Boundary Logic and Cross-Sectional Projection

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This section reformulates the core statements of the Dimensional Memorandum (DM) framework in the standard language of quantum field theory (QFT), general relativity (GR), and holographic effective field theory. DM’s key claim—that each lower-dimensional layer of physics is a boundary projection of a higher-dimensional coherence field—is expressed using familiar constructs: actions, boundary terms, path integrals, traces, and Noether currents. We show how 5D coherence fields Φ(x,t,s) reduce to 4D effective wavefunctions Ψ(x,t) and 3D observables ρ(x) via integration over an extra coherence coordinate s and over time, reproducing standard collapse, decoherence, and conservation laws as boundary conditions. All of this can be written as a single master field equation whose boundary projections yield Maxwell, Schrödinger, Dirac, and Einstein equations as special cases.

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1. Boundary Logic in Physical Terms

In conventional physics, a boundary corresponds to a hypersurface on which field values or fluxes are constrained. In field theory, the dynamics of a field Φ are derived from a variational principle applied to an action S[Φ] defined over a manifold M, with boundary conditions imposed on ∂M. The Dimensional Memorandum (DM) rephrases this in geometric language: each lower-dimensional layer is literally the boundary of the one above.

In DM, the fundamental object is a 5D coherence field Φ(x,t,s) defined on a 5D manifold M₅, where x ∈ ℝ³ are spatial coordinates, t is physical time, and s is a coherence coordinate. The 4D spacetime field Ψ(x,t) that we normally work with in QFT/GR is treated as an effective field obtained by integrating out the extra coordinate s:

Ψ(x,t) = ∫ Φ(x,t,s) e^{−s/λₛ} ds.

Here, λₛ is a coherence length along the s-axis. This construction is directly analogous to holographic setups and Kaluza–Klein reductions: integrating out the extra dimension generates lower-dimensional kinetic terms, masses, and couplings.

In GR/QFT language:
• The Φ-field is a higher-dimensional parent field whose action S[Φ] lives on M₅.
• The Ψ-field is the 4D effective field obtained after compactification/projection along s.
• The ρ-domain corresponds to a 3D equal-time hypersurface on which observables are defined.

The DM boundary statement,
“Each dimension is the boundary of the one above,”
can be written as: physical laws are the Euler–Lagrange conditions of the higher-dimensional action S[Φ], evaluated on its boundary.

Formally, with an action S[Φ] = ∫_{M₅} 𝓛[Φ,∂Φ] d⁴x ds, the stationary condition δS[Φ] = 0 yields both bulk equations of motion and boundary terms. Imposing that the boundary variations vanish,
(δ𝓛/δΦ) |_{∂M₅} = 0,
produces, under appropriate projections, the familiar equations of motion: Einstein, Schrödinger, and Maxwell emerge as lower-dimensional Euler–Lagrange conditions on ∂M₅. In other words, what we usually call “fundamental equations” are, in DM, boundary conditions of a single higher-dimensional coherence action.

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2. Cross-Sections as Measurement and Collapse

In standard quantum theory, a measurement is modeled either as projection in Hilbert space or as environment-induced decoherence. Operationally, both can be represented as a trace over unobserved degrees of freedom or as restriction to an equal-time hypersurface. DM makes this geometric and explicit.

The 3D observed density ρ(x) at a fixed laboratory time t₀ is written in DM as
ρ(x,y,z) = ∫ Ψ(x,y,z,t) δ(t − t₀) dt.

This is simply the statement that a detector records the field configuration on a t = t₀ slice. In QFT language, this is a spacelike hypersurface Σ_{t₀}, and ρ is the restriction of the state to Σₜ₀. The projection from Φ to Ψ is similarly written as an integral over the extra coordinate:
Ψ(x,y,z,t) = ∫ Φ(x,y,z,t,s) e^{−s/λₛ} ds.

Taken together, these two projections are equivalent to tracing over unobserved degrees of freedom—first over s (coherence depth), then over t (except for one chosen time slice). In density-matrix language, one can write schematically:
ρ₍₃D₎ = Trₛ,ₜ ( |Φ⟩⟨Φ| ),

where Trₛ,ₜ denotes a trace over the s-coordinate and over time evolution except for the chosen measurement hypersurface. This is exactly how decoherence and collapse are modeled in open quantum systems: by tracing out unobserved environment variables, one obtains an effective mixed state for the subsystem.

Thus, in standard physics terms, DM’s “cross-section” language corresponds to:
• equal-time slicing (Σₜ) familiar from canonical quantization,
• path integration over hidden coordinates (s),
• and partial tracing that produces an effective, reduced density matrix.

What DM adds is a concrete geometric interpretation: the degrees of freedom we trace out are not abstract environments but components of a higher-dimensional coherence field. 

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3. Conservation Laws as Boundary Conditions

Conservation laws in physics are generated by continuous symmetries of the action via Noether’s theorem. In GR, this appears as ∇_μ T^{μν} = 0; in electromagnetism, ∂_μ J^μ = 0; in fluid dynamics, as divergence-free fluxes. In DM, these conservation laws are interpreted geometrically as the manifestation of higher-dimensional invariance on a lower-dimensional boundary.

Let M₅ be the 5D manifold and ∂M₅ its boundary. For a differential form F defined on M₅, Stokes’ theorem states:
∫_{∂M₅} F = ∫_{M₅} dF.

If dF = 0 in the bulk (closed form), then the boundary integral vanishes, corresponding to a conserved flux. In DM language, the coherence field Φ has an associated higher-dimensional current Jᴬ (with A running over the 5D indices), whose divergence vanishes in M₅:
∇ᴬ J_A = 0.

When projected to 4D and 3D, this becomes the familiar conservation equations:
∇_μ T^{μν} = 0,   ∂_μ J^μ = 0.

In other words: conservation in observed spacetime is the shadow of a higher-dimensional conservation law in the coherence manifold. DM encapsulates this as:

“Conservation is the shadow of higher-dimensional invariance.”

Mathematically, this is just Stokes plus Noether in one dimension higher: the invariance of the 5D action under a continuous symmetry implies a divergence-free current in M₅; upon restriction to ∂M₅, that becomes the familiar conservation laws we measure.

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4. Wavefunction Collapse in Physical Units

In standard decoherence theory, a quantum system loses phase coherence on a timescale τ_φ determined by its coupling to the environment. Collapse is effectively complete when τ_φ is much shorter than the internal oscillation period 1/ƒ꜀ of the system, so that phase information cannot be recovered.

DM parameterizes coherence using a depth coordinate s and a scaling law
ƒ(s) = ƒₚ e^{−s/λₛ},

where ƒₚ is the Planck frequency and λₛ is a coherence length along s. For a particle of mass m, the internal frequency is the Compton frequency
ƒᴄ = mc² / h,

which DM identifies with f(s) at the appropriate coherence depth s = s(m). When the environment drives effective decoherence such that the relevant τ_φ ≪ 1/ƒ꜀, the projection from Φ → Ψ → ρ becomes effectively irreversible: the system appears collapsed in the ρ-domain.

Thus, the DM picture is fully compatible with decoherence theory, but recast geometrically:
• The coherence frequency ƒ꜀ is just the internal phase rotation rate of the 5D coherence field Φ.
• The environment pushes the system deeper in s (larger effective s/λₛ), reducing f(s) and destroying recoverable phase relations.
• Collapse is the limiting case where Φ’s off-diagonal coherence in s and t has been traced out, leaving a 3D mixed state ρ(x) consistent with measurement statistics.

In physical units, nothing exotic is introduced: DM simply reinterprets ƒ꜀ = mc²/h and τ_φ as geometric parameters of a single higher-dimensional coherence manifold.

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5. Unified Master Equation and Projections

The unifying statement of DM can be written as a single 5D master field equation whose lower-dimensional projections reproduce known field equations. A convenient schematic form is:

iħ ∂Φ/∂t = −(ħ²/2m) ∇²Φ − (ħ²/2mₛ) ∂²Φ/∂s² + μ Φ + g |Φ|² Φ,

where m is an effective 4D mass, mₛ is an effective mass associated with the s-direction, μ is a potential term, and g encodes nonlinear self-interaction. This equation is structurally a 5D nonlinear Schrödinger/Klein–Gordon type equation.

The observed 4D and 3D fields are then defined by the projection relations:
Ψ(x,t) = ∫ Φ(x,t,s) e^{−s/λₛ} ds,
ρ(x) = ∫ Ψ(x,t) δ(t − t₀) dt.

Under appropriate limits and choices of parameters, the master equation reduces to:
• Schrödinger equation for nonrelativistic wavefunctions (when ∂ₛ terms are negligible and μ, g are tuned),
• Dirac-type equations when spinor structure and Clifford generators are included in Φ,
• Maxwell’s equations when Φ is restricted to gauge-like components and ∂ₛΦ ≈ 0,
• Einstein-type equations when Φ is coupled to a metric and the action S[Φ,g] is varied with respect to g.

From the perspective of QFT and GR, DM is therefore not adding a competing theory but providing a higher-dimensional coherence field whose boundary restrictions generate all familiar dynamics as effective theories. The “collapse” is just the geometric act of restricting the global field configuration on M₅ to a 3D hypersurface within a 4D section of that manifold.

In summary, rewritten in your language:
• DM postulates a single higher-dimensional field Φ(x,t,s) living on M₅.
• Effective fields (Maxwell, Schrödinger, Dirac, Einstein) are boundary projections on ∂M₅.
• Collapse, measurement, and decoherence correspond to cross-sectional restriction and tracing over s and t.
• Conservation laws are Noether currents in M₅ whose shadows are the familiar 4D and 3D conservation laws.

What DM adds is a clean geometric picture: all of this is simply boundary logic and cross-sectional projection from a coherent higher-dimensional action.