docs: field-reasoner wedge design + falsification dossier
Source-grounded design of the field<->symbol coherence-gate wedge, corrected by an 11-agent adversarial review into a falsifiable experiment with a sanctioned negative outcome (field stays a servant -> C3 two code-disjoint symbolic readings). Records the substrate ledger (the CL(4,1) field has one exact strength relevant to reasoning -- the conformal distance metric -- and no solver/incidence machinery), the corrected design, the three adversarial verdicts, the falsifiable experiment (ablation + per-class diversity), and the phased path.
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<!-- CANONICAL | docs/analysis/field-reasoner-wedge-design-and-falsification-2026-06-04.md | 2026-06-04 | research/design+falsification | the source-grounded design of the field↔symbol coherence-gate wedge, corrected by adversarial review into a falsifiable experiment with a sanctioned negative outcome | verified: all load-bearing claims confirmed against source (file:line cited); produced by an 11-agent adversarial research workflow + first-hand substrate read -->
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# The field-reasoner wedge — design, adversarial correction, and the falsification it must pass
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This is the load-bearing research record that turns the deferred keystone of the
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universal-structure plan
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([`universal-structure-and-field-symbol-coherence-gate-2026-06-04.md`](./universal-structure-and-field-symbol-coherence-gate-2026-06-04.md))
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into a concrete, *falsifiable* experiment. It is planning-and-research only: it
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assigns no ADR number, changes no serving path, and introduces no capability claim.
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**Provenance.** Compiled from (a) a first-hand read of the CL(4,1)/field/binding-graph/
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reliability-gate substrate, and (b) an 11-agent adversarial research workflow (4 substrate
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mappers + 2 external-literature researchers → 1 design synthesis → 3 adversarial verifiers →
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1 integrator). Every load-bearing claim below is confirmed against source with a `file:line`
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citation. The recommended design returned **"revise" on all three adversarial lenses**
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(decoration/independence, sub-enumeration, wrong=0); their required fixes are folded in.
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**The one-sentence finding.** The CL(4,1) field has exactly **one** exact native strength
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relevant to reasoning — the conformal distance metric — and **none** of the incidence,
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solver, or independent-reading machinery a metric reasoner needs; so the inevitable next
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move is not "make the field the reasoner," it is **the cheapest honest test of whether a
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metric *encoding* buys any genuine reading-independence at all**, wired so a negative result
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cleanly lands the field back as a servant without polluting `wrong = 0`.
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---
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## 1. The inevitable next level
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The architecture has exhausted the cheap moves and pinned the next one. Four constraints
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intersect to a single design shape:
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1. The only verified reasoning capability today is **propositional entailment** (ROBDD,
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`generate/proof_chain/entail.py`, holdout 500/500 `wrong=0` vs an independent
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truth-table oracle). A clean geometric decoder of *logic* was already shown to be
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`O(2^n)` truth-table enumeration in algebraic clothing — **the field cannot earn its
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reasoning role on combinatorial logic**. Redirect to a **metric-continuous** domain
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(distance, proportion) where CGA has real purchase.
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2. **Independence must live in the *reading*, not the *solving*.** Two solvers over the
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same extracted structure is fake independence; the `wrong=0` risk lives in
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comprehension. For quantitative/NL problems the *reading* (text → structure) is the
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unsound step — unlike logic, whose input is already a formula (independence there is
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cheap: two parsers).
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3. **`wrong = 0` is structural, not aspirational.** Whatever the field produces is admitted
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only through a *checkable* gate.
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4. The dormant reliability-gate lever `checker="t2_precision"` (`core/reliability_gate/gate.py:22,48-53`)
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can fire only when **two structurally-distinct derivations converge on one
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`commitment_key`** — and no genuinely-independent second derivation exists.
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Their intersection collapses the design space to one shape: **an independent metric
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*reading* of quantitative-relational problem text, packaged as the field-side
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`OperatorEvidence` for the existing `verify_tier2_agreement` seam, scored against a third,
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code-disjoint gold registered as a new `INDEPENDENT_GOLD_LANE`.** And the inevitable *first
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step* is the cheapest falsifiable version of it — because every heavier design **and the
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null** branch off its result.
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---
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## 2. What the substrate actually supports today (source-verified ledger)
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| Capability | Status | Evidence |
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|---|---|---|
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| Exact conformal distance metric `cga_inner(embed_point(a), embed_point(b)) = −½‖a−b‖²` | **REAL, exact** | `algebra/cga.py:30-37, 65-84`; docstring `:11-17` ("the ONLY distance metric") |
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| Null-preserving conformal transforms (versors act correctly on encoded points) | **REAL** (dual-path: null input → raw sandwich) | `algebra/versor.py:160-186, 149-157` |
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| Parameterized conformal motions (rotor slerp / transition rotor) | **REAL** but wired to word→word transitions, not point transforms | `algebra/rotor.py:75-144, 147-179` |
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| `embed_point` precision | **f32-HARDCODED** — unusable at GSM8K scale; needs an f64 reimplementation | `algebra/cga.py:72,76,82,83`; v=12345 → 12344.99996 in f32 |
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| Incidence/flat/round constructors (line/plane/sphere via wedge) | **ABSENT** | `algebra/cga.py` exports only `cga_inner / outer_product / is_null / null_project / embed_point` |
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| `meet / join / dual / cross_ratio / betweenness` | **ABSENT** (zero construction hits across `algebra/field/generate/core`) | substrate audit |
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| A constraint-propagation **solver** | **ABSENT** — `field/propagate.py` is a fixed-rotor walk ("no correction, no branching"); `field/operators.py::ConstraintCorrectionOperator` is **damped diffusion to a fixed prompt centroid**, NOT algebraic constraint satisfaction (the name is a trap) | `field/propagate.py:1-20,47`; `field/operators.py:153-281` |
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| `versor_condition` | **unit-versor closure residual**, NOT a constraint-satisfaction measure | `algebra/versor.py:189-204` |
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| Any runtime path embedding problem quantities as conformal points | **ABSENT** — `embed_point` used only in `evals/lab/` probes | substrate audit |
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| Reasoning readers import field/algebra | **NONE** (`generate/derivation`, `generate/proof_chain`, `core/reliability_gate` import zero `versor_apply/cga_inner/holonomy/field.propagate`) | grep, confirmed |
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| Binding-graph interlingua (`SemanticSymbolicBindingGraph`) | **EXISTS, frozen, INERT** — 6 sub-collections, refusal-first, acyclic, unit-aware, `SourceSpanLink` provenance, INV-26 neutral; derived FROM `MathProblemGraph` by `adapter.py` (so it carries no *independent* reading) | `generate/binding_graph/model.py:397-507`; `adapter.py` |
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| Agreement-gate seam | **EXISTS** — `verify_tier2_agreement` admits iff ≥2 distinct `structural_signature`s collapse to one `commitment_key` | `core/reasoning/evidence.py:169-205` |
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| Independence firewall on the gate's two readers | **DOES NOT EXIST** — the seam checks only `len(set(signatures)) < 2`, a free-form **label-string** test; INV-25 governs only the *oracle*, and its import check is **single-level / non-transitive** | `core/reasoning/evidence.py:180-187`; `tests/test_architectural_invariants.py:1115-1136` |
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| `t2_precision` lever | **WIRED, dead** — nothing populates `t2_verified`; no serving call site passes `checker="t2_precision"`; no serving consumer of proposals | `core/reliability_gate/gate.py:22,48-53`; `propose.py:66-103`; `core/learning_arena/engine.py:80-99` |
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| Independent-gold template | **EXISTS, proven** — `deductive_logic` + `dimensional` lanes (oracle shares no SUT code; gold frozen at generation; runner counts correct/wrong/refused) | `evals/deductive_logic/{oracle,generate,runner}.py`; `evals/dimensional/{oracle,runner}.py`; `INDEPENDENT_GOLD_LANES` at `tests/test_architectural_invariants.py:1088-1106` |
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**Ledger in one line:** one real native strength (the conformal distance metric); everything
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else load-bearing must be built, and the precision substrate must move to f64.
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---
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## 3. The recommended wedge — corrected form
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**Design C1 — the number-line incidence reader, sealed to forward-substitutable relations,
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packaged as the field-side evidence for the existing t2 seam.** It is the smallest design
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that exercises the one real substrate strength. The adversarial review did not kill it; it
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corrected three things that, left unfixed, would have made it decoration or breached
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`wrong=0`.
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### 3.1 The independent reading — and the honest size of the claim
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A **new allowlisted bridge** `generate/binding_graph/field_adapter.py` maps **raw problem
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text** to conformal points. Its number detection, clause segmentation, unit assignment, and
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relational-cue classification are an **independent reimplementation keyed off relational
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phrase schemas** ("more than", "times as many", "in total", "ratio of"), importing **none**
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of `generate.derivation`, `generate.math_candidate_parser`, `generate.math_roundtrip`,
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`WORD_NUMBERS`, or `state/bind`.
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**Honest downgrade (required by all three lenses):** this is a *second hand-written reader*,
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**not a categorically different comprehension mechanism.** There is no existing
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quantity-preserving text→geometry map. By Knight–Leveson, two architecturally-disjoint
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readers of the same text are exactly the regime where **correlated comprehension failure
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persists** (~half of N-version faults correlate through the shared specification).
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Therefore:
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- The ADR/code states independence is **a hypothesis the experiment is designed to refute**,
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never a property held by construction. **Do not cite McGilchrist as evidence** — design
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metaphor only; the only falsifiable form is the checkable gate.
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- The independence firewall is made real and **transitive**: a new architectural invariant
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follows the import graph transitively and applies to **both** readers feeding the t2 gate
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(today's `_module_imports` is single-level and would miss a helper that re-pulls
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`math_roundtrip`). The existing `len(set(signatures)) < 2` check is a label test and
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proves nothing.
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The geometric encoding does add **one genuine marginal check the symbolic regex lacks**: for
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an *over-determined* collinear configuration the read points must form a consistent set with
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non-contradictory distances, so a relation the regex silently accepts can produce an
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*incoherent* configuration the field refuses. That signal is **real but narrow** — vacuous
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on a single unary relation, and it does **not** catch a *coherent* misreading. The margin is
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measured, not asserted.
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### 3.2 The sub-enumeration argument (sealed)
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For the **forward-additive / part-whole** core the encoding is genuinely sub-enumeration and
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not the logic wedge's `O(2^n)` trap:
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- "A is k more than B" → translation versor `T_k`, `A = versor_apply(T_k, B)`.
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- part-whole → collinear metric sum.
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- a chain of unary relations (`A = B+3; C = 2A; …`) is **one-shot closed-form forward
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composition** of versors — `O(1)` per constraint, no Boolean assignment lattice. Values
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live in a continuous metric space; that is the categorical difference from logic.
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Required fixes that keep this true:
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1. **Seal the micro-domain to forward-substitutable (triangular) relations.** The property
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holds only where each relation pins one new unknown from already-resolved points. A
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**coupled/over-determined** system is meet-of-hyperplanes = Gaussian elimination =
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`O(n³)` and needs `meet/join/dual` primitives that **do not exist**. The generator and
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gold exclude coupled systems.
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2. **Drop the "propagate to `versor_condition<1e-6`" framing.** A forward chain has no
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iterative solve and no residual; `versor_condition` measures the *operator's* closure,
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not constraint satisfaction. Conflating them invites a **forbidden** hot-path normalizer
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(CLAUDE.md forbids drift repair in `field/propagate.py`).
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3. **The win is a sub-*enumeration encoding*, never a *solving* speedup.** Never pitch the
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field as faster than Gaussian elimination.
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### 3.3 Correct answer read-back (the wrong=0 bug the verdicts caught)
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The naive read-back — "answer = signed distance from origin via `cga_inner`" — is
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**mathematically wrong** for any weight-changing operator. A dilation (the multiplicative
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case) yields a **weighted** null vector: dilating x=2 by k=4 gives e1=2.0, n_o-weight 0.25,
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i.e. `0.25·embed_point([8,0,0])`; naive distance-from-origin returns 4.0 for true coordinate
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8.0. **Fix:** read by **projective dehomogenization** — `answer = e1_coefficient /
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n_o_weight` — verified exact (8.0). This is the read-back for **all** operators, since
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composed motors change weight.
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Two further metric hazards the *read side* must own (not the geometry):
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- **`cga_inner` is sign-/orientation-blind** (`−d²/2` cannot distinguish A=2B from A=−2B, nor
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enforce betweenness order). Orientation is a separate read-side decision the geometry does
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not verify. Either add an explicit signed-coefficient orientation invariant **or refuse
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all sign-ambiguous configurations.**
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- **"k times" via dilation-about-origin / cross-ratio needs a non-degenerate 4-point frame**;
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collinear configs are ill-conditioned. The additive/part-whole core is the load-bearing
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claim; the multiplicative case is **fenced** as the conditioning risk.
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### 3.4 The wrong=0 admission contract (honest)
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The "three loud-failing layers" was inflated; two are decoration:
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- **"Geometric coherence"** on one number line is collinearity-by-construction + non-
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contradictory distances — it catches *contradictory* readings, not a *single coherent
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wrong* relation.
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- **"Exact integer round-trip"** is decoration as a *comprehension* guard (confirmed:
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`embed_point([103]), [300], [7]` each round-trip residual 0.0). A coherent misreading (300
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vs 103) passes trivially. Per CLAUDE.md's schema-defined-proof-obligation rule, demote it
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to a **float-sanity guard**, not a `wrong=0` layer.
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**The real, load-bearing protection is exactly two checks:**
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1. **field⟂symbol agreement** on the **exact** `commitment_key` = (integer value +
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`UnitVector` dimension). Float-tolerant equality is **never** the admit criterion — that
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is precisely what sank `resolve_pooled` (right/wrong float readings indistinguishable).
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2. **Scoring vs an independent gold** — a third oracle code-disjoint from **both** readers,
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importing no field/algebra so it cannot agree with the field by construction.
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Plus **f64 + a pinned magnitude ceiling**: an f64 reimplementation of `X = x + n_o +
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½|x|²·n_inf`, refusing any quantity above an *empirically pinned* bound (GSM8K reaches
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thousands; ½|x|² ≈ 1e7; f64 recovers integers exactly to ~1e7, but the ceiling is pinned by
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measurement, not assumed). Above the ceiling → refuse.
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### 3.5 The asymmetric refinement (the right second step, not the first)
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The literature's strongest guarantee is **producer/checker asymmetry** (certifying
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algorithms / translation validation): an untrusted producer emits a *witness* a simpler,
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code-disjoint *checker* validates without redoing the work. Symmetric "two decodings agree"
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is weaker (exposed to correlated error). CORE already has an asymmetric checker for the
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dimensional sub-check — `generate/binding_graph/units.py`'s `unit_proof` is free and
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code-disjoint. **Use the asymmetric check where one exists** (a C2-style certifying-witness
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for the dimensional component); reserve symmetric agreement only where no independent checker
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is constructible. This is the correct step *after* the wedge proves the field lands a
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coherent configuration at all — **not** premature surface area to build first.
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### 3.6 If the experiment fails — the sanctioned alternative (C3)
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The field stays a servant in its current cognition-turn role, and quantitative-relational
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capability is carried by a **second fully-symbolic, code-disjoint reading** (a
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relational-schema parser) agreeing with the existing candidate-graph reader at exact-integer
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`commitment_key`s. C3 sidesteps every float/precision/orientation hazard and is the cleanest
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`wrong=0` story. Its honest weakness: two symbolic readers are the closest analogue to
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Knight–Leveson's refuted-independence case. **C3 is the correct landing iff the wedge shows
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the field shares the symbolic reader's blind spots** — a *success*, because it answers the
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field-as-reasoner question with evidence instead of deferring it.
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---
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## 4. The falsifiable experiment (cheapest decisive test)
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**Build only:** the f64 number-line reader + four constraint versors (additive/part-whole
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load-bearing; multiplicative/ratio fenced) + projective-dehomogenization read-back + the
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ablation/diversity instrument. **Do not** build the C2 certifying checker yet.
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**Dataset:** ~150 seeded two/three-quantity problems ("k more than", "k times", part-whole),
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gold from a **third hand-authored arithmetic oracle** sharing no code with either reader,
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registered in `INDEPENDENT_GOLD_LANES`. Split into generation and a **held-out** set; measure
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on held-out only (anti-overfit firewall — never tune to the 150 templates).
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**Three pass/fail measurements, wired as the gate (not diagnostics):**
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1. **Field-alone correctness** — does the field's projective-dehomogenized integer answer
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equal the independent gold on committed cases, **`wrong=0`** on held-out?
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2. **Ablation** — run the `wrong=0` gate **with the field vote removed** vs **with it**. The
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admitted set **must CHANGE** (the field must refuse ≥1 comprehension error the symbol path
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alone admits). Unchanged ⇒ **the field is decoration → land C3.**
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3. **Diversity, reported PER relation class** — double-fault rate / Q-statistic /
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coincident-failure diversity must clear a pinned threshold. **Aggregate is forbidden** (it
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could net out signal while the field is decoration on the cases that matter). Field credit
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only on classes where it demonstrably refuses an error the symbol path admits.
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**A negative result is sanctioned and expected-possible.** If (1) breaches `wrong=0` at
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scale, or (2) shows an unchanged admitted set, or (3) shows high coincident failure (Q≈1),
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the field is a servant here → land C3, and that result *confirms* the thesis that genuine
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reading-diversity needs a categorically different mechanism — justifying a later deeper
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research bet rather than this cheap wedge. No unfalsifiable "the field knows" claim is ever
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permitted.
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---
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## 5. Phased path
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> Each phase: **entry gate · exit gate · independent gold · `wrong=0` guard.** No phase
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> begins until the prior phase's exit gate is green. A lookback review runs at each boundary.
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### Phase 0 — foundation (build regardless of the wedge result)
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- **Build:** (a) f64 `embed_point` reimplementation + magnitude-ceiling refusal; (b) the
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**transitive reader-disjointness invariant** applied to *both* readers feeding the t2 gate;
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(c) replace the `verify_tier2_agreement` label-string distinctness check with a mechanistic
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one (distinct signatures must be provably unable to come from the same decoding pathway);
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(d) the independent-gold lane scaffolding (third arithmetic oracle, `INDEPENDENT_GOLD_LANES`
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registration).
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- **Exit:** all four shipped, full-suite green, INV-25/INV-26 still green; no serving path
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touched. These are net hardening even if the field never reasons.
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### Phase W — field-alone metric reader + decoration instrument
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- **Entry:** Phase 0 green; micro-domain sealed to forward-substitutable relations;
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projective-dehomogenization read-back implemented.
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- **Exit:** the three §4 measurements pass on held-out (field-alone `wrong=0`; ablation
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changes the admitted set; per-class diversity above threshold) **OR** they fail → land C3
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and stop.
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- **Independent gold:** third hand-authored arithmetic oracle, code-disjoint from both
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readers, imports no field/algebra.
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- **`wrong=0` guard:** exact integer + `UnitVector` `commitment_key` (no float tolerance);
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f64 + refuse-above-ceiling; refuse sign-ambiguous configs.
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### Phase 3 — activate `t2_precision` (the dormant lever)
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- **Entry:** Phase W exit green; field and symbol each emit `OperatorEvidence` with the same
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exact `commitment_key` and **genuinely distinct** signatures (proven by the non-decoration
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test, not by label).
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- **Exit:** in sealed practice, `ClassTally.t2_precision` rises past `N_MIN`+ to the chosen θ
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(PROPOSE=0.85 staging before SERVE=0.99); `propose_from_ledger(..., checker="t2_precision")`
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emits a proposal; **and** a concrete serving consumer of `checker="t2_precision"` exists and
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**refuses on disagreement** (close the ratify-vs-consume gap with an eval delta, not an
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artifact append).
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- **Independent gold:** the GoldTether earning t2_precision is independent of **both** readers
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(INV-25 generalized to two SUTs). Re-using the symbolic answer as gold collapses the
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firewall.
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- **`wrong=0` guard:** practice scoring is gold-gated, so agree-on-wrong scores
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`t2_verified=1, t2_agrees_gold=0` and **lowers** the license. The residual hazard lives
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in the unbuilt serving consumer: pin its contract now — admit iff (agreement AND class
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licensed); disagreement or field-refusal → deterministic refuse; no float tolerance
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anywhere. The Wilson floor at θ=0.99 demands ~657 perfect committed agreements/class — the
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coverage wall is the binding constraint.
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### Phase 4 — GSM8K on the structure (held-out / sealed)
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- **Entry:** Phase 3 exit green; the field reader generalizes from seeded templates to **real
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GSM8K** forward-substitutable relational cases (validate on `train_sample`/`holdout_dev`,
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never the 150 templates).
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- **Exit:** on a **sealed** GSM8K split, the agreement gate admits with **`wrong=0`**;
|
||||
ablation still shows the field changing the admitted set on real cases.
|
||||
- **Independent gold:** GSM8K reference answers; the gate's gold kept independent of both
|
||||
readers.
|
||||
- **`wrong=0` guard:** the serving-frozen lane (`scripts/verify_lane_shas.py`, `CLAIMS.md`) —
|
||||
no bridge re-enables a reader without a sealed/independent `wrong=0` run. The standing
|
||||
warning is `resolve_pooled` (2 right / 87 wrong): never admit a reader that cannot
|
||||
distinguish its right readings from its wrong ones.
|
||||
|
||||
### Phase 5 — cross-domain anti-overfit panel
|
||||
- **Entry:** Phase 4 green; the field reader is a **third** golded domain alongside
|
||||
`deductive_logic` and `dimensional`.
|
||||
- **Exit:** registered in `INDEPENDENT_GOLD_LANES` with a code-disjoint oracle
|
||||
(forbidden-prefix entries for **both** readers); INV-25 sub-properties hold; INV-26 green
|
||||
(field→graph translation lives in the bridge, never in `model.py/units.py/admissibility.py`).
|
||||
- **`wrong=0` guard:** full panel `wrong=0` across all three domains; a deliberately-unsound
|
||||
engine must disagree with each oracle (non-vacuity).
|
||||
|
||||
### And beyond
|
||||
The deferred keystone remains deferred: a **categorically different comprehension mechanism**
|
||||
(Gentner-style structure-mapping over a path that does not pass through any lexeme extractor)
|
||||
that would make field⟂symbol independence *real by construction* rather than measured. The
|
||||
wedge does **not** deliver that; it tests whether a metric *encoding* buys any independence on
|
||||
the one real substrate strength. If Phase W lands C3, the "beyond" is a dedicated research
|
||||
track on the comprehension mechanism — out of the near-term sequence, never licensed to
|
||||
displace it.
|
||||
|
||||
---
|
||||
|
||||
## 6. Risks, honest tensions, and STOP conditions
|
||||
|
||||
**Tensions (named, not waved):**
|
||||
- **Independence is architectural-only, not mechanistic.** A second hand-written parser is
|
||||
precisely where correlated comprehension failure survives (Knight–Leveson). The claim is a
|
||||
hypothesis the experiment refutes, not a property; the current firewall (a label-string
|
||||
distinctness test) does not enforce it.
|
||||
- **Symmetric agreement is weaker than producer/checker asymmetry.** Use the asymmetric
|
||||
(certifying-witness) check where one exists (dimensional `unit_proof`); reserve symmetric
|
||||
agreement only where no checker is constructible.
|
||||
- **The metric is sign-/orientation-blind and the substrate has no incidence primitives.**
|
||||
"Incidence" is faked via distance arithmetic; orientation is a read-side decision the
|
||||
geometry does not verify — thinning the independent signal exactly on multiplicative/ratio
|
||||
cases.
|
||||
|
||||
**STOP (and land C3 or the servant null):**
|
||||
1. Ablation shows an **unchanged admitted set** — the field is decoration.
|
||||
2. Per-class diversity **collapses** (double-fault high / Q≈1) — fake independence.
|
||||
3. **f64 + ceiling cannot preserve exact distance at GSM8K scale** — restrict to a band so
|
||||
narrow it is not the comprehension layer, or stop.
|
||||
4. **The seal cannot hold** — if real cases force coupled/over-determined systems, the
|
||||
sub-enumeration claim degrades to `O(n³)` on primitives that do not exist; do **not**
|
||||
fabricate `meet/join/dual`.
|
||||
5. **A float coherence threshold gets reconciled with the exact admit gate by tolerance** —
|
||||
the `resolve_pooled` failure mode; stop immediately, this breaches `wrong=0`.
|
||||
6. **Any "the field knows" claim, or a McGilchrist-as-evidence citation, appears in code/ADR
|
||||
without the checkable gate behind it** — that imports an unfalsifiable claim; revert.
|
||||
|
||||
**Bottom line, without overclaim:** the field has one real, exact strength (the conformal
|
||||
distance metric) and none of the reasoning machinery around it. The wedge is the cheapest
|
||||
honest test of whether a metric *encoding* buys genuine reading-independence on
|
||||
quantitative-relational structure. If it does — `wrong=0`, ablation-positive, diverse — it
|
||||
activates the dormant `t2_precision` lever as the genuine second derivation. If it does not,
|
||||
**the field stays a servant, a second code-disjoint symbolic reading carries the next level,
|
||||
and that negative result is the success.**
|
||||
Loading…
Reference in a new issue