The Discovery Project · Founding Volume
The Discovery Project — Day 0
Thinking differently as a route to thinking better
A canonical record of the project's founding hypothesis, research culture, discovery roles, first heuristic, experimental architecture, and the complete Discovery Question Library v0.1.
Document control
- Purpose
- Preserve the decisions, language, and first instruments of the Discovery Project so that work can continue coherently across sessions and versions.
- Canonical principle
- Future documents may revise this volume explicitly, but should not silently drift from it.
- Stable identifiers
- Heuristics use H-numbers; questions use Q-numbers; experiments use E-numbers; documents use DP-D identifiers.
- Version rule
- A question may be refined, split, merged, retired, or restored, but its history remains visible.
- Primary audience
- The project participants now; future collaborators and readers later.
| Document ID | DP-D000-V0.1 |
| Status | Canonical founding document |
| Date | 21 July 2026 |
| Format | Living monograph |
The project begins from the hypothesis that many advances arise not from stronger deduction applied to an unchanged problem, but from changing the representation, observable, direction of approach, or question before deduction begins.
1Executive summary
Day 0 began with a recent example of a compact mathematical counterexample and quickly moved away from the glamour of any single result. The central interest became the process that could make such discoveries more likely: not blind requests for proofs or counterexamples, but the generation of questions, reframings, candidate mechanisms, sceptical attacks, exact tests, and reusable lessons.
The project therefore treats mathematical discovery itself as an object of study. Its first target is not Fermat's Last Theorem, nor any other celebrated problem. Its first target is the quality of its own thinking.
Three decisions define the founding programme:
- Thinking differently is often the route to thinking better. The problem may need a new language, observable, scale, branch, or geometry before it needs a stronger proof.
- False ideas can contain valid frames. An incorrect conclusion may introduce a useful coordinate system or analogy. The first heuristic, H001, is therefore Salvage the Frame.
- Questions are research instruments. Before choosing a conjecture, the project creates a problem-independent library of questions. This prevents the questions from being unconsciously tailored to a known answer and allows the same instrument to be tested across domains.
The Discovery Question Library v0.1 contains fifty questions. Each has a stable identifier and four accompanying notes: why it exists, what kind of change it induces, how to use it, and how it can mislead. The library is deliberately provisional. It is a first best guess, not scripture.
Success is not restricted to proving or disproving the chosen conjecture. A session succeeds if it produces a materially better question, a useful representation, a falsifiable new conjecture, an exact counterexample, or a reusable heuristic about discovery.
2The founding hypothesis
The conventional image of mathematical work is linear:
That image records the final validation stage but often omits the stage in which the problem changes shape. The Discovery Project adopts a richer working model:
Deduction remains indispensable. The claim is not that proof matters less, but that the quality of the object presented to proof may depend on earlier acts that are less formal and less systematically supported.
2.1 Thinking differently and thinking better
“Thinking better” can mean greater accuracy, discipline, depth, or technical power within a fixed frame. “Thinking differently” changes the frame itself. The two are complementary, but the order can matter. More force applied inside a poor representation may only produce a more sophisticated dead end.
Day 0 identified recurring forms of difference:
- change the mathematical language;
- change the observable;
- change the scale or normalisation;
- change a static object into a process;
- approach a singular point from another branch;
- enlarge the parameter space and return;
- search over construction rules rather than finished objects;
- preserve the frame of a false idea while rejecting its conclusion.
The founding conversation recognised a long-standing research habit: an apparently singular route may become regular when approached from another side, even through a region dominated by imaginary or otherwise non-physical solutions. The reusable lesson is not the original physics result; it is the question, “Is the singularity intrinsic, or only a property of the path by which I approached it?”
3A vocabulary for the project
The project uses terms deliberately.
- Observation
- A pattern or fact actually seen in examples, calculations, proofs, or experiments.
- Feeling
- A pre-formal intuition: “this feels like Pythagoras”, “these near-solutions seem to converge”. It is admissible evidence for generating interpretations, never for asserting truth.
- Interpretation
- A precise mathematical meaning that a feeling might encode.
- Representation
- A language, coordinate system, construction, or object that encodes all or part of the problem.
- Question
- An instrument designed to induce a specific change in thought.
- Conjecture
- A precise statement whose truth conditions are clear enough to test or prove.
- Experiment
- A reproducible computation, enumeration, symbolic test, or structured comparison.
- Verdict
- True, false, unsupported, unresolved, or superseded, with the basis stated.
- Heuristic
- A reusable rule for generating or evaluating research moves. It is not a theorem and carries a record of successes and failures.
- Reflection
- An explicit statement of uncertainty, assumptions, and what would change the current view.
4The research culture
4.1 Ideas are not possessions
Once an idea is recorded, nobody is obliged to defend it. Explorer may extend it; Sceptic may destroy it; Interpreter may change its meaning; Curator may recognise an old form; Verifier may reject the evidence. The objective is to improve the idea or learn from its failure, not protect authorship.
4.2 Curiosity before correctness; verification before conclusion
Exploration must permit statements that are imprecise, extravagant, or probably false. Otherwise the search space collapses to what is already respectable. But exploration and conclusion are different phases. Curiosity receives temporary priority during generation; exact verification receives absolute priority before a claim is retained.
4.3 Failure is data
A failed conjecture should leave more than a deletion. It should record why it was attractive, how it failed, whether the failure was structural or accidental, what frame survives, and which future question it improves.
When an idea is false, do not ask only why it is false. Ask what new representation, analogy, decomposition, observable, or direction of approach made it attractive. Reject the conclusion; preserve any frame that produces distinct and testable mathematics.
4.4 The methodology is falsifiable
The Discovery Project is not exempt from its own scepticism. If the role system, question library, or documentation burden fails to improve the quality of questions or discoveries, it should be simplified, replaced, or abandoned. Elegance of method is not evidence of effectiveness.
5The discovery roles
The roles are cognitive functions, not personalities. One person or model may occupy several roles at different times, but separation reduces premature convergence.
| Role | Does | Note |
|---|---|---|
| Explorer | Generates mechanisms, constructions, conjectures, and bold possibilities. | Rewarded for diversity and generative reach, not immediate correctness. |
| Interpreter | Converts feelings and analogies into several precise mathematical interpretations. | Prevents vague intuition from being either worshipped or discarded. |
| Reframer | Changes language, scale, object, observable, or direction without attempting to solve the original statement. | Protects representation search from being dominated by proof attempts. |
| Sceptic | Assumes attractive ideas are wrong and tries to explain them away. | Must state what survives; destructive criticism alone is incomplete. |
| Curator | Recognises known shapes, theories, and historical analogues. | Uses prior knowledge as weak guidance without collapsing exploration into received answers. |
| Verifier | Reproduces every material claim with exact arithmetic, proof, or independently checked computation. | Has no incentive to preserve the narrative. |
| Codex / Laboratory | Builds code, search algorithms, tests, databases, and reports. | Implements questions chosen by the research process; it does not decide mathematical meaning by itself. |
| Synthesiser | Compresses surviving work into the smallest defensible statement and updates the project memory. | Separates genuine result from the larger cloud of exploration. |
6The discovery loop
The loop need not begin with a conjecture. It may begin with a feeling, a failed calculation, a near-solution, an analogy, or a statement known to be wrong. The crucial discipline is conversion: each informal input must eventually produce a precise object that can survive or fail cleanly.
7What counts as progress
The project uses five success levels:
- Question improvement: the problem is now asked more precisely or fruitfully.
- Representation improvement: a new language or observable creates genuine leverage.
- New falsifiable statement: an intuition has become testable mathematics.
- Counterexample or proof component: an exact object or lemma resolves part of the landscape.
- Resolution: the original conjecture is proved, disproved, or correctly reformulated.
Levels 1–3 are not consolation prizes. They are the principal measurements of whether the discovery system is functioning.
8Why the question library comes first
Generating the questions before selecting the first conjecture reduces a subtle bias. If the target is already known, a knowledgeable model may unconsciously choose questions that point toward known mathematics or a remembered counterexample. A fixed, versioned battery allows later comparisons:
- Which questions generate genuinely different representations?
- Which produce only generic prose?
- Which are useful across unrelated fields?
- Which repeatedly generate false excitement?
- Which questions work best in sequence rather than alone?
The library is therefore both a tool and an experimental object.
Do not answer all fifty questions mechanically. First run a broad pass, marking each as productive, neutral, inapplicable, or dangerous. Then choose a diverse subset for deep treatment. Record not only answers but whether the question changed the research state.
9Discovery Question Library v0.1
The following fifty questions are the project's first best guess. Their numbering is stable. Their wording and notes may change by explicit version update.
9.1 Representation
Q001 — Is this really the fundamental object, or merely one representation of something else?
Why this question exists. Researchers often inherit the nouns used in the original statement and unconsciously treat them as ontologically privileged. The question interrupts that inheritance and asks what structure survives a change of language.
What it changes. It shifts attention from the presented object to equivalence classes, invariants, maps, and relations. A graph may become a matrix, a walk, a code, or an energy landscape.
A disciplined use. List at least three objects that encode the same information and three that deliberately discard some information. Ask which loss is useful.
Caution. A new representation can look profound while merely renaming the original problem. Demand a concrete gain: simpler constraints, new invariants, or a tractable verifier.
Q002 — What is the simplest representation that preserves everything important?
Why this question exists. Complex notation may conceal that only a small structural core matters. Compression forces a decision about what is essential.
What it changes. It induces minimal modelling: quotienting redundancies, choosing sufficient statistics, and searching for canonical forms.
A disciplined use. Write the problem in the fewest independent variables you can justify. Then reconstruct the original object from them or state exactly what cannot be reconstructed.
Caution. Oversimplification can delete the obstruction itself. Preserve a ledger of what was discarded and test whether counterexamples live in the discarded part.
Q003 — What completely different mathematical language could express this problem?
Why this question exists. A problem can be hard because the native language makes its useful operations awkward. Translation exposes tools and intuitions from another field.
What it changes. It encourages category changes: algebra to geometry, geometry to topology, combinatorics to probability, or analysis to dynamics.
A disciplined use. Make a translation table: objects, relations, invariants, and valid operations in the old and proposed languages.
Caution. Analogy is not equivalence. Identify precisely which statements translate both ways and which survive only as suggestive metaphors.
Q004 — Can this become geometry?
Why this question exists. Geometry makes constraints visible as shape, distance, intersection, curvature, or obstruction. What is opaque symbolically may become obvious spatially.
What it changes. It invites embeddings, configuration spaces, polytopes, manifolds, and pictures of feasible and forbidden regions.
A disciplined use. Ask what the points, distances, angles, boundaries, and geodesics would be. Seek a picture that predicts a calculation rather than merely illustrates one.
Caution. A seductive diagram can encode only low-dimensional intuition. Check whether the geometric picture respects dimension, topology, and degeneracies.
Q005 — Can this become algebra?
Why this question exists. Algebra can replace case-by-case structure with identities, ideals, modules, or transformations that compose cleanly.
What it changes. It directs attention to generators, relations, factorisation, elimination, and invariant rings.
A disciplined use. Name the operations under which the problem is closed. Search for a presentation in which the desired property is an equation, divisibility statement, or rank condition.
Caution. Algebraisation may introduce many formal solutions with no meaning in the original domain. Track reality, positivity, integrality, and admissibility conditions.
Q006 — Can this become optimisation?
Why this question exists. Existence and extremal statements often hide an objective function. Once exposed, near-solutions and gradients can guide both thought and computation.
What it changes. It produces feasible sets, loss functions, dual bounds, relaxations, and adversarial examples.
A disciplined use. Define a scalar violation score that is zero at the claimed boundary and positive for a counterexample. Examine its landscape before choosing an algorithm.
Caution. The chosen objective can manufacture structure. Test several inequivalent losses and distinguish a genuine mathematical effect from an artefact of scoring.
Q007 — Can this become probability?
Why this question exists. Randomisation can replace rigid universal structure with typical behaviour, concentration, coupling, or expectation identities.
What it changes. It promotes random models, probabilistic existence proofs, martingales, and distributions over constructions.
A disciplined use. Specify what is random and why that distribution is natural. Compare typical objects with extremal objects rather than assuming they coincide.
Caution. Probability may explain why examples are common while missing the rare configuration that decides a universal conjecture.
Q008 — Can this become dynamics?
Why this question exists. A static object may be better understood by how it is generated, deformed, iterated, or attracted to a limiting state.
What it changes. It changes nouns into processes and invites fixed points, stability, flows, recurrences, and renormalisation.
A disciplined use. Invent at least one evolution that preserves admissibility. Ask whether the conjectured property is monotone, invariant, or eventually forced under that evolution.
Caution. An artificial dynamic can impose its own answer. Justify why the process reveals the original structure rather than merely selecting a convenient subclass.
Q009 — Can this become information?
Why this question exists. Many constraints concern distinguishability, compression, uncertainty, or the amount of structure required to describe an object.
What it changes. It suggests entropy, coding length, mutual information, sufficient statistics, and minimum-description constructions.
A disciplined use. Ask what must be encoded to verify the property and how many bits distinguish admissible from forbidden objects.
Caution. Information language can become decorative. Use it only when it yields a bound, a compression argument, or a testable trade-off.
Q010 — If I forgot the original notation, how would I rediscover the problem?
Why this question exists. Notation carries historical commitments. Re-deriving the problem from its operational content can expose assumptions that became invisible through familiarity.
What it changes. It encourages first-principles reconstruction and separates the phenomenon from its conventional presentation.
A disciplined use. Describe the problem without symbols, then introduce variables only when a distinction becomes necessary. Compare the reconstructed formulation with the inherited one.
Caution. Reconstruction can waste mature theory. The goal is not permanent amnesia but temporary freedom followed by reconciliation with known structure.
9.2 Hidden observables
Q011 — What quantity should exist but has not yet been defined?
Why this question exists. Progress often begins with a new observable: a quantity that aligns more directly with the mechanism than inherited measurements do.
What it changes. It moves the search from proving statements about existing variables to inventing variables that make the statement natural.
A disciplined use. Complete the sentence: ‘If I could measure one thing directly, it would be …’. Then demand invariance, computability, and discriminating power.
Caution. Invented quantities can overfit examples. Test on negative controls and ask whether the definition has meaning before the desired result is known.
Q012 — What observable would nature choose?
Why this question exists. Human coordinates may reflect convenience rather than mechanism. A physical or generative process often singles out a more intrinsic measurement.
What it changes. It encourages operational definitions: quantities tied to interventions, evolution, energy, flux, or observable consequences.
A disciplined use. Imagine the object is produced by a mechanism. What would an observer with limited access actually record?
Caution. Anthropomorphising nature can conceal arbitrary modelling choices. State the mechanism explicitly and compare alternative observers.
Q013 — What quantity is almost conserved?
Why this question exists. Exact invariants may not exist, yet slow drift can reveal hidden structure, perturbative regimes, or the correct scaled variable.
What it changes. It directs attention to adiabatic invariants, approximate symmetries, residuals, and defect measures.
A disciplined use. Compute candidate quantities along examples, deformations, or iterations and rank them by stability after appropriate scaling.
Caution. Almost-conservation can be a finite-range illusion. Extend the range, separate deterministic drift from noise, and seek an error law.
Q014 — What quantity is almost monotone?
Why this question exists. A nearly one-directional quantity can organise a complicated search even when strict monotonicity fails.
What it changes. It suggests potentials, Lyapunov-like functions, order parameters, and progress measures.
A disciplined use. Plot or tabulate the quantity along natural transformations. Study the exceptions instead of smoothing them away.
Caution. Near-monotonicity may come from selection bias or sorting. Test arbitrary paths and adversarial transformations.
Q015 — What is changing that nobody is measuring?
Why this question exists. Static levels can miss trajectory, instability, curvature, or rearrangement. The overlooked signal may live in the change itself.
What it changes. It induces derivatives, variation, turnover, path dependence, and higher-order descriptors.
A disciplined use. Replace each level variable by first differences, dispersion through time, ordering, curvature, and time spent in regimes.
Caution. More dynamic features mean more opportunities for noise and multiple testing. Require a mechanism and a null model.
Q016 — What is staying constant that nobody is measuring?
Why this question exists. A hidden invariant can collapse a large space of possibilities and explain why apparently different examples behave alike.
What it changes. It directs attention to conserved topology, parity, rank, determinant, homology, or combinatorial type.
A disciplined use. Compare diverse examples and search for exact equalities or equivalence classes that survive all admissible operations.
Caution. Constancy over observed examples is not invariance. Prove preservation under generators of the allowed transformations or find a counterexample.
Q017 — What derived quantity becomes simpler than the original one?
Why this question exists. The original variables may interact nonlinearly while a transform, ratio, residual, logarithm, or spectrum obeys a simpler law.
What it changes. It promotes change of variables and the search for linearised or additive structure.
A disciplined use. Generate ratios, logs, ranks, projections, residuals, and transforms. Prefer quantities with a clear inverse or interpretation.
Caution. A derived quantity may simplify by erasing decisive information. Track fibres: which distinct original objects map to the same value?
Q018 — What is the shadow of the real object?
Why this question exists. Sometimes the exact object is inaccessible or nonexistent, but near-solutions, projections, traces, or boundary behaviour retain organised structure.
What it changes. It invites study of defects, silhouettes, limits, marginal distributions, and approximate objects.
A disciplined use. Define the shadow operationally and ask what features of the hidden object it could or could not determine.
Caution. Shadows are underdetermined. Do not infer uniqueness without showing that alternative objects cannot cast the same shadow.
9.3 Boundaries and continuations
Q019 — What happens in the smallest non-trivial example?
Why this question exists. The smallest case strips away accidental complexity and often reveals the first obstruction, exceptional symmetry, or missing hypothesis.
What it changes. It promotes exact enumeration and structural classification rather than asymptotic intuition.
A disciplined use. Identify the first size at which the conclusion is not automatic. Classify all objects there, including degenerate ones.
Caution. Small cases may be unrepresentative because exceptional isomorphisms and low-dimensional coincidences disappear later.
Q020 — What happens at infinity?
Why this question exists. Large-scale behaviour can reveal limiting shape, dominant balance, phase transitions, and which terms are genuinely structural.
What it changes. It encourages scaling limits, compactification, asymptotics, and renormalised variables.
A disciplined use. Choose a scale, normalise the objects, and ask what subsequences or distributions can converge.
Caution. Infinity depends on the route and normalisation. Compare multiple scalings and identify which conclusions are coordinate-dependent.
Q021 — What happens at zero?
Why this question exists. Zero often marks a change of regime: disappearance, degeneracy, linearisation, or singularity. It can expose the skeleton of a problem.
What it changes. It suggests perturbation, tangent objects, first-order structure, and exact base cases.
A disciplined use. Set each meaningful parameter to zero separately and together. Record which operations cease to commute at the limit.
Caution. The zero case can be singular and qualitatively unlike all nearby cases. Distinguish evaluation at zero from a limit toward zero.
Q022 — What happens at the singular or degenerate cases?
Why this question exists. Pathologies often reveal the theorem's load-bearing assumptions more clearly than generic examples do.
What it changes. It turns excluded cases into diagnostic probes and searches for boundary strata.
A disciplined use. List every denominator, rank assumption, strict inequality, and genericity condition. Violate each one deliberately.
Caution. Degenerate examples may be dismissed too quickly or overgeneralised. Determine whether they can be perturbed into nearby admissible counterexamples.
Q023 — Can I approach the critical point from another direction or branch?
Why this question exists. A singularity may belong to one path through parameter space rather than to the destination itself. Another branch can regularise the limit and reveal an analytic continuation.
What it changes. It induces directional limits, signed defects, complex continuation, homotopy, and branch-sensitive analysis.
A disciplined use. Map the local parameter space and compute limits along qualitatively different paths, including apparently unphysical or complex branches.
Caution. A value reached by continuation need not solve the original real or admissible problem. State precisely what returns to the original domain.
Q024 — Can I enlarge the parameter space?
Why this question exists. Embedding a rigid problem in a richer family can create room to deform around obstructions and then return with new information.
What it changes. It suggests complexification, relaxation, added dimensions, auxiliary variables, and deformation families.
A disciplined use. Introduce the smallest extra parameter that makes movement possible. Track which original objects form a distinguished slice.
Caution. The enlarged problem may be easier only because it changed the question. A valid return map or restriction argument is essential.
Q025 — Can I solve a nearby problem first?
Why this question exists. A neighbouring statement may isolate the mechanism, supply a continuation path, or reveal which features are robust.
What it changes. It promotes interpolation between known and unknown cases and the construction of solvable toy models.
A disciplined use. Define ‘nearby’ explicitly: weakened hypothesis, changed dimension, perturbed exponent, approximate equality, or restricted class.
Caution. Toy problems can become comfortable detours. State in advance what result would transfer back and what would not.
Q026 — Which limit is actually the natural one?
Why this question exists. Apparent convergence can be manufactured by an arbitrary normalisation or order of limits. The correct limit should respect the mechanism and units.
What it changes. It encourages dimensional analysis, commuting-limit checks, and comparison of absolute, relative, and projective scales.
A disciplined use. Write every plausible normalisation and order of limits. Ask which produces a non-trivial, stable, interpretable object.
Caution. Choosing the limit after seeing the answer invites overfitting. Pre-register candidate scalings or validate on held-out regimes.
9.4 Symmetry and invariance
Q027 — What symmetries are obvious?
Why this question exists. Visible symmetries reduce duplicate work and identify transformations under which any proposed quantity should behave coherently.
What it changes. It induces group actions, orbit reduction, canonical representatives, and invariant features.
A disciplined use. List transformations that leave hypotheses and conclusions unchanged. Quotient the search space by their orbits.
Caution. Obvious symmetries can distract from asymmetric extremisers. Do not assume an optimal or counterexample object inherits all symmetries of the statement.
Q028 — Which symmetries are hidden?
Why this question exists. A problem may be invariant under transformations obscured by its coordinates. Discovering them can expose the right representation.
What it changes. It promotes conjugacy, duality, gauge freedom, and unexpected automorphisms.
A disciplined use. Search for transformations that preserve computed invariants or map solutions to solutions despite changing their appearance.
Caution. Coincidental agreement on a dataset is not a group action. Verify closure, inverses, and preservation exactly.
Q029 — Which symmetry is being broken?
Why this question exists. Interesting structure often appears not in perfect symmetry but in how a family departs from it.
What it changes. It directs attention to order parameters, bifurcations, defects, and asymmetric branches.
A disciplined use. Start from the most symmetric object and perturb in irreducible directions. Record which perturbation changes the target property first.
Caution. The language of symmetry breaking can be metaphorical. Identify the exact group and stabiliser or use plainer terminology.
Q030 — Can I quotient out unnecessary symmetry?
Why this question exists. Redundant copies inflate search spaces and obscure the genuinely distinct mechanisms.
What it changes. It induces canonical labelling, gauge fixing, normal forms, and orbit-space geometry.
A disciplined use. Choose a representative rule and prove every admissible object has at least one representative. Handle objects with extra symmetry separately.
Caution. A poor gauge can introduce singularities or bias. Check whether the quotient loses continuity or creates artificial boundaries.
Q031 — Can I add symmetry to reveal a solvable core?
Why this question exists. Imposing symmetry can collapse a high-dimensional problem into a tractable family and expose the algebraic mechanism.
What it changes. It suggests ansatz construction, invariant subspaces, and symmetric seed objects.
A disciplined use. Solve the maximally symmetric subclass, then relax one symmetry at a time and observe which conclusions survive.
Caution. Symmetric families can miss generic or extremal behaviour. Treat them as mechanisms, not evidence of universality.
Q032 — Is asymmetry the interesting part?
Why this question exists. A theorem may be driven by the residual after symmetric structure is removed rather than by the symmetric component itself.
What it changes. It promotes decomposition into invariant and symmetry-breaking parts and focuses on residual degrees of freedom.
A disciplined use. Project onto symmetric subspaces, calculate the remainder, and test whether the target property lives primarily in that remainder.
Caution. Residualisation depends on the chosen symmetry and metric. Compare decompositions and avoid declaring the residual intrinsic too soon.
Q033 — Is the statement really about invariance?
Why this question exists. Many theorems phrased as inequalities or existence claims are consequences of something being unchanged under a hidden operation.
What it changes. It directs the search toward conserved structure rather than direct estimation.
A disciplined use. Ask what transformations would make the conclusion inevitable if some quantity remained fixed.
Caution. Not every regularity is invariant. Forcing an invariant interpretation may obscure monotonicity, convexity, or order structure.
9.5 Counterexamples and assumptions
Q034 — What would the smallest counterexample look like?
Why this question exists. Minimality converts an unbounded search into structural constraints: every deletion, contraction, or simplification must restore the theorem.
What it changes. It induces minimal-counterexample arguments and targeted finite search.
A disciplined use. Assume a smallest counterexample exists and derive what it cannot contain. Turn each consequence into a pruning rule.
Caution. The actual counterexample may be small in description length but large in raw size, or no minimal object may exist under the chosen order.
Q035 — What properties must every counterexample possess?
Why this question exists. Necessary conditions focus search and often become lemmas even when no counterexample is found.
What it changes. It encourages contrapositive reasoning, obstruction theory, and constraint propagation.
A disciplined use. Prove easy subclasses satisfy the conjecture; negate their defining properties to build a counterexample profile.
Caution. A long profile can reflect only the limits of current proofs. Distinguish proved necessities from intuitions and computational observations.
Q036 — Which properties definitely cannot occur in a counterexample?
Why this question exists. Excluding familiar structures prevents search algorithms and human imagination from repeatedly returning to safe regions.
What it changes. It creates forbidden-pattern lists and certified pruning rules.
A disciplined use. For each common family, either prove the conjecture there or record a verified computational bound.
Caution. Over-pruning based on unproved beliefs can remove the true witness. Label every exclusion by confidence and proof status.
Q037 — If the statement is false, where is it most likely to fail?
Why this question exists. Universal claims often fail at interfaces: low rank, high asymmetry, boundary parameters, mixed scales, or transitions between regimes.
What it changes. It directs adversarial search toward stress points rather than uniform random sampling.
A disciplined use. Make a failure atlas with axes for size, sparsity, symmetry, rank, sign, and degeneracy. Seek corners and regime boundaries.
Caution. Human expectations about ‘interesting’ failure can be wrong. Preserve a baseline random search and simple enumerations.
Q038 — Which assumption is probably unnecessary?
Why this question exists. Removing an assumption tests whether it is structural or merely inherited from a proof technique.
What it changes. It generates stronger conjectures and clarifies the theorem's conceptual core.
A disciplined use. Delete one hypothesis at a time, construct near-boundary examples, and ask what part of the proof first breaks.
Caution. A proof's inability to proceed does not show the assumption is needed; nor does finite evidence show it can be removed.
Q039 — Which assumption is secretly doing all the work?
Why this question exists. Some hypotheses appear routine but carry the entire obstruction. Identifying them can simplify both proof and counterexample design.
What it changes. It induces ablation studies, sensitivity analysis, and theorem decomposition.
A disciplined use. Hold all but one assumption fixed and deliberately weaken the remaining one. Compare the space of examples before and after.
Caution. Interactions matter. An assumption may be powerless alone but decisive jointly; avoid purely one-factor attribution.
Q040 — Can I construct an object rather than search randomly?
Why this question exists. Elegant counterexamples usually arise from mechanisms, not isolated lottery wins. A construction can explain and generalise the failure.
What it changes. It shifts search from object space to program space: products, lifts, substitutions, gadgets, and recurrences.
A disciplined use. Write a grammar of admissible construction moves and search over short programs that maximise violation while penalising complexity.
Caution. A restrictive grammar can encode current imagination and miss alien constructions. Keep mutation at both program and object levels.
9.6 False ideas and analogies
Q041 — What false statement feels strangely compelling?
Why this question exists. Intuition often arrives as an overstatement. Naming it without defending it creates raw material for interpretation.
What it changes. It licenses bold analogy while separating imaginative generation from truth assessment.
A disciplined use. State the strongest version that captures the feeling, mark it explicitly as probably false, and ask what made it attractive.
Caution. The emotional appeal of a false statement can bias later evaluation. Hand it to an independent sceptic before investing in rescue.
Q042 — What new viewpoint made that false statement attractive?
Why this question exists. This is the operational core of H001, Salvage the Frame: the proposition may fail while the coordinate system succeeds.
What it changes. It extracts representation, decomposition, observable, or process from a rejected claim.
A disciplined use. Separate nouns, transformations, and conclusion. Keep each novel noun or transformation and formulate weaker claims around it.
Caution. Sometimes nothing survives. Do not force value into every mistake; salvage must produce a distinct, testable viewpoint.
Q043 — What survives after removing the false conclusion?
Why this question exists. A failed implication can leave lemmas, special cases, inequalities, algorithms, or useful questions.
What it changes. It turns binary failure into a graded map of surviving structure.
A disciplined use. Negate only the exact failed clause. Inventory definitions, intermediate observations, and restricted cases that remain valid.
Caution. Patchwork survival can become rationalisation. Re-verify each retained claim independently of the original narrative.
Q044 — If this analogy were literally true, what mathematics would follow?
Why this question exists. Taking an analogy seriously for a moment reveals its hidden commitments and translates a feeling into candidate statements.
What it changes. It creates a bundle of precise consequences that can be tested separately.
A disciplined use. Write five consequences at increasing levels of strength: vocabulary, invariant, equation, equivalence, and theorem.
Caution. Literalisation can caricature a useful analogy. Failure of the strongest consequence does not invalidate the weaker structural resemblance.
Q045 — Which completely different problem feels the same?
Why this question exists. Structural analogy can transfer representations and heuristics across domains before a formal bridge is known.
What it changes. It encourages cross-domain mapping based on roles and relations rather than surface notation.
A disciplined use. Build a correspondence table and identify one prediction the analogy makes in the target problem.
Caution. Superficial resemblance produces cargo-cult mathematics. Require at least one preserved operation or invariant.
Q046 — Is this intuition trying to describe a structure I have not formalised?
Why this question exists. A vague feeling may be compressed recognition of order, curvature, hierarchy, or interaction that lacks a name.
What it changes. It turns intuition into a search for definitions rather than an immediate claim.
A disciplined use. Generate multiple formal interpretations and design examples that distinguish them. Let the intuition choose only after comparison.
Caution. The interpreter can project familiar structures onto ambiguous language. Preserve several alternatives long enough for evidence to discriminate.
9.7 Meta-discovery
Q047 — Why has nobody solved this already?
Why this question exists. The answer determines strategy: perhaps the problem is computationally vast, representation-poor, technically deep, falsely believed, or simply neglected.
What it changes. It encourages obstacle diagnosis before effort allocation.
A disciplined use. List historical attempts, known barriers, missing data, verification costs, and sociological reasons. Distinguish ‘hard’ from ‘unfashionable’.
Caution. Speculation about others' failures can become arrogance. Treat this as hypothesis generation and verify against the literature when the project is selected.
Q048 — What assumptions am I making without noticing?
Why this question exists. Unstated assumptions constrain imagination more powerfully than explicit hypotheses because they are never tested.
What it changes. It promotes premise auditing, adversarial reading, and domain shifts.
A disciplined use. Rewrite the problem as a formal contract: objects, domains, equivalence, admissibility, and conclusion. Highlight everything imported by convention.
Caution. Complete assumption discovery is impossible. Revisit the question after each reframing and invite an independent reader.
Q049 — What question should I be asking instead?
Why this question exists. A stubborn problem may be a symptom of asking at the wrong level: proof instead of mechanism, value instead of structure, object instead of process.
What it changes. It authorises problem replacement while preserving the motivating phenomenon.
A disciplined use. Write the original motivation separately from the formal question. Generate alternatives that serve the motivation by different routes.
Caution. Changing the question can become avoidance. Record what relationship the replacement retains to the original and what would count as return.
Q050 — If I could ask only one more question before working for a year, what would it be?
Why this question exists. Scarcity forces prioritisation. It reveals which uncertainty is most load-bearing and which answer would most alter the research path.
What it changes. It induces value-of-information thinking and guards against diffuse curiosity.
A disciplined use. List candidate questions and estimate how each possible answer would change decisions. Choose the one with the largest expected redirection.
Caution. The highest-value question may be unanswerable. Prefer a question whose answer can actually be approached, not merely wished for.
10The first experimental protocol
The first working example should be small, finite, exactly verifiable, representation-rich, and safe to fail on. A historically false conjecture with a known compact counterexample is a strong calibration target, provided the active Explorer and Interpreter are insulated from the answer as far as practical.
10.1 Blindness and contamination
A knowledgeable AI cannot honestly erase prior knowledge. The project therefore treats contamination as an experimental variable.
- A selector chooses the conjecture and supplies only its statement to a naive session, potentially Myra.
- The Question Library is applied without literature access.
- Explorer, Interpreter, and Reframer produce a structured notebook.
- Sceptic attacks the ideas without being told the known route.
- Only after the exploration is frozen does Curator compare it with known mathematics and the historical counterexample.
The objective is not to pretend ignorance perfectly. It is to measure how much useful structure the process generates before the answer is revealed.
10.2 Experiment record
Every experiment should end with the same four-page conceptual record, even when the actual document is longer:
- What changed? The precise difference between the initial and final problem representation.
- New heuristics. Candidate additions or modifications to the discovery system.
- Wrong ideas worth remembering. False claims whose frames remain fertile.
- Questions for the next session. The smallest set of unresolved, high-value questions.
11Repository and canonical memory
Chat is the whiteboard, not the archive. The repository is the durable memory. A proposed structure is:
discovery-project/ canonical/ founding volume and current methodology questions/ Q-library source, metadata, and change history heuristics/ H001, H002, ... with evidence logs experiments/ E000, E001, ... conjectures/ precise statements and status intuitions/ pre-formal feelings and interpretations failures/ failed claims and salvaged frames code/ exact verifiers and search tools syntheses/ periodic reflections and version releases
11.1 House rules for continuity
- Never silently renumber questions or heuristics.
- Preserve old wording in version history when a question changes meaning.
- Separate observation, interpretation, conjecture, and verdict in every record.
- Record negative controls, null models, and failed replications.
- Keep raw intuition; do not retrospectively rewrite it to look more prescient.
- Treat AI model version, prompts, and context as part of experimental provenance.
12Roadmap from Day 0
| Stage | Purpose | Exit condition |
|---|---|---|
| Day 0 | Establish language, culture, roles, H001, and Question Library v0.1. | Canonical document compiled and read. |
| Experiment 0 | Calibrate on a conjecture with a known counterexample under partial blindness. | Structured record of what the method generated before reveal. |
| Version 0.2 | Revise questions based on observed usefulness and failure. | Explicit change log; no silent expansion. |
| Experiments 1–3 | Apply the same framework to different domains. | Evidence of transfer or evidence that the method is domain-specific. |
| Version 0.5 | Stabilise experiment templates, scoring, and repository practice. | Reproducible workflow suitable for repeated use. |
| Version 1.0 | State only principles that have survived meaningful tests. | A methodology supported by a body of documented experiments. |
13Open reflections at the end of Day 0
Reflection 1 — Are questions enough?
A library of questions may improve breadth without improving depth. The experiments must test whether questions lead to precise representations and falsifiable statements, not merely interesting conversation.
Reflection 2 — Can roles become theatre?
Named roles can create the appearance of methodological rigour while all sessions converge on the same model habits. Independence, prompt diversity, withheld context, and exact verification matter more than role labels.
Reflection 3 — How will we recognise better thinking?
The project needs evidence beyond subjective excitement. Candidate measures include representation diversity, rate of conversion from intuition to testable conjecture, survival under sceptical attack, heuristic reuse, and transfer across domains. None is yet canonical.
Reflection 4 — What must remain human?
Human taste may remain essential in choosing which feelings matter, which questions are beautiful, and which failures are worth carrying forward. The project does not assume that all discovery should be automated; it studies the partnership.
14What changed on Day 0
- The objective changed from finding a counterexample to studying and improving the process of discovery.
- “Thinking differently” was distinguished from merely applying more effort inside a fixed frame.
- False ideas were reclassified as possible carriers of useful representations.
- H001, Salvage the Frame, was established.
- Explorer, Interpreter, Reframer, Sceptic, Curator, Verifier, Laboratory, and Synthesiser were defined.
- Questions were promoted to versioned research instruments.
- The first fifty questions were written before selection of the first conjecture.
- Chat was designated the whiteboard; the versioned repository and this monograph were designated canonical memory.
15Questions for the next session
- Which historically false conjecture provides the best blind calibration example?
- Which subset of the fifty questions should be compulsory in Experiment 0?
- How should Myra's naive exploration be separated from literature-aware curation?
- What scoring scheme can distinguish a genuine reframing from polished paraphrase?
- What information must be preserved so that future models can reproduce the intellectual path without inheriting the answer?
The objective of the Discovery Project is not merely to solve mathematical problems, but to understand and improve the process by which mathematical discovery itself occurs.
Version history
| Version | Date | Change |
|---|---|---|
| 0.1 | 21 July 2026 | Founding volume. Establishes the hypothesis, vocabulary, roles, H001, experimental architecture, canonical-memory rules, and Discovery Question Library Q001–Q050 with explanatory notes. |