---
slug: baseline_plus_correction
name: Baseline Plus Correction
one_liner: "Solve an intractable problem by splitting it into an exactly solvable baseline plus a small deviation, then recovering the true answer as the baseline corrected by successive terms in that deviation."
type: prime
schema_version: 2
status: density_v2_draft
categories: [structural, computational, physical]
origin_domain: applied_mathematics
alternate_origin_domains: [physics, numerical_analysis, engineering]
aliases: [baseline_and_correction, solvable_core_plus_deviation, series_expansion_around_a_solution, reference_plus_residual, expansion_in_a_small_parameter]
related: []
solution_archetypes: []
review_flags:
- "OPERATOR-DERIVED (op-05 generalization) — the proposed GENUS of `perturbation_theory` (lift: strip the *physics/Hamiltonian-operator* frame — the baseline need not be a solvable Hamiltonian and the expansion need not be quantum — up to any solve-a-baseline-plus-a-small-correction decomposition). Discovered in the op-05 pilot sweep 2026-07-03. Tier A candidate; `linearization` absent, and the existing `approximation` prime is a different (static) structure. STATUS: HOLD pending owner keep/cut."
- "Created 2026-07-03; STUB ONLY (frontmatter + Core Idea + Notes). SI/SF/similarity are PROPOSER ESTIMATES pending triangulation. Body sections (Structural Signature onward) NOT written."
- "Phase-C action: re-similarity-check vs `perturbation_theory`, `approximation`, and the named siblings (Taylor-series linearization, regular perturbation in ODEs, first-order structural approximation); confirm it is a genuine genus (covers the input AND the siblings) and not a domain instance or a restatement of the input, and that it is distinct from the static `approximation` prime; draw a subsumption edge perturbation_theory→baseline_plus_correction + sibling edges at incorporation; sync one_liner to a v1 draft per CLAUDE.md before merge."
substrate_independence:
  composite_substrate_independence: 4
  domain_breadth: 4
  structural_abstraction: 5
  transfer_evidence: 4
  reasoning: 'The bare structure — decompose an intractable problem into an exactly solvable reference plus a small deviation, then reconstruct the true answer as a series of corrections ordered by the deviation''s size — is a pure mathematical relation with maximal structural abstraction (5): it makes no reference to any physical medium and applies to any quantity that can be expanded around a solvable point. Domain breadth and transfer sit at 4 because, while it recurs across physics, numerical analysis, structural engineering, and economics (comparative statics), its natural home is quantitative/analytic settings — it needs a notion of "small parameter" and a solvable baseline, which not every domain affords. It scores well above the framed input `perturbation_theory`, whose vocabulary (Hamiltonian, eigenstate, coupling) is physics-bound; the lift removes exactly that vocabulary.'
similarity_to_nearest_existing_prime:
  slug: perturbation_theory
  similarity: 0.6
structural_framed:
  label: structural
  aggregate: 0.14
  criteria: {vocab_travels: 0.9, evaluative_weight: 0.05, institutional_origin: 0.15, human_practice_bound: 0.05, import_vs_recognize: 0.2}
  rationale: 'The pattern is a pure structural/computational decomposition carrying no evaluative content and no dependence on human practice: it is recognized as a bare mathematical form (solvable baseline + expanded correction), and its skeleton travels transparently across physics, numerical methods, and engineering under different names (perturbation, linearization, comparative statics). It reads as one of the cleaner structural primes on the scale.'
phase_b_passes_completed: []
---
# Baseline Plus Correction

## Core Idea

Baseline-plus-correction is the strategy of *solving an intractable problem by decomposing it into an exactly solvable baseline plus a small deviation, then recovering the true answer as the baseline corrected by successive terms ordered in the size of that deviation*. Bare structure: the hard problem H is written as H = H₀ + (small)·V, where H₀ is chosen precisely because it can be solved exactly and V captures everything the baseline omits; the quantities of interest are then expanded as a series whose leading term is the baseline answer and whose higher terms are corrections that shrink with each order. The constraint stripped from the input `perturbation_theory` is *the physics operator frame*: perturbation theory is the special case where H₀ is a solvable Hamiltonian or Lagrangian, V is a physical coupling, and the expanded quantities are eigenvalues, eigenstates, and cross-sections. Lift away "the baseline is a solvable physical operator" and what remains is the general decomposition — solvable reference plus expanded small correction — of which quantum perturbation theory is one species.

The load-bearing anatomy has three parts. First, a *solvable baseline*: a nearby problem chosen not because it is the truth but because it can be answered exactly. Second, a *small deviation*: the gap between the real problem and the baseline, parameterized by a quantity assumed small. Third, an *ordered correction series*: the true answer expressed as the baseline plus terms of increasing order in the deviation, each smaller than the last. The characteristic payoff is *tractability with a convergence guarantee in the small-deviation regime*: an otherwise unsolvable problem yields an answer as accurate as one cares to compute, and truncating after the first correction already captures the leading behavior. The characteristic cost is that the whole method is *conditional on the deviation actually being small* — when the correction is not small (strong coupling, large residual) the series diverges or converges uselessly, and the baseline, chosen for solvability rather than fidelity, can sit so far from the truth that no finite number of corrections rescues it. This distinguishes it sharply from a static `approximation`, which replaces the problem with a nearby easier one and stops; baseline-plus-correction keeps the exact problem and reconstructs it as an ordered, in-principle-completable expansion around the baseline.

Read structurally rather than as a physics technique, the same move recurs wherever a hard problem sits near a solvable one. In *dynamical systems*, Taylor-series linearization writes a nonlinear system as its solvable linear part plus higher-order corrections, valid near an operating point. In *differential equations*, regular perturbation expands the solution of a hard ODE around the solution of a simpler one in a small parameter. In *structural engineering*, first-order analysis takes the exactly-solvable linear-elastic response as the baseline and adds corrections for geometric and material nonlinearity. These siblings are structurally unlike the quantum input — a linearized pendulum is not a perturbed Hamiltonian eigenstate — yet each is the same skeleton: pick a solvable baseline, measure the small deviation from it, and rebuild the true answer as the baseline plus an ordered series of corrections.

## Notes

**Provenance.** Produced by operator 05 (generalization / lift) as the proposed genus of `perturbation_theory`, reached by stripping the physics/operator frame up to any solvable-baseline-plus-expanded-correction decomposition. Surfaced in the op-05 pilot sweep on 2026-07-03. Claimed siblings: Taylor-series linearization of nonlinear dynamics, regular perturbation in singular ODEs, first-order approximation in structural engineering. Manifest tier: A. Flags note `linearization` is absent and that the existing `approximation` prime is a *different, static* structure.

**For owner evaluation (keep/cut).** The case FOR: this is a clean, high-abstraction mathematical genus (structural score ~5) whose siblings are genuinely unlike the quantum input — Taylor linearization and first-order structural analysis share the skeleton without sharing any physics vocabulary, which is strong evidence the lift is real and not `perturbation_theory` renamed. It captures a distinct problem-solving strategy the corpus lacks. The specific RISKS Kurt should weigh: (1) **collapse into `approximation`** — the sharpest boundary question. The flag itself insists these differ: `approximation` (static) *replaces* the problem with an easier nearby one; baseline-plus-correction *keeps* the exact problem and reconstructs it as an ordered, completable series. Kurt should confirm this distinction holds and is worth a separate node. (2) **overlap with `linearization`** — if `linearization` is later drafted it is a *sibling species* (first-order-only), not a parent; guard against parenting inversion. (3) **restatement of the input** — similarity ~0.6; keep only if the non-physics siblings are judged genuinely covered. Sharp keep/cut question: *is "solvable baseline + ordered correction series" distinct enough from `approximation` to earn its own prime, or should it be folded in as an approximation sub-mode?*

**At incorporation.** Re-run similarity against `perturbation_theory` (the input), `approximation` (the collapse risk), and any `linearization` or `series_expansion` prime to confirm no collapse. Draw a subsumption edge `perturbation_theory` → `baseline_plus_correction` and sibling edges to linearization / regular-perturbation primes if present. Missing-edge note: `linearization` (currently absent) should attach here as a sibling species when added; verify the boundary edge to `approximation`. Write a v1 draft and sync the one_liner across v1 and v2 per CLAUDE.md before merge.
