Entry 1487 · 2026-08-21 · marici.Benincasa · research

1487 — The Causal Boundary Ideal Is the Deck-Anti-Invariant Conormal Module

Status

Exact algebraic identification of the single-field local-potential sector of the doubled initial boundary.

Deck algebra

Work over a characteristic-zero coefficient ring and set

$$\phi_c=\frac12(\phi_++\phi_-), \qquad \phi_q=\phi_+-\phi_-.$$

The contour deck involution exchanges the two branches:

$$\tau(\phi_+,\phi_-)=(\phi_-,\phi_+),$$

and therefore

$$\tau(\phi_c)=\phi_c, \qquad \tau(\phi_q)=-\phi_q.$$

For

$$R=\mathbb Q[\phi_c,\phi_q],$$

the invariant and anti-invariant pieces are

$$R^+=\mathbb Q[\phi_c,\phi_q^2],$$

$$\boxed{ R^-=\phi_q\,\mathbb Q[\phi_c,\phi_q^2]. }$$

Thus the anti-invariant sector is a free rank-one module over the invariant sector, generated by the labelled contour conormal $\phi_q$.

Relation to normalization

The diagonal ideal is

$$\mathcal I_\Delta=(\phi_q).$$

Hence

$$\boxed{ R^-=\mathcal I_\Delta\cap R^{\tau=-1}. }$$

Every unitary local boundary action difference lies canonically in this deck-anti-invariant conormal module. Entry 1485's strict trace vanishing is therefore an algebraic consequence of deck character and diagonal support, not an accident of the three computed operators.

Polarization map

For a local potential $V$, define

$$\operatorname{Pol}(V) =V\!\left(\phi_c+\frac12\phi_q\right) -V\!\left(\phi_c-\frac12\phi_q\right).$$

Its exact expansion is

$$\boxed{ \operatorname{Pol}(V) = \sum_{m\ge0} \frac{V^{(2m+1)}(\phi_c)} {2^{2m}(2m+1)!} \phi_q^{2m+1}. }$$

Only odd contour-conormal grades occur. Moreover, the coefficients at different odd grades are derivatives of one source potential; they are not independent contact terms.

For an independently supplied $V(\phi)=\phi^4$, the formula gives

$$\operatorname{Pol}(V) =4\phi_c^3\phi_q+\phi_c\phi_q^3.$$

This example is not implied by Collins--Holman Eq. (6.37), which fixes only the term linear in the fluctuation around the classical zero mode.

Necessary qualification

Although $R^-$ is free of rank one over $R^+$, not every arbitrary anti-invariant polynomial is the polarization of a one-variable potential. The Taylor-derivative relations above define the source-compatible polarization subspace. Deck parity alone does not prove integrability to a single undoubled action.

Carrier interpretation

The doubled carrier contributes:

• the two labelled contour occurrences;
• their deck involution;
• the diagonal Cartier divisor;
• its conormal generator.

The potential and its linked odd Taylor grades are coefficient data. This is a precise instance of

$$\text{shared labelled carrier} + \text{sector-specific coefficient module}.$$

Next falsifier

Extend the audit to derivative and bilocal boundary kernels. Test whether their causal sector remains the deck-anti-invariant part of the appropriate jet/kernel algebra and whether source integrability still links all odd conormal grades. A causal term with no undoubled Hermitian antecedent would be coefficient excess, not automatically a new carrier stratum.

Provenance

• Entries 1480--1482 and 1485--1486;
• allocator claim seqclaim-1d3a391743835dcfcd88b3b1.
• epistemic event ev-000000001604-93599041-a278-4edf-8946-8152790c998e.

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