core/docs/analysis/universal-structure-and-field-symbol-coherence-gate-2026-06-04.md

Universal structure + the field↔symbol coherence gate

A plan for the most robust, clean, logical, and masterful path to breaking reading and inputs of any modality into a logical structure — so the engine can comprehend, articulate, and solve a problem in any field for which it has adequate packs. GSM8K (math logic) is the first proving ground; the same spine must lift GSM8K and be confirmed by independent problem-solving in other fields.

This document is planning-only. It assigns no ADR number, changes no serving path, and introduces no capability claim. It records the architecture so later, gated PRs can build it without re-deriving the rationale.

0. End goal

input (any modality)
  -> comprehended into ONE logical structure
  -> reasoned over to produce a solution AND an articulated "understanding"
       (what the problem is, the approach)
  -> verified against INDEPENDENT gold
  -> refuses when grounding is incomplete

Success = held-out GSM8K rises drastically with wrong == 0, and the identical machinery clears an independent gold in ≥2 other fields.

1. The honest grounding (what exists today)

The bottleneck is comprehension→structure, not arithmetic. The GSM8K composer is epistemically unsafe on real data (resolve_pooled = 2 right / 87 wrong on holdout); bespoke per-shape readers do not compose. Capability is real only where the conclusion is checkable against independent gold — deductive propositional entailment (ROBDD) vs. an independent truth-table oracle: dev 200/200, holdout 500/500, wrong = 0.

There are several disjoint problem-structure representations:

• generate/math_problem_graph.py::MathProblemGraph — the GSM8K candidate-graph solver IR.
• generate/derivation/model.py::GroundedDerivation — the (unsafe-as-serving) derivation reader IR.
• generate/binding_graph/SemanticSymbolicBindingGraph (ADR-0132/0133/0134/0135) — a field-agnostic, unit-aware, provenance-carrying, acyclic, refusal-first semantic-symbolic DAG. It already unifies two domains: math (bind_math_problem_graph) and proofs (generate/proof_chain/builder.py, ADR-0204 — "the binding graph's first consumer").
• generate/proof_chain + generate/logic_canonical.py (ROBDD) — the deductive decision IR.
• generate/graph_planner.py::PropositionGraph -> ArticulationTarget — the articulation IR (for saying the answer), separate from reasoning.

The reasoning readers have zero field/algebra dependency. generate/derivation, generate/proof_chain, math_solver, math_candidate_graph import no versor_apply / cga_inner / holonomy / field.propagate. The CL(4,1) field engine participates in the cognition turn loop (capture field state, ratify intent, recall, identity) — not in problem reasoning. Any claim of "field-as-reasoner" is therefore currently unverified.

The dormant lever. The reliability gate (ADR-0175) already exposes checker="t2_precision" ("widening past gold"), but it is inert: no genuinely structurally-distinct second derivation exists, and nothing consumes it.

2. The synthesis — field comprehends, structure bridges, symbol verifies

The correct architecture is not "field OR symbol." It is a precise division of labor with a checkable handshake:

• Field/manifold engine = comprehension & constraint-resolution master (right-mode). A problem's entities/relations embed as geometric objects in CGA; solving = propagating constraints via versor_apply until the configuration is coherent (versor_condition < 1e-6). The converged field is the holistic understanding — apprehended as a whole, context-first, in CORE's native operations.
• Binding-graph DAG = interlingua / corpus callosum. The typed, unit-checked, provenance-carrying form the field configuration is read into and out of: the inspectable, checkable commitment where holistic apprehension hands off to analysis and analysis hands corrections back. Not master, not mere servant.
• Symbolic servants (parser, ROBDD, solver, oracle) = analyze & dispose. They verify the structure against independent gold, refute incoherence, articulate. They never originate truth.

2.1 Why this is correct, not a compromise

The field-derivation and the symbolic-derivation are two structurally-distinct canonical decodings of the same problem. Their agreement is the wrong = 0 gate.

1. It operationalizes the deepest axiom — "Truth is coherent." Two independent canonical decodings (geometric field + algebraic ROBDD) agreeing is coherence across vantage points. Agreement → admit; disagreement → refuse. This is exactly McGilchrist's "right apprehends, left analyzes, right re-integrates."
2. It is the genuine second derivation that is currently missing. The Tier-2 spine and t2_precision lever are inert because solver-vs-verifier shares structure (decoration) and the two GSM8K readers never co-fire. Field ⟂ symbol is that second derivation. Activating the field as a reasoner is what unblocks the dormant capability lever — mechanism, not metaphor.
3. It makes the field earn its role checkably (conforms to wrong = 0 / no decoration). The field is load-bearing only where its answer agrees with an independent symbolic/gold derivation. It can never serve an unagreed answer — disagreement refuses. This is INV-25 (Phase 0) generalized.

2.2 Conformance to the governing axes

Axis	Conformance
Philosophical ("decoding, not generating")	Field & symbol are two canonical decodings of a reality that already is; agreement = it tracks the canonical form.
Axiomatic	Reasoning is versor_apply + cga_inner + versor_condition (the actual primitives); exact recall stays exact.
Ethical / wrong = 0	The field never serves an unagreed answer; refusal is the correction surface.
Mechanical sympathy	Field does vectorized geometric propagation (native strength, Rust-parity-ready); symbol does discrete checks.
Logical per "intelligence"	Holistic grasp + analytic check + refuse-on-incoherence is what integrated understanding is, with the non-negotiable twist that both modes must cohere or it declines.

3. The earning path (the field must prove it reasons)

Today the readers have zero field dependency, so field-as-reasoner is a research arc that earns its role on the safest checkable micro-domain first. A negative result is an acceptable, honest outcome: where the field cannot earn agreement, it stays a servant and the symbolic path carries that domain, refusing where it cannot verify. No unfalsifiable "the field knows" claim ever enters the codebase.

The wedge (Phase 1.5) — FINDING (2026-06-04): logic is the wrong first domain. The original wedge asked whether the field could decide propositional entailment geometrically and agree with the ROBDD oracle. A bounded experiment settled it: the cleanest geometric/algebraic encoding (the commuting-idempotent function-algebra, each formula → its function on the 2^n minterms) agrees 716/716 with the oracle but is O(2^n) — it is truth-table enumeration in algebraic clothing, not a genuine sub-enumeration reduction the way the ROBDD is. So a geometric decoder for logic would either re-encode enumeration (adding no independence the oracle does not already provide) or be decoration. Propositional logic is combinatorial (all-assignments), not geometric/metric — the field engine's native strength (distance, incidence, proportion, betweenness) has no purchase there.

Corrected wedge — quantitative-relational structure. The field must first earn its reasoning role where the structure is genuinely metric: quantitative-relational problems (A is twice B, A is 3 more than B, part-whole, ratios) — exactly where GSM8K comprehension lives. There, the field derivation (solve the linear/metric relation system by propagation) and the symbolic derivation (step-by-step arithmetic) are two genuinely distinct decodings, and their agreement on the dataset answer is a real second derivation. Logic keeps its independent gold (ROBDD ⟂ truth-table oracle) as a symbolic second derivation; the field earns its role in the metric domains. This is the plan working as designed: the field earned nothing it did not deserve, and we learned the right domain cheaply.

4. Phased plan

• Phase 0 — independent-gold discipline. ✅ SHIPPED. INV-25 (tests/test_architectural_invariants.py): no capability claim without an independent gold sharing no code with the SUT; deductive lane SHA-pinned (deductive_logic_v1). The foundation every later phase rides on.
• Phase 1 — canonize the universal structure. ✅ SHIPPED. SemanticSymbolicBindingGraph is the documented problem-structure interlingua; INV-26 enforces its neutrality (the interlingua imports no field/eval/runtime, and its core imports no domain reader — only allowlisted bridges may). Deferred: resolving the closed-vocab placeholder (semantic_role="unknown") until a load-bearing consumer defines it; the stronger "servant may not bypass consistency checks" rule (the checkable neutrality half shipped; the rest stays doctrine until a checkable form exists).
• Phase 1.5 — the keystone wedge. ✅ RUN (logic ruled out; wedge redirected). The propositional-logic wedge was settled by experiment (§3): a clean geometric encoding is enumeration-class, so logic cannot prove field-as-reasoner. The wedge is redirected to quantitative-relational structure, where the field (metric/proportion propagation) and the symbol (step arithmetic) are genuinely distinct decodings. The dedicated quantitative wedge is the next field-reasoner experiment.
• Phase 2 — comprehension compiler → binding graph. ✅ FIRST SLICE SHIPPED: the finite-entity grounding compiler (evals/deductive_logic/grounding.py) lowers a typed finite-entity problem (entities + unary predicates + single-var universal rules) into the propositional regime, refusal-first, gated by engine == oracle == gold — the first reader proving a different problem shape compiles into the same checkable substrate. The diversity panel starts here (see cross-cutting): finite-entity is the second golded domain, so the compiler is validated against ≥2 structurally-distinct domains from its first commit and cannot overfit to one shape. Remaining Phase 2: target the binding-graph interlingua directly and add field ⟂ symbol agreement as the admission gate once the quantitative wedge lands.
• Phase 3 — activate t2_precision. The field⟂symbol agreement is the t2 signal; wire the propose-loop consuming checker="t2_precision". No invented second derivation needed — it is real now.
• Phase 4 — GSM8K on the universal structure (the drastic-lift phase), gated by field⟂symbol agreement + independent gold + held-out/sealed validation (never train_sample — the overfit firewall). Safe by construction.
• Phase 5 — cross-domain confirmation. Systems/software arena (execution = a third independent decoding; ADR-0199's first non-math arena), then a second non-executable field with a proof-checker gold. The generalization proof.
• Phase 6 — multimodal grounding. Connect the sensorium modality compilers (sensorium/compiler CompilerLike/CompilationUnitLike) → binding graph. A modality becomes "just another reader" grounding into the universal structure; stays afferent/gated per ADR-0198.

Cross-cutting (every phase) — diversity panel, checkable budgets, adjoints, phenomenology

• Structurally-diverse checkable panel (the anti-overfit instrument) — woven in from Phase 2, not deferred to Phase 5. Multi-domain diversity is not a morale tool bolted onto the plan; it is the experiment that confirms or refutes the central thesis (that different subjects compile into the same universal structure and are handled by the same solve/verify machinery). The defining failure of this project was a single gameable ruler (the 50-case train_sample) hiding unsoundness for weeks; the antidote is a panel of structurally distinct domains, each with a genuine independent gold (the INV-25 bar — a verifier sharing no code with the engine). Adding a domain without real independent gold is worse than useless: it is another gameable ruler. The cheap, strong starting panel (logic / grounding / dimensional / execution / constraint, all with real gold):

  • deductive logic (ROBDD vs. truth-table oracle) — shipped.
  • finite-entity grounding (ROBDD/oracle gold; Phase 2).
  • units / dimensional analysis — nearly free: the binding graph already carries dimensional algebra, so "does the answer's dimension check out" is an independent structural gold.
  • systems/software (execution = gold; Phase 5) — the strongest verifier.
  • a small constraint/scheduling domain (brute-force/SMT gold).

  The discipline (adopt now): a capability change must move ≥2 structurally- distinct domains, or it is suspected overfitting until proven otherwise. Make the panel a first-class CI signal alongside GSM8K so progress is never read from one flat number. Honesty note: a domain with no adequate pack yet simply refuses everything — that is a coverage signal (a breadth map), not a capability signal. Capability credit counts only where there is a real pack and a real gold and committed (non-refused) agreement.

• Distortion budgets as first-class CI gates, but only checkable ones: grounding completeness %, unit-proof coverage, acyclicity, and structural perturbation stability (paraphrase/reorder the input → the binding graph and answer are invariant, or it refuses). Regressions gate the build as hard as accuracy. This is the antidote to metric-capture (the train_sample trap).

• Adjoint-per-operator. Every new transform ships with its verifier/correction pass + a test that it reduces a structural error, not just a string.

• Phenomenology harness. Sample full structured articulations (the "understanding": problem restatement + approach) for periodic trusted review; qualitative depth as an explicit but subordinate signal (never a new truth criterion — that would re-open left-mode capture under a different name).

5. Risks & honest tensions

• HIGH — Phase 4 wrong = 0. GSM8K serving breached it before. Mitigation: agreement gate + independent gold + held-out/sealed + pre-push verification (auto-merge pipeline).
• MEDIUM — field-reasoning may not pan out in a given domain. Expected and acceptable: it stays a servant there; the symbolic path carries it, refusing where unverifiable.
• MEDIUM — McGilchrist over-reach. "Field as master of reasoning" is realized only through the checkable agreement gate; never asserted as unfalsifiable fact.
• MEDIUM — IR convergence cost. Converge the disjoint reasoning IRs incrementally (adapters first, direct targeting later), never a big-bang rewrite.

6. What this supersedes

The earlier 554/555/556 sequence is reframed: 554 (Tier-2 verifier) was inert → Phase 3 done right (field⟂symbol is the genuine second derivation); 555 (target-slot composition) was unsafe via the composer → Phase 4 (comprehension on the universal structure, gated); 556 (systems/software arena) → Phase 5.