{"schema_version":1,"experiment_id":"eoa_inverse_innovation_exp06_four_proposal_generalization60_20260803","cell_id":"ritualized_meaning_and_commitment_enactment__mathematics","arm":"COMPLETE_PROPOSAL_PORTFOLIO","candidate_id":"math_axiom_changeover_assembly_v0","proposal_index":2,"version":0,"title":"Axiom Changeover Assembly for Comparative Geometry","problem":"Students studying Euclidean and non-Euclidean geometries can state that conclusions depend on axioms yet continue importing familiar theorems across systems without identifying the required assumptions. Lectures explain axiom dependence propositionally, but transitions between formal systems lack a shared, memorable occasion that makes the withdrawal of one premise and the resulting change in permissible conclusions socially and physically salient.","actors":["Students enrolled in a comparative geometry course","Course instructor","Teaching assistants or proof reviewers","Students using nonvisual or asynchronous participation paths","Curriculum coordinator responsible for learning requirements","Future instructors receiving the practice record"],"observable_state":"After the course changes from one geometric axiom system to another, ungraded proof attempts or seminar explanations cite a result that requires the former system, label an axiom-dependent step as obvious, or omit the active axiom set even though students can define the systems when asked directly.","consequence":"A derivation may appear locally plausible while being invalid in its stated system, and students may leave the course treating axioms as background facts rather than explicit conditions governing what can be proved.","affected_objective":"Support accurate axiom-relative reasoning: students should identify the active formal system, trace conclusions to permitted premises, and flag imported results before relying on them.","intervention":"At each planned transition between geometric axiom systems, run a voluntary, governed Axiom Changeover Assembly. Beforeward, publish its mathematical purpose, script, accessibility options, nonparticipation rule, and a warning that enactment neither proves nor refutes any proposition. The instructor prepares removable axiom cards and conclusion cards connected by visible dependency paths for one worked theorem. A rotating facilitator marks the opening and names the system being left. Participants inspect the dependency display silently; a plain turn marker then circulates, allowing each person to identify a dependency, pose a question, submit a written observation, or pass. With equivalent visual, verbal, and digital options, willing participants remove or replace the changing axiom and turn over every conclusion whose displayed justification no longer survives. A witness reads the surviving, suspended, and undecided claims without inviting collective assent. Participants may then renew, revise, or decline concrete reasoning practices for the next module, such as labeling the active system or citing the axiom used at a contested step. The instructor closes by publishing corresponding rubric cues, office-hour support, and unresolved questions. An immediate meaning debrief and end-of-module drift review determine whether to adapt, pause, or retire the assembly.","structural_mapping":[{"archetype_element":"Ritual purpose and meaning claim","domain_realization":"The assembly enacts the claim that mathematical conclusions inherit conditions from named axioms; it does not require students to prefer a system or affirm a theorem."},{"archetype_element":"Participant and stakeholder boundary","domain_realization":"Students, instructor, teaching assistants, accessibility-path users, curriculum coordinators, and future instructors are distinguished, including absent students affected by resulting rubric changes."},{"archetype_element":"Legitimacy, consent, and exit","domain_realization":"Attendance at the mathematical lesson may be required, but ritual speech, object handling, synchronized action, personal commitment, and debrief disclosure remain optional and ungraded, with an ordinary observation or asynchronous path available."},{"archetype_element":"Ritual time, place, and cadence","domain_realization":"The assembly occurs only when the course formally changes axiom systems, making recurrence responsive to conceptual transitions rather than routine scheduling."},{"archetype_element":"Symbolic frame and shared narrative","domain_realization":"The removable premise and dependent conclusion cards portray a formal system as a structure whose results remain answerable to explicit starting commitments."},{"archetype_element":"Ritual script and sequence","domain_realization":"The recognizable sequence is opening, naming the departing system, silent inspection, distributed dependency reading, axiom changeover, witnessed classification of conclusions, practical commitment, and return to ordinary proof work."},{"archetype_element":"Role, witness, and facilitation structure","domain_realization":"Facilitator, dependency witness, access monitor, instructor, and participants have separate functions, with student roles rotating and no ritual role empowered to judge proof validity."},{"archetype_element":"Marked media and embodied participation","domain_realization":"Removable cards, reversible dependency paths, and optional turning gestures make a change in inferential permission perceptible while preserving verbal, written, digital, and no-touch alternatives."},{"archetype_element":"Attention and synchrony choreography","domain_realization":"A brief marked opening, silent inspection, and optional simultaneous turning of affected conclusion cards focus attention on the exact moment an axiom change alters the displayed dependency structure."},{"archetype_element":"Recognition, commitment, and closure","domain_realization":"Closure distinguishes surviving, suspended, and unresolved claims and translates the enacted meaning into chosen student practices, rubric cues, support, and named follow-up questions."},{"archetype_element":"Plural meaning and accessibility guardrail","domain_realization":"Students may interpret the exercise historically, structurally, visually, or formally and may contest the displayed dependencies; no shared philosophical belief about mathematical foundations is required."},{"archetype_element":"Emotional safety envelope","domain_realization":"No student is asked to expose a prior mistake, defend a belief, move physically, touch an object, or answer publicly; uncertainty is attributed to displayed claims rather than to individuals."},{"archetype_element":"Memory and intergenerational handoff","domain_realization":"A short record preserves the active axiom sets, dependency corrections, script revisions, access accommodations, and reasons for retaining or changing the practice across course offerings."},{"archetype_element":"Renewal and drift signal","domain_realization":"Immediate debriefs and review of subsequent proof work test whether the assembly remains interpretable and connected to axiom-labeling conduct rather than becoming theatrical repetition."},{"archetype_element":"Repair and retirement path","domain_realization":"Misleading dependency displays are corrected openly, pressured participation triggers repair, and the assembly can be replaced by an ordinary instructional method without changing course membership or assessment standing."}],"mechanism_mapping":[{"mechanism_slug":"ritual_design_canvas","role":"Places the mathematical meaning, selected axiom transition, symbolic dependency display, participation paths, sequence, operational follow-through, evidence, and retirement rule on one reviewable surface.","counterfactual_removal":"Without the canvas, the physical changeover could be designed for spectacle before its mathematical boundary, consent conditions, and assessment consequences are coherent."},{"mechanism_slug":"ritual_access_and_consent_review","role":"Uses pre-session veto points for compelled speech, public error exposure, inaccessible card handling, conspicuous refusal, recording, or hidden grading consequences.","counterfactual_removal":"Without the review, visible participation could become an informal assessment or belonging test, making the assembly's apparent engagement unreliable."},{"mechanism_slug":"opening_marking_and_threshold_gesture","role":"Marks the transition from routine work inside one system to focused examination of what changes when the governing premises change.","counterfactual_removal":"Without a reserved threshold, the axiom replacement may register as another diagram edit rather than a recognizable occasion for reconsidering inherited conclusions."},{"mechanism_slug":"silence_reflection_and_lament_interval","role":"Provides bounded time to inspect dependencies and privately notice which familiar results must be relinquished or reconsidered before discussion begins.","counterfactual_removal":"Without protected inspection, confident speakers may classify the consequences before other students can trace or question the dependencies."},{"mechanism_slug":"symbolic_object_circulation","role":"Moves a plain turn marker through the group so opportunities to identify a dependency, question the display, contribute asynchronously, or pass are visibly distributed.","counterfactual_removal":"Without the circulation rule, participation may revert to self-selection by the quickest or highest-status speakers."},{"mechanism_slug":"synchronized_movement_or_rhythm","role":"Offers a low-intensity, optional simultaneous turning or digital marking of conclusions affected by the axiom change, embodying the common change of inferential context rather than agreement with a proposition.","counterfactual_removal":"Without the bounded coordinated action, the intervention loses part of its enacted changeover structure, although the dependency analysis can still proceed instructionally."},{"mechanism_slug":"witnessing_and_public_recognition","role":"A named witness reads which displayed claims survive, become suspended, or remain disputed and recognizes corrections to the dependency map with contributor consent.","counterfactual_removal":"Without witnessed classification, the activity can close ambiguously, allowing students to remember the visual action without its precise mathematical consequence."},{"mechanism_slug":"collective_commitment_renewal","role":"Allows participants to renew, revise, or decline concrete practices for reasoning under the next axiom set while requiring the instructor to state the matching support and rubric obligations.","counterfactual_removal":"Without revisable practical commitments, the symbolic changeover may remain memorable but detached from subsequent proof-writing conduct."},{"mechanism_slug":"after_ritual_meaning_debrief","role":"Immediately compares the intended meaning with experiences of mathematical clarity, pressure, exclusion, ambiguity, and confusion between enactment and proof.","counterfactual_removal":"Without the debrief, organizers may mistake smooth choreography for conceptual clarity or legitimate participation."},{"mechanism_slug":"ritual_drift_and_harm_audit","role":"Reviews whether repetition has become rote, whether the displayed dependencies remain accurate, whether participation affects grading informally, and whether promised instructional support occurs.","counterfactual_removal":"Without skeptical review, an inaccurate or coercive assembly could persist because it appears established and engaging."},{"mechanism_slug":"ritual_retirement_or_repair_ceremony","role":"Provides a transparent way to correct or withdraw a misleading display, acknowledge participation harm, archive useful mathematical lessons, and release the script and objects from expected use.","counterfactual_removal":"Without a legitimate ending path, instructors may preserve the form after its instructional meaning, accuracy, or consent has failed."}],"causal_chain":["Students can repeat definitions of multiple geometries while familiar conclusions retain the felt status of context-free truths.","A consent-reviewed assembly creates a marked occasion specifically for crossing between axiom systems.","The displayed axiom-to-conclusion paths concentrate shared attention on the conditional structure of one worked theorem.","Silent inspection and distributed, pass-permitted turns allow dependencies and doubts to become legible without rewarding only rapid public responses.","Removing or replacing an axiom and reversibly marking affected conclusions converts abstract conditionality into a repeatable social performance.","Witnessed classification prevents the performance from being remembered as indiscriminate destruction or as a proof of the replacement system.","Revisable student practices and reciprocal instructor obligations carry the symbolic change into proof labels, citations, rubric cues, and support during the next module.","Debrief, work-sample inspection, repair, and retirement keep mathematical accuracy and consent authoritative over ritual continuity."],"baseline":"Explain each axiom system through lectures, textbook tables, worked examples, and ordinary exercises, then correct illicit imports when they appear in submitted proofs. This transfers formal information but does not create a separately marked, recurring occasion for relinquishing conclusions tied to the former system and renewing context-labeling practices.","nearest_rivals":["An inquiry-based lesson in which students derive or refute propositions under alternative axiom sets; it supports discovery but need not create a recurring symbolic changeover with consent, recognition, practical closure, and retirement governance.","A proof assistant or formal logic environment that rejects steps unavailable in the active context; it enforces formal scope but does not itself make the community's transition between systems a shared, interpretable renewal occasion.","A dependency-graph or concept-map exercise that visualizes axiom relationships but may remain a one-time representational task without marked recurrence, plural participation, witnessed closure, or commitments outside the exercise.","A grading rubric requiring every proof to name its axiom system; it can incentivize compliant notation but does not enact why the condition matters or permit collective examination and revision of the associated practice.","A historical lecture about the parallel postulate and non-Euclidean geometry; it supplies narrative context but does not embody the withdrawal of a premise or connect that act to current reasoning obligations."],"remaining_contrastive_claim":"The proposal is limited to combining a recurring, symbolically marked change of formal context with reversible dependency enactment, nonretaliatory participation, witnessed mathematical classification, reciprocal learning obligations, and governed review. It does not claim that ritual action establishes validity or that embodied instruction is necessary for understanding axiomatic systems.","authority_safety":{"decision_authority":"The instructor may authorize a bounded pilot within the approved curriculum; the curriculum coordinator retains authority over learning requirements, and ordinary mathematical argument and assessment rules determine correctness. An access monitor may halt the ritual form but may not change mathematical content or grades.","authorized_first_step":"Run one ungraded, 20-minute pilot at a single planned transition from Euclidean geometry to a specified alternative system, using one instructor-checked dependency display and offering observation-only, written, digital, and pass options.","excluded_actions":["Treating removal, turning, synchrony, silence, or witnessed classification as proof or refutation","Grading attendance, speech, gesture, personal commitment, debrief participation, or refusal","Requiring students to disclose previous errors or defend philosophical beliefs about mathematics","Inferring agreement, comprehension, loyalty, or competence from visible participation","Changing required course content or assessment standards during the pilot","Using sacred, culturally borrowed, humiliating, or emotionally intense symbols","Recording identifiable participation or feedback without separate explicit consent","Replacing formal derivation, countermodels, proof review, or corrective instruction with the assembly"],"halt_rollback":"Stop the assembly if the dependency display is mathematically disputed in a way the session cannot resolve, any participation path becomes conspicuous or inaccessible, students report grading pressure, or enactment is confused with proof. Restore every card and digital artifact to a neutral teaching state, mark the displayed classification as withdrawn, correct any mathematical error through ordinary instruction, disregard ritual participation in assessment, and either redesign under independent review or retire the practice."},"negative_tests":{"strongest_counterevidence":"If an ordinary dependency-map exercise with identical mathematical content, participation distribution, and follow-up produces equally accurate axiom identification and is experienced as clearer or less pressuring, the symbolic assembly has no supported added function and should not continue.","problem_falsifier":"The problem is unsupported in the selected course if students consistently name the active axiom system, distinguish system-relative conclusions, and identify imported premises in fresh ungraded proofs despite transitions between systems.","intervention_falsifier":"The intervention fails its bounded test if participants confuse enacted withdrawal with mathematical refutation, cannot accurately classify the displayed conclusions afterward, experience passing as costly, or show the same illicit imports in the selected follow-up task without being able to use the assembly to diagnose them.","risks":["A memorable gesture may oversimplify dependencies that are indirect, contested, or model-relative.","Students may infer that changing axioms is arbitrary or that conclusions become false rather than underivable in the new system.","Visible participation may function as informal grading despite written safeguards.","Synchronized action may manufacture apparent agreement or exclude students using alternative modalities.","The instructor's dependency map may acquire unwarranted authority and suppress legitimate objections.","Symbolic closure may conceal unresolved mathematical questions.","Repeated assemblies may become rote and consume time better used for derivation or countermodel construction.","The practice may privilege physical and visual metaphors over formal, verbal, or nonvisual understanding."]},"next_evidence_step":"For the single pilot transition, collect anonymous immediate responses about perceived choice, accessibility, mathematical meaning, and any confusion between enactment and proof. Then use one ungraded fresh proof-classification task asking students to name the active system, identify an unavailable premise, and state whether a conclusion is false, unproved, or undecided there. Compare these bounded observations with the preceding transition taught through the course's ordinary dependency-map method, inspect follow-through on promised rubric cues and support, and make only a pilot-specific keep, adapt, or retire decision.","prior_art_status":"UNSEARCHED","diversity_from_prior_proposals":"Earlier proposal 1 addressed loss of social ownership for verification obligations within a long-lived collaborative theorem under researcher turnover. This proposal addresses students' illicit transfer of conclusions between different axiom systems during instruction. Its intervention is a recurring enacted change of formal context using removable premises and reversibly suspended conclusions, rather than a renewal of responsibility for checking an existing theorem. Its causal path runs from marked axiom replacement to salient conditionality and context-labeled proof behavior, not from witnessed task acceptance to durable proof stewardship. It is independently adoptable in a comparative geometry course with no collaborative research project, theorem-maintenance ledger, publication workflow, or researcher handoff.","revision_record":{"parent_version":null,"progress_targets_addressed":["Generate one additional complete candidate for proposal index 2","Address a problem materially different from proposal 1","Preserve the governed ritual lifecycle and authored mechanism logic","Specify operational authority, safeguards, falsifiers, rivals, and bounded first evidence","Explain diversity from every earlier sealed proposal"],"conceptual_changes":["Initial version; no parent proposal","Centered the candidate on axiom-relative reasoning during formal-system transitions rather than stewardship of a collaborative proof","Separated enacted suspension of derivability from claims of falsity or proof"],"operational_changes":["Specified a voluntary 20-minute changeover pilot using removable axiom and conclusion dependencies","Added multimodal participation, pass rights, rotating roles, reciprocal instructor obligations, mathematical-error rollback, and retirement rules"],"evidence_changes":["Defined anonymous experience checks and one fresh ungraded proof-classification task","Included a bounded comparison with the course's preceding ordinary dependency-map transition","Kept prior-art status unsearched"],"claim_changes":["Restricted the functional claim to governed symbolic enactment of formal-context change and its connection to reasoning practices","Excluded claims of novelty, prevalence, demand, effect size, or mathematical validity through ritual"]}}