Basis Expansion and Projection¶
Analytic projection — instantiates Superposition Modeling and Interference Analysis
Expresses a state as coordinates in a chosen basis by projection, then checks how completely the basis reconstructs it.
Basis Expansion and Projection runs the inverse of construction: rather than building a composite from known weights, it takes a given state and solves for its coordinates in a declared basis, then reports how much of the state that basis actually captures. Its defining idea is the basis as an accounting frame — the mechanism's whole burden is choosing a representation space, projecting the state onto each basis element to read off a coefficient, and quantifying the residual left over. A coefficient means nothing without the basis and normalization convention that produced it, so this mechanism treats the basis and its completeness as the object to be certified, not merely a convenience.
Example¶
A geodesy team needs a compact, computable description of Earth's gravity field. The field over the globe is a function on a sphere, and they represent it by spherical-harmonic expansion: the basis elements are the spherical harmonics Yₗₘ, indexed by degree ℓ and order m, and every longer-wavelength harmonic captures a coarser feature of the geoid. To find the coefficients they project — integrate the measured field against each harmonic — yielding one Stokes coefficient per basis element. Truncating the sum at, say, degree 2159 (as high-resolution models such as EGM2008 do) reconstructs the field down to features roughly ten kilometres across; everything finer falls into the truncation residual, the "omission error" they report alongside the model. Because the harmonics are orthonormal under the standard convention, the total power of the field equals the summed squared coefficients, so the team can say exactly how much of the field's energy each degree band accounts for and where the representation stops resolving. Change to a different basis and the coordinates change entirely, though the represented field does not.
How it works¶
- Declare the space and basis. Fix the representation space and the basis functions, and record their character — complete or truncated, orthogonal or not, and under which normalization convention.
- Project. Compute each coefficient by taking the inner product of the state with the corresponding basis element; for an orthonormal basis this reads the coordinate off directly.
- Reconstruct and measure residual. Sum the retained terms and compare to the original; the residual quantifies how completely the chosen (possibly truncated) basis represents the state.
- Check the conserved quantity. Verify that the invariant tying the state to its coefficients — total energy or norm — is preserved across the two representations, confirming the projection lost nothing it should not have.
Tuning parameters¶
- Basis choice — which functions span the space; a basis matched to the state's structure concentrates energy in few coefficients, a mismatched one smears it across many.
- Truncation degree — how many terms are retained; more terms shrink the residual but raise cost and can amplify noise in the high-order coefficients.
- Orthogonality — an orthonormal basis makes projection a clean per-element read; a non-orthogonal one couples the coefficients and complicates interpretation.
- Normalization convention — the scaling that fixes what a coefficient's magnitude means; changing it rescales every coefficient, so it must be declared and held fixed.
When it helps, and when it misleads¶
Its strength is compression and interpretability: a well-matched basis turns an unwieldy state into a short, ranked list of coefficients whose residual is explicit, so you know precisely what the representation keeps and what it drops.
Its failure mode is basis amnesia — reporting coefficients as if they were intrinsic properties of the state rather than artifacts of a chosen frame. Because the same state has different coordinates in every basis, a coefficient quoted without its basis and normalization is uninterpretable, and comparing coefficients across two conventions is meaningless. The invariant that anchors the mechanism, that projection preserves total energy between the state and its coefficients,[n1] is also the standard check that a basis is behaving: if the summed coefficient power drifts from the state's own, either the basis is incomplete or the normalization has slipped. The guarding discipline is to carry the basis, convention, and residual with every coefficient, and never to present basis-dependent numbers as invariant physical quantities.
How it implements the components¶
representation_space_and_basis— its core output: a declared basis, its completeness or truncation, and the coordinates a state takes within it.normalization_conservation_and_invariant_contract— the energy/norm equality between state and coefficients is the invariant it verifies to confirm the projection is faithful.superposition_purpose_and_claim— it states the representation's purpose (compression, computation) and the evidence burden that discharges it: a bounded, reported residual.
It does not build a composite forward from chosen constituents and weights (constituent_state_inventory, linearity_and_combination_rule) — that is its nearest twin Vector Linear-Combination Construction, which assembles rather than projects — and it does not adjudicate whether a recovered decomposition is the only one that fits (decomposition_identifiability_and_recovery_profile), the concern of Mode Decomposition and Recomposition.
Related¶
- Instantiates: Superposition Modeling and Interference Analysis — this mechanism supplies the coordinate representation and completeness evidence the archetype's decomposition and reconstruction claims rest on.
- Sibling mechanisms: Vector Linear-Combination Construction · Mode Decomposition and Recomposition · Phasor or Complex-Amplitude Addition · Wave Superposition Simulation · Interference Pattern Mapping · Response-Addition Linearity Test · Boundary-Condition Superposition Test · Nonlinear Breakdown Review · Coherence and Dephasing Sweep
Editorial Notes¶
Form Classification¶
Form family: Analysis, Modeling & Optimization
Rationale: Expresses a state as coordinates in a chosen basis by projection, then checks how completely the basis reconstructs it, making its operative form a computation or analytic transformation that produces an inference, comparison, or optimized result.
Independent corroboration: The frozen evidence defines Basis Expansion and Projection as 'Expresses a state as coordinates in a chosen basis by projection, then checks how completely the basis reconstructs it', so its operative form is Analysis, Modeling & Optimization.
Review outcome: Independent reviewer agreement; high confidence.
Origin Attribution¶
Primary origin: Mathematics
Origin pattern: Single lineage
Present-day reach: Specialized
Rationale: Linear and functional analysis express a state in basis coordinates by projection and quantify the residual outside the chosen span.
Related originating lineages:
- Engineering & Design — Signal decomposition uses reconstruction error to test basis completeness.
- Physics — Fourier and quantum expansions apply projection to physical states.
Review resolution: Mathematics is the agreed primary through functional analysis, inner-product projection, and residual reconstruction. Physics and engineering are materially formative applications within the established specialized lineage.
Review outcome: Reconciled after independent review; high confidence.
Notes¶
[n1] Parseval's theorem — for an orthonormal basis, the total energy (squared norm) of a state equals the sum of the squares of its expansion coefficients. It guarantees that projection redistributes but does not create or destroy the state's energy, which is why a mismatch between the two is a reliable signal of an incomplete basis or a slipped normalization convention. ↩