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DnonDn: restore the fold block for odd n >= 5 (invariant was incomplete) - #55
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For odd n the last chain block beta_{rho1, rho_n2d} is the fold:
rho_1 x rho_n2d = rho_{n2d-1} + rho_{(n+1)/2}, and rho_{(n+1)/2} is
equivalent to rho_n2d. The constructor dropped it for odd n >= 5
(n3 = n2d - 1, commented as a "g-invariance bug" fix), which left the
selective bispectrum blind to the continuous twist
F_k -> R(k theta) F_k: the Jacobian had rank 2n - 1 instead of 2n,
and invert() returned a signal on that one-parameter family rather
than in the D_n orbit of the input (roundtrip of the selected entries
still passed, which is why the existing tests did not catch it).
n3 = n2d for all n now. invert() generalizes the n = 3 fold resolution
to every odd n: the fold block re-extracts rho_n2d in a frame rotated
by R(-n theta), so the angle is measured against the chain value of
F_n2d (which is S itself at n = 3), with fold_scale = n and 2n
candidate shifts. Even n is unchanged. output_size grows by 16 for odd
n >= 5 (n=5: 37, n=7: 53).
Tests: Jacobian rank == 2n for n = 3..12, a generic twist changes the
bispectrum while theta = 2 pi / n does not, and orbit recovery for all
n = 3..12 (previously even n only). Sweep n = 3..32 x 5 seeds x 64:
worst roundtrip 4e-8, all reconstructions in the orbit.
Codecov Report✅ All modified and coverable lines are covered by tests. Additional details and impacted files@@ Coverage Diff @@
## main #55 +/- ##
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- Coverage 95.46% 95.45% -0.01%
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Files 11 11
Lines 2227 2224 -3
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- Hits 2126 2123 -3
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Problem
n3 = n2d if n % 2 == 0 else max(n2d - 1, 1)dropped the block β_{ρ₁, ρ_{n2d}} for odd n ≥ 5. For odd n, ρ₁ ⊗ ρ_{n2d} = ρ_{n2d−1} ⊕ ρ_{(n+1)/2}, and ρ_{(n+1)/2} folds back onto ρ_{n2d} — so this block is the fold, and without it the continuous twist F_k → R(kθ)·F_k preserves every selected entry. The invariant was incomplete (Jacobian rank 2n−1 instead of 2n) andinvertcould not recover the orbit for odd n ≥ 5.Fix
n3 = n2dfor all n.invert: the fold branch is generalised from n = 3 to every odd n (fold block = the 2D block labelledn2din the last decomposition,fold_scale = n,n_shift = 2n, rotation measured from the chain-extracted F̂). Even-n path untouched; n = 3 behaviour unchanged.Breaking
output_sizegrows by 16 and the layout changes for odd n ≥ 5 (n=5: 21 → 37, n=7: 37 → 53). Even n and n = 3 unchanged.Tests
Jacobian rank == 2n for n = 3…12 (was 2n−1 for odd n ≥ 5); continuous twist changes β (odd n); twist by 2π/n is invariant; orbit recovery for n = 3…12 (1e-8). Sweep n = 3…32 × 5 seeds × 64 signals: worst roundtrip 4e-8, every reconstruction in the D_n orbit.
Part of the v0.4.0 stack (C).