induce

Periagoge Protocol

Crystallize in-process abstraction by aligning concrete cases first and naming last, with the live alternative readings held visible throughout. Type: (AbstractionInProcess, AI, INDUCE, A) → CrystallizedAbstraction.

Definition

Periagoge (περιαγωγή): A dialogical act of turning an in-process abstraction toward its crystallized form, where AI detects when an instance set has converged toward an unnamed essence, puts the two most alignable cases side by side for the user to correspond, extracts the invariant relation that correspondence carries together with the readings it leaves open, probes those open readings apart against further cases and near-misses the user judges, and only then proposes a name and rule for what survived — so the abstraction is located by the correspondences the user built rather than steered from a candidate offered ahead of them (the Greek dialectical vocabulary supplies the source terms).

── FLOW ──
Periagoge(A) → Detect(A) →
  InProcess(Iᵢ, E, L?): Pair(Iᵢ, E) → (i₁, i₂) →
         Align(i₁, i₂, ctx) → Stop → Aₐ →
         (AsShown: M committed
          | Correct(slot, value): re-present Align
          | Repartner(ref): re-enter Align with ref as partner
          | Abandon: declare(alignment_trace, open_trace) → AlignmentSuspended) →
         Extract(M, Iᵢ, H?) → (R, H) →
         Probe(H, Iᵢ, ctx, gap?) → p → Qp(p, H) → Stop → W → narrow(H, W) → H' →
         loop until (settled(H) ∧ probed(Λ)) ∨ budget_spent(Λ) →
         Name(H, R, L?, Λ) → Qn → Stop → Nₐ →
         (Confirm: declare(alignment_trace, open_trace) → CrystallizedAbstraction
          | NotYet(gap) ∧ ¬budget_spent(Λ): re-enter Probe with gap seeded
          | NotYet(gap) ∧ budget_spent(Λ): declare(alignment_trace, open_trace) → AlignmentSuspended with gap Deferred
          | Abandon: declare(alignment_trace, open_trace) → AlignmentSuspended)
         or unalignable(Λ) → declare(alignment_trace, open_trace) → AlignmentSuspended
  NotInProcess: deactivate

── MORPHISM ──
A
  → detect(instances, essence, locator)   -- verify in-process abstraction exists
  → pair(instances, essence)              -- select the two cases that align most readily
  → align(pair, slots)                    -- user constructs the correspondence; AI supplies the slots; the answer keeps it, corrects a slot, repartners, or abandons
  → extract(correspondence)               -- invariant relation, plus the axes that correspondence leaves live, over the space already built
  → probe(space, instances)               -- select the case that separates live axes rather than confirming the leading one
  → narrow(space, answer)                 -- per axis: rule it out, bound it, or leave it undecided; or, for the run, redraw the case, extend the language, or repartner
  → name(space, relation, label, record)  -- AI proposes name + rule, reading the boundary off the probe record, after the space has narrowed and not before
  → declare(trace, open_trace)            -- terminal evidence trace + open-item disposition
  → CrystallizedAbstraction
requires: in_process(A)                    -- runtime checkpoint (Phase 0)
deficit:  AbstractionInProcess              -- activation precondition (Layer 1/2)
preserves: instance_set(A)                  -- Iᵢ read-only; an axis leaves H.ruled_out only when the user's own AxisMissing names it again, so AI never re-proposes one
invariant: Correspondence Before Naming through Maintained Alternatives over Single-Candidate Steering

── TYPES ──
A              = AbstractionSeed (in-process state: instances + essence intuition + optional user concept label)
Detect         = A → DetectResult
DetectResult   ∈ {InProcess(Iᵢ, E, L?), NotInProcess}
                 -- closed: Phase 0's two exits; the payload rides the InProcess branch, so no verdict and payload
                 -- can disagree, and InProcess is what witnesses requires: in_process(A)
Iᵢ             = Set(Instance)                             -- instance set observed; |Iᵢ| ≥ 2 is what Pair requires, since a correspondence needs two terms
Instance       = { content: String, context: String }       -- concrete case observed
E              = EssenceIntuition                           -- variation-stable core signal from conversation
L              = Option(TentativeLabel)                     -- user-provided provisional name or concept, if any
ctx            = DomainContext                              -- user's domain context gathered via artifact read, artifact search, and a conditional external fetch
Pair           = (Iᵢ, E) → (i₁, i₂)                        -- the readiest alignment, not the most distant; distance is what Probe is for
Slot           = { role: String, in_first: String, in_second: Option(String) }
                 -- in_second is None exactly where the second case carries no counterpart, which is what unmatched(M) collects;
                 -- filling it reads off the cases rather than choosing between readings, which is what H.live and Probe carry
M              = Correspondence { slots: List(Slot) }
unmatched(M)   = {s.role : s ∈ M.slots, s.in_second = None}
                 -- what one case carries and the other does not, read off the slots rather than kept beside them; it is evidence, not failure
Align          = (i₁, i₂, ctx) → M                          -- constructed at the Phase 1 gate; the user fills or corrects the slots
Aₐ             = AlignAnswer ∈ {AsShown, Correct(slot, value), Repartner(ref), Abandon}
                 value        = what the user says the cases carry in that slot -- the filling is replaced, no reading is chosen
                 -- the Phase 1 answer. AsShown commits M; Correct re-presents with the slot replaced; Repartner and Abandon
                 -- are the run-level moves W also carries, reachable here so a wrong pairing or a withdrawal need not wait
                 -- for a probe. ref is as defined under W
Axis           = { relation: String }                       -- one reading of what the correspondence carries
H              = HypothesisSpace { live: Set(Axis), ruled_out: Map(Axis, Ground) }
                 -- live is what the correspondence has not yet decided between; ruled_out records why each was dropped
                 -- invariant: live ∩ dom(ruled_out) = ∅ — an axis is in exactly one of them, which is what makes settled(H) total
Ground         = String                                     -- the probe answer that ruled the axis out, in the user's own terms
R              = InvariantRelation { statement: String, carried_by: List(Slot) }
                 -- carried_by cites the slots the relation reads off, so the relation stays traceable to the correspondence
Extract        = (M, Iᵢ, H?) → (R, H')                      -- internal; no gate. On re-entry H is the space already
                 -- built: ruled_out carries over untouched and axes already supplied stay live, so re-extraction
                 -- extends the language rather than resetting it. Absent H, H' is derived from the correspondence alone
Probe          = (H, Iᵢ, ctx, gap?) → p               -- gap? is the NotYet payload on re-entry from Phase 5, seeding the draw; absent on every other entry
p              = ProbeCase { content: String, separates: Set(Axis) }
                 -- separates names which live axes this probe tells apart; a probe separating none is not a probe
V              = AxisVerdict ∈ {Refutes(ground), Bounds(ground), Undecided}
                 ground       = the user's reason, recorded verbatim as the Ground for whatever it rules out or bounds
                 -- one axis judged against one probe case. Refutes rules the axis out; Bounds places the case outside
                 -- what that axis claims, so the axis survives; Undecided settles neither and leaves it live.
                 -- These three are the axis-local judgments; every other constructor of W is run-level
Vs             = AxisVerdicts = Map(Axis, V)
                 -- invariant: dom(Vs) = p.separates — every axis the probe tells apart carries exactly one verdict,
                 -- which is what makes narrow total over the probe's separating set
W              = ProbeAnswer ∈ {Judged(Vs), Redraw(correction), AxisMissing(description), Repartner(ref), Abandon}
                 correction   = what the user says the probe case actually is -- the case is drawn again, no axis is edited
                 description  = a dimension the live set does not contain -- extends the language rather than editing a candidate
                 ref          = another instance or a neighbouring abstraction to align against instead
narrowed_none(Vs)    = ∄ a ∈ dom(Vs) : Vs(a) = Refutes(_)
                 -- the round ruled no axis out, whether every axis was kept by scoping the case out of it or left
                 -- undecided. Both are theory-preserving, so the announcement after Phase 4 is owed on either
ProbeRecord    = { case: ProbeCase, answer: W }
                 -- one probe with the answer it received; a Redraw round appends none, since no axis was judged
boundary(Λ, H) = [(r.case, g) : r ∈ Λ.probes, r.answer = Judged(Vs), a ∈ H.live ∩ dom(Vs), Vs(a) = Bounds(g)]
                 -- every case the user placed outside a surviving axis's claim, read back from the probe record where
                 -- the case and its ground both survive. No field carries a boundary, so narrow writes none, and the
                 -- derivation is total whether Phase 5 fired on settled(H) or on budget_spent with several axes live
narrow         = (H, W) → H'                                -- Judged(Vs): each Refutes axis moves to ruled_out with its ground,
                                                            -- each Bounds and each Undecided axis stays live;
                                                            -- AxisMissing adds the axis the user described to live, and where the description names an axis in ruled_out
                                                            -- that axis moves back to live, and the ground that ruled it out is read from the probe record that carried the Refutes
                                                            -- and shown at its return; Redraw, Repartner and Abandon leave H unchanged
settled(H)     = |H.live| = 1
probed(Λ)      = |Λ.probes| ≥ 1                             -- Phase 5 opens on settled(H) only past this floor: a space that arrives settled from
                 -- Extract is still probed once, so the rule is named against at least one case the user judged rather than
                 -- against the correspondence alone
unalignable(Λ) = H.live = ∅                                 -- every axis ruled out and none supplied; the seed did not carry one
                 -- settled(H) is false whenever live is empty, so no second conjunct is doing work here
Name           = (H, R, L?, Λ) → (N, Rule)                  -- Λ is what boundary(Λ, H) is read from; the rule is read off one live axis, which named(Λ) cites
N              = AbstractionName { name: String, label_basis: Option(L) }
Rule           = { statement: String }                      -- presented together with boundary(Λ, H), so what the abstraction excludes is shown, not asserted;
                 -- no field carries it, so none can drift from the probe record it is read from
Nₐ             = NameAnswer ∈ {Confirm, Rename(name), RuleWrong(correction), NotYet(gap), Abandon}
                 gap          = what the user says is still missing; seeds the next Probe while the budget holds, and at budget_spent
                                is recorded as a Deferred open item instead, so no answer at the cap draws another probe
Qp             = Probe judgment interaction [Tool: Constitution interaction]
Qn             = Naming interaction with name + rule + boundary [Tool: Constitution interaction]
max_probes     = the probe cap LOOP fixes per abstraction seed
budget_spent(Λ) = |Λ.probes| ≥ max_probes
                 -- a resource bound on user attention, not a sufficiency criterion: reaching it says the run stopped,
                 -- never that the abstraction formed. Phase 4 still fires, and whatever stayed live goes to the open trace
crystallized(Λ) = Λ.crystallized ≠ None                     -- written at Confirm and nowhere else
OpenDisposition ∈ {None, Nonblocking, Deferred}
                 None        = no live axis, no unmatched slot, and no gap remains; explicitly declared
                 Nonblocking = an item remains visible but does not block Confirm
                 Deferred    = user routes an item to later work via free response
OpenItemDisposition = OpenDisposition \ {None}             -- per-item value space; None is a whole-trace verdict only
named(Λ)       = {the live axis Rule.statement was read off} when Λ.crystallized = Some(_); ∅ otherwise
                 -- so at AlignmentSuspended every live axis is open, and at Confirm only the axes the rule did not take are
OpenItems(Λ)   = (H.live \ named(Λ)) ∪ ⋃{unmatched(M) : M ∈ Λ.correspondences} ∪ {gap : Λ.naming = Some(NotYet(gap)) ∧ budget_spent(Λ)}
                 -- what a run can still owe at its terminal, from all three carriers; the gap enters with disposition Deferred
                 -- by construction, since the run it would have seeded is the one the cap stopped; the union runs over every
                 -- correspondence built, since a slot left unmatched before a repartnering is still unmatched after it.
                 -- H.live reads as ∅ while Λ.space is None, so a run abandoned at Phase 1 owes nothing but declares that
OpenTrace      = { items: Map(String, OpenItemDisposition) }
                 -- invariant: dom(items) = OpenItems(Λ) — every open item carries exactly one disposition, which makes status(O) total
status(O)      = None if dom(O.items) = ∅; Deferred if ∃ i ∈ dom(O.items) : O.items(i) = Deferred; otherwise Nonblocking
AlignmentTrace = List<(Slot | ProbeRecord | (N, Rule))>
                 -- the run in the order it happened: the correspondence the user built, each probe with the verdict
                 -- every separated axis received, and the naming it terminated on.
                 -- Derived from Λ.correspondences, Λ.probes, and Λ.naming
CrystallizedAbstraction = (N, Rule) where confirmed via Nₐ = Confirm ∧ alignment_trace_declared(AlignmentTrace) ∧ open_disposition_declared(OpenTrace)
AlignmentSuspended = (R?, H?) where alignment_trace_declared(AlignmentTrace) ∧ open_disposition_declared(OpenTrace)
                 -- the non-crystallizing terminal. It carries what the run established so a later run resumes from it
                 -- rather than restarting; R and H are absent only when Phase 2 was never reached, which is what
                 -- Abandon at Phase 1 produces

── A-BINDING ──
bind(A) = explicit_arg ∪ recent_instance_cluster ∪ surfaced_essence
Priority: explicit_arg > recent_instance_cluster > surfaced_essence

/induce "theme"              → A = AbstractionSeed with theme label
/induce (alone)              → A = most recent instance cluster in session
"the pattern across..."      → A = instance cluster under discussion

If no essence signal is detectable (neither user sensing language nor AI-inferrable core pattern): pause activation and surface the scan result before Phase 0, inviting the user to either name what feels in-process or withdraw. If |Iᵢ| < 2, scan the accumulated session context and the user's artifacts for cases that could correspond with the one in hand, and present what the scan found as candidates to recognize or replace before Phase 1. Where it finds nothing, say what was searched and invite a second case.

── PHASE TRANSITIONS ──
Phase 0: A → Detect(A) → InProcess(Iᵢ, E, L?) | NotInProcess               -- detection checkpoint (silent)
Phase 1: (Iᵢ, E) → Pair(Iᵢ, E) → (i₁, i₂) → Align(i₁, i₂, ctx) → Stop → Aₐ ; on AsShown, Λ.correspondences := Λ.correspondences ++ [M]   -- correspondence Constitution interaction [Tool]
         -- on re-entry from Repartner(ref), ref is the second term and Pair is skipped: the user already chose the partner
Phase 2: (M, Iᵢ, Λ.space) → Extract(M, Iᵢ, Λ.space) → (R, H') ; Λ.relation := Some(R) ; Λ.space := Some(H')   -- relation + live-axis derivation (track)
Phase 3: H → Probe(H, Iᵢ, ctx, gap?) → p → Qp(p, H) → Stop → W                   -- probe Constitution interaction [Tool]
Phase 4: W → narrow(H, W) → H' ; Λ.space := Some(H') ; Λ.probes := Λ.probes ++ [ProbeRecord(p, W)] where W ≠ Redraw(_)   -- space update + probe record (track)
         -- the append is what budget_spent counts, so a constructor that declares it spends no probe must not reach it
Phase 5: (H, R, L?, Λ) → Name(H, R, L?, Λ) → (N, Rule) → Qn → Stop → Nₐ ; Λ.naming := Some(Nₐ)   -- naming Constitution interaction [Tool]

── LOOP ──
After Phase 1: evaluate the alignment answer.
If Aₐ = AsShown: M is committed; proceed to Phase 2.
If Aₐ = Correct(slot, value): replace that slot's filling with value and re-present at Phase 1 within the same round; the correspondence M then carries is the corrected one.
If Aₐ = Repartner(ref): re-enter Phase 1 with ref as the alignment partner and Pair skipped; the pairing left behind commits nothing.
If Aₐ = Abandon: Λ.alignment_trace := derive(Λ), Λ.open_trace := derive(OpenItems(Λ), free_response), declare both, terminate as AlignmentSuspended with R and H as Λ carries them (absent when Phase 2 was never reached).

After Phase 4: evaluate the probe answer.
If W = Judged(Vs): each Refutes axis is ruled out with its ground, each Bounds and each Undecided axis stays live; return to Phase 3. If narrowed_none(Vs), say that this round ruled no axis out, and which axes were kept by scoping the case out of their claim and which were left undecided, before drawing the next probe.
If W = Redraw(correction): the probe is drawn again with the correction taken up; H is unchanged, no ProbeRecord is appended, and no probe is spent from the cap since no axis was judged; return to Phase 3.
If W = AxisMissing(description): the axis the user described enters H.live; where the description names an axis ruled out earlier, that axis returns to live by the user's word with the ground that ruled it out shown beside it; return to Phase 2 (relation re-extracted over the extended language).
If W = Repartner(ref): return to Phase 1 with ref as the alignment partner and Pair skipped; H and its ruled_out survive the repartnering, and the correspondence already built stays in Λ.correspondences.
If W = Abandon: Λ.alignment_trace := derive(Λ), Λ.open_trace := derive(OpenItems(Λ), free_response), declare both, terminate as AlignmentSuspended.
Continue Phase 3 until: (settled(H) ∧ probed(Λ)) ∨ budget_spent(Λ) ∨ unalignable(Λ).
If unalignable(Λ): declare as AlignmentSuspended — every axis was ruled out and none supplied, which is a finding about the seed and is reported as one.
Otherwise proceed to Phase 5.

After Phase 5: evaluate the naming answer.
If Nₐ = Confirm: Λ.crystallized := Some((N, Rule)), Λ.alignment_trace := derive(Λ), Λ.open_trace := derive(OpenItems(Λ), free_response), declare both, terminate as CrystallizedAbstraction.
If Nₐ = Rename(name): N.name := name, re-present at Phase 5 within the same round.
If Nₐ = RuleWrong(correction): Rule.statement := correction, re-present at Phase 5 within the same round.
If Nₐ = NotYet(gap) ∧ ¬budget_spent(Λ): return to Phase 3 with gap seeding the next Probe; the budget is spent by this round like any other.
If Nₐ = NotYet(gap) ∧ budget_spent(Λ): Λ.alignment_trace := derive(Λ), Λ.open_trace := derive(OpenItems(Λ), free_response) with gap Deferred, declare both, terminate as AlignmentSuspended; the gap is what the next activation resumes from.
If Nₐ = Abandon: declare as AlignmentSuspended.
Cap: max_probes = 5 probes per abstraction seed. This bounds user attention; it is not a claim that five probes suffice to form an abstraction, and reaching it never converts a run into a crystallized one. What makes it a bound is that no answer at budget_spent draws another probe: Confirm crystallizes, NotYet suspends with the gap on record, Abandon suspends.
Convergence evidence: At crystallized(Λ), present the alignment trace — the correspondence the user built slot by slot, then each probe with the verdict every separated axis received, then the name and rule that survived — plus the boundary derived from the Bounds verdicts standing against the axes still live, plus the OpenTrace. Show every axis that was ruled out beside the ground that ruled it out, so the surviving axis is seen to have won rather than asserted to have. OpenTrace status is None when nothing stayed open, Deferred when any item is routed to later work, and Nonblocking otherwise. Convergence is demonstrated, not asserted.

── CONVERGENCE ──
crystallized(Λ): see TYPES (Λ.crystallized ≠ None, written at Confirm)
settled(H) ∧ probed(Λ): see TYPES (exactly one live axis, at least one probe recorded) — Phase 5's fire condition, not a terminal on its own

── TOOL GROUNDING ──
-- Realization: Constitution → TextPresent+Stop; Extension → TextPresent+Proceed
Phase 0 Detect     (sense)   → Internal analysis (no external tool)
Phase 1 Pair+Align (constitution) → present the two cases side by side with every slot filled from the cases themselves and unmatched roles marked as such, and the four ways the answer can go (mandatory); artifact read, artifact search for the cases' own context; external fetch (conditional: the cases' domain lies outside the user's artifacts)
Phase 2 Extract    (track)   → Internal state update: writes Λ.relation and Λ.space, so Phase 3 has a space to probe and the terminal has a relation to name
Phase 3 Probe+Qp   (constitution) → present the probe case with every live axis it separates on screen together, each axis's support and the case that breaks it beside that axis's own verdict slot with what each verdict does to that axis, and before the gate what the live set becomes on each way the round can close (mandatory); external fetch (conditional: a probe drawn from outside the user's domain)
Phase 4            (track)   → Internal state update: writes Λ.space and appends Λ.probes, so the cap can advance and the alignment trace has per-probe material to build from
Phase 5 Name+Qn    (constitution) → present name, rule, boundary, whatever stayed live, and whether the budget is spent (mandatory)
converge           (extension)   → TextPresent+Proceed (alignment trace + open disposition; proceed with the crystallized abstraction)
seam               (extension)   → TextPresent+Proceed (fires at deactivation/handoff: a user-declared chain naming the next protocol settles the next move — proceed directly to it, citing that settling source. This protocol declares no wired outbound continuation edge: its only cross-protocol link is an inbound misfit absorption (`Upstream misfit absorption`), not a post-crystallization handoff, so the second trigger is vacuously absent. Every Constitution gate inside this protocol and inside the next protocol fires unchanged)

── MODE STATE ──
Λ = { phase: Phase, A: AbstractionSeed, Iᵢ: Set(Instance), E: EssenceIntuition,
      correspondences: List(M), relation: Option(R), space: Option(H),
      probes: List(ProbeRecord), naming: Option(Nₐ),
      crystallized: Option((N, Rule)),
      alignment_trace: Option(AlignmentTrace),
      open_trace: Option(OpenTrace),
      active: Bool, cause_tag: String }

── COMPOSITION ──
*: product — (D₁ × D₂) → (R₁ × R₂). Dimension resolution emergent via session context.