substrate-membership.md

What belongs in fungeom — membership in a decidability substrate

Audience: an agent working in the fungeom repo (~/GitHub/functional_api). From: the standalone_contact_detection / markovlib session. Status: a settled reframing of fungeom's purpose and of the rule for what belongs in it — meant to be baked into fungeom's canonical knowledge, not left as a side note. Action for this session: internalize this, then propagate it — (1) update AGENTS.md §"What this is" and README.md so fungeom describes itself as a general decidability substrate (geometry = instance #1, time = instance #2) whose partiality is first-class and graded-capable; (2) retire the geometry-era slogan — "exact/closed-form ⇒ fungeom; statistical/iterative ⇒ parked" — as the membership rule wherever it is cited (it appears in markovlib-bridge.md and retarget/docs/fungeom-needs-for-substrate.md), replacing it with the two bright lines below; (3) link this note from README's deep-dives and from CHECKLIST.md's add-a-primitive guidance, so the test is applied at authoring time. This file is the canonical rationale — keep it. Read with: time.md (whose opening — "fungeom is a decidability engine; geometry is just instance #1" — is the seed of all of this) and free-variables.md.

The spine: fungeom is a decidability substrate, not a geometry library

fungeom's invariant is immutable, lazily-evaluated, composable, decidable computation graphs over typed values, with partiality first-class. Geometry is instance #1; time is instance #2; "anything" is the goal. So the rule for what belongs in fungeom must be stated in terms of the substrate's invariant — honest, referentially-transparent decidability — and not in terms of any one instance's norms. The rule we have today is stated in geometry's norms. That is the bug.

The rule we are retiring (and why it was locally right)

The standing rule — "exact / closed-form / combinatorial ⇒ fungeom; statistical / iterative / smoothing ⇒ parked numeric, consumer-side" (retarget/docs/fungeom-needs-for-substrate.md) — is a good geometry rule: in geometry, exact/closed-form is the norm, and "statistical" reliably correlates with "doesn't belong." It mis-cuts the instant geometry stops being the only instance, because exactness is not the property a substrate selects on.

Why the slogan mis-cuts: it is four cuts wearing one shirt

"Statistical/iterative" silently bundles four properties that co-vary in geometry but come apart in general:

1. referential transparency — pure function of the graph; no hidden RNG / iteration state;
2. closed-form vs. limit-of-a-sequence;
3. honest resolution — a clean decide() verdict vs. a returned best-effort / silent NaN;
4. forced-by-the-inputs vs. a baked modeling opinion (a prior, a kernel, a threshold).

Once they separate, the slogan returns wrong answers:

• Forward–backward, Viterbi, Kalman/RTS are exact, deterministic, referentially-transparent scans — no more "iterative" than a fold or a matmul — carrying zero modeling opinion once the model is given. The slogan parks them with RANSAC; they share nothing with RANSAC.
• Savitzky–Golay is a linear convolution — strictly closed-form — yet parked as "smoothing." It is correctly excluded, but for a reason the slogan never names: its window/order are the consumer's taste.
• RANSAC / MCMC are excluded for hidden randomness, which a substrate cures by reifying the seed as an input, not by banning the method.

Three different reasons, one shirt. The slogan happens to give geometry-intuitive answers while tracking none of the properties a substrate cares about.

The rule that replaces it — two bright lines

Stated in the substrate's own terms:

Admit an op iff it is (1) referentially transparent — a pure function of its resolver graph, with every seed / initial guess / iteration budget reified as an explicit input value (so the graph stays deterministic, memoizable, equational) — and (2) honestly resolvable — its success, failure, and approximation character are expressed through decide()'s lattice, never a silent NaN or a best-effort return.

Park an op iff it bakes a modeling commitment fungeom cannot make for the consumer — a prior, a kernel bandwidth, a loss/regularizer, a stopping tolerance whose right value is domain taste, not a property of the inputs — and that commitment is hidden rather than an explicit input.

How the same examples re-sort (admit = no longer excluded by category; actual residency is still gated by domain-size / dependency-floor / performance — see §"What this changes in practice"):

operation	old rule	new rule	why
forward–backward / Viterbi / Kalman–RTS	parked ("iterative")	admit	exact, deterministic, pure; zero opinion given the model
tolerance / ε-geometry (parallel-within-ε; curve-as-polygon)	ad hoc (approx_equal, parked Scalar.approx/X2)	admit — as a graded resolution	the value-level atol lifted into decide() where it belongs
EM / coordinate ascent	parked	admit once the lattice is graded; tol an explicit input	deterministic given inputs; honest output = "converged to tol / hit cap"
particle filter / MCMC	parked	admit once seed reified + lattice graded	pure given (data, seed, N); honest output = "approximate, O(1/√N)"
Savitzky–Golay / smoothing	parked	park if the window is hidden; admit if it is an explicit input and the result is honest	exclusion is about hidden taste, not "smoothing"
RANSAC (hidden RNG + hidden inlier threshold)	parked	park as-is; admit only if seed + threshold are reified	hidden randomness and hidden opinion
signed-distance, convex hull, Kabsch (exact)	admit	admit (unchanged)	pure, exact, opinion-free

Note the rule is stricter in places (RANSAC stays out unless reified) and more permissive in others (exact inference stops being excluded by category). It is not "let the numerics in" — it is "select on honesty, not on the kind of math."

Exactness is a grade, not a gate

The deepest move: demote exactness from a membership gate to a resolution-quality attribute. exact / approximate-to-ε-by-method-X / stochastic-O(1/√N) become grades of resolution reported by decide(), not in/out verdicts. This is precisely the parked depth-B RFC — a third constructor on the lattice,

Approximate(value, method, error_character)        #  + promote  ok: bool  →  a quality ordering

— and it is the mechanism that makes the new membership rule expressible. Seen this way, the old slogan is mechanically a workaround for a binary lattice that cannot represent an approximate result honestly: "we can't say it, so we park anything that produces it." Give the lattice grades and the workaround dissolves — not by getting permissive, but because the substrate can finally say what such an op returns.

Why this is fungeom's own thesis, not a graft: partiality ⊇ uncertainty

The reason this is not "bolting statistics onto a geometry library": fungeom's partiality already is the bottom of an uncertainty lattice. Binary Resolvable / Unresolvable is the two-point quotient of a single order — how much is known about the value:

• partiality with a measure = "resolvable to within ε" (tolerance geometry — your parked Scalar.approx / X2 want exactly this);
• partiality with a distribution = "resolvable to this posterior" (inference).

exact → graded → distributional is one refinement of fungeom's own native notion of partiality, not a foreign axis. A decidability engine "for anything" is already committed to this axis; it merely truncates it at two points today. Lifting the truncation is the substrate growing into itself, not taking on a new mission.

Keep a bright fence — we only moved it

A caution worth canonizing. The old rule's hidden virtue was being a bright line: "statistical → out" is crude but enforceable — a Schelling fence against kitchen-sink accretion. And "use it for anything" makes accretion more likely, not less, so do not trade the fence for case-by-case litigation. The replacement is still a bright line, just better placed:

Reify every opinion and every seed as an explicit input, and surface every approximation in the resolution lattice. If an op cannot be expressed that way, it stays out.

Crisp, refusable, and it fences along honesty and transparency — what actually protects the substrate's value — instead of along "kind of math," a correlate that stops holding the moment geometry isn't the only instance.

What this changes in practice (and what it does not, yet)

• Changes: the criterion and the self-description. Exact, pure inference (FB/Viterbi/RTS) is no longer excluded by category; tolerance/ε-geometry gets a first-class home; the road to graded/distributional resolution is recognized as fungeom's own roadmap (depth B), not a favor to a consumer.
• Does NOT change (yet): placement is still staged, for reasons independent of the rule — a belief/numerical layer is a large additive domain (primitives/<name>/ depending only on core), gated like any domain on a real in-substrate consumer; consumers on older Python floors still bind fungeom-free cores (so a substrate-native engine serves only the newer-floor side); and even substrate-native, an engine schedules at model/query granularity (one opaque numerical node), never a resolver per message. So this note is a principle + identity reframe, not a mandate to absorb numerics tomorrow.

The action, restated

1. AGENTS.md §"What this is" / README — lead with general decidability substrate; geometry is instance #1, time #2; partiality is first-class and graded-capable (the uncertainty lattice).
2. Retire the geometry-era slogan as the membership rule wherever cited; point to this note. Preserve its spirit (anti-accretion) via the relocated bright line.
3. The depth-B graded-Resolvability RFC (markovlib-bridge.md §"the one genuine fungeom-core RFC") is the enabling lattice change; this note is its motivation, generalized past markovlib — graded resolution is for tolerance geometry first, inference second.