• Definitions / Notation used
• Square-root logic
  • 1) If $\Upsilon_\omega = 0$, then $\omega$ is a stationary point of $I_2$.
  • 2) If $\omega$ is a stationary point of $I_2$, then $\Upsilon_\omega = 0$.
• One technical lemma (structure of the second-order equation)
  • Lemma (Second-order Euler–Lagrange has the form “adjoint derivative of $\Upsilon$”).
  • Proof sketch.
• What you gain by squaring
  • 1) A functional that is naturally suited to numerics.
  • 2) A direct bridge to EFT thinking.
  • 3) A cleaner path to quantization heuristics (without claiming success yet).
• What you lose by squaring
  • 1) You enlarge the solution space.
  • 2) You obscure the geometric meaning.
  • 3) You inherit the ambient signature problem in a sharper form.
• Assumptions vs consequences
  • Assumptions:
  • Consequences:
• Why this matters
• Key takeaway
• Technical takeaway

First-order equations are incisive: they define a tight solution space and often come with good deformation theory. They are also unforgiving: small changes in modeling choices (boundary conditions, gauge-fixing, numerical discretization) can push you off the solution set in a way that is hard to control. Squaring the first-order object $\Upsilon_\omega$ is the standard move that trades “sharp constraints” for “energy-like control.”

§Definitions / Notation used

• $\Upsilon_\omega := \bullet _\varepsilon(F_B) − \kappa_1 T$ is the first-order field object on $Y$ ($\mathrm{ad}(P_H)$-valued).
• Define the squared action:

$$ I_2(\omega) := \int_Y ⟨\Upsilon_\omega, \ast_Y \Upsilon_\omega⟩. $$

§Square-root logic

There are two logically distinct statements:

§1) If $\Upsilon_\omega = 0$, then $\omega$ is a stationary point of $I_2$.

This is immediate: the integrand is quadratic in $\Upsilon_\omega$, so if $\Upsilon_\omega$ vanishes pointwise, any first variation of $I_2$ vanishes.

§2) If $\omega$ is a stationary point of $I_2$, then $\Upsilon_\omega = 0$.

This is false in general. Stationary points of a square include $\Upsilon_\omega = 0$ solutions, but can also include configurations where $\Upsilon_\omega$ is nonzero yet satisfies the second-order Euler–Lagrange equation (a covariant “divergence-free” condition). This is the precise sense in which “first-order implies second-order,” but not conversely.

So the square-root slogan here is not mystical: it is a strict inclusion of solution sets: ${\Upsilon_\omega = 0} ⊂ {EL(I_2) = 0}$.

§One technical lemma (structure of the second-order equation)

§Lemma (Second-order Euler–Lagrange has the form “adjoint derivative of $\Upsilon$”).

Under variations of $\omega$ that respect the boundary conditions, the first variation of $I_2$ can be written schematically as

$$ \delta I_2(\omega) = 2 \int_Y \langle \delta \omega, 𝓛_\omega^†(\Upsilon_\omega)\rangle, $$

so the Euler–Lagrange equation for $I_2$ is

$$ 𝓛_\omega^†(\Upsilon_\omega) = 0, $$

where $𝓛_\omega$ is the linearization of the map $\omega ↦ \Upsilon_\omega$ (hence depends on $A_0$, $\varepsilon$, the Shiab operator, and the $\sigma$-split metric through $\ast_Y$, and $𝓛_\omega^†$ is its formal adjoint with respect to the $⟨·,·⟩/\ast_Y$ pairing.

§Proof sketch.

$I_2 = \int ⟨\Upsilon, \ast_Y \Upsilon⟩$. Varying gives $\delta I_2 = 2 \int ⟨\delta \Upsilon, \ast_Y \Upsilon⟩$. But $\delta \Upsilon = 𝓛_\omega (\delta\omega)$ by definition of the linearization. Move $𝓛_\omega$ off $\delta\omega$ by adjunction to obtain the displayed form (plus boundary terms that vanish under the assumed support/decay or imposed boundary conditions). No componentwise “Ricci tracing” occurs: everything is packaged in the covariant linearization of $\Upsilon$.

§What you gain by squaring

§1) A functional that is naturally suited to numerics.

Even in split signature, $I_2$ is the canonical “least-squares” objective: it measures failure to satisfy the first-order equation. This is exactly the structure you want if you are doing continuation methods, Newton–Krylov solvers, or constrained minimization.

§2) A direct bridge to EFT thinking.

Expanding $I_2$ around a background solution $\omega_0$ gives a quadratic form governed by the linearized operator $𝓛_{\omega_0}$. That is the entry point to propagators, effective operators, and mode suppression.

§3) A cleaner path to quantization heuristics (without claiming success yet).

Path integrals over $\omega$ weighted by $exp(−I_2)$ (or its Lorentzian analogue) are the standard story. In a split-signature ambient space, the real work is to identify the correct involution/projection that yields a well-behaved quadratic form on the propagating sector. Squaring is necessary, not sufficient, but it is the move that makes the question well-posed.

§What you lose by squaring

§1) You enlarge the solution space.

The first-order equation $\Upsilon_\omega=0$ is a strong geometric constraint. The second-order equation $𝓛_\omega^†(\Upsilon_\omega)=0$ allows “harmonic” $\Upsilon_\omega$ configurations: nonzero, but divergence-free in the appropriate covariant sense. Whether those extra branches are physically relevant, gauge artifacts, or pathological depends on boundary conditions and the sector ($E$-block vs everything). You do not get to ignore this.

§2) You obscure the geometric meaning.

$\Upsilon_\omega=0$ is a direct balance law: Shiab-contracted curvature equals $\kappa_1$ times torsion. The second-order equation reads like “a differential operator applied to that balance law vanishes.” That is less interpretable. It is not worse; it is just further from the conceptual anchor.

§3) You inherit the ambient signature problem in a sharper form.

On a $(7,7)$ manifold, inner products are not automatically positive. If you want $I_2$ to behave like an energy, you must specify the pairing/involution that selects the physical sector (and, in our instantiation, you will do that through the gravitational block $E$ and the pullback-visible modes). Until that is spelled out, any positivity language is propaganda. Here we keep it neutral: $I_2$ is the natural square; its analytic character depends on the sector.

§Assumptions vs consequences

§Assumptions:

• Same geometric and gauge setup as before ($\mathrm{Spin}(7,7)$, $\sigma$-split, $A_0$ fixed, $\bullet_\varepsilon$ fixed via $E/\Theta_E$, $D\Theta_E=0$).
• Boundary conditions that kill integration-by-parts terms (compact $Y$, or decay, or explicit boundary term choices).
• A specified adjoint/inner product structure for defining $𝓛_\omega^†$ (this is where split signature matters).

§Consequences:

• First-order solutions $\Upsilon_\omega=0$ are automatically solutions of the second-order EL equation. • Second-order solutions include (possibly many) additional branches with $\Upsilon_\omega \neq 0$ but $𝓛_\omega^†(\Upsilon_\omega)=0$. • The squared action is the right object for perturbation theory and numerical “residual minimization,” but it does not replace the conceptual primacy of the first-order equation.

§Why this matters

If the project is going to produce an EFT corner, a numerical fitting program, or any credible discussion of fluctuations, you will end up linearizing something and controlling error norms. $I_2$ is that control functional. The first-order equation is the geometric statement; the square is the engineering interface. Keeping both, and being explicit about what each one buys you, is how we avoid overpromising.

§Key takeaway

Squaring gives you a robust second-order theory whose solutions include all first-order solutions, but also potentially more.

§Technical takeaway

$I_2(\omega) = \int_Y \langle \Upsilon_\omega, \ast_Y Υ_\omega⟩$, with $\Upsilon_\omega = 0 ⇒ EL(I_2): 𝓛_\omega^†(\Upsilon_\omega)=0$, but not conversely.