q3/run_all.sh

The published range is still $40\le\tau_5\le 44$ . This folder hunts either an exact Gegenbauer dual that is nonpositive on the whole interval $[-1,1/2]$ and has value $<44$ , or an explicit 41-point spherical code.

Script	What it does
`sphere_types.py`	extras kiss at most 36 $D_5$ roots; same-missed groups
`complete_slices.py`	complete $n_1\ge 33$ emptiness on the 1480-point graph
`union_slices.py`	generating-set scan of missed-root unions
`t5_repair.py`	edit-distance repairs of the published 35-cliques
`t5_36.py`	assembles the $T^5$ 36-clique hunt
`clique.c`	coloured branch-and-bound; `t5_36_c.json` is the 40M-node run
`a4_continuous.py`	extras over a fixed $A_4$ at height $\\|s\\|>2$
`expand_T.py`	$T^5$ pools on larger exact angle sets
`golden_pool.py`	$D_5$ plus $(\varphi,1,1/\varphi,0,0)/\sqrt{2}$
`dual_gap.py`	unrestricted and gapped Delsarte; Sturm certification
`verify.py`	replay any dual JSON under `certs/` and any claimed 41-set

A numerical SDP that does not become an exact positivity certificate is an incomplete search. The unrestricted range remains $40\le\tau_5\le 44$ unless a verifier-plus-certificate pair here says otherwise.