Research: collision response calculus after Phase 1I

[mountain]

Context

PR #121 is merged at 51bc934 and closes the long Phase 1E--1I branch. Its terminal result is an exact problem-local target/fibre response ledger,

$$
\Phi(\pi U F)-\Phi(p)

[\Phi(B_\delta p)-\Phi(p)]
+
[\Phi(B_\delta p+R_\sigma(F))-\Phi(B_\delta p)],
$$

including an exact sign-reversal witness in which target dissipation is outweighed by the retained fibre response.

The Phase 12C maturity audit changes the interpretation of this result. The renewed target semigroup supplies a target-local action skeleton and retaining $F$ supplies exact one-step response reconstruction. Phase 1I does not yet supply a native collision-fibre action, a codomain response monoid, a regular response cocycle, future-task adequacy, or effective compression. After $F$ is forgotten, admissible vertical responses may form a non-singleton response fibre.

References:

• Phase 1I contract
• Phase 1I result
• Phase 12C maturity audit
• Merged research branch

Historical Phase 1I filenames containing “calculus” remain for link stability; they are not a maturity claim.

Work package 1J-A — finite collision response/covector gate

Use a new branch and PR in the existing Boltzmann--BBGKY Sonnet.

• Freeze a genuinely collision-derived, order-sensitive or noncommutative finite fixture.
• Declare separately the target/process action and the response action or relation; do not infer the latter from the former.
• Expose the response fibre that remains after the microscopic carrier is forgotten, including nonuniqueness and set-valued failure modes.
• Test the Phase 12C gates in order: finite skeleton (C0), response existence/reconstruction (C1), then regular reconstruction or a cocycle law (C2).
• Only after those gates, compare a frozen chart atlas across dynamics, composition, covector, decoder, and residual costs.
• Ask whether the collision-product character selects a closed one-form; integration to an H-like potential is downstream and held out.
• Preserve exact counterexamples whenever a native response composition, cocycle, or chart-independent selector fails.

Acceptance:

• Sonnet-local contract, result essay, exact executable certificate, and cross-referenced README/phase ledger.
• Process action, response relation, retained data, and forgetting map are separately typed.
• Any C0/C1/C2 claim is matched by a proof or exact certificate; absent C3/C4 gates are named.
• No uniqueness of H, generic fibre calculus, arithmetic-rank promotion, or Core/Map/API change.

Work package 1J-B — continuum H-response budget

Use another branch and PR; do not extend the merged #121 branch.

• Read the Deng--Hani--Ma construction at the level needed to identify candidate molecule/cumulant composition, the cutting map, stopped/continued branches, and the topology in which residuals are controlled.
• Keep candidate roles separate from proved identifications.
• Formulate the continuum analogue of the Phase 1I horizontal/vertical response ledger in the weak collision-flux topology fixed by Phases 1E/1F.
• State the exact extra hypotheses needed to pair the response with the unbounded $\log f$ covector: positivity, weighted integrability, truncation/renormalization, or an alternative duality.
• Separate trace control, history-tail control, response reconstruction, and the final H-response estimate.
• Record precisely where the Deng estimates do or do not supply these obligations.

Acceptance:

• Primary-paper citations are exact and claims are tagged as proved, formal, or record-only.
• A typed candidate H-response inequality names its spaces, ruler, horizon, and error terms.
• At least one failure mode shows why bulk $L^1$ or target monotonicity alone is insufficient.
• No claim to reproduce the Deng theorem, derive Boltzmann from mechanics, or prove a continuum hard-sphere H theorem.

Sequencing and branch policy

1J-A freezes the finite response interface first. 1J-B may reuse that vocabulary but may not borrow its finite conclusion as continuum evidence. Each package receives its own branch, PR, validation ledger, and closure decision. The present issue is the umbrella tracker; implementation must not reopen or expand #121.

[mountain]

Phase 1J-B has now advanced one independent gate further in #132.

B2 proves only the fixed-molecule one-mark operator lemma: root-visible C-atom pushforward contracts total variation, the aggregate local mark cost is linear in eligible atoms, and the inserted observable commutes with exact Fubini cutting. It also corrects the topology ledger: the full bounded test ball is total variation and may be stronger than the H-adapted or weak-flux class actually needed.

The remaining analytic target is now isolated in #133: marked operation-sequence summation with an explicit test class and retained gain in every molecule/error subcase. Physical flux identification, logarithmic tail, entropy chain rule, and H theorem remain later gates.