{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp03_full320_20260801","cell_id":"negative_space_design__mathematics","trajectory_id":"R","attempt_index":0,"archetype_slug":"negative_space_design","domain_slug":"mathematics","decision":"CANDIDATE","problem_id":"crowded_formal_proof_state_obscures_active_obligations","causal_lever_id":"protected_separation_of_active_goal_from_recoverable_context","proposal":{"problem":"Interactive formal-proof displays can present the active goal, hypotheses, side conditions, prior goals, and automation traces with nearly equal visual weight. The resulting crowding can obscure which obligation is current and which context it depends on, even though the necessary information is present.","actors_substrate":["Mathematicians and students constructing machine-checked proofs","Interactive theorem-prover proof-state interface","Active goals, hypotheses, side conditions, and tactic traces","Screen-reader and keyboard-only users"],"observable_state":"Users repeatedly scan or search a dense proof-state panel before identifying the active obligation, overlook changed side conditions, or mistake a blank/failed panel for successful goal discharge.","consequence":"Attention is spent locating rather than reasoning about the current obligation; users may try irrelevant tactics or misread whether the proof is complete.","affected_objective":"Accurate, timely comprehension and discharge of formal proof obligations without weakening verifiability.","structural_mapping":[{"archetype_element":"Competing elements crowd the field","domain_realization":"Goals, hypotheses, metadata, and traces occupy one continuously dense proof-state surface with insufficient perceptual separation.","claim_kind":"HYPOTHESIS"},{"archetype_element":"Important positive form","domain_realization":"The current goal and its load-bearing dependencies are the form that must remain immediately legible.","claim_kind":"INFERENCE"},{"archetype_element":"Deliberate omission and protected absence","domain_realization":"Secondary or stale material is collapsed into recoverable groups while fixed spacing surrounds the active goal and separates dependency classes.","claim_kind":"INFERENCE"},{"archetype_element":"Meaningful empty state","domain_realization":"A cleared goal region explicitly distinguishes proof completion from loading failure, permission failure, or filtering.","claim_kind":"INFERENCE"},{"archetype_element":"Validated perceptual effect","domain_realization":"Users must identify the current obligation and relevant dependencies more accurately or quickly without losing access to hidden context.","claim_kind":"HYPOTHESIS"}],"component_map":[{"component":"Omission Candidate","status":"adapted","domain_realization":"Stale goals, routine metadata, duplicate type information, and verbose traces are candidates for reversible collapse, never deletion from the proof record."},{"component":"Protected Empty Space","status":"direct","domain_realization":"A reserved visual or auditory interval separates the active goal from contextual groups."},{"component":"Positive Form Relationship","status":"direct","domain_realization":"The reserved region frames the active goal and makes dependency-group boundaries perceptible."},{"component":"Attention Competition Map","status":"adapted","domain_realization":"Task observations classify which proof-state elements compete with versus support recognition of the active obligation."},{"component":"Absence Boundary","status":"direct","domain_realization":"Layout rules prohibit secondary panels and annotations from entering the active-goal margin."},{"component":"Clarity or Effect Test","status":"direct","domain_realization":"A crossover test measures goal/dependency identification accuracy, latency, and context-recovery failures."},{"component":"Rest and Pacing Zone","status":"adapted","domain_realization":"Stable separation between discharged and newly generated goals provides a brief orientation boundary without delaying interaction."},{"component":"Meaning-of-Absence Check","status":"direct","domain_realization":"Explicit labels distinguish no remaining goals, collapsed context, loading, and error states."},{"component":"Reintroduction Trigger","status":"direct","domain_realization":"A visible command, keyboard action, or screen-reader control immediately expands hidden context."},{"component":"Accessibility and Recoverability Guardrail","status":"direct","domain_realization":"All collapsed material remains keyboard and screen-reader discoverable; errors and proof obligations cannot be hidden."},{"component":"Context Preservation Frame","status":"direct","domain_realization":"The theorem, current goal identifier, dependency provenance, and proof-status shell remain visible while secondary detail is collapsed."}],"mechanism_dispositions":[{"slug":"architectural_void","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Its figure-ground idea is already realized by spacing; architectural orientation and monumentality add no distinct proof-state mechanism.","counterfactual_removal":"Removing it leaves the proposed causal chain unchanged."},{"slug":"blank_or_rest_frame","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"A timed blank between goals could interrupt expert proof flow or resemble a rendering fault.","counterfactual_removal":"Removing it avoids delay without eliminating protected separation."},{"slug":"editorial_cut","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Reversibly collapses redundant or premature context while preserving every formal obligation and its provenance.","counterfactual_removal":"Without selective collapse, competing material continues to occupy the active field."},{"slug":"empty_state_design","disposition":"selected_supporting","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Labels completion, loading, error, filtered, and collapsed states distinctly.","counterfactual_removal":"Crowding may still improve, but blank states could falsely signal proof completion."},{"slug":"facilitation_silence","disposition":"incompatible","contribution_type":"NONE","adaptation_or_rejection":"Withholding speech by a powerful facilitator has no corresponding actor in the static proof-state view.","counterfactual_removal":"Removing it has no effect."},{"slug":"focus_mode_or_control_hiding","disposition":"selected_load_bearing","contribution_type":"SAFETY_GUARDRAIL","adaptation_or_rejection":"Temporarily hides only noncritical interface chrome and guarantees obvious, immediate recovery; goals and errors are exempt.","counterfactual_removal":"Without recoverable hiding, omission either fails to reduce competition or risks concealing necessary context."},{"slug":"margin_and_gutter_system","disposition":"selected_supporting","contribution_type":"OPERATIONAL","adaptation_or_rejection":"Defines consistent protected spacing among the active goal, hypotheses, and traces, with responsive variants.","counterfactual_removal":"The intervention becomes vulnerable to inconsistent spacing and clutter creep."},{"slug":"negative_space_logo","disposition":"incompatible","contribution_type":"NONE","adaptation_or_rejection":"A hidden secondary image has no role in identifying or verifying proof obligations.","counterfactual_removal":"Removing it has no effect."},{"slug":"pause_in_speech","disposition":"incompatible","contribution_type":"NONE","adaptation_or_rejection":"Live rhetorical pacing does not address a persistent interactive proof-state surface.","counterfactual_removal":"Removing it has no effect."},{"slug":"sparse_layout","disposition":"considered_rejected","contribution_type":"NONE","adaptation_or_rejection":"Permanent thinning risks removing expert-needed context; reversible editorial collapse is the safer causal implementation.","counterfactual_removal":"Removing it preserves the proposal's reversible rather than permanently sparse design."},{"slug":"whitespace","disposition":"selected_load_bearing","contribution_type":"CORE_CAUSAL","adaptation_or_rejection":"Uses separation around the active goal and between dependency classes as the direct perceptual signal.","counterfactual_removal":"Without perceptible gaps, retained and collapsed groups again read as an undifferentiated proof state."}],"causal_chain":["Dense, equally weighted proof-state elements compete for attention.","Reversible collapse removes selected competitors from the immediate field while preserving access.","Protected spacing isolates the active goal and groups its dependencies.","Explicit state labels and reveal controls prevent absence from being mistaken for completion or loss.","Users identify the current obligation with fewer search errors while retaining full proof verifiability."],"baseline":"The prover continuously displays all available goals, hypotheses, metadata, and traces in one dense panel; users scroll, search, or mentally filter it.","nearest_rival":"Focal-emphasis styling that keeps every element visible but highlights the active goal through color, typography, or borders. It may improve salience without using protected absence.","authority_safety":{"affected_parties":["Proof authors","Students and instructors","Maintainers of shared prover interfaces","Screen-reader and keyboard-only users","Reviewers relying on complete obligations"],"decision_authority":"The interface maintainer may offer the experimental view; each participant controls activation and may instantly return to the standard view.","authorized_first_step":"Run a reversible crossover test on archived, non-production proof tasks with 12–20 consenting users. Compare baseline and experimental views on active-goal identification, relevant-dependency identification, time, and recovery errors; do not alter proof objects or tactic execution.","excluded_actions":["Deleting hypotheses, goals, errors, or provenance from the proof record","Hiding safety- or correctness-critical obligations","Changing tactic execution or proof semantics","Mandatory deployment before accessibility testing","Treating faster responses alone as success when accuracy declines"],"halt_rollback":"Stop the test and restore the standard view if any obligation becomes inaccessible, any participant mistakes hidden content for absent content, accessibility navigation fails, or identification accuracy is materially worse in the experimental condition."}},"negative_tests":{"strongest_counterevidence":"Expert users may exploit dense side-by-side context efficiently, and observed proof failures may stem from missing mathematical knowledge rather than attention competition. The nearest-rival highlight-only view may perform as well without hiding anything.","analogy_break":"In mathematics, visually secondary notation can be logically indispensable. Empty space cannot establish an inference, and omission that works perceptually can still conceal a dependency needed for verification.","failure_condition":"The intervention fails if users lose necessary context, spend comparable time reopening collapsed groups, misinterpret empty states, or gain speed at the cost of correctness or accessibility.","problem_falsifier":"The diagnosed problem is falsified if representative users already identify active goals and dependencies accurately and promptly under the dense baseline, and errors instead track mathematical-knowledge gaps rather than display competition.","intervention_falsifier":"The intervention is falsified if the bounded crossover test shows no improvement over baseline or highlight-only presentation in identification accuracy or latency, or shows increased context-recovery, completion-state, or accessibility errors.","risks":["Collapsed context may hide an unexpectedly relevant hypothesis.","Whitespace may reduce information visible on small displays.","Experts may incur extra reveal actions.","Screen readers may not convey spatial grouping unless equivalent structure is encoded.","A completion label could be wrong during asynchronous loading.","Interface emphasis may bias users toward the displayed goal while neglecting global proof strategy."]},"null_rationale":null,"classification":{"candidate_kind":"DOMAIN_TRANSFER","prior_art_status":"UNSEARCHED","evidence_maturity":"HYPOTHESIS"},"revision_change_log":{"revision_kind":"ORIGINAL","prior_problem_id":null,"prior_causal_lever_id":null,"problem_changed":false,"causal_lever_changed":false,"conceptual_changes":[],"operational_changes":[],"repairs_addressed":[]},"confidence":0.78,"generator_notes":"The candidate targets proof-state presentation, not mathematical validity itself. Empirical claims about attention, latency, and error reduction remain hypotheses pending the bounded test."}