#Mathematical Lens Adequacy (MLA)

C.29 · Cluster C.IV - Composite & Macro-Scale · Stable · Architectural pattern · Normative unless explicitly marked informative · Part C - Kernel Extension Specifications

#About this pattern

This is a generated FPF pattern page projected from the published FPF source. It is canonical FPF content for this ID; it is not a fpf-memory product feature page.

#How to use this pattern

Read the ID, status, type, and normativity first. Use the content for exact wording, the relations for adjacent concepts, and citations to keep active work grounded without pasting the whole specification.

Type: Architectural pattern Status: Stable Normativity: Normative unless explicitly marked informative

Plain-name. Mathematical lens adequacy.

Governed object. C.29 governs only mathematical-lens adequacy claims carried by FPF prose, pattern examples, method notes, review records, PublicationUnits, decision-facing text, comparison-facing text, bridge-facing text, or assurance-input text that use a mathematical object, formalism, learned representation, simulation substrate, or mathematical family as a lens for a stated use. It does not govern those objects themselves: PublicationUnits, decision records, comparative review units, bridges, work records, evidence paths, and assurance inputs remain with their own FPF loci; C.29 contributes only the bounded adequacy of the mathematical lens used inside them.

Output posture. C.29 outputs are claim-supporting notes, not actors, approvals, gates, work records, or release decisions. They state what the mathematical lens can support for one declared use and which neighboring FPF locus carries any live claim outside lens adequacy.

No new U.* from MLA. MLA.OneLine, MLA.MiniCard, MLA.FullCard, MLA.Card@Context, MLAOutputRef, and CC-MLA-* are C.29-local instruments. They do not mint U.MathematicalLens, U.MLARecord, LensKind, MLACompliance, or a durable record family. Durable names, kinds, or records require explicit FPF support through F.18, C.3, F.8, and E.9.

Use this card before the full card. It is enough for the first reading unless publication, bridge, assurance input, benchmark, model selection, prediction, formal pattern claim, or repeated cross-case use is live.

#Keywords

• mathematical lens
• structure-preserving representation
• lens mapping mode
• preserved structure
• lost structure
• invariants
• stop condition
• scale window
• coarse-graining
• rival lens
• LensSupportPosture
• validation posture
• learned lens
• ontology smuggling.

#Relations

#Content

#First-use card

Use this card before the full card. It is enough for the first reading unless publication, bridge, assurance input, benchmark, model selection, prediction, formal pattern claim, or repeated cross-case use is live.

Non-use comes first. Do not open C.29 merely because mathematics appears. Open it only when the mathematical structure changes explanation, decision, prediction, comparison, publication, bridge, assurance input, reusable transfer, or the next admissible repair. If state, transition, measurement, causal use, bridge semantics, temporal adequacy, assurance, selector, benchmark, or release is the live claim, name the neighboring FPF locus and keep C.29 to the mathematical-lens adequacy part.

#Problem frame

FPF already uses mathematical structures in several local patterns. A.6.P asks for stable mathematical substrate during relation precision restoration; A.3.3 governs dynamics; A.19 governs characteristic spaces and structural overlays; C.18.1 and C.19.1 govern scale-law and Bitter-Lesson claims; C.26 contains the separation of a quantum-like lens from physical quantum ontology; F.9 governs cross-context bridges and loss.

The positive need is as important as the guard. In working projects, first-principles mathematical thinking starts from the smallest declared structure that can make a next move derivable, inspectable, or honestly blocked. A queue can expose waiting and bottlenecks, a state space can expose variables and transitions, a graph can expose dependencies, a metric-space distance or topology can expose comparability limits, a symmetry can expose invariants, a variational principle or constrained optimization functional can expose an extremal condition, admissible variation space, boundary condition, conservation link, or trade-off, an information or probability measure can expose uncertainty, a resource bound can expose realizability limits, and an obstruction can expose where a transfer or simplification stops.

The missing FPF rule is general but narrow: when FPF prose, a pattern example, method note, review record, PublicationUnit, or neighboring-pattern note uses or plausibly needs a mathematical object, formalism, or family as the basis for explanation, decision, prediction, comparison, publication, bridge, assurance input, or reusable transfer, the C.29 application records the useful first-principles modeling basis and its boundary. It names the candidate mathematical object or family, what structure is preserved, what structure is lost, what invariant, supported distinction, obstruction, diagnostic boundary, or constructive limit becomes visible, which LensSupportPosture value is declared for that use, and where the transfer stops.

C.29 is not opened because mathematics appears. It is opened when either a mathematical object is used for explanation, decision, prediction, comparison, publication, bridge, assurance input, or reusable transfer, or a stable working problem is under-lensed and a cheap candidate lens could expose useful structure for the next move.

The first move is not a full-card demand. It is a first-principles entry decision: choose the smallest mathematical structure that changes the next admissible move, keep ordinary prose when no mathematical basis changes the move, or send the live claim to the neighboring FPF pattern. The result records what the lens preserves, what it loses, what it makes visible, what remains blocked, and where the use stops.

Selected compact formulation:

A useful mathematical lens is compression with invariants and declared losses.

This compact line is retained as a Plain-register orientation, not as a substitute for the card. It keeps the useful metaphor of a lens: a mathematical object can make a hidden structure visible, but only by carrying some structure and dropping other structure. The first reader questions are: what survives the transfer, what is lost, what can now be done, and where does the lens stop?

#First-minute working situation

An FPF author, reviewer, or practitioner faces a working situation where ordinary prose can hide useful structure, or where a mathematical phrase is already doing work:

• waiting, backlog, bottleneck, or throughput can call for a queue or flow lens;
• state change, stabilization, control pressure, or forecast can call for state-space or dynamics vocabulary;
• dependency, interface, composition, or transfer failure can call for graph, hypergraph, category, operad, or compositional vocabulary;
• similarity, distribution shift, population movement, or shape change can call for metric-space distance, topology, embedding, or optimal-transport vocabulary;
• scale transition, coarse behavior, universality, knee, or scaling pressure can call for coarse-graining, RG, or scaling-law vocabulary;
• probe effects, order effects, context effects, or incompatible frames can call for quantum-like or contextual-probability vocabulary.

The useful first-minute intuition is not “hunt for overclaim.” It is “find the structure that would improve the next move, then name the limits.” A vivid phrase can remain when the C.29 output records what the lens lets the reader see, what it does not license, and which neighboring pattern carries any causal, evidence, bridge, dynamics, scale, measurement, assurance, or release claim.

Without a general adequacy discipline, the reader cannot tell whether the phrase is a bounded structure-preserving representation, an analogy-only prompt, an unsupported ontology import, a local domain model, or prestige language.

#Minimum scenario / anti-case basis

Positive scenario. A production line is represented as a queueing network. The lens preserves flow, bottlenecks, service rates, and waiting times; it loses human meaning, contractual obligations, rare failure modes, and causal interventions not represented by the network; the stop condition says that the queueing lens supports throughput and latency reasoning, not a full organizational ontology.

Anti-case. “The organization is a quantum system” is written without a candidate mathematical object, probe/readout distinction, preserved structure, lost structure, LensSupportPosture, or stop condition. The C.29 result is either a downgrade to local metaphor or a repaired use through C.29 and, where relevant, C.26.

Under-lensed anti-case. “The work stream has dynamics” or “this portfolio is a network” is used for a diagnosis that affects prediction, comparison, repair, or stop conditions, but no mathematical object changes what can be predicted, compared, diagnosed, repaired, or stopped. The repair is to choose a cheap candidate lens that exposes useful structure, or keep the sentence as ordinary prose.

False-positive scenario. A Markov kernel appears inside accepted local reliability modeling. If no contested lens-transfer, publication, assurance, bridge, or reusable explanation claim is live, the claim stays under A.3.3 and does not open C.29.

#Intended FPF use-value

C.29 gives a cheap use path before any full card or boundary table. Its first job is to help the working reader introduce, choose, repair, bound, or decline a mathematical lens: choose no MLA, a candidate note, a one-line repair, a mini-card, a full card only when publication, bridge, assurance-input, benchmark, model-selection, prediction, or reusable-explanation use requires it, or a named neighboring locus when the live claim belongs to evidence, causal, bridge, assurance, work, decision, publication, or admission governance rather than lens adequacy. Use it only when the mathematical lens affects a claim or next move; ordinary local math and decorative prose stay outside C.29. A successful C.29 result makes useful mathematical compression available to FPF as a disciplined modeling move while reducing ontology smuggling, prestige vocabulary, loss-free transfer, causal laundering, bridge duplication, evidence laundering, and assurance laundering.

#Problem

Current FPF math-lens adequacy criteria are distributed and local:

• A.6.P governs relation precision restoration, but not every mathematical-object transfer.
• A.3.3 governs state, transition, observation, validity, constraints, and calibration for dynamics, but not all mathematical representation choices.
• A.19 governs characteristic spaces, structural overlays, comparability, normalization, and bridge-aware state comparison, but not the adequacy of all mathematical lenses.
• C.18.1 and C.19.1 govern scale-law and BLP claims, but not non-scale mathematical lenses.
• C.26 is the local precedent for mathematical-lens detachment, but only for quantum-like modeling.
• F.9 governs cross-context semantic bridges, but does not decide whether a mathematical substrate is adequate inside one context or as a domain-transferring lens.

There are two symmetric failure modes.

The first failure mode is mathematical under-lensing: a working situation needs a mathematical lens that changes prediction, comparison, diagnosis, repair, or stop conditions, but the record carries only ordinary prose, familiar school math, or a broad family name such as graph, field, space, score, trend, dynamics, or quantum-like without a useful invariant, obstruction, state variable, mapping, scale behavior, rival lens, or action-changing payoff.

The second failure mode is mathematical overread:

a mathematical phrase begins as a helpful representation and then silently becomes ontology, evidence, causality, comparability, assurance, or admissibility.

#Forces

#Solution and selected answer

#Selected answer in one paragraph

C.29 — Mathematical Lens Adequacy (MLA) is the general FPF discipline for mathematical lenses used in explanation, decision, prediction, publication, comparison, assurance input, bridge, or reusable transfer. It handles two first-use cases, with the positive case first: an under-lensed situation where the next admissible move can benefit from a cheap first candidate lens; and an existing candidate lens ready for application, repair, bounding, replacement, or rejection. Its job is to help the reader introduce, choose, apply, limit, replace, or remove a mathematical lens so that a useful admissible next move survives. A mathematical lens is admissible for a declared use when it compresses a phenomenon by preserving declared structure, exposing useful invariants, and producing lens-supported predictions, distinctions, obstructions, or diagnostic boundaries inside a bounded context. It is inadmissible for an undeclared or unsupported use when it imports source-domain ontology, hides loss under metaphor, treats source prestige as evidence, or licenses claims outside its declared scale, context, validation, bridge, causal, or assurance boundary.

C.29 does not mint MathematicalLens, U.MathematicalLens, LensKind, or any universal FPF lens object. In this pattern, “mathematical lens” names a declared use of a mathematical object, formalism, learned representation, simulation substrate, or mathematical family under declared mapping, preserved/lost structure, LensSupportPosture, admissible use, and stop condition; the target phenomenon and any claim outside lens adequacy keep their own FPF kinds.

Admission guard: C.29 governs mathematical-lens adequacy claims. It does not mint mathematical-lens kinds, and it does not govern or create the described entity, Bridge, evidence path, causal support, assurance score, measurement construction, dynamics semantics, decision record, work record, explanation rendering, comparative review unit, representation transition, coarsened rendering, selector, benchmark, or scale audit. Its outputs are local adequacy outputs unless a separate FPF naming and admission decision makes one durable.

#Mathematical Lens Adequacy Principle

Mathematical Lens Adequacy Principle. A mathematical lens is admissible for a declared use when it compresses a phenomenon by preserving declared structure, exposing useful invariants, and producing lens-supported predictions, distinctions, obstructions, or diagnostic boundaries inside a bounded context. It is inadmissible for an undeclared or unsupported use when it imports source-domain ontology, hides loss under metaphor, treats source prestige as evidence, or licenses claims outside its declared scale, context, validation, bridge, causal, or assurance boundary.

Compact plain form:

A useful mathematical lens is compression with invariants and declared losses.

Register policy: Tech exactness below, Plain metaphor above. Plain phrases such as “structures that survive transfer,” “what the lens makes visible,” and “where the lens stops” are admissible as recognition aids. When a sentence carries FPF-kind, relation, evidence, admissibility, causal, assurance, bridge, gate, work, decision, or pattern-application claim force, the corresponding C.29 output recovers the exact fields and receiving patterns.

Zero/first-principles compatibility note: E.1 and E.2 govern the mission and pillar authority. C.29 supports them by making mathematical first-principles support inspectable for one declared use: candidate mathematical object, preserved structure, lost structure, visible payoff, admissible move, neighboring-pattern boundary, and stop condition. It does not replace pillar authority, neighboring governing loci, ordinary FPF reasoning, or E.9 design-rationale support for normative changes.

Mathematics is not a prerequisite for FPF use. Ordinary prose is valid when no mathematical structure changes the next admissible move. C.29 earns its place only when a mathematical object, formalism, learned representation, simulation substrate, or mathematical family changes explanation, decision, prediction, comparison, publication, bridge, assurance input, reusable transfer, or the next admissible repair.

Plain/Tech bridge:

State, scale, and dynamics trigger: if the lens carries state, transition, forecast, rate, temporal window, scale window, observation, measurement, comparison, or causal implication, the cheapest honest output either names the minimal relevant field or names the receiving FPF locus. State and transition semantics stay with A.3.3; characteristic spaces and overlays stay with A.19; measurement construction and direct comparability stay with C.16; temporal-use adequacy stays with C.27; scale-law and scale-preference claims stay with C.18.1 and C.19.1; causal-use support stays with C.28.

#Mathematicalization Utility Principle

A mathematical lens is worth introducing only when it changes the working reader's next admissible move by making at least one first-principles modeling basis visible:

• a declared signature, structure, state variable, transition, or observation map;
• a symmetry, invariant, conservation-like constraint, equivalence, or composition rule;
• a local-global relation, boundary relation, scale variable, coarse-graining rule, scale window, or correspondence condition;
• a variational principle, action, energy, free-energy, loss, or value functional, Euler-Lagrange or stationarity condition, constrained optimization target, dual view, objective vector, or resource trade-off;
• an uncertainty, probability, information, typicality, approximation, sensitivity, or validation boundary;
• an algorithmic, constructive, resource, realizability, implementation, or adversarial limit;
• a bottleneck, obstruction, impossibility, consistency boundary, or failed transfer in the candidate-model space;
• a rival-lens distinction that changes model choice;
• a causal, intervention, or counterfactual preservation question governed by C.28;
• a bridge or export loss governed by F.9;
• a measurement or comparability condition governed by C.16.

If no next admissible move changes, keep the text as ordinary prose, downgrade it to a didactic metaphor, or return NoMLANeededNote. A lens that merely makes prose more impressive is not a successful C.29 result.

#First-principles lens-family support

C.29 supports first-principles use only when the principle family changes what the working reader can derive, inspect, compare, observe, or honestly block. The family name is never enough. Each row below is a discovery and recovery discipline: it tells the reader what must be named before the mathematical lens can carry claim force.

This table is normative as a recovery guide, not as a mandatory taxonomy. A local project may name a closer family, but it must recover the same kind of load-bearing structure: mathematical substrate, preserved structure, lost structure, visible payoff, support posture, and stop condition.

#Use boundary

This boundary prevents C.29 from being over-applied.

Use C.29 when a mathematical object, formalism, learned representation, simulation substrate, or mathematical family is used as a lens for explanation, decision, prediction, publication, comparison, assurance input, bridge, or reusable transfer over a physical, organizational, epistemic, social, computational, scientific, or methodological phenomenon, or when a phenomenon, decision, explanation, comparison, model-selection, diagnosis, or method-choice problem is stable enough that the first useful move is to choose a cheap candidate lens that makes relevant structure visible.

Do not use C.29 as the governing pattern when:

• the mathematics is ordinary local domain theory already governed by a domain pattern;
• the phrase is a purely didactic analogy that is not reused for decisions, evidence, assurance, publication, bridge, comparison, or transfer;
• the live question is causal-use support, which is governed by C.28;
• the live question is measurement construction, scale legality, direct comparability, or evidence-stub adequacy, which is governed by C.16;
• the live question is cross-context meaning or substitution safety, which is governed by F.9;
• the live question is dynamics semantics without a separate lens-transfer claim, which is governed by A.3.3;
• the live question is a CharacteristicSpace overlay with no domain-transfer, prediction, assurance, publication, or reusable explanation claim, which stays under A.19.
• the live object is a ChoiceResult, local choice record, selected-set publication, selected method, U.WorkPlan, performed U.Work, work-result record, or work-relevant source restoration; those claims stay with C.11, G.5/G.9, A.15, A.15.1, or A.15.4 as appropriate.
• the live object is an explanation-facing rendering, bounded comparative review unit, same-described-entity representation-scheme transition, or controlled semantic coarsening; those claims stay with E.17.EFP, E.17.ID.CR, A.6.3.RT, or A.6.3.CSC, with MLA fields carrying only mathematical-lens adequacy when the mathematical lens affects the stated admissible use.
• the live claim is about forecast, rate, trajectory, rhythm, recovery, convergence, stabilization, speed, temporal window, or rate-change as sufficient for use; temporal-claim adequacy stays with C.27.

This boundary keeps mathematical-lens adequacy from becoming a shadow record for neighboring work.

Lexical rule: use structure-preserving representation rather than structure-preserving identification in discoverability-bearing prose, unless equivalence or identity is explicitly the declared LensMappingMode.

#Action path before the full card

Begin with action guidance, not with the full card.

First action choices: keep ordinary prose, introduce a cheap candidate lens, name a substrate that fits the stated use more directly, add visible payoff, add loss, choose the principal rival lens, add validation posture, narrow an existing claim, downgrade an overclaim, or move any evidence, causal, bridge, assurance, work, decision, publication, or admission claim to the exact neighboring FPF locus.

Memory hook: a successful C.29 application can raise or lower the mathematical claim force. It can introduce a first candidate lens, keep ordinary domain prose, remove a mathematical lens, repair relation wording through A.6.P, declare a CharacteristicSpace through A.19, use C.16 for measurement and comparability, open F.9 for bridge semantics, ask the C.28 causal-use question, restore work or source responsibility through A.15, or send temporal-use adequacy to C.27.

No-lens cheap path: name the ProblemStructureCue, choose the cheapest candidate lens family that makes it visible, test whether that lens changes the next admissible move, and if no move changes, keep ordinary prose or collect more observations before using mathematical-lens wording.

First neighboring-locus map:

Math-apparatus boundary: C.29 coordinates the lens-adequacy part across relation substrate, state/characteristic spaces, measurement, dynamics, scale, bridge, causal, evidence, assurance, selector, and benchmark patterns. It does not replace any one of them.

1. Find the claim-bearing phrase. Mark the exact mathematical phrase that affects explanation, decision, prediction, comparison, publication, bridge, assurance-input, or reusable transfer.
2. Choose the smallest output class that preserves honesty. The output-class decision happens before any full-card fields.
3. Name the concrete mathematical object or structure. Family labels such as category theory, field, graph, quantum, RG, or geometry are entry prompts, not adequate substrates for the stated use by themselves.
4. State the lens mapping mode. Use the least committing honest C.29-local lens mapping mode: analogy-only prompt, representation, empirical fit, simulation, quotient, abstraction, coarse-graining, embedding, homomorphism, isomorphism, functor-like transfer, cross-context lens-transfer candidate, or accepted local theory. If cross-context meaning, substitution, CL, sense cells, or Bridge-supported use is live, F.9 governs that claim; the MLA fields record only mathematical-lens adequacy for the declared transfer.
5. State preserved structure and lost structure. This is the central repair move.
6. State what becomes visible. Name the invariant, obstruction, fixed point, symmetry, conservation law, diagnostic boundary, lens-supported distinction, model-selection consequence, or other payoff.
7. State the supported use and blocked use. Say what is now admissible, what remains blocked, and which named neighboring FPF locus governs any live claim outside lens adequacy.
8. If the claim does not pass, repair rather than merely fail. Downgrade, narrow, switch to a principal rival lens, add LensSupportPosture or validation posture, split out bridge, dynamics, measurement, causal, temporal, decision, work, explanation, comparison, representation, scale, or assurance claims to the neighboring governing locus, or remove the mathematical phrase from claim-bearing use.

Application output classes:

Micro-template examples:

MLA.LensCandidateNote example := {
  TargetPhenomenon: slow Product-X team flow,
  ProblemStructureCue: waiting and work-in-progress look more important than individual task difficulty,
  CandidateLensFamily: queue or flow lens,
  CandidateMathObject?: single-server or multi-server queue candidate,
  WhyThisLensCouldHelp: arrivals, service time, WIP, and waiting time could expose the bottleneck,
  ExpectedVisiblePayoff: decide whether delay is arrival-rate, service-rate, batching, or WIP-boundary pressure,
  ObservableOrControllableCue?: arrivals, service time, wait time, WIP limit,
  AdmissibleNextMove: observe the variables before claiming queue adequacy,
  OrdinaryRivalOrFallback: ordinary process narrative without queue assumptions,
  StopCondition: no claim about motivation, obligation, blame, or full team ontology,
  NextMLAOutput: NoMLANeededNote or MLA.OneLine after observation
}

MLA.OneLine example := {
  TargetPhenomenon: Product-X backlog delay,
  CandidateMathObject: queue model over arrivals, service time, waiting time, and work in progress,
  LensMappingMode: representation,
  PreservedStructure: flow, bottleneck candidates, wait, WIP, service-rate pressure,
  LostStructure: motivation, priority politics, contractual duties, skill learning, quality of work,
  VisiblePayoff: identify whether delay is arrival-rate, service-rate, batching, or WIP-boundary problem,
  AdmissibleNextMove: observe arrivals, service, wait, and WIP; test one local WIP-limit or batching hypothesis,
  ObservationOrReadoutNeeded?: service-time and wait-time readings,
  OrdinaryRivalOrFallback: process narrative without queue assumptions,
  StopCondition: do not infer team obligation, motivation, blame, or organizational ontology
}

MLA.MiniCard example := {
  TargetPhenomenon: production-line throughput and latency,
  CandidateMathObject: queueing network with stated stations and service-rate assumptions,
  LensMappingMode: representation,
  PreservedStructure: flow, bottlenecks, service rates, waiting times,
  LostStructure: human meaning, contractual obligations, rare failure modes, causal interventions not represented by the network,
  InvariantsExposed: bottleneck station and queue-length sensitivity under stated assumptions,
  LensSupportPosture: accepted local theory plus local observations,
  admissibleUse: throughput and latency reasoning inside the declared line model,
  nonAdmissibleUse: motivation, duty, causal intervention, full organization ontology, or release assurance,
  PrincipalRivalLens?: direct empirical dashboard reading,
  RivalLensRelation?: complementary,
  StopCondition: no inference about motivation, obligation, rare-event causality, or full organizational ontology
}

MLA.OneLine := {
  TargetPhenomenon,
  CandidateMathObject,
  LensMappingMode,
  PreservedStructure,
  LostStructure,
  VisiblePayoff,
  AdmissibleNextMove,
  ObservationOrReadoutNeeded?,
  OrdinaryRivalOrFallback,
  StopCondition
}

For MLA.OneLine, VisiblePayoff says what the lens makes visible, such as a bottleneck, invariant, obstruction, incompatibility, loss boundary, or diagnostic split. AdmissibleNextMove says the now-admissible user move, such as compute a local quantity, compare only inside a declared structure, run a validation slice, apply a neighboring pattern, keep the phrase as local metaphor, or remove the phrase from claim-affecting use. ObservationOrReadoutNeeded? names the missing observable, readout, assignment, outcome, validation slice, or scale point needed before the repaired line can support the stated move. OrdinaryRivalOrFallback says what the reader would use without this mathematical lens: ordinary prose, accepted local domain theory, direct measurement, a causal model, a queueing model instead of a quantum-like metaphor, an [A.19](/generated/patterns/A.19) space declaration instead of [C.29](/generated/patterns/C.29), or an [F.9](/generated/patterns/F.9) bridge instead of category-like wording. If two mathematical lenses already change the next move at this cheap-output class, add one ordinary-language note about the disagreement and move to MLA.MiniCard or MLA.FullCard before claiming a reusable rival-lens relation.

MLA.LensCandidateNote := {
  TargetPhenomenon,
  ProblemStructureCue,
  CandidateLensFamily,
  CandidateMathObject?,
  WhyThisLensCouldHelp,
  ExpectedVisiblePayoff,
  ObservableOrControllableCue?,
  AdmissibleNextMove,
  OrdinaryRivalOrFallback,
  StopCondition,
  NextMLAOutput
}

MLA.LensCandidateNote is not evidence, assurance, a bridge, a decision record, a selector result, a literature survey, or a full adequacy card. It is a cheap first-candidate lens selection note. Its successful next outputs are NoMLANeededNote, MLA.OneLine, or a named neighboring governing-locus note.

Name guard for this note: ProblemStructureCue is a recognition cue, not a FPF signature; CandidateLensFamily is a family prompt, not a kind; AdmissibleNextMove is action guidance, not a work record; NextMLAOutput is the next C.29 output class, not a new record family.

Do not use MLA.OneLine with an empty CandidateMathObject. If the candidate object has not yet been named, use MLA.LensCandidateNote first or exit to ordinary prose or a neighboring governing locus.

Cheap stop: if the mathematical phrase does not affect any claim beyond orientation, do not open the full card. If the first honest output is NoMLANeededNote, that is a successful [C.29](/generated/patterns/C.29) result, not an underfilled card.

#Output set and use-rights

After applying C.29, the output is one of these:

Positive warning: a successful C.29 output makes the mathematical lens honest for its declared use. It does not make the claim true, safe, released, benchmark-superior, decision-ready, or causally supported. Truth, safety, release, benchmark, decision, and causal-use claims need their governing neighboring FPF patterns.

LensMappingMode, LensSupportPosture, and use posture are separate readings.

LensMappingMode names construction, not permission. Typical local values include representation, abstraction, quotient, coarse-graining, embedding, homomorphism, isomorphism, functor-like transfer, simulation substrate, and learned or fitted representation. A broad family name such as graph, field, category, geometry, quantum-like, variational, or Bayesian is only a prompt until the concrete construction and preserved/lost structure are named.

LensSupportPosture grants only limited use-rights:

Use posture is not inferred from elegance, familiarity, source prestige, or mapping type. It is stated in admissibleUse, nonAdmissibleUse, and StopCondition. Mathematical adequacy is not empirical truth, causal support, bridge substitution, assurance, release confidence, decision sufficiency, or benchmark superiority; those claims need their governing neighboring FPF patterns.

#From lens to local action

Local action change from a mathematical lens is limited to these cases unless a neighboring pattern supports the needed non-C.29 use:

1. observe or measure a newly named variable or relation;
2. compare only under a declared structure and loss boundary;
3. diagnose a bottleneck, obstruction, mismatch, invariant, or failed transfer;
4. choose or reject a principal rival lens for the current local use;
5. narrow, downgrade, or block a tempting overread;
6. open the exact neighboring FPF locus when the live claim is causal, bridge, evidence, assurance, measurement, temporal, decision, work, scale, selector, or benchmark.

Each item closes either as a local C.29 output or as a named neighboring-pattern opening. If the needed result is a work plan, choice result, selector output, benchmark, or evidence record, publish that neighboring result in its governing pattern rather than from this list.

#No-lens entry: choosing a first candidate lens

Use this when the next admissible move can benefit from a mathematical lens but no adequate mathematical object has been named. The output is MLA.LensCandidateNote, not MLA.OneLine and not a full card. State the ProblemStructureCue, choose one cheap CandidateLensFamily, say what it could make visible, name the ObservableOrControllableCue? when available, state the AdmissibleNextMove, compare it with the OrdinaryRivalOrFallback, and stop if no action changes. If the cue is still pre-articulation and no stable ProblemStructureCue can be named, do not mathematize it; preserve cue plurality through C.2.LS, A.16, A.16.1, B.4.1, B.5.2.0, or the relevant language-state locus before returning to C.29.

Candidate guidance rows are examples for first recognition. Use the row that fits the working cue, or state a closer local cue using the same fields.

MLA.LensCandidateNote is local first-candidate guidance. It does not replace G.2 SoTA synthesis, tradition mapping, or broad lens-family review. Use G.2 when the live work is tradition-scale source synthesis; use C.29 when the local need is to choose one cheap candidate lens that changes the next admissible move. The cheap observation and control check does not open C.16 or A.10 by default; it only asks what the user can observe, read out, assign, vary, or validate now. Measurement construction, evidence strength, intervention support, or validation still moves to the neighboring pattern when live.

#First honest C.29 entry cases

For E.11-style first-entry recognition, distinguish the working entry case before choosing an output:

#False-positive bank and entry stops

Do not open C.29 for these non-use cases unless a separate lens-transfer, publication, assurance, bridge, comparison, or reusable-explanation claim becomes live:

• ordinary ODE inside accepted physics or local engineering model;
• Markov kernel inside accepted stochastic dynamics;
• graph used as a local data structure;
• metric-space distance, topology, order, product, subspace, or embedding declared inside A.19 CharacteristicSpace with no domain-transfer claim;
• category-theoretic proof internal to a domain where that formalism is the local theory;
• one-off pedagogical metaphor not reused for decision, evidence, assurance, publication, bridge, comparison, or transfer.

False-negative bank: open C.29 even when no polished mathematical buzzword appears if the working problem has a structure that changes an admissible next move and ordinary prose is currently hiding it.

Entry guidance states when C.29 is the first governing locus and when another pattern is first:

Admissible entry stops are: no MLA needed, MLA one-line opened, or neighboring governing pattern selected.

#Governing-locus boundary table

A receiving C.29 application uses this governing-locus discipline so mathematical-lens adequacy stays in the C.29 discipline rather than becoming a second authority over neighboring claims.

Positive governed claim:

A C.29 application gives a pattern-local adequacy discipline for claims that use a mathematical object, formalism, learned representation, simulation substrate, or mathematical family as a mathematical lens for a stated use. The application asks for candidate mathematical object, lens mapping mode, preserved and lost structure, visible invariant or distinction, LensSupportPosture or validation posture, admissible use, non-admissible use, and stop condition.

Boundary transfer rule: when the live claim is a choice result, work plan, evidence path, assurance tuple, explanation rendering, comparative review unit, representation shift, temporal claim, bridge, causal-use claim, measurement claim, scale-law claim, selector, or benchmark, the NeighborGoverningLocusNote names the exact receiving FPF locus and exact project-side record. A C.29 application can contribute a lens-supported prediction, distinction, obstruction, diagnostic boundary, or rival-lens note that the receiving record can cite; it does not create that neighboring record.

#MLA.Card@Context shape

MLA.Card@Context is a pattern-local card in C.29. It is not U.MLACard, U.LensAdequacyRecord, or any universal U.* kind.

Namespace note: MLA.Card@Context, MLAOutputRef, MLA.OneLine, MLA.MiniCard, MLA.FullCard, and CC-MLA-* are C.29-local instruments unless they cite existing FPF kinds or refs. MLAOutputRef references the applicable C.29 output for the stated use; it is not a demand for MLA.FullCard. Do not mint generic suffixes such as SystemMLA, MLAQuality, or MLACompliance. Durable cross-pattern MLA names, records, or refs require explicit mint/reuse and naming/admission support through F.8, F.18, C.3, and E.9; otherwise they remain pattern-local labels.

Read the MLA card through three aspects:

Validity boundary: mathematical validity of the object under its assumptions is not the same as representational adequacy to the phenomenon; representational adequacy is not empirical validation for a use; empirical validation is not causal-use support; causal-use support is not assurance, release confidence, decision sufficiency, or benchmark superiority.

MLA.FullCard base fields:
MLA.Card@Context := {
  TargetPhenomenon,
  describedEntityRef?,
  BoundedContext,
  CandidateMathObject,
  LensMappingMode,
  PreservedStructure,
  LostStructure,
  InvariantsExposed,
  LensSupportedPredictionOrDistinction?,
  LensSupportPosture,
  admissibleUse,
  nonAdmissibleUse,
  StopCondition
}

Conditional fields apply only when the corresponding neighboring claim, claim-bearing use, or publication use is live:

MLA.FullCard conditional fields := {
  DynamicsRef?,
  TransitionLawRef?,
  ObservationMapRef?,
  ScaleWindow?,
  CoarseGrainingRule?,
  SourceReturnCondition?,
  PublicationUsePosture?,
  PrincipalRivalLens?,
  RivalLensSet?,
  RivalLensRelation?,
  ValidationUseOverlayRef?,
  LearnedLensOverlayRef?,
  BridgeRefSet?,
  CausalUseDisposition?,
  AssuranceUseDisposition?,
  ExportPolicyRef?
}

Plain reading of the card. A useful mathematical lens says: what phenomenon is being seen, through which mathematical object, by what mapping, what survives, what is lost, what becomes visible, what support posture and validation boundary support this use, the now-admissible user move, the blocked user inference, and where the lens stops.

#Conditional overlays

The base card stays light. These overlays are used only when their use is live. Ordinary C.29 use does not fill this block; it escalates here only when the claim is already publication-facing, assurance-input, benchmark, bridge, model-selection, prediction, scientific/model, learned-lens, or causal-use facing.

MLA.ValidationUseOverlay@Context :=
⟨
  ClaimUse,
  ValidationRegime,
  EvaluationSlice,
  ApproximationOrUncertaintyNote,
  KnownFailureCaseOrCounterexample,
  SensitivityOrRobustnessNote?,
  DomainOfApplicability,
  OutputChangeCondition?
⟩

Use the validation overlay when the lens supports prediction, publication, assurance input, benchmark use, model selection, or scientific/model claim. LensSupportPosture alone is then insufficient. Keep the neighboring notions separate: verification is proof or formal checking under stated assumptions; validation is fit for a declared use and regime; calibration aligns model parameters or readouts with observations; explanation states why the lens makes a distinction intelligible. The C.29 output does not let any one of these four labels silently stand in for the others.

MLA.LearnedLensOverlay@Context :=
⟨
  DataOrTrainingRegime,
  ObservationMapRef,
  GeneralizationClaim,
  DiscretizationOrResolutionPolicy?,
  ValidationRegime,
  ApproximationOrUncertaintyNote,
  StopCondition
⟩

Use the learned-lens overlay when the mathematical object is fitted, learned, latent, simulation-trained, data-derived, a neural operator, a surrogate solver, an embedding, or a world-model representation.

Learned-lens stop variants are named explicitly when they are tempting:

MLA.CausalAbstractionCheck@Context :=
⟨
  LensMappingMode,
  InterventionStructureStatus ∈ {preserved, approximated, notClaimed},
  CounterfactualUseStatus ∈ {preserved, approximated, notClaimed},
  C28ApplicationRef?
⟩

This is not a first-class causal abstraction card. It is a lightweight check: when LensMappingMode is abstraction, quotient, coarse-graining, macro-model, or simulation, and admissibleUse would include intervention, policy, counterfactual, or causal explanation, apply [C.28](/generated/patterns/C.28) for causal-use support.

#Repair decision table

#Field meanings

| OrdinaryRivalOrFallback | Ordinary prose, accepted local theory, direct measurement, or simpler neighboring-pattern exit the reader would use without this lens. | Required for cheap outputs; prevents prestige bias before broad rival review. | | PrincipalRivalLens? | Default ordinary or most relevant rival lens. | Preferred over a broad literature survey. | | RivalLensSet? | Broader comparison set only when publication, selection, or claim-bearing comparison is live. | Not a G.5 selector, benchmark harness, or parity result. | | RivalLensRelation? | Declared relation between the current lens and the principal rival or live rival set. Allowed local relation values include ordinaryFallback, complementary, sameUseLowerCost, morePreservedStructureHigherCost, lowerErrorOnDeclaredEvaluationCriterion, clearerExplanationForDeclaredReader, bridgeNeedsF9, causalUseNeedsC28, differentScaleWindow, differentLossProfile, incomparableForCurrentUse, blockedByStopCondition, and unresolved. Examples: a queueing lens and a causal lens can be complementary for different moves; a latent manifold and a causal graph can conflict when latent axes are read causally; an RG-like lens and a micro-dynamics lens can have different scale windows. | Names disagreement only; a C.29 output is not a winning-lens choice, literature review, selector result, benchmark result, or parity result. Any superiority claim names the evaluation criterion, reader, cost, scale window, or receiving pattern that makes the comparison admissible. |

| LensSupportPosture | Local support-posture label. | Not evidence, an EvidenceGraph, a PathId, or an assurance score. | | BridgeRefSet? | Reference to F.9 Bridge material when context crossing is live. | Bridge semantics stay with F.9. | | CausalUseDisposition? | One of noCausalUseClaim, causalUseBlocked, C28ApplicationRef, or C28SupportRecordRef. | No causal-reference shortcut; no causal verdict from C.29. | | AssuranceUseDisposition? | One of noAssuranceUseClaim, assuranceUseBlocked, evidenceInputOnly, A10Ref, or B3ApplicationRef. | No assurance verdict from mathematical elegance. | | admissibleUse | Admissible current use of the lens. | Matches evidence and validation posture. | | nonAdmissibleUse | Tempting neighboring use that is blocked or handed to another governing locus. | Names the neighboring pattern when live. | | StopCondition | Most tempting nearby claim the lens does not license. | Main anti-overread output; not boilerplate. | | ExportPolicyRef? | Governed reuse or export policy when publication or downstream reuse is live. | Not required for local orientation or mini-card use. |

#Neighboring-pattern boundaries

Neighboring patterns remain necessary and are not displaced. A retained neighboring-pattern relation note answers the working question for the neighboring pattern being used: what does the reader do with the mathematical lens now? State the neighboring-pattern trigger and the first admissible move for that neighboring pattern. If a note only repeats that C.29 does not replace a neighbor, keep that boundary in the C.29 governing-locus table instead of copying generic boundary prose into the neighboring pattern:

• A.6.P handles relation precision restoration.
• A.3.3 handles dynamics semantics.
• A.19 handles characteristic spaces, overlays, normalization, and comparability.
• F.9 handles cross-context semantics and Bridge loss.
• C.18.1 and C.19.1 handle scale-law and BLP claims.
• C.26 handles one specific quantum-like lens family.
• C.28 handles causal-use admissibility.
• A.10 and B.3 handle evidence and assurance.
• C.11, A.15, A.15.1, and A.15.4 handle choice results, method/work separation, work plans, performed work, and work-relevant source restoration.
• E.17.EFP, E.17.ID.CR, A.6.3.RT, and A.6.3.CSC handle explanation-facing renderings, bounded comparative review units, same-described-entity representation-scheme transitions, and controlled semantic coarsening.
• C.27 handles temporal-claim adequacy.

Use the C.29 discipline when the live question is: Is this mathematical lens adequate for this declared use, and where does it stop?

#Naming, ontology, and semantic-rewrite account

#Name

Name: C.29 — Mathematical Lens Adequacy (MLA).

Abbreviation: MLA = Mathematical Lens Adequacy. No prior temporary code is reused; the pattern code, card prefix, reference prefix, and checklist IDs use only MLA.

The stable name is Mathematical Lens Adequacy because C.29 governs adequacy for a declared use, not strength on an unnamed scale. Plain prose can still say that a useful mathematical lens compresses many cases while preserving declared distinctions; load-bearing use is recovered through CandidateMathObject, LensMappingMode, PreservedStructure, LostStructure, LensSupportPosture, and StopCondition.

#C.29-local naming guard

MLA.* instruments are C.29-local unless separately admitted. They are not U.* kinds, not durable FPF record families, and not substitutes for U.Kind, KindSignature, KindBridge, BridgeCard, EvidenceGraph, ChoiceResult, U.WorkPlan, U.Work, or assurance records.

Do not mint LensKind, MathematicalLensKind, MLAQuality, MLACompliance, or MLARecord from C.29 use.

When one C.29 application needs a mathematical-lens name to become reusable outside that application, use F.18 local-first naming; when it quantifies over a class of described entities, use C.3 Kind-CAL; when it creates or reuses a durable concept or record family, use F.8 mint/reuse and E.9 design-rationale support.

#Tempting wrong names rejected

#Ontology guard selected for FPF

A physical, organizational, or epistemic phenomenon is not directly identified with a mathematical object; it is represented through a mathematical object by an explicitly declared mapping that preserves some structures and loses others.

#E.10.SEMIO recoveries applied

#Rationale

#Why this improves FPF

The selected first-principles posture in C.29 is operational, not metaphysical. It treats first-principles mathematical thinking as local construction discipline: declare the smallest structure, rule, invariant, resource condition, observation, or consistency boundary from which the next move follows or is blocked. In that sense, a C.29 application puts mathematical construction before adequacy control: the reader can introduce a queue, graph, state space, measure, topology, algebraic structure, variational quantity, simulation substrate, or learned representation when that structure improves the work, and then record the mapping, preserved structure, lost structure, support posture, and stop condition.

First-principles support can come from several families without turning any one family into an FPF-wide foundation: signatures, logics, axioms, type or abstraction distinctions, symmetries, invariants, compositional structure, local-global relations, scale relations, boundary conditions, variational principles, action, energy, free-energy, loss, or value functionals, constrained optimization structure, probability, information, typicality, algorithmic construction, resource bounds, implementation constraints, consistency boundaries, causal or intervention-preservation questions, operator or function-space mappings, and admissible observation maps. Each use still needs declared mapping, preserved structure, lost structure, validation or support posture, and stop condition.

This fits FPF because FPF already commits to state explicitness, bounded contexts, evidence and assurance, cross-context bridges, open-ended evolution, SoTA alignment, notational independence, and avoidance of ornamental formalism.

C.29 makes an existing discipline explicit: when FPF uses a mathematical substrate as a mathematical lens for a stated use, the C.29 application declares what the substrate preserves, what it loses, what it makes visible, which rival lenses remain live, and where its admissible use stops.

The compact Plain line remains useful because it points to a real heuristic: good mathematical lenses are not decoration; they are compact ways of seeing structures that survive transfer. The Plain line stays readable, while the card and checklist carry the exact FPF commitments.

#Alternatives rejected

#Pillar impact analysis

#Principle-taxonomy balance

#Consequences and validation harness

#Validation harness for Stable admission and material refresh

For Stable admission or material refresh of C.29, run a small MLA validation harness. The harness is not a benchmark mandate and not a tool requirement. It is a repeatable admission check that the pattern yields correct first outputs, avoids false positives, preserves neighboring-pattern writing boundaries, and keeps the first useful move visible.

This subsection governs steward-side validation, not the ordinary C.29 user path. A working user applies the action path and chooses the cheapest honest output; they do not run the harness merely to decide between ordinary prose, MLA.OneLine, MLA.MiniCard, or a neighboring governing locus.

C.29 output-change conditions:

AI-assisted thin-echo result rule:

C.29 edge-case boundary results:

Harness shape:

Minimum harness cases:

Reader-fit checks for admission or material refresh:

#Archetypal grounding

Worked micro-cases by failure mode:

Vanchurin-style universe-as-learning is not an ordinary first grounding archetype. Keep it in the validation harness and SoTA posture as a candidate stress test: it can teach overclaim control and adapt-not-adopt discipline, but it does not ground accepted physics, assurance, quantitative law, or routine lens adequacy.

#Bias annotation

#Conformance checklist

C.29 checklist verifies the action path without replacing the Solution. Candidate-lens guidance belongs in C.29:4.4.3 or worked grounding, not in this checklist; the checklist verifies only that the cheapest honest output and next admissible move remain visible.

| CC-MLA-10a Validation posture | If the lens supports prediction, publication, assurance input, benchmark, model selection, or scientific/model claim, add validation regime, evaluation slice, uncertainty or approximation note, failure case, domain of applicability, and output-change condition when needed. | Keeps prediction-bearing and model-bearing uses SoTA-aligned. | | CC-MLA-10b Source-basis role | If a source carries C.29 force, name its SourceBasisRole; do not let source prestige silently become evidence, causal support, bridge semantics, assurance, release, selector, benchmark, or accepted law. | Separates source use from adoption posture. |

| CC-MLA-11 LensSupportPosture | Label LensSupportPosture as analogy-only prompt, diagnosticOnly, formal derivation, simulation, empirical fit, accepted domain theory, SoTA-echo candidate, or mechanized proof, with matching use-rights. | Prevents evidence laundering. | | CC-MLA-12 No ontology smuggling | Do not import source-domain ontology without separate proof/evidence and receiving pattern. | Protects FPF from metaphysical collapse. | | CC-MLA-13 Stop condition | State the most tempting nearby claim the lens does not license. | Makes misuse locally visible. | | CC-MLA-14 Bridge discipline | Cross-context mathematical transfer cites F.9; Bridge and MLA fields agree without duplicate writing. | Keeps semantics bounded. | | CC-MLA-15 Causal-use discipline | Causal-use claims apply C.28; MLA cannot make causal use admissible by itself. | Blocks causal laundering. | | CC-MLA-16 Assurance discipline | Assurance, release, reliability, and engineering-justification claims apply A.10, B.3, and relevant G patterns. | Prevents elegance from raising assurance directly. | | CC-MLA-17 E.10.SEMIO recovery | Broad heads, source/target wording, mapping wording, pattern-application wording, and Plain metaphors are recovered to exact FPF kinds, fields, neighboring patterns, or explicit non-transfer dispositions. | Keeps the pattern from minting parallel ontology. | | CC-MLA-18 Plain/Tech balance | A Plain sentence can remain when it aids recognition; if it carries ontology, evidence, causal, assurance, bridge, gate, work, decision, or admissibility claim force, that claim force is recovered through the Tech fields or neighboring pattern. | Preserves didactic force without shadow semantics. | | CC-MLA-19 Non-use and false-positive bank | The pattern includes non-use examples for ordinary local domain equations, local graph data structures, A.19 overlays, local category proofs, and one-off metaphors. | Prevents MLA-everywhere. | | CC-MLA-20 Repair matrix | Failed checks map to repair outputs: downgrade, narrow, add loss, add evidence, choose rival lens, apply neighbor, or block overread. | Keeps MLA as repair pattern. | | CC-MLA-21 Validation harness | Stable admission requires the small harness cases in §8.1 or an admitted equivalent validation carrier. | Makes repeatable readiness visible. |

#Common anti-patterns

#SoTA-echoing account

SoTA support for C.29 is accepted only when it changes action guidance. A citation that only decorates the file does not carry C.29.

C.29 separates source-basis roles from adoption posture. Adopt, Adapt, Reject, and candidate-stress-test disposition say what FPF does with the source; SourceBasisRole says what work the source may perform inside a C.29 application.

#Sandberg thread / structural sameness examples

Adopt the Sandberg thread as a recognition cue, with two distinct source roles retained: the original X post is the source identity locator, while the Axis of Ordinary Math section is the checked text carrier used here.

The source-basis examples are not a proof source and not an exhaustive taxonomy. They are a checked example carrier for InvariantsExposed: generalized Stokes and boundary-exterior derivative duality; de Rham, cohomology, and topological obstruction; CLT as RG or fixed-point viewpoint; Lawvere-style diagonal family; Noether and symmetry-conservation; and Legendre, potential-duality, and tropical-limit family.

Plain reading: the thread illustrates mathematical compression that makes hidden structure visible. Tech reading: every FPF use still needs CandidateMathObject, LensMappingMode, PreservedStructure, LostStructure, LensSupportPosture, and StopCondition.

Do not adopt the thread as a proof source, peer-reviewed taxonomy, or authority for all mathematical details.

#Vanchurin 2026 as candidate lens

Adopt/adapt the following as a candidate lens family:

learning dynamics → coarse-graining → effective geometry → gauge fields, metric-tensor fields, or distance-like structure → variational/thermodynamic optimality

Use this source as the case for why FPF needs a lens adequacy card: the source carries many mathematical structures, but its claims are broad and speculative. The candidate-lens stress-test value comes through trainable variables, local update rules, Legendre transforms, thermodynamic potentials, gauge-field, metric-tensor-field, and distance-language claims, memory and processing trade-offs, and RG-like re-optimization of compressed representations.

The Plain lesson is “this is a useful candidate lens, not a new FPF cosmology.” Selected posture:

Adoption stance: Adapt, not Adopt.

Adapt:

• learning dynamics as a general language of change,
• resource constraints as a possible source of effective laws,
• coarse-graining as a mechanism for simple macrodescriptions,
• thermodynamic or variational potentials as links between cost, memory, processing, and geometry,
• RG-like re-optimization as a scale-transition discipline.

Do not adopt as FPF norm:

• “the universe really is a neural network,”
• “physics has already been proven from learning,”
• “quantum, GR, or gauge theory reduce to a learning rule or learning dynamics” as established fact.

Known limitations from the checked source stance remain material for MLA use: non-Abelian gauge fields are not treated as a landed FPF result; thermodynamic RG flow is not treated as a quantitative FPF law; quantitative predictions require explicit learning-algorithm specification.

#Plural foundations stance

Adopt the plural-foundations stance: several structural families can be reusable across domains, and their adequacy depends on declared mapping, local use, mutual interpretability, and recoverable loss.

Source-basis use: Rodin supports the positive stance that several structurally useful families recur across domains. C.29 carries this as local adequacy discipline: select the family that fits the declared use, state the mapping, and publish recoverable loss.

#Applied category theory

Adopt applied category theory as one major organizer for cross-domain transfer, especially composition, interfaces, views, transformations, and bridges. Retain the concrete source-basis examples: databases, electric circuits, and dynamical systems as application families; adjoint functors, enriched categories, and toposes as categorical structures that organize transfer.

In C.29, category-theoretic material is used through the same local adequacy fields as any other lens: stated use, named structure, preserved composition or interface, lost structure, failed transfer, and neighboring-pattern exits. It is especially useful when composition, interfaces, views, transformations, or bridges matter to the admissible move.

#Obstructions to compositionality

Adapt the obstructions and failures-of-compositionality perspective into LostStructure and StopCondition: a lens can be useful precisely because it exposes where transfer fails, not only where it succeeds. In Plain language, a good lens does not only say “this travels”; it also names the boundary where transfer stops.

#Source locators and source-basis guard

SoTA materials are not nameless background. Exact substantive basis and governing inheritance remain recoverable by value, and SoTA rows shape action guidance rather than decorate the file. The source locators and the source-basis role of each external source are retained here.

#Exact source and governing basis posture

#Source correction notes retained as selected basis

1. VAN-SELF-LEARNING-2026 is treated as a ResearchGate preprint / early-stage source. Peer-review status is not established from the checked page. C.29 therefore does not state that its physics is accepted or that its derivations are FPF law.
2. SAND-THREAD-MATH-LINKS-2026-05-12 is a recognition cue, not a mathematical proof source or FPF law.
3. CLT-as-RG/fixed-point wording is retained only as a structural modeling viewpoint. A safe formulation is: under the usual normalization, the Gaussian is an attractive fixed point for finite-variance distributions; other stable laws are other fixed points under suitable normalization.
4. Claims that the Vanchurin preprint derives Schrödinger, Klein–Gordon, Dirac, Einstein, or Maxwell equations are represented as claims made by that preprint, not as accepted FPF facts.
5. The intake correction from direct identification to structure-preserving representation is selected and becomes a central ontology guard.

#Informative taxonomy seed

Use this recognition menu only to identify a possible lens family and likely neighboring-pattern exits. After selecting a row, state the local C.29 fields that make the lens adequate or stop the use.

#Relations

• Builds on: A.1.1, A.6.P, A.3.3, A.19, A.10, A.15, B.3, C.16, E.17.EFP, E.17.ID.CR, A.6.3.RT, A.6.3.CSC, F.9.

• Constrained by: E.8, E.10, E.10.SEMIO, E.19.

• Decision basis: E.9 design-rationale discipline and the source-basis rows in C.29:13a.

• Supports: E.2 pillar-impact analysis when a pillar argument relies on mathematical first-principles structure; only lens adequacy is in scope, with no amendment to pillar content, priority, or constitutional authority.

• Coordinates with: C.11, A.15.1, A.15.4, C.18.1, C.19.1, C.26, C.27, C.28, G.5, G.9, G.2, G.10.

• Specialization relation: C.26 is selected as an MLA-compatible specialization for quantum-like modeling, with affordability qualifications.

• Does not replace: F.9 bridges, C.28 causal-use discipline, A.3.3 dynamics semantics, A.19 characteristic-space governance, C.16 measurement construction, scale legality, direct comparability, and evidence-stub adequacy, A.10 and B.3 evidence and assurance, C.11 decision records, A.15/A.15.1/A.15.4 method and work records, E.17.EFP explanation-use discipline, E.17.ID.CR comparative review units, A.6.3.RT representation transitions, A.6.3.CSC coarsening, C.27 temporal-claim adequacy, C.18.1 and C.19.1 scale-law and BLP support, or G-pattern selector and benchmark work.

#C.29:End

Last Updated: 2026-05-17 — this section last modified in upstream FPF commit 46d5d585 (github.com/ailev/FPF)