Composition, crossover and gap product: verified calculations for A08, A09 and G01

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Independent Claude Fable 5.1 session, 2026-09-08, after the six-direction review. Every displayed identity is checked exactly in scripts/six_direction_checks.py; the bridge table comes from an exact sampler of the C033 construction. These are working derivations for the A08, A09 and G01 tasks. They carry no claim IDs until their per-result audits.

1. Composition inside the A01 class (A08)

Result. Let two independent finite irreducible reversible velocity chains carry masses \(m_1,m_2\) and plateaus \(H_1,H_2\) in the sense of C019. The centre-of-mass velocity is again a finite irreducible reversible chain, so C019 applies to it, and with \(M=m_1+m_2\)

\[H_{\rm cm}=\frac{m_1H_1+m_2H_2}{M},\qquad H_{\rm rel}=\frac{m_2H_1+m_1H_2}{M},\qquad H_{\rm cm}+H_{\rm rel}=H_1+H_2.\]

Proof. The pair \((J^1,J^2)\) has generator \(Q=Q_1\otimes I+I\otimes Q_2\) and invariant law \(\pi_1\otimes\pi_2\). Detailed balance holds factorwise and irreducibility of the product follows from that of the factors. The centre velocity is \(v_{\rm cm}=(m_1/M)\,v\otimes1+(m_2/M)\,1\otimes w\), with mean zero. Since \(Q_2\mathbf1=0\), \((-Q)^{-1}(v\otimes1)=((-Q_1)^{-1}v)\otimes1\) on the mean-zero subspace, and the cross term \(\langle v\otimes1,\,1\otimes(-Q_2)^{-1}w\rangle\) vanishes because \(\langle v,\mathbf1\rangle_{\pi_1}=0\). Hence \(2M\langle v_{\rm cm},(-Q)^{-1}v_{\rm cm}\rangle =(m_1/M)\,2m_1\langle v,(-Q_1)^{-1}v\rangle +(m_2/M)\,2m_2\langle w,(-Q_2)^{-1}w\rangle\). The relative coordinate uses the reduced mass and \(v\otimes1-1\otimes w\). \(\square\)

The composite law changes while the coefficient closes. Two \(\pm u\) particles give a four-state centre chain with velocities \(\pm u\) and \(\pm u(m_1-m_2)/M\), as the review anticipated.

Gaussian class and many bodies. With \(\operatorname{Var}\Delta X_i=\kappa_i\Delta/m_i\), a collective coordinate \(y=\sum_ic_iX_i\) with kinetic mass \(\mu_y=(\sum_ic_i^2/m_i)^{-1}\) has

\[\kappa_y=\frac{\sum_i(c_i^2/m_i)\,\kappa_i}{\sum_ic_i^2/m_i},\]

a convex combination of the constituent coefficients. Over a mass-orthogonal basis the coefficients add to \(\sum_i\kappa_i\); the three-body Jacobi basis is checked explicitly. The review’s cross covariance \(\operatorname{Cov}(\Delta R,\Delta r)=\Delta(\kappa_1-\kappa_2)/M\) is confirmed, so independence of centre and relative motion is equivalent to \(\kappa_1=\kappa_2\) in this class.

Additive-function step. If \(\kappa(m)\ge0\) depends only on mass and \(\kappa(m_1+m_2)\) equals the composite value, then \(f(m)=m\kappa(m)\) is additive and nonnegative, hence monotone, hence \(f(m)=Km\) on the positive reals. Two splits of one mass already give \(\kappa(2)=\kappa(1)\). The value \(K=0\) remains admissible.

Countertest. The A02 bath coefficient \(H(m)=(m+M_b)s^2/\nu\) violates the composition premise by an exact amount:

\[H(m_a+m_b)-\frac{m_aH(m_a)+m_bH(m_b)}{m_a+m_b} =\frac{2m_am_bs^2}{\nu(m_a+m_b)}>0.\]

A rigid composite immersed as one tracer fluctuates more than the centre of mass of two separately immersed tracers. This is the precise sense in which a reservoir coefficient depends on the preparation, which is the review’s separate preparation obligation.

2. Crossover and the conditioned midpoint (A09)

Coordinator update, A09a/B17: the exact crossover proof now gives the full beta mixture and plateau limit. It corrects the earlier unscaled-mean interpretation below; the sampler remains numerical evidence.

Necessary window. C018 and C031 give, for any coefficient \(K\) realized at window \(\Delta\) under speed bound \(u\), the condition \(\Delta\ge\Delta_*:=K/(mu^2)\). For the stationary telegraph observable of C020 the bound reads \(\mathsf h(\Delta)\le H_*\Delta/\Delta_*\) with \(\Delta_*=1/\lambda\), and \(\mathsf h\) saturates it at leading order: \(x-f(x)=2x^2/3+O(x^3)\) for \(f(x)=1-(1-e^{-2x})/(2x)\).

Conditioned midpoint of the C033 bridge. With the count weights and simplex durations of the return-bridge note, the \(k=1\) component has \(\mathbb E[Y]=uT/6\) and \(\mathbb E[(Y-uT/2)^2]=u^2T^2/6\). Expanding the mixture in \(z=\lambda T\),

\[\kappa_{\rm mid}:=\frac{4m\operatorname{Var}(Y)}{T}=H_*\frac{z^3}{6}+O(z^5), \qquad \frac{2m\,\mathbb E[Y^2]}{T}\to\frac{mu^2T}{2}\ (z\to0).\]

The exact sampler confirms the cubic onset and the plateau, with \(m=u=1\):

The variance coefficient converges to \(H_*\); the second-moment version converges to \(H_*/2\). The unscaled mean tends to \(u/(2\lambda)\), while its contribution to the action divided by \(T\) vanishes. Monotonicity is not established here; the second-moment table already shows an overshoot. An earlier Monte Carlo with position-only conditioning, which admits both terminal velocities, showed a quadratic onset. The small-window exponent therefore depends on the conditioning protocol; \(\Delta_*\) and the plateau are the same in both.

Scales. At \(K=\hbar\) and \(u=c\), \(\Delta_*=\hbar/(mc^2)\) and \(u\Delta_*=\hbar/(mc)\), the reduced Compton time and length; the substitution adds the identification \(K=\hbar\), as the review notes. For A07, an acceleration ceiling \(a\) and the crossover are related by \(a=u\lambda=u/\Delta_*\); at \(u=c\) and \(\lambda=mc^2/\hbar\) this is \(mc^3/\hbar\), the scale of Caianiello’s maximal-acceleration proposal. That attribution is a lead to verify.

3. Gap product and slow modes (G01)

From C019’s spectral form, \(H_*\gamma_{\min}\le2m\sigma_v^2\le H_*\gamma_{\max}\), with equality \(H_*\gamma=2mu^2\) for two states; both are checked on fixed reversible three-state chains. The review’s slow-mode countertest is confirmed exactly: appending an independent two-state label with rate \(\epsilon\), with velocity depending only on the telegraph sign, gives \(-Q\) eigenvalues \(\{0,2\epsilon,2\lambda,2\lambda+2\epsilon\}\) and leaves \(H_*=mu^2/\lambda\) unchanged. The susceptibility bounds the gap from above only. With a supplied action unit \(K\), the two-state gap in energy units is \(2K\lambda\); at \(u=c\), \(K=\hbar\) and \(\lambda=mc^2/\hbar\) it equals \(2mc^2\), the separation of the free Dirac branches at zero momentum. In Monte Carlo language the bound is \(\tau_{\rm int}\le\tau_{\rm exp}\), with \(H_*=2m\sigma_v^2\tau_{\rm int}\) and \(\gamma_{\min}=1/\tau_{\rm exp}\).

4. Continuation check (B15 obligation)

Both routes agree symbolically. Substituting \(u=e^{-i\omega t}\psi\) into \(p_{tt}+2i\omega p_t=c^2p_{xx}\) gives \(\psi_{tt}-c^2\psi_{xx}+\omega^2\psi=0\). Substituting \(p=e^{-\lambda t}\phi\) into the real telegraph equation gives \(\phi_{tt}-c^2\phi_{xx}-\lambda^2\phi=0\), and \(\lambda=i\omega\) recovers the same Klein–Gordon form. The continued object is an amplitude; the Dirac two-component construction remains the next obligation recorded in B15.

5. Source leads at metadata level

• Kac, A stochastic model related to the telegrapher’s equation, Rocky Mountain J. Math. 4, 497–509 (1974), DOI 10.1216/RMJ-1974-4-3-497, open access at Project Euclid and a reprint of 1956 Magnolia Petroleum lectures. This route removes the access block recorded in B09.
• Sokal, Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, in Functional Integration (Springer, 1997), DOI 10.1007/978-1-4899-0319-8_6, for the integrated-versus-exponential autocorrelation inequality.

Both await passage-level reading in the A09 and G01 audits.

6. Status and reproduction

python3 scripts/six_direction_checks.py writes out/six-direction-checks.json and runs inside make check. The sampler is optional and prints the table above from a fixed seed. formal/Crossover.lean drafts the crossover lemma for F01 and is uncompiled. The P02 handoff records the session and its limits.