# Work Note

## Task

Proceed with ranked direction 1 from the ARC idea run:

**Collapsed Phase of the Exact Strong-Mixing Primordial Trispectrum.**

The scientific target is a tree-level, $s$-channel scalar four-point function in quasi-single-field inflation with a massive internal scalar and exact quadratic mixing. The concrete first-pass goal is not yet the full numerical phase scan. It is to derive a source-aware analytic setup for the calculation:

1. the Hamiltonian/flow inputs for the simplified exact-mixing model;
2. the collapsed-limit nonanalytic clock-frequency form that motivates fitting
   $$\Delta \mathcal T(x) \sim x^\alpha \mathcal A_4 \cos\!\left(2\mu_{\rm eff}\log x+\delta_4\right).$$

The later numerical target is the phase $\delta_4(\rho)$ after analytic-background subtraction on the slice $k_1=k_2=k_3=k_4=k_S$, $x=k_{12}/k_S$, benchmark $(m/H,\rho/H,c_s)=(2,2,0.3)$.

## Physics Background And Logic Flow

The field anchored by arXiv:0911.3380 studies massive isocurvature fields mixed with the inflaton. The selected idea moves beyond the weak-mixing trispectrum benchmark by resumming the quadratic mixing $\rho\dot\pi_c\sigma$ and using the cubic spatial-derivative interaction $(\partial_i\pi_c)^2\sigma/\Lambda$ as the perturbative tree-level exchange vertex.

The logic flow is:

- Use F1-F6 to define the model and kinematics.
- Calculate D1 to obtain Hamiltonian variables and interaction tensors for the exact-mixing flow problem.
- Use accepted D1 plus the standard collapsed soft-propagator logic to calculate D2, the nonanalytic clock-frequency form.
- Later, after D1-D2 are accepted, promote needed results and plan numerical flow integration, subtraction stability, and phase extraction.

The calculation must separate the nonanalytic massive-exchange signal from analytic local/contact background. The coefficient and phase of the residual are model- and subtraction-dependent; the first ready calculations only establish the setup and the frequency form, not the final phase value.

## Notation And Conventions

- Background: quasi-de Sitter with scale factor $a(t)$ and approximately constant $H$.
- Fields: $\pi_c$ is the canonically normalized Goldstone fluctuation and $\sigma$ is the additional massive scalar.
- Curvature perturbation: use $\zeta\simeq -H\pi$ at linear order. Conversion factors between $\pi_c$ and $\zeta$ must be tracked before comparing reduced correlators.
- Soft internal momentum: $s=k_{12}=|\mathbf k_1+\mathbf k_2|$.
- Equal-external-momentum slice: $k_i=k_S$ for $i=1,\dots,4$ and $x=s/k_S$.
- Benchmark: $(m/H,\rho/H,c_s)=(2,2,0.3)$.
- Principal-series notation: when a massive mode has effective frequency $\mu$, write the collapsed clock as a function of $\log x$ with angular frequency $2\mu$.
- Reduced trispectrum convention from the selected idea:
  $$
  \mathcal T(x)=\frac{(k_1k_2k_3k_4)^3}{(2\pi^2P_\zeta)^3}\,
  \langle\zeta_{\mathbf k_1}\zeta_{\mathbf k_2}\zeta_{\mathbf k_3}\zeta_{\mathbf k_4}\rangle_c' .
  $$

## Axioms And Starting Points

F1. Work in the two-field fluctuation system $\boldsymbol\varphi^\alpha=(\pi_c,\sigma)$ with exact quadratic mixing and perturbative cubic interactions.

F2. Use the simplified direction-1 Lagrangian density
$$
\frac{\mathcal L}{a^3}
=\frac12\dot\pi_c^2-\frac{c_s^2}{2a^2}(\partial_i\pi_c)^2
+\frac12\dot\sigma^2-\frac{1}{2a^2}(\partial_i\sigma)^2-\frac12m^2\sigma^2
+\rho\dot\pi_c\sigma
-\frac{1}{2\Lambda}\frac{(\partial_i\pi_c)^2}{a^2}\sigma .
$$
This truncates other cubic operators such as $\dot\pi_c^2\sigma$, $\sigma^3$, $\dot\pi_c\sigma^2$, and pure $\pi_c$ self-interactions. Those omitted operators are later systematic-error checks, not allowed inputs for D1-D2.

F3. Treat $\rho\dot\pi_c\sigma$ exactly in the quadratic system. Treat the cubic vertex proportional to $\Lambda^{-1}$ perturbatively at tree level.

F4. The target tree diagram is the connected $s$-channel four-point function with two insertions of the same cubic $(\partial_i\pi_c)^2\sigma$ vertex.

F5. The collapsed scan uses $x=k_{12}/k_S\ll1$ but should exclude the first few extrema when fitting analytic background.

F6. The analytic subtraction family starts from
$$
\mathcal T_{\rm an}(x)=a_0+a_1x+a_2x^2+b_2x^2\log x+\cdots,
$$
and the residual is $\Delta\mathcal T(x)=\mathcal T_{\rm full}(x)-\mathcal T_{\rm an}(x)$.

## Accepted Derived Results

ADR1. From the truncated exact-mixing Lagrangian F2, the canonical momenta and velocities are
$$
P_\pi=a^3(\dot\pi_c+\rho\sigma),\qquad P_\sigma=a^3\dot\sigma,
$$
$$
\dot\pi_c=\frac{P_\pi}{a^3}-\rho\sigma,\qquad \dot\sigma=\frac{P_\sigma}{a^3}.
$$
The accepted quadratic Hamiltonian is
$$
H_2=\frac{P_\pi^2+P_\sigma^2}{2a^3}-\rho\sigma P_\pi
+\frac{a c_s^2}{2}(\partial_i\pi_c)^2+\frac{a}{2}(\partial_i\sigma)^2
+\frac{a^3}{2}(m^2+\rho^2)\sigma^2.
$$
The accepted perturbative cubic Hamiltonian is
$$
H_3=-\frac{a}{2\Lambda}\int_{\mathbf k_1\mathbf k_2\mathbf k_3}(2\pi)^3
\delta^{(3)}(\mathbf k_1+\mathbf k_2+\mathbf k_3)
(\mathbf k_1\cdot\mathbf k_2)\pi_{c,\mathbf k_1}\pi_{c,\mathbf k_2}\sigma_{\mathbf k_3},
$$
with explicit identical-leg $1/2$ retained. On the equal-external $s$-channel slice,
$$
\mathbf k_1\cdot\mathbf k_2=\mathbf k_3\cdot\mathbf k_4=\frac{k_S^2}{2}(x^2-2).
$$

ADR2. Under the preliminary late-time effective-mass approximation from ADR1,
$$
\mu_{\rm eff}=\sqrt{\frac{m^2+\rho^2}{H^2}-\frac94}.
$$
After analytic-background subtraction, the accepted collapsed residual frequency form is
$$
\Delta\mathcal T(x)\sim x^3\mathcal A_4
\cos\!\left(2\mu_{\rm eff}\log x+\delta_4\right).
$$
For $m/H=2$ and $\rho/H=2$,
$$
\mu_{\rm eff}=\frac{\sqrt{23}}{2}\simeq2.40,\qquad
2\mu_{\rm eff}=\sqrt{23}\simeq4.80.
$$
This result does not determine $\delta_4$ or $\mathcal A_4$.

## Validation-Only References

V1. arXiv:2312.06559, "The cosmological flow: a systematic approach to primordial correlators". Use for validating the Hamiltonian formulation, exact quadratic-mixing treatment, strong-mixing terminology, and flow implementation. Its equations are not final answers for D1 unless independently derived from F1-F3.

V2. arXiv:1803.04412, "Quantum Standard Clocks in the Primordial Trispectrum". Use for validating the weak-mixing collapsed trispectrum clock structure and doubled collapsed frequency. Its coefficients and formulas are validation targets, not premises for deriving D2.

V3. arXiv:0911.3380, "Quasi-Single Field Inflation and Non-Gaussianities". Use only as the domain anchor and physical motivation for massive scalar exchange in QSFI.

V4. Project domain summary at `/arc-dev/arc-tests/0911.3380/0911.3380_domain_summary.md`. Use as domain context, especially solved-case warnings; do not treat it as a derivation source.

## Detailed Steps Ready To Calculate

### D1

id: `d1_hamiltonian_inputs`

status: accepted

target quantity: Derive the canonical momenta, quadratic Hamiltonian/flow input matrices, and the perturbative cubic source tensor or equivalent vertex factor for the truncated model F2 with exact quadratic mixing.

allowed premises: F1-F4, notation/conventions above, standard Legendre transform, Fourier-space replacement of spatial derivatives, and algebra derived in the response.

forbidden inputs: Do not copy arXiv:2312.06559 Eq. 4.8 or Eq. 4.9 as the final result. Do not include omitted F2 interactions. Do not use the future desired phase ansatz as an input.

proposer-visible context: The task, notation, F1-F4, and the fact that V1 may be used only for validation or source orientation.

reviewer-only target ids: none.

expected form without target formula: The answer should give canonical momenta, solve velocities in terms of momenta, identify the quadratic Hamiltonian structure for $\boldsymbol X=(\pi_c,\sigma,p_{\pi_c},p_\sigma)$ or equivalent flow variables, and state the momentum dependence/sign convention of the $(\partial_i\pi_c)^2\sigma$ cubic vertex.

acceptance standard: Two independent proposers must agree on the momentum definitions, the $\rho$-dependent quadratic mass/mixing structure, and the cubic vertex normalization up to explicitly stated symmetrization conventions. Reviewer must confirm the result follows from F2 and not from copying V1.

source-discrepancy handling: If the result differs from V1 because of the intentional truncation or notation, record `ambiguous_convention` only if it affects future calculations. Otherwise record no source discrepancy and state the mapping.

why follows: D1 is the minimum analytic input required before a flow or numerical tree-level four-point computation can be specified.

At the end of this step, calculate the Hamiltonian/flow input for the truncated F2 model in terms of $(m,\rho,c_s,\Lambda,a,k)$ and Fourier momenta. Forbidden input: the final source expressions in V1.

accepted result:

Canonical momenta and velocities:
$$
P_\pi=a^3(\dot\pi_c+\rho\sigma),\qquad P_\sigma=a^3\dot\sigma,
$$
so
$$
\dot\pi_c=\frac{P_\pi}{a^3}-\rho\sigma,\qquad \dot\sigma=\frac{P_\sigma}{a^3}.
$$

The exact quadratic Hamiltonian is
$$
H_2=\frac{P_\pi^2+P_\sigma^2}{2a^3}-\rho\sigma P_\pi
+\frac{a c_s^2}{2}(\partial_i\pi_c)^2+\frac{a}{2}(\partial_i\sigma)^2
+\frac{a^3}{2}(m^2+\rho^2)\sigma^2.
$$
Equivalently,
$$
\frac{H_2}{a^3}=\frac12(p_\pi-\rho\sigma)^2+\frac12p_\sigma^2
+\frac{c_s^2}{2a^2}(\nabla\pi_c)^2+\frac{1}{2a^2}(\nabla\sigma)^2+\frac12m^2\sigma^2.
$$

The corresponding first-order quadratic flow input is
$$
\dot\pi_{c,\mathbf k}=\frac{P_{\pi,\mathbf k}}{a^3}-\rho\sigma_{\mathbf k},\qquad
\dot\sigma_{\mathbf k}=\frac{P_{\sigma,\mathbf k}}{a^3},
$$
$$
\dot P_{\pi,\mathbf k}=-a c_s^2 k^2\pi_{c,\mathbf k},\qquad
\dot P_{\sigma,\mathbf k}=\rho P_{\pi,\mathbf k}-a k^2\sigma_{\mathbf k}-a^3(m^2+\rho^2)\sigma_{\mathbf k}.
$$

For the perturbative spatial-derivative vertex, with Fourier convention $f(\mathbf x)=\int_{\mathbf k}e^{i\mathbf k\cdot\mathbf x}f_{\mathbf k}$,
$$
H_3=\frac{a}{2\Lambda}\sigma(\partial_i\pi_c)(\partial_i\pi_c)
=-\frac{a}{2\Lambda}\int_{\mathbf k_1\mathbf k_2\mathbf k_3}(2\pi)^3
\delta^{(3)}(\mathbf k_1+\mathbf k_2+\mathbf k_3)
(\mathbf k_1\cdot\mathbf k_2)\pi_{c,\mathbf k_1}\pi_{c,\mathbf k_2}\sigma_{\mathbf k_3}.
$$
Keeping the explicit identical-leg $1/2$ in $H_3$, the symmetric kernel is
$$
V_{\pi\pi\sigma}=-\frac{a}{2\Lambda}(\mathbf k_1\cdot\mathbf k_2).
$$
If the identical-leg symmetry factor is absorbed into the vertex definition, use
$$
V_{\pi\pi\sigma}^{\rm absorbed}=-\frac{a}{\Lambda}(\mathbf k_1\cdot\mathbf k_2).
$$
On the equal-external $s$-channel slice,
$$
\mathbf k_1\cdot\mathbf k_2=\mathbf k_3\cdot\mathbf k_4=\frac{k_S^2}{2}(x^2-2).
$$

status note: Accepted by consensus in `calc_direction1_20260531_100009`; reviewer found no source discrepancy and selected proposer 002 as the clearest writeup.

### D2

id: `d2_collapsed_clock_frequency`

status: accepted

target quantity: Derive the collapsed-limit nonanalytic frequency form of the residual scalar-exchange trispectrum, and compute the preliminary benchmark value of the effective principal-series frequency under the late-time effective-mass assumption accepted from D1 or derived locally.

allowed premises: F1-F6, accepted D1 output, standard small-argument Hankel expansion for a massive scalar mode, and algebra derived in the response.

forbidden inputs: Do not use the exact V2 collapsed trispectrum coefficients as premises. Do not claim a numerical phase $\delta_4$. Do not assume the analytic subtraction fixes the phase uniquely. Do not use unaccepted numerical flow output.

proposer-visible context: The target is only the functional clock form and benchmark frequency under the stated approximation. Coefficients, amplitude, and phase remain later numerical targets.

reviewer-only target ids: none.

expected form without target formula: The answer should show why the collapsed massive internal line gives conjugate nonanalytic powers in $x$, how principal-series mass turns these powers into an oscillation in $\log x$, why the collapsed exchange frequency is doubled relative to one soft massive leg, and what the benchmark frequency is under the effective-mass approximation.

acceptance standard: Proposers must agree on the exponent/frequency relation and the benchmark arithmetic. Reviewer must reject any answer that promotes the phase $\delta_4$ or amplitude $\mathcal A_4$ to a calculated result.

source-discrepancy handling: V2 is only a weak-mixing validation reference. If D2 gives the same doubled-frequency structure but with $\mu_{\rm eff}$ rather than the weak-mixing $\widetilde\nu$, record no source discrepancy; it is a regime change, not a contradiction.

why follows: D2 converts D1's massive mixed eigenmode input into the observable collapsed-fit target used by the selected direction.

At the end of this step, calculate the residual's clock-frequency form and the benchmark preliminary $\mu_{\rm eff}$ in terms of $m/H$ and $\rho/H$. Forbidden input: V2 final coefficients or any numerical phase.

accepted result:

For the collapsed residual scalar-exchange trispectrum after analytic-background subtraction,
$$
\Delta\mathcal T(x)\sim x^\alpha\mathcal A_4
\cos\!\left(2\mu_{\rm eff}\log x+\delta_4\right),\qquad x=\frac{s}{k_S}.
$$
The accepted collapsed-limit derivation gives
$$
\alpha=3,\qquad \omega_{\log}=2\mu_{\rm eff}.
$$
Under the preliminary late-time effective-mass approximation inherited from D1,
$$
\mu_{\rm eff}=\sqrt{\frac{m^2+\rho^2}{H^2}-\frac94}.
$$
For the benchmark $m/H=2$, $\rho/H=2$,
$$
\mu_{\rm eff}=\frac{\sqrt{23}}{2}\simeq2.40,\qquad
2\mu_{\rm eff}=\sqrt{23}\simeq4.80.
$$
No value is assigned to $\delta_4$ or $\mathcal A_4$.

status note: Initial D2 run failed because the reviewer marked `special_limit_only=true` while also marking `all_agree`. Root cause: the reviewer treated the collapsed target as a special-limit proxy rather than the full target. The retry `calc_direction1_d2_retry_20260531_100009` clarified that the collapsed asymptotic limit is the target, and D2 was accepted with no source discrepancy.

## Rough Steps For Later Planning

R1. Implement or identify an executable four-point cosmological-flow solver that can evolve the exact quadratic system from ADR1 and compute the tree-level two-vertex $s$-channel four-point function. This is now a blocked implementation prerequisite, not an executable consensus calculation step.

R2. After a solver exists, specify a reproducible scan over $x=k_{12}/k_S$ at $(m/H,\rho/H,c_s)=(2,2,0.3)$.

R3. Fit and subtract $\mathcal T_{\rm an}(x)$ over multiple collapsed windows and analytic bases. Acceptance requires stability of $\delta_4$ under basis/window changes.

R4. Validate against V2 in a weak-mixing/sound-speed benchmark before claiming strong-mixing novelty.

R5. Vary $\rho/H$ at fixed $m/H,c_s$ and test whether $\delta_4(\rho)$ contains information not reconstructible from the bispectrum alone.

## Reviewer-Only Targets

None. This is a new calculation, not a blind check of a hidden target formula.

## Calculation Status

- `d1_hamiltonian_inputs`: accepted in `/arc-dev/arc-tests/0911.3380/calculate/massive_scalar_tree_correlator_20260531_093913/execute/calc_direction1_20260531_100009/`.
- `d2_collapsed_clock_frequency`: accepted in retry run `/arc-dev/arc-tests/0911.3380/calculate/massive_scalar_tree_correlator_20260531_093913/execute/calc_direction1_d2_retry_20260531_100009/`.
- No ready detailed calculation steps remain.
- Stop condition: full numerical phase extraction for $\delta_4$ is blocked until an executable four-point cosmological-flow solver is implemented or supplied. Repository search found no local ARC package solver for the required four-point flow system.

## Open Questions

Open implementation prerequisite: an executable four-point cosmological-flow solver is required before calculating $\mathcal T_{\rm full}(x)$, fitting/subtracting $\mathcal T_{\rm an}(x)$, and extracting $\delta_4$. The current ARC packages provide paper/domain/LLM/workflow tooling but no local numerical solver for this physics system.

## Revision History

- v001: Created work note for ranked direction 1. Added two ready analytic setup steps and deferred the full numerical phase scan to rough planning.
- v002: Replaced `\bm` with `\boldsymbol` for portable XeLaTeX PDF export. No physics content changed.
- v003: Recorded accepted D1 and D2 consensus results. Noted D2 retry after reviewer-schema invariant failure. No D1/D2 results are yet promoted by `plan.md`.
- v004: Promoted accepted D1 and D2 results into Accepted Derived Results. Recorded solver prerequisite as stop condition for the full $\delta_4$ scan.

## Journal

- 2026-05-31T10:00:09Z: Selected ranked idea #1 from `/arc-dev/arc-tests/0911.3380/ideas/massive_scalar_tree_correlator_20260531_093913/ranked-ideas.json`.
- 2026-05-31T10:00:09Z: Planned PDF export for `/arc-dev/arc-tests/0911.3380/work-note.md` after mirroring this version.
- 2026-05-31T10:00:09Z: First PDF export failed because `\bm` was unavailable in the default XeLaTeX template. Wrote v002 with portable `\boldsymbol`.
- 2026-05-31T10:05:00Z: Ran `calculate_runner.py` with `calculate.config.json`. D1 accepted. D2 failed due reviewer output invariant `all_agree` plus `special_limit_only=true`.
- 2026-05-31T10:07:00Z: Ran D2 retry with `calculate.d2-retry.config.json`; D2 accepted. Root cause was reviewer interpretation, not physics disagreement.
- 2026-05-31T10:08:00Z: Searched repo for four-point cosmological-flow implementation; no local solver package found.
- 2026-05-31T10:09:00Z: Applied `plan.md` promotion pass from `planning-request.md`; no further ready calculation step was created because the next scientific target requires an executable solver.

## Source Audit Trail

- `/arc-dev/arc-tests/0911.3380/ideas/massive_scalar_tree_correlator_20260531_093913/ranked-ideas.json`: selected idea title, summary, and calculation plan.
- `/arc-dev/arc-tests/0911.3380/0911.3380_domain_summary.md`: domain solved-case warnings and methodology context.
- `arc-paper get-metadata arXiv:0911.3380 --json`: seed/foundation metadata.
- `arc-paper get-metadata arXiv:2312.06559 --json`: source metadata for cosmological flow and strong mixing.
- `arc-paper get-section arXiv:2312.06559 --section S4.SS1.SSS0.Px4 --json`: validation context for coupled $\pi_c$-$\sigma$ Lagrangian and strong-mixing terminology.
- `arc-paper get-section arXiv:2312.06559 --section S4.SS1.SSS0.Px5 --json`: validation context for Hamiltonian flow implementation.
- `arc-paper get-section arXiv:2312.06559 --section S4.SS2.SSS1 --json`: validation context for effective mass in the exact-mixing quadratic sector.
- `arc-paper get-section arXiv:1803.04412 --section S3.SS2 --json`: validation context for collapsed-limit trispectrum clock structure.
- `arc-paper get-section arXiv:1803.04412 --section S2.SS1 --json`: validation context for mixed propagator and mode-function conventions.