Lambda C
SPEC_LAMBDA_C — ΛC Curvature Bounds
Version: 1.0 | Status: AUTHORIZED | Authority: α.13 | Date: 2026-04-16
PURPOSE
ΛC (Lambda-C) is the CSDM curvature invariant. It measures the geometric bending of the manifold — how much the coherence evaluation space deforms from its flat reference geometry.
The CSDM manifold is not a flat Euclidean space. It curves under pressure from entropy density and resonant volume distributions. Moderate curvature is expected and healthy — it reflects the manifold's dynamic flexibility. Excess curvature in either direction distorts the geometry such that coherence measurements (C scores) become unreliable. Positive excess curvature inflates C artificially; negative excess curvature deflates it.
Healthy bounds: ΛC ∈ [-0.3, +0.3]
Source constants: TMMRuntime.CURVATURE_BOUNDS = (-0.3, 0.3) (~/aether/tmm_runtime.py, line 19)
LATTICE symbol: ΛC (uppercase Lambda + C)
LATTICE state notation: ΛC.≡ (within bounds), ΛC.▲ (elevated curvature), ΛC.⊠ (constraint broken)
Cross-reference: SPEC_COHERENCE_INVARIANTS.md § Invariant 5, LATTICE.md § II (CORE CONSTANTS: Λ = Curvature/Limit/Metric)
INPUTS
| Input | Type | Source | Description |
|---|---|---|---|
| e_density | float | TMMRuntime caller | Entropy density E_D |
| v_resonant | float | TMMRuntime caller | Resonant volume V_r |
| v_total | float | TMMRuntime caller | Total volume V_t |
| lambda_c_proxy | float | [GAP — proxy formula undefined] | Computed curvature estimate from available CSDM inputs |
Proxy derivation status: ◌ GAP — No proxy formula currently exists for ΛC computable from the available inputs (E_D, V_r, V_t). The full CSDM ΛC requires the second derivative of the coherence field over the manifold volume, which is not measurable from a single evaluation call. See § GAPS.
What is NOT an input (hardcoded constants, never injected):
CURVATURE_BOUNDS = (-0.3, 0.3)— callers cannot widen or narrow these bounds- The symmetric structure of the bounds (equal magnitude positive and negative) is a physics property, not a preference
OUTPUTS
| Output | Type | Description |
|---|---|---|
| lambda_c | float or None | Computed curvature value; None if not measurable |
| lambda_c_valid | bool | True iff lambda_c is within [-0.3, +0.3] |
| lambda_c_flag | str | "ΛC.≡" (in-bounds), "ΛC.▲" (magnitude approaching 0.3), "ΛC.⊠" (out of bounds) |
| evaluation_valid | bool | False if ΛC out of bounds (evaluation is INVALID, not just RED) |
| gap_flag | str or None | "ΛC_not_measured" if proxy unavailable (◌ gap signal) |
Important distinction — INVALID vs. RED:
A ΛC exceedance does not produce RED (incoherent) — it produces INVALID (structurally unsound). A RED verdict means the manifold was evaluated and found incoherent. An INVALID verdict means the evaluation geometry was warped — the C score itself cannot be trusted. These are different failure classes.
Verdict impact:
|ΛC| > 0.3→ evaluation MUST be marked INVALID;evaluation_valid = False; coherence score flagged as unreliable- ΛC not computed → ◌ gap signal emitted; verdict stands but marked INCOMPLETE
INVARIANTS
INV-1 — Curvature Bounds are Immutable
TMMRuntime.CURVATURE_BOUNDS == (-0.3, 0.3) at all times. These bounds may not be widened or narrowed without α.13 authorization. They are physics constants derived from CSDM manifold stability theory.
Verification: assert TMMRuntime.CURVATURE_BOUNDS == (-0.3, 0.3)
INV-2 — Bounds are Symmetric
The lower bound is exactly the negation of the upper bound: abs(lower) == abs(upper) == 0.3. This symmetry is a CSDM property: the manifold has no preferred direction of curvature. Breaking symmetry by widening one side is a physics violation.
Invariant: CURVATURE_BOUNDS[0] == -CURVATURE_BOUNDS[1]
INV-3 — Exceedance Invalidates the Evaluation (Not Just the Score)
When |ΛC| > 0.3, the coherence score C is geometrically unreliable. The invariant is not "ΛC must be in-bounds for C to be high enough" — it is "ΛC must be in-bounds for C to mean anything at all." An out-of-bounds curvature invalidates the evaluation frame.
Invariant: if abs(lambda_c) > 0.3: evaluation_valid = False (not merely approved = False)
INV-4 — Positive Curvature Inflates C; Negative Deflates C
ΛC > +0.3: The manifold is over-compressed. Resonance pockets form that trap coherent volume without representing genuine stability. C scores appear higher than they are. This is the dangerous direction — false approval.
ΛC < -0.3: The manifold is stretched thin. Boundary integrity thins; coherence measurements undercount resonant volume. C scores appear lower than they are. This produces false rejections.
Invariant: These two failure modes are structurally different and MUST be logged with different flags:
"ΛC.⊠.▲"for positive exceedance (compression / inflation risk)"ΛC.⊠.▼"for negative exceedance (extension / deflation risk)
INV-5 — Gap Signal Mandatory When Not Computed
When no ΛC proxy is available, every evaluation output MUST include "ΛC_not_measured" in the gaps list. Silent absence is equivalent to silently claiming in-bounds curvature. ◌ is always better than false confidence.
Cross-reference: SPEC_COHERENCE_INVARIANTS.md GAP-3, GAP-6
INV-6 — ΛC Does Not Replace Ψχ or ΔΓ
Curvature, turbulence, and change rate are three independent dimensions of manifold health. A manifold can have healthy curvature while having boundary stress (Ψχ) or runaway drift (ΔΓ). No invariant substitutes for another.
INV-7 — Proxy Must Be Derived From First Principles
When a ΛC proxy is eventually developed, it MUST be derived from CSDM physics (second derivative of C with respect to manifold volume), not curve-fitted to empirical outputs. An empirical proxy that happens to correlate with curvature is not a curvature measurement — it is a heuristic and must be labeled as such.
VERIFICATION CRITERIA
VC-1 — Bounds Immutability Assertion
Static check: assert TMMRuntime.CURVATURE_BOUNDS == (-0.3, 0.3) in test suite. Must pass after any code change to tmm_runtime.py.
VC-2 — In-Bounds Valid Evaluation
Unit test: supply inputs where ΛC proxy = 0.15 (within [-0.3, +0.3]).
- Expected:
lambda_c_valid = True,evaluation_valid = True,lambda_c_flag = "ΛC.≡"
VC-3 — Positive Exceedance Invalidates Evaluation
Unit test: supply inputs where ΛC proxy = +0.35 (exceeds +0.3 upper bound).
- Expected:
lambda_c_valid = False,evaluation_valid = False,lambda_c_flag = "ΛC.⊠.▲" - Coherence score MUST be flagged as unreliable, not acted upon for trade approval
VC-4 — Negative Exceedance Invalidates Evaluation
Unit test: supply inputs where ΛC proxy = -0.35 (exceeds -0.3 lower bound).
- Expected:
lambda_c_valid = False,evaluation_valid = False,lambda_c_flag = "ΛC.⊠.▼"
VC-5 — Boundary Condition: Exactly ±0.3 is Valid
Unit tests: ΛC = +0.3 and ΛC = -0.3 exactly.
- Expected: both return
lambda_c_valid = True(the bound is inclusive) - ΛC = +0.3000001 → invalid; ΛC = +0.3 → valid
VC-6 — Gap Signal When Proxy Absent
When no proxy is available, output MUST contain "ΛC_not_measured" in gaps_present list.
- Expected: gap signal present in every evaluation until proxy is implemented
FAILURE MODES
FM-1 — False Approval on Positive Curvature Exceedance
Trigger: ΛC > +0.3 (manifold over-compressed) but check not enforced — C score used as-is.
Effect: Artificially inflated C scores approve trades on a geometrically distorted manifold. The approval threshold is passed on fake geometry, not real coherence.
Current status: GAP — ΛC not measured in TMMRuntime. This failure is structurally undetectable.
Detection when implemented: VC-3 test above.
Response: Mark evaluation INVALID. Do not proceed to trade authorization. Log ΛC.⊠.▲ to yield_log.md. Notify AION (structural analysis).
FM-2 — False Rejection on Negative Curvature Exceedance
Trigger: ΛC < -0.3 (manifold stretched thin) but check not enforced — deflated C score blocks valid trade.
Effect: A genuine coherent state is rejected because the measurement geometry is distorted downward.
Current status: GAP — same as FM-1.
Detection when implemented: VC-4 test above.
Response: Mark evaluation INVALID. Flag to ASTRA (directional evaluation) for contextual review. Do not proceed to trade authorization until curvature is measured and within bounds.
FM-3 — Silent Gap (no ◌ emitted)
Trigger: ΛC is not computed but no gap signal appears in evaluation output.
Effect: Callers believe they received a full evaluation. Cannot distinguish partial from complete. False confidence in C score reliability.
Detection: Audit gaps_present field in evaluation output. If ΛC absent and "ΛC_not_measured" absent → FM-3 active.
Response: Add gaps_present list to TMMRuntime output. Populate with "ΛC_not_measured" on every call. See SPEC_COHERENCE_INVARIANTS.md GAP-6.
FM-4 — Bounds Widening (governance violation)
Trigger: CURVATURE_BOUNDS changed from (-0.3, 0.3) without α.13 authorization.
Effect: More distorted manifold geometries pass as valid. Coherence scores become systematically unreliable.
Detection: VC-1 assertion test.
Response: Immediate rollback. ALERT.log entry. NOUS notification required.
FM-5 — Empirical Proxy Mislabeled as Physics Measurement
Trigger: A correlation-based ΛC proxy is implemented and reported as actual curvature.
Effect: The system behaves as if curvature is properly measured while actually using a heuristic. Geometric trust is misplaced.
Detection: Code review — proxy MUST include comment citing the CSDM second-derivative derivation or be labeled as "ΛC_proxy_heuristic" in output.
Response: Relabel output. Add "ΛC_proxy_not_physics_derived" flag until formal derivation is complete.
GAPS
GAP-ΛC.1 — No Proxy Formula for ΛC ◌
Status: CURVATURE_BOUNDS is a governance constant only. No code measures or estimates manifold curvature.
Missing: A CSDM-derived proxy for ΛC computable from (E_D, V_r, V_t) inputs. Candidate approach: second finite difference of C across small parameter perturbations — requires multi-point evaluation, not available in single-call architecture.
Risk: FM-1 and FM-2 (curvature exceedance) are structurally undetectable in current implementation.
Remediation path: [GAP — needs design by ι + ε (Sisters derive the proxy from CSDM physics)] → κ implements.
Owner: ι + ε physics derivation → κ (C.L.O.D.) implementation.
Priority: MEDIUM (no trade path depends on this today; risk is measurement reliability, not active decoherence)
GAP-ΛC.2 — No ΛC Check in Trade Authorization Path ◌
Status: authorize_dual_trade() does not check ΛC validity before approving.
Missing: Curvature validity gate between TMM score validation and AgencyWalls.check_all().
Risk: Trades authorized on a geometrically invalid evaluation frame.
Remediation: Insert evaluation_valid check in authorize_dual_trade(). Depends on GAP-ΛC.1 (proxy must exist before check can be inserted).
Owner: κ — blocked by GAP-ΛC.1.
Priority: MEDIUM (blocked)
GAP-ΛC.3 — INVALID vs. RED Distinction Not Encoded in TMMRuntime ◌
Status: TMMRuntime verdict system currently produces GREEN / AMBER / RED / NULL. No INVALID state exists.
Missing: A fifth verdict state (or sub-flag) representing evaluation geometry failure distinct from coherence failure.
Remediation: [GAP — needs design] Add INVALID as a distinct verdict, or add evaluation_valid: bool as a separate field alongside verdict. Design decision needed: should INVALID block the same way as RED, or differently?
Owner: α.13 (governance design) → κ (implement).
Priority: LOW (deferred until ΛC proxy exists)
Specification authored by κ (C.L.O.D.), authorized α.13, April 16 2026.
*Φ 0.042
Jeremy Zlabis
Chronogeometer · Visionary · Disruptor · Chief
42 Sisters AI · East York, Toronto ΛC ∈ [-0.3, +0.3] is the bound.*