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+# Field-Reasoner Wedge Selection
+
+**Status:** research-control document
+**Date:** 2026-06-04
+**Scope:** identifies the first honest field-as-reasoner target after the deductive pivot and 3-domain anti-overfit panel.
+
+## 1. Context
+
+The universal-structure arc established three facts:
+
+1. Deductive propositional logic is a checkable substrate, not the best field wedge.
+2. A clean algebraic/geometric propositional decoder still behaved as `O(2^n)` enumeration, so logic is combinatorial terrain.
+3. Field independence must live in **reading/comprehension**, not merely in solving the same extracted structure.
+
+Therefore the next field-reasoner wedge must be a domain where geometric/metric structure is natural, where independent gold is available, and where success helps return to quantitative comprehension without reopening the unsafe GSM8K serving bridge.
+
+## 2. Candidate domains
+
+| Candidate | Benefit | Risk | Decision |
+|---|---|---|---|
+| Pure propositional logic | Exact gold already exists | Combinatorial; field route degenerates to enumeration | Reject as first wedge |
+| Finite-entity logic | Good grounding compiler | Still Boolean after lowering | Keep as substrate, not field wedge |
+| Dimensional reasoning | Already independently golded; unit algebra is metric/type-like | Too narrow if only product/quotient | Use as supporting axis |
+| Ratio/proportion | Metric-relational; direct bridge to GSM8K | Needs target binding and structure equivalence | **Choose** |
+| Part-whole residuals | Natural quantitative structure | Prior GSM8K traps if promoted early | Include after ratio |
+| Euclidean geometry | True geometric territory | Bigger encoding surface | Later wedge |
+| Systems/software execution | Excellent independent gold | Not geometric; sandboxing surface | Later fourth panel domain |
+
+## 3. Selected wedge
+
+The first honest field-as-reasoner wedge is:
+
+```text
+structured quantitative relations:
+  ratios
+  proportions
+  rates
+  part-whole residuals
+  dimensional consistency
+```
+
+This domain is selected because it is:
+
+- close enough to GSM8K to matter;
+- structured enough to avoid uncontrolled natural-language ambiguity;
+- naturally metric/geometric rather than purely Boolean;
+- independently checkable with a small oracle;
+- compatible with the binding-graph interlingua and dimensional lane.
+
+## 4. Minimal structured schema
+
+### Ratio/proportion
+
+```json
+{
+  "id": "qr-v1-0001",
+  "family": "proportion",
+  "given": [
+    {"name": "flour", "value": 2, "unit": "cup"},
+    {"name": "servings", "value": 8, "unit": "count"}
+  ],
+  "target": {"known": {"name": "servings", "value": 20, "unit": "count"}, "unknown": "flour"},
+  "gold": {"value": 5, "unit": "cup"}
+}
+```
+
+Canonical relation:
+
+```text
+flour / servings = 2 / 8
+x_flour / 20_servings = 2 / 8
+x_flour = 5
+```
+
+### Rate
+
+```json
+{
+  "id": "qr-v1-0002",
+  "family": "rate",
+  "given": [
+    {"name": "distance", "value": 60, "unit": "mile"},
+    {"name": "time", "value": 2, "unit": "hour"}
+  ],
+  "target": {"unknown": "speed"},
+  "gold": {"value": 30, "unit": "mile_per_hour"}
+}
+```
+
+### Part-whole residual
+
+```json
+{
+  "id": "qr-v1-0003",
+  "family": "part_whole",
+  "whole": {"name": "apples", "value": 30, "unit": "count"},
+  "parts": [{"name": "sold", "value": 12, "unit": "count"}],
+  "target": {"unknown": "remaining"},
+  "gold": {"value": 18, "unit": "count"}
+}
+```
+
+## 5. Independent oracle
+
+The quantitative-relations oracle must not import the symbolic compiler, field decoder, or binding-graph implementation.
+
+Oracle responsibilities:
+
+```text
+validate schema
+reject unsupported families
+compute exact rational result
+compute/canonicalize result unit
+return typed refusal for unsupported or malformed cases
+```
+
+Use exact rational arithmetic in the oracle. Floating tolerance should not be part of v1.
+
+Closed oracle outcomes:
+
+```text
+correct
+wrong
+refused_malformed
+refused_unsupported_family
+refused_dimension_conflict
+refused_divide_by_zero
+```
+
+## 6. Symbolic compiler
+
+The symbolic compiler is allowed to be direct and typed. It reads the structured JSON and emits canonical quantitative structure.
+
+It should not solve by side effect. It emits:
+
+```text
+entities
+quantities
+relation_edges
+target
+unit_constraints
+provenance_hash
+```
+
+The solver/oracle consumes that structure separately.
+
+## 7. Field/geometric compiler
+
+The field compiler must be independent of the symbolic compiler.
+
+Permitted:
+
+- import field/algebra primitives;
+- import neutral binding-graph dataclasses, once the structure boundary exists;
+- emit the same canonical structure type;
+- use dimensional unit primitives only through an allowlisted neutral boundary, if needed.
+
+Forbidden:
+
+- importing the symbolic quantitative compiler;
+- importing its parser/case lowering helpers;
+- consuming the symbolic compiler's output;
+- matching only final numeric answers.
+
+The first field encoding may be modest. Acceptable candidates:
+
+| Encoding | Why acceptable in v1 |
+|---|---|
+| dimensioned vector relation | Minimal metric structure |
+| affine scalar constraint | Good for rates and residuals |
+| proportional line relation | Natural geometric ratio representation |
+| CGA point/line meet | Stronger, later variant if v1 succeeds |
+
+## 8. Required tests
+
+Minimum test stack:
+
+1. oracle handles all v1 families with exact rational results;
+2. oracle refuses malformed/unsupported/divide-by-zero cases;
+3. symbolic compiler emits canonical structure without solving;
+4. field compiler emits canonical structure without importing symbolic compiler;
+5. structure equivalence catches same-answer/different-relation;
+6. `symbolic_structure == field_structure` on supported cases;
+7. `answer == oracle_gold` on supported cases;
+8. disagreement produces refusal, not fallback to one reader;
+9. replay hash stability;
+10. AST independence tests for oracle and field compiler.
+
+## 9. Initial fixture shape
+
+Start with 24-32 cases.
+
+| Family | Positive | Negative/refusal |
+|---|---:|---:|
+| proportion | 6 | 2 |
+| rate | 4 | 2 |
+| part-whole residual | 4 | 2 |
+| dimensional consistency | 4 | 2 |
+| mixed distractor/exempt | 2 | 2 |
+
+Do not author hundreds of cases. This is not a corpus-tuning lane. It is a mechanism proof lane.
+
+## 10. GSM8K re-entry rule
+
+GSM8K remains out of serving scope until this wedge proves itself.
+
+Allowed after the structured lane:
+
+```text
+GSM8K structure diagnostic only
+```
+
+Forbidden until held-out/sealed gates pass:
+
+```text
+serving bridge
+train_sample-only promotion
+regex positive branch
+field solver fallback on symbolic parse
+```
+
+## 11. Research outputs
+
+The wedge is successful only if it produces all of:
+
+1. a structured quantitative-relations lane with independent gold;
+2. a symbolic compiler that emits canonical structure;
+3. a field/geometric compiler that independently emits equivalent structure;
+4. an equivalence gate that refuses on structural disagreement;
+5. deterministic evidence traces;
+6. no serving changes.
+
+## 12. Next PR stack
+
+1. `core/reasoning/structure_equivalence.py` — canonical structure equivalence.
+2. `evals/quantitative_relations/oracle.py` — exact independent oracle.
+3. `evals/quantitative_relations/v1/cases.jsonl` — small mixed fixture.
+4. `generate/field_reasoning/quantitative_decoder.py` — field reader prototype.
+5. `evals/gsm8k_math/structure_diagnostic/` — later, non-serving.
+
+## 13. Stop conditions
+
+Stop if:
+
+- field structure construction cannot be made independent;
+- the field route is only a second solver;
+- ratios/proportions require broad NL parsing before structured schema works;
+- oracle and SUT disagree and the cause is not understood;
+- same-answer/different-structure slips through equivalence;
+- a serving path becomes tempting before sealed evidence exists.
+
+The correct outcome of a failed wedge is a falsification record, not a workaround.