Heterogeneous Boundary Transfer Scales
Date: 17 October 2026 Branches: B02, B03, B07 and B08 Claim status: exact piecewise-free transfer analysis; physical normalization remains nonunique
1. Transfer matrix
For a Euclidean segment with mass \(m\) and length \(L\), propagate
with
It satisfies
For a piecewise interface,
Subdivision of any constant segment leaves this ordered product unchanged.
2. Boundary response from transfer data
Write
The Dirichlet-to-Neumann response is
and
This gives the determinant scale
It is intrinsic to the sewn boundary response whenever \(b,c>0\).
3. Effective decay scale
For a hyperbolic transfer matrix, define
This is the effective Floquet or decay exponent per unit length.
For a homogeneous interface,
so
4. Heterogeneous splitting
For piecewise masses \(m_i\), both scales remain invariant under refinement of the same physical profile:
when
Therefore \(\mu_{\det}\) and \(\mu_{\rm F}\) are both subdivision-independent.
But generally,
Homogeneity had hidden this ambiguity by making the two invariants coincide.
5. Reversal
For the tested positive scalar segments, reversing the layer order leaves both displayed scalar invariants unchanged, although it exchanges the two diagonal boundary responses. More general matrix-valued or nonreciprocal channels need not share this property.
The important point is not order sensitivity but nonuniqueness: transfer data contains more than one natural dimension-one invariant.
6. Consequence for carrier normalization
The homogeneous result
is robust and exact. In a heterogeneous environment, selecting \(\mu_{\det}\) rather than \(\mu_{\rm F}\) requires a physical principle.
Possible selectors are:
- direct normalization by boundary flux response, favoring \(\mu_{\det}\);
- long-distance decay, favoring \(\mu_{\rm F}\);
- a gluing law for coupled matrix channels;
- an observed pole or threshold;
- a stress-energy functional.
Subdivision invariance alone does not choose.
7. Stop-everything criteria
Advance determinant scale: the dressed carrier couples directly to the boundary Hessian, making \(\sqrt{\det X}\) operational.
Advance Floquet scale: the carrier is defined by asymptotic transfer/decay rather than local boundary flux.
Reject uniqueness: both scales affect observables independently and no symmetry relates them.
Generalize: find a unique symplectic spectral invariant for interacting matrix-valued transfer that reduces to both in the relevant limit.
8. Certificate
The heterogeneous transfer certificate compares both scales for homogeneous and piecewise masses, verifies exact refinement invariance and checks the reversal behavior.