Physics theoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)verified 2026-11-04 (gpt-6.1-sol)

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A homogeneous harmonic axion has a controlled averaged cold-matter limit

Physical scope

Conditional leading-limit result for one homogeneous, canonically normalized axion in a prescribed expanding flat-FLRW background, under the time-independent-mass harmonic models and the explicit small- $H_{n} / m_{a, n}$ , total-variation, and centered-window limits in the Statement. It predicts neither an unconditional relic abundance nor a global string compactification.

Statement

Use natural units $ℏ = c = 1$ . Let $J \subseteq R$ be an open time interval and let $I = [t_{-}, t_{+}] \subset J$ be a fixed nondegenerate compact interval. For each family index $n \geq 1$ , let $A_{n} \in C^{\infty} (J, (0, \infty))$ be the scale factor in a product chart of a prescribed expanding spatially flat FLRW background, with metric $d s^{2} = - d t^{2} + A_{n} (t)^{2} δ_{i j} d x^{i} d x^{j}$ , and set $H_{n} = {\dot{A}}_{n} / A_{n} > 0$ on $I$ . For each $n$ , supply a mass $m_{a, n} > 0$ constant in time; it may vary across the family. The harmonic potential and cutoff regime are supplied for each family member, with no common finite cutoff asserted for the entire limiting family. Let $θ_{n} \in C^{\infty} (J, R / (2 π Z))$ be one homogeneous axion field with the convention of the earlier axion EFT datum, with $f > 0$ constant on the model patch and homogeneous canonical field $χ_{n} \in C^{\infty} (J, R)$ given on the chosen harmonic branch by $χ_{n} = f {\tilde{θ}}_{n}$ , where ${\tilde{θ}}_{n} \in C^{\infty} (J, R)$ is a specified real lift of $θ_{n}$ . Assume $χ_{n} (J)$ lies in a selected harmonic branch on which $V_{n} (χ) = \frac{1}{2} m_{a, n}^{2} χ^{2}$ up to an irrelevant additive constant, and assume the source-free model equation ${\ddot{χ}}_{n} + 3 H_{n} {\dot{χ}}_{n} + m_{a, n}^{2} χ_{n} = 0$ on $J$ . This model includes no energy-transfer term or backreaction on the prescribed backgrounds. Define $ρ_{a, n} = \frac{1}{2} {\dot{χ}}_{n}^{2} + \frac{1}{2} m_{a, n}^{2} χ_{n}^{2}$ and $p_{a, n} = \frac{1}{2} {\dot{χ}}_{n}^{2} - \frac{1}{2} m_{a, n}^{2} χ_{n}^{2}$ . For $Q_{n} \in {ρ_{a, n}, p_{a, n}}$ and window lengths $Δ_{n} > 0$ , define the centered uniform average ${\overset{ˉ}{Q}}_{n} (t) = Δ_{n}^{- 1} \int_{t - Δ_{n} / 2}^{t + Δ_{n} / 2} Q_{n} (s) d s$ whenever the window is contained in $I$ . Put $W_{n} (t) : = [t - Δ_{n} / 2, t + Δ_{n} / 2]$ and call $t$ an admissible center when $W_{n} (t) \subseteq I$ . In natural units, $χ_{n}, m_{a, n}, H_{n}$ have mass dimension one, $t, Δ_{n}$ have dimension minus one, and $ρ_{a, n}, p_{a, n}$ have dimension four; $A_{n}$ follows the scale-factor unit convention of the declared FLRW chart.

Consider a controlled family of nonzero solutions with ${\overset{ˉ}{ρ}}_{a, n} (t) > 0$ at admissible centers and

$\sup_{I} (H_{n} / m_{a, n}) \to 0, \sup_{I} (∣ {\dot{H}}_{n} ∣ / m_{a, n}^{2}) \to 0, \sup_{I} (∣ {\ddot{H}}_{n} ∣ / m_{a, n}^{3}) \to 0, m_{a, n}^{- 1} \int_{I} ∣ {\dot{H}}_{n} (s) ∣ d s \to 0.$

Use admissible windows satisfying

$m_{a, n} Δ_{n} \to \infty, Δ_{n} \sup_{I} H_{n} \to 0, Δ_{n} \sup_{I} ∣ {\dot{H}}_{n} ∣ / m_{a, n} \to 0, Δ_{n} \sup_{I} ∣ {\ddot{H}}_{n} ∣ / m_{a, n}^{2} \to 0.$

Fix a reference center $t_{0} \in I$ with $W_{n} (t_{0}) \subseteq I$ for all sufficiently large $n$ . For every such fixed reference center, uniformly over admissible centers $t$ , the limits are ${\overset{ˉ}{p}}_{a, n} (t) / {\overset{ˉ}{ρ}}_{a, n} (t) \to 0$ and $A_{n} (t)^{3} {\overset{ˉ}{ρ}}_{a, n} (t) / (A_{n} (t_{0})^{3} {\overset{ˉ}{ρ}}_{a, n} (t_{0})) \to 1$ . These conclusions apply only to families satisfying the displayed limits. No finite-parameter averaging error, transition solution, string/domain-wall contribution, or relic abundance is asserted.

Statement references

Periodic axion EFT data and sources, A quadratic modulus has leading averaged dust behavior

Facts & Assumptions

Given: The natural-unit scalar EFT conventions, a periodic axion with a fixed local canonical chart, a positive time-independent mass $m_{a, n}$ for each family member, the prescribed interval/backgrounds, and the controlled family and admissible windows of the Statement.

[F1]

Periodic axion EFT data and sources: on a fixed single-field patch the angle $θ$ has period $2 π$ , $f > 0$ is constant, and $χ = f θ$ is a local canonical scalar; a harmonic approximation is an additional range-restricted model input.

[F2]

A quadratic modulus has leading averaged dust behavior: the actual draft statement assumes a family of homogeneous canonical oscillators, centered windows, the stated pointwise background/window limits, and $m_{n}^{- 1} \int_{I} ∣ {\dot{H}}_{n} ∣ d s \to 0$ . It concludes ${\overset{ˉ}{p}}_{n} / {\overset{ˉ}{ρ}}_{n} \to 0$ and $a_{n}^{3} {\overset{ˉ}{ρ}}_{n}$ is constant relative to a reference center, uniformly over admissible centers. Its current Proof 4.2 establishes $R_{n} (t_{0}) > 0$ from the assumed positive average using the comparison in Proof 3.1. Its Step 1 decision remains escalated for a separate transitive plan-interface issue; that status is not acceptance of the result.

[A1]

The source-free homogeneous scalar equation in the statement is the adopted dynamical model on this patch; it is not inferred from the periodic identification alone.

Proof

1.1givenF1F2A1algebra

By [F1], each chosen angle map, with the specified real lift, has the local canonical scalar $χ_{n} = f {\tilde{θ}}_{n}$ with the stated constant $f$ . The harmonic-range assumption gives $V_{n} (χ_{n}) = m_{a, n}^{2} χ_{n}^{2} / 2$ on every field value visited, and [A1] supplies exactly the oscillator equation used by [F2] after setting $φ_{n} = χ_{n}$ and $m_{n} = m_{a, n}$ . The energy and pressure definitions also coincide.

1.2givenA1algebra

For each $n$ , $A_{n} \in C^{\infty} (J, (0, \infty))$ makes $H_{n} = {\dot{A}}_{n} / A_{n}$ continuous. Put $y_{n} = (χ_{n}, {\dot{χ}}_{n}) : J \to R^{2}$ and $B_{n} (t) = (\begin{array}{cc} 0 & 1 \\ - m_{a, n}^{2} & - 3 H_{n} (t) \end{array})$ , so $y_{n}^{'} = B_{n} y_{n}$ . If $y_{n} (t_{0}) = 0$ , choose any compact interval $K \subset J$ containing $t_{0}$ and let $M = \sup_{s \in K} ∥ B_{n} (s) ∥ < \infty$ for the Euclidean operator norm. For $u \in K$ with $u \geq t_{0}$ , the integral equation gives $∣ y_{n} (u) ∣ \leq M \int_{t_{0}}^{u} ∣ y_{n} (s) ∣ d s$ ; setting $v (u) = \int_{t_{0}}^{u} ∣ y_{n} (s) ∣ d s$ gives $(e^{- M (u - t_{0})} v (u))^{'} \leq 0$ and $v (t_{0}) = 0$ , so $v = 0$ and $y_{n} = 0$ on $[t_{0}, \max K]$ . For $u \in K$ with $u \leq t_{0}$ , set $v (u) = \int_{u}^{t_{0}} ∣ y_{n} (s) ∣ d s$ ; then $∣ y_{n} (u) ∣ \leq M v (u)$ and $(e^{M (u - t_{0})} v (u))^{'} \geq 0$ , while $v (t_{0}) = 0$ , so nonnegativity gives $v = 0$ and $y_{n} = 0$ on $[\min K, t_{0}]$ . Varying $K$ shows $y_{n} \equiv 0$ on $J$ , contrary to the assumed nonzero solution. Thus $(χ_{n} (t), {\dot{χ}}_{n} (t)) \neq (0, 0)$ for all $t \in J$ . Since $m_{a, n} > 0$ , $ρ_{a, n} (t) = \frac{1}{2} {\dot{χ}}_{n} (t)^{2} + \frac{1}{2} m_{a, n}^{2} χ_{n} (t)^{2} > 0$ , in particular $A_{n} (t_{0})^{3} ρ_{a, n} (t_{0}) > 0$ before the reference ratio supplied by [F2] is used.

2.1givenF2step 1.1step 1.2algebra

Set $a_{n} = A_{n}$ , $φ_{n} = χ_{n}$ and $m_{n} = m_{a, n}$ in [F2]. The smoothness assumptions now agree exactly. The total-variation limit is an explicit hypothesis; it is not inferred from pointwise derivative bounds when the mass varies with $n$ . Step 1.2 gives nonzero pointwise and averaged energies, and the reference windows are admissible eventually. The prescribed background, source-free equation and all pointwise and window limits are precisely those of [F2].

3.1givenF2step 1.2step 2.1∎

Under the full matched hypotheses in step 2.1, the actual conclusion of [F2] gives ${\overset{ˉ}{p}}_{a, n} (t) / {\overset{ˉ}{ρ}}_{a, n} (t) \to 0$ and $A_{n} (t)^{3} {\overset{ˉ}{ρ}}_{a, n} (t) / (A_{n} (t_{0})^{3} {\overset{ˉ}{ρ}}_{a, n} (t_{0})) \to 1$ , uniformly over admissible centers. This supplies no finite-parameter error bound.

A homogeneous harmonic axion has a controlled averaged cold-matter limit

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources