Ternary Routing Column: Design Decisions (0.4.34)

Why the fractal router's ternary column is scoped the way it is. Companion to BALANCED_TERNARY_CONTRACT.md (which proves the domain this column routes into). Decisions ratified by the owner 2026-08-14.

Context

Since v0.4.0 the router's classifier has computed TERNARY_BIT (shadow denominator stripped of 3s), but no routing-table column consumed it: ternary-exact values fell through to Binary (where they are inexact) or to the symbolic rational fallback. The 0.4.33 contract suite plus the post-0.4.33 gap-closing (tiers 1–6 adversarial coverage, two sign-extension fixes, checked storage narrowing) made the domain formally defended; this column makes it reachable.

Target class: values exact in ternary but not in binary/decimal: denominator 3^k, k ≥ 1 (1/3, 2/3, 100/3…). Domain preference order becomes Binary > Decimal > Ternary > Symbolic, ranked by cost: integers keep routing Binary; only the 3-adic class moves. The win is replacing rational-pair arithmetic (two multiplies + add + gcd normalization per op on num/den pairs up to I512, not BigInt, which is feature-gated behind infinite-precision and absent from default builds) with single integer adds on scaled raws.

Decision 1: the column covers add/sub (+neg) only, not mul/div

This is a correctness argument, not caution.

For operands representable at scale 3^F (denominators dividing 3^F):

• add/sub: the result's denominator divides lcm(3^a, 3^b) = 3^max(a,b) ≤ 3^F. Routed addition is always exact. Same for subtraction and negation.
• mul: denominators multiply: 3^a · 3^b = 3^(a+b), which may exceed 3^F. When it does, ternary truncates toward zero (contract §3) while today's symbolic path is exact. Routing such a multiply would make routing change results: violating the router's fail-safe invariant: wrong routing costs performance, never correctness.
• The static table indexes 4-bit exactness classes; it cannot see denominator exponents, so it cannot separate safe multiplies (a+b ≤ F) from unsafe ones. A compile-time table has no correct mul column to offer.
• div produces non-3-adic quotients in general; same argument, stronger.

Mul routing can become viable later via (a) a dispatch-time shadow-exponent guard (route to ternary only when a+b ≤ F, checked against the actual shadows, falling back otherwise), or (b) an explicit owner re-scoping of the fail-safe invariant. Option (a) is the natural follow-up; neither blocks this release. This scoping also decouples the column from the pending mul/div rounding decision (toward-zero vs the theorem's tie-free nearest): that decision can now be taken on its own schedule.

Decision 2: coercion failure falls back silently to the previous route

Gap-closing proved (and pinned by test) that narrow profiles cannot hold Tier-2+ ternary raws in FASC storage: realtime caps at TQ10.10/i32, compact at TQ20.20/i64. Coercing a large 3-adic value into ternary on those profiles is a legitimate TierOverflow.

If the router propagated that error, routing would introduce failures where the pre-column path succeeded: the other way to violate the fail-safe invariant. So: on coercion failure, dispatch proceeds exactly as if the column did not exist (rational fallback).

Implementation note: this costs nothing structural. The routing seam (try_route_coerce) already returns Option; None is the rational fallback. A failed convert_to_ternary maps to None.

Scope honesty: what the column does NOT change

• Same-domain pairs bypass the router. 1/3 + 1/3 parses both operands Symbolic and takes the native symbolic path untouched. The column fires on cross-domain pairs (Ternary ⊕ Symbolic, Ternary ⊕ Binary-integer, Symbolic-3-adic ⊕ integer, …). Intercepting the symbolic same-domain fast path would tax every symbolic add with a classify; it stays deliberately out of scope (revisit only with a measured consumer case).
• Transcendentals on ternary operands keep routing to the binary engines (contract: ternary has no native transcendentals).
• Conversion exactness: convert_to_ternary computes num·3^F / den in exact integer arithmetic; for the routed class (den = 3^k, k ≤ F) the division is remainder-free: coercion is exact by construction. The same function truncates for non-3-adic dens, but the table never routes those to Ternary.

Found during implementation

• convert_to_ternary's tier-3 and fallback arms stored raws through the unchecked to_binary_storage: the same silent-wrap class fixed in 0.4.33's ternary_to_storage, still live here and reachable today via set_gmath_mode("…:ternary") on narrow profiles. Fixed in this release (routed through the checked conversion); pinned by test.
• Symbolic operands were invisible to the classifier: StackValue:: Symbolic carries no shadow (it IS exact), so classify() fell back to SYMBOLIC_ONLY and a symbolic 1/3 could never trigger the column. The classifier now reads the rational's own denominator: richer than any shadow. Deliberate side effect: symbolic operands with 2-adic/10-adic denominators can now coerce into Binary/Decimal wherever both operands are exact there; exactness is preserved in every such case by the intersection rule.
• A pre-existing, unrelated realtime defect surfaced while validating: integer literals beyond binary storage range (32768+ at Q16.16) fail PARSE with Overflow instead of falling back to Symbolic: bisected to ≤0.4.32, filed as ROADMAP 0.5.0 item 0. Not addressed here (out of the column's scope).

Measured cost (embedded, release, full evaluate pipeline)

0t2 + 1/3 (routed to ternary): 334 ns/eval; 0t2 + 1/7 (rational fallback): 355 ns/eval: ~1.06×. Honest reading: at single-expression scale the win is modest because literal parsing dominates both paths. The column's value is architectural: the 3-adic exactness class finally reaches the domain that represents it, results stay in fixed-point form instead of rational pairs, and the seam is in place for the guarded mul extension.

Validation

• Routing-table pins: Add/Sub of ternary∩symbolic classes → Ternary; Mul/Div of the same classes → Symbolic (unchanged); integer∩integer → Binary (unchanged precedence).
• End-to-end: cross-domain 3-adic add routes into the ternary domain and equals the plain-literal computation exactly (router-difference zero).
• Fallback: a 3-adic pair whose coercion overflows narrow storage still evaluates correctly on every profile (exercises the silent fallback on realtime/compact, the ternary route on wide profiles).
• Wrap-fix pins for the two convert_to_ternary arms.
• Measured before/after cost of the routed case in the CHANGELOG.