Block elimination and constrained minimization give explicit effective laws. Their coefficients depend on supplied gaps, couplings, and response operators; controlled dynamics additionally needs a remainder bound and a declared preparation sector.
An auxiliary can matter without being observed
A retained description often omits fast, costly, or otherwise inaccessible variables. Those variables can still connect configurations through virtual excursions or constrained responses. Eliminating them produces an effective interaction among the retained variables, even when the larger Hamiltonian contains only simple local couplings.
This is a constructive SMM/DTT possibility: inner generative work could mediate relations among completed records. The explanation becomes source-level when the same update determines both the retained dynamics and the auxiliary response. If their coefficients are independently programmed, elimination is still valid, but their common origin remains a separate question.
Exact energy-dependent block elimination
Split a finite Hilbert space into retained and auxiliary sectors. Write an eigenvector as (x,y) and the Hamiltonian in blocks with diagonal entries HPP, HQQ and coupling V. The auxiliary equation is V†x+(HQQ−E)y=0. Whenever HQQ−E is invertible:
Heff(E)=HPP−V(HQQ−E)−1V†.
The resulting retained equation is exact, but the effective operator depends on the unknown energy E. Replacing E by a reference value is an approximation that needs a spectral separation and an error estimate. The negative sign explains the downward energy shift from coupling to a sufficiently high positive sector.
Check the approximation in two dimensions
Consider H with matrix entries [[0,g],[g,Δ]], where Δ>0 and g is real. Its lower eigenvalue is E−=(Δ−√(Δ²+4g²))/2. Expanding for |g|/Δ≪1 gives:
The leading term agrees with setting E=0 in the eliminated denominator. For g=0.1Δ, it predicts −0.01Δ, while the exact result is approximately −0.009902Δ. The example shows how the gap and coupling set both the induced energy scale and the expansion parameter. Merely having a positive auxiliary gap does not make every truncated series accurate.
A virtual closed loop produces a holonomy potential
In the four-position singlet model, a complete oriented excursion around the loop first appears at fourth order in hopping. The small-hopping expansion of the exact static energy is:
The second-order term is independent of holonomy. The fourth-order coefficient multiplies a familiar invariant loop expression. Terms accounting for intermediate returns and energy normalization must also be retained; they cancel a holonomy-independent fourth-order contribution at W=I. The coefficient is selected by this apparatus and its parameters, not by gauge symmetry alone.
This is a static finite-model expansion. Turning it into a field Hamiltonian requires controlling electric variables, many cells, preparation, and omitted operators.
A classical constraint yields a different effective law
On a connected graph let B be an edge-by-vertex incidence matrix, L=BᵀB its Laplacian, and G=L⁺ its inverse on neutral sources. For η > 0, minimize (η/2)||F||² over edge fluxes with BᵀF=ρ and Σρ=0. The minimizer is F*=BGρ.
Any other feasible flux is F*+z with Bᵀz=0, and ⟨F*,z⟩=0 proves the minimum. Opposite source signs produce a negative cross term after expanding the quadratic form. This is a quasistatic mediator mechanism; it does not by itself introduce quantum photons or determine η relative to matter propagation.
Translate an operator error into a time interval
For self-adjoint finite effective and target Hamiltonians with ||Heff−Htarget||≤r, Duhamel's formula gives a unitary evolution error bounded by min(2,|T|r/ħ). To guarantee error at most ε, the sufficient interval is |T|≤εħ/r.
The bound applies to the declared retained or dressed sector. Starting from a bare auxiliary state can add preparation error. Finite link truncations can also spoil exact unitarity and produce boundary corrections before the desired loop order. A useful approximation report states the regime, preparation, operator remainder, and useful observation time together; a formally valid estimate larger than two supplies no practical accuracy certificate.
Units must remain consistent in this estimate: r is an energy and T is a time, so |T|r/ħ is dimensionless. Replacing ħ by one in a simulation is a unit convention, not a source derivation of the action scale or its laboratory value.
Derive a relation, not just an available mechanism
The positive result is that hidden work can induce a calculable interaction. The stronger SMM/DTT target is to constrain its strength and form through the same nested source dynamics that creates the record geometry and matter band. For example, one could derive a fixed relation between an auxiliary gap, a hopping rate, and an independently measured clock response.
Such a relation would survive after the auxiliary variables were eliminated and could be checked across several protocols. The six inner roles should enter through explicit dependencies that enforce it. Until then, the effective calculations provide rigorous model mechanisms and a clear list of coefficients still awaiting source-level selection.
Continue with the articles
- Controlled effective gauge Hamiltonians
Introduce the finite-cell effective Hamiltonian and its explicit remainder. Explain why a controlled approximation needs a stated norm bound and an evolution-error estimate.
- An exact singlet-loop response
Introduce the four-position mobile carrier and stationary dual used in the solvable model. Explain how the holonomy-dependent resolvent determines the ground-state response under the supplied Hamiltonian.
- A protected gap in the finite singlet model
Explain the model’s unique ground state and gap bound. Distinguish a finite-system spectral statement from a continuum mass gap in a physical gauge theory.
- Eliminating a mediator to obtain attraction
Describe the local quadratic mediator and its exact Green-function interaction. Explain how the chosen source signs and response operator enter the effective pair energy.
- Why binding does not fix the coupling
Introduce the dilation argument showing that a family of positive couplings remains compatible with binding and completion. Identify the extra source-selection law needed to obtain a specific interaction strength.