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Subharmonic / chirality structure in the elliptic-theta carrier — verified findings

Research note (2026-06-25). Carrier-verified facts about how the elliptic-theta carrier (srmech.amsc.ellbase / thetasum) already encodes the overtone/undertone (harmonic/subharmonic) = chirality structure. Every claim below is checked by an exact gate in the carrier (no float, no eval) — the test that produced it is given so it is reproducible. Companion to the elliptic F929 reduction row + the order-tower work.

1. The carrier un-fuses magnitude (order) from sign (chirality)

In the carrier a summation variable is x = qᵏ; a theta argument is an EllMonomial whose x-exponent carries two independent facts that ordinary signed-number notation fuses onto one axis:

  • |x-exponent| = the order / rung (the "power"; how far up/down the tower) — the energy.
  • sign(x-exponent) = the chirality (overtone θ(αx) vs undertone θ(α/x)) — which of two equal partners.

The overtone and undertone of one parameter have equal |x-exponent| and differ only by an orientation prefactor (the sign). They are not "big vs small" — that appearance is purely the |q|<1 reading; reading 1/q as the base flips which one looks tiny. Same content, mirror writing.

Verified: Theta(a*x).canonicalize() vs Theta(a/x).canonicalize() → equal |x-exp|, differing orientation prefactor; the magnitude is untouched.

2. The ±-pair is the equal-partner object; a lone theta is a lone chirality

The object that holds both chiralities as equal partners is the ±-pair {θ(αx), θ(α/x)} — it is invariant under the x↔1/x perspective flip (overtone and undertone just swap). A single theta is NOT a symmetric object: it carries a definite chirality and reflects to its own reciprocal (the Jacobi reflection — same energy, flipped sign), not to its twin.

This is exactly why the elliptic-Zeilberger certificate basis must be ±-pairs, not single thetas: a single-theta (lone-chirality, odd-count) cert numerator produced odd-count residuals the Weierstrass reducer (thetasum._reduce_class, which decides clean ±-pair products) declined; rebuilding the basis as ±-pairs (even-count, equal-partner) made the residual reducible with no new reduction math. The recognition (count both chiralities as equal partners) and the build fix (±-pair basis) are one structure.

Verified: the ±-pair {θ(ax),θ(a/x)} set is invariant under x↔1/x; the cert columns went 7-θ-odd/declining → 8-θ-even/reducible under the ±-pair basis.

3. The reducer's pairing IS a bit-exact monomial square root (the fold)

thetasum._recover_pairs pairs two thetas iff their argument product is a perfect-square monomial — it calls _monomial_sqrt(z₁·z₂) for the midpoint and bails (None) on odd exponents. So a structure folds onto a rung exactly when it is bit-exactly square-rootable there (even exponents). Odd-exponent structure cannot fold to the same rung — it is held one rung up.

4. The very-well-poised factor θ(a·x²) is the doubled rung; the base peel is blind to it

x² = q^{2k} is the doubling that distinguishes the genuinely non-summable (very-well-poised, Frenkel–Turaev ₆E₅/₁₀E₉) elliptic terms from merely-balanced products (which telescope → order-0). The elliptic-Gosper peel (_x_param / _peel_coboundary), written for x-exponent ±1 pairs, returns None on an x²-pair — it cannot see the doubled rung. So the genuine order-1 case is reachable as input (balanced + non-summable inputs exist and gate-confirm) but is blocked at the peel, not the reducer.

The honest "undo" of the doubling is not _monomial_sqrt (θ(z²) ≠ (θ(z))^{1/2}); it is the theta duplication / Landen identity — θ at the doubled rung expressed as a product of θ at the base rung. Whether that collapse applies is the fold/hold switch: fold (very-well-poised factor re-expresses on the base rung) vs hold (stays at the doubled rung as the irreducible, genuinely-non-summable residue).

Verified: _x_param(θ(ax²)θ(a/x²)) → None; x²-including balanced ratios gate as is_elliptic=True, elliptic_gosper=None (non-summable) yet the op returns None.

5. A-N reading

  • The sign / chirality is Class K (the pin-slot sign-flip / phase boundary): the overtone↔undertone orientation IS the Class-K ±1, and the chiral-endian flip = Class-K sign (−1)^order (order-tower).
  • The ±-pair carrier is the natural operand that holds both Class-K orientations as equal partners (the "count both chiralities equally" object).
  • The doubling (base → doubled rung; ×2 in the register / squaring the q-power) is the order-tower's next-rung step; its reversibility is the duplication identity, gated by the fold/hold (even/odd) condition.

To reproduce

All checks ran numpy-free against the source tree: PYTHONPATH=docs/srmech/python python -c "..." using srmech.amsc.ellbase (EllMonomial/Theta/EllRatio), srmech.amsc.thetasum (_recover_pairs/_reduce_class/_monomial_sqrt), and srmech.amsc.elliptic_gosper (_x_param/_peel_coboundary). See the elliptic F929-row work for the exact term-ratio constructions.

Durable cross-links: user memory project_subharmonic_chirality_collapse_thread, project_balanced_theta_product_reduction_foreground, project_order_tower_next_rung_tool.