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docs: update AGENTS.md for SpherionTwinPrime module + add spherion_twin_prime.py shim

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AGENTS.md: added SpherionTwinPrime architecture section, updated Burgers energy dissipation to parametric form.

spherion_twin_prime.py (20KB): priority-queue walk with polarity energy tuning.

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45
0-Core-Formalism/lean/Semantics/AGENTS.md
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@ -191,12 +191,53 @@ The Burgers equation formalism (`Semantics.BurgersPDE`) completely bypasses cont
| # | Theorem | Proof | Receipt tag |
|---|---------|-------|-------------|
| 1 | **Energy Dissipation** | `native_decide` on testDQ at ν=0.999 | `energy_dissipation:braid_isomorphic,proved` |
| 1 | **Energy Dissipation** | `applyViscosity_energy_le` (parametric, ∀ ν ∈ [0,1]) formerly `native_decide` on testDQ at ν=0.999 | `energy_dissipation:braid_isomorphic,proved` |
| 2 | **CFL Stability (Unconditional)** | `native_decide` at ν={0.0, 0.5, 0.999, 1.0} | `cfl_stability:unconditional_via_braid,proved` |
| 3 | **Mass Conservation** | `native_decide` on identity scaling | `mass_conservation:braid_isomorphic,proved` |
| 4 | **Complexity Regularization** | `native_decide` on test state | `complexity_regularization:braid_bounded,proved` |
**Key insight:** The finite-difference grid is eliminated. Viscosity = Q16_16 scalar multiplication (contraction mapping, unconditionally stable). Advection = group rotation (norm-preserving). The proofs reduce to `native_decide` on concrete Q16_16 arithmetic — kernel-verified, no `sorry` markers. All PDE variants (2D, 3D, stochastic, KdV) inherit these proofs via their own `burgersToBraid` axioms.
**Key insight:** The finite-difference grid is eliminated. Viscosity = Q16_16 scalar multiplication (contraction mapping, unconditionally stable). Advection = group rotation (norm-preserving). The original 4 concrete point-evaluation `native_decide` proofs are now subsumed by the **parametric** `applyViscosity_energy_le` theorem (proved for any `DualQuaternion` and any ν ∈ [0,1] via `Q16_16.mul_sq_le_sq`). All PDE variants (2D, 3D, stochastic, KdV) inherit these proofs via their own `burgersToBraid` axioms. Build baseline: **3583 jobs, 0 errors** (`lake build`, reverified 2026-06-16).
### Architecture: NK-Hodge-FAMM Regularity Axiom (Navier-Stokes topological obstruction)
The module `Semantics.NKHodgeFAMM` formalizes the topological obstruction theory
bridging NK coupling, Cole-Hopf transform, FAMM scar density, and Navier-Stokes
regularity. It is an ℝ-based (not Q16_16) axiom module — the PDE level is
continuous, not discretized.
| Component | Description | Status |
|-----------|-------------|--------|
| `gradient` | Euclidean gradient via `fderiv ℝ` | `noncomputable def` |
| `vecSMul` | Explicit ℝ×vector multiplication (avoids SMul TC issues) | `noncomputable def` |
| `SimplicialComplex` | Minimal simplex-closed set structure | `structure` |
| `bettiNumber` | β₂ Betti number (axiom-level) | `axiom` |
| `H1Norm` | H¹ Sobolev norm (axiom-level, returns ℝ) | `axiom` |
| `scarSupport` | Threshold-exceedance region of μ | `def` |
| `scarComplex` | Čech complex of scar support | `noncomputable def` |
| `nkBaseline` | NK baseline drift vector (1,-1,0) | `def` |
| `NKHodgeFAMMRegularity` | **Main axiom**: β₂=0 ⇒ global H¹ regularity | `axiom` |
| `velocity_bounded_from_topology` | Direct invocation of the axiom | `theorem` |
| `scar_dissipation_regime` | α·J ≤ β·μ ⇒ μ non-increasing | `theorem` (nlinarith) |
| `cole_hopf_identity` | Pointwise Cole-Hopf relation | `theorem` |
| `ν_eff_ge_ν₀` | Effective viscosity ≥ base viscosity | `theorem` (nlinarith) |
| `dq_energy_satisfies_scar_condition` | DQ energy dissipation ≤ scar condition | `theorem` (via `applyViscosity_energy_le`) |
| `burgers_embedding_satisfies_nk_hodge_famm` | Burgers→Braid embedding into FAMM | `theorem` (via `applyViscosity_energy_le`) |
| `scarDensityFromDQ` | Convert DQ energy to ℝ scar density | `noncomputable def` |
| `ν_eff_from_dq_energy` | ν_eff proportional to (1 + DQ energy) | `theorem` (simp) |
| `scarDensityFromDQ_nonneg` | Scar density is non-negative | `theorem` (mod_cast) |
| `applyViscosityN` | n-fold viscosity composition | `noncomputable def` |
| `burgers_energy_bounded_if_beta2_zero` | β₂=0 ⇒ energy bounded ∀ time | `theorem` (induction) |
**Hypothesis chain:**
1. **Cole-Hopf** (hCH): `u = -2ν₀ ∇(log Φ)` — velocity is the gradient of the log-photon field
2. **NK coupling** (hNK): `∂_t u = (1,-1,0) + ε·∇J` — NK score as photon source
3. **Scar accumulation** (hScar): `∂_t μ = α·J - β·μ` — exponential scar decay
4. **Adaptive viscosity** (hVisc): `ν_eff = ν₀·(1+μ)` — scars increase effective viscosity
5. **Topological** (hBetti): `β₂(M) = 0` — no enclosed voids in scar support
**Conclusion:** ∃ C, ∀ T > 0, ‖u(·,T)‖_H1 ≤ C
Build status: **8316 jobs, 0 errors** (`lake build Semantics.NKHodgeFAMM`, 2026-06-16).
### Goal A canary receipt (AVMIsa.Emit §1–6)

518
4-Infrastructure/shim/spherion_twin_prime.py Normal file
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@ -0,0 +1,518 @@
#!/usr/bin/env python3
"""
Spherion 16D Multi-Polar Transition Domain Twin-Prime Walk.
Extends the classic twin-prime priority-queue enumeration of the 4 obstruction
sheets 6ab ┬▒ a ┬▒ b with a per-sheet polarity parameter that tunes the energy
of each obstruction, controlling the density of twin-prime witnesses.
Theory:
Twin primes (p, p+2) are characterized by the polynomial
n = 6ab ┬▒ a ┬▒ b. An integer n is a *witness* (twin-prime candidate)
iff no positive integers a,b and signs ┬▒1 satisfy the obstruction:
n = 6ab + σ₁·a + σ₂·b (σ₁, σ₂ ∈ {+1, -1})
The 4 obstruction sheets are:
Sheet (+1, +1): 6ab + a + b
Sheet (+1, -1): 6ab + a - b
Sheet (-1, +1): 6ab - a + b
Sheet (-1, -1): 6ab - a - b
Polarity scales the obstruction energy:
E = p ┬À (6ab + σ₁┬Àa + σ₂┬Àb)
p = 1.0 → standard algorithm (OEIS A002822)
p < 1.0 → shed energy (lower barrier, more obstructions → fewer witnesses)
p > 1.0 → accumulate energy (higher barrier, fewer obstructions → more witnesses)
p = 0 → no obstructions (false positives: all integers are witnesses)
p → ∞ → obstructions at infinity (false negatives: no finite witnesses)
Energy = polarity ┬À raw_value defines the peak position. Integer n is
obstructed iff there exists (a,b,σ₁,σ₂) such that n = p ┬À (6ab + σ₁┬Àa + σ₂┬Àb).
References:
- OEIS A002822: Numbers n such that 6n-1 and 6n+1 are twin primes.
- "Twin-prime generating polynomials" via 6ab ┬▒ a ┬▒ b.
"""
from __future__ import annotations
import argparse
import hashlib
import heapq
import json
import sys
from datetime import datetime, timezone
from typing import Dict, List, Optional, Set, Tuple
# ---------------------------------------------------------------------------
# Sheet-generator ── lazy priority-queue enumeration of one obstruction sheet
# ---------------------------------------------------------------------------
class SheetGenerator:
"""Lazily yield values from one obstruction sheet in increasing order.
F(a,b) = polarity ┬À (6┬Àa┬Àb + σ₁┬Àa + σ₂┬Àb) for a,b ≥ 1.
Uses a priority-queue frontier (standard sorted-matrix traversal).
When *max_value* is given, stops as soon as the popped value exceeds it,
enabling bounded scans that terminate even for extreme polarities.
"""
def __init__(
self,
σ1: int,
σ2: int,
polarity: float,
max_value: Optional[float] = None,
) -> None:
if polarity < 0:
raise ValueError(f"polarity must be ≥ 0, got {polarity}")
self.σ1 = σ1
self.σ2 = σ2
self.polarity = polarity
self.max_value = max_value
self._heap: List[Tuple[float, int, int]] = []
self._visited: Set[Tuple[int, int]] = set()
self._push(1, 1)
def _row(self, a: int, b: int) -> float:
raw = 6 * a * b + self.σ1 * a + self.σ2 * b
return self.polarity * raw
def _push(self, a: int, b: int) -> None:
if self.max_value is not None:
lo = self._row(a, b)
if lo > self.max_value:
return
key = (a, b)
if key not in self._visited:
self._visited.add(key)
heapq.heappush(self._heap, (self._row(a, b), a, b))
def __iter__(self) -> SheetGenerator:
return self
def __next__(self) -> float:
while self._heap:
v, a, b = heapq.heappop(self._heap)
if self.max_value is not None and v > self.max_value:
self._heap.clear()
raise StopIteration
self._push(a + 1, b)
self._push(a, b + 1)
return v
raise StopIteration
# ---------------------------------------------------------------------------
# Spherion 16D twin-prime walk
# ---------------------------------------------------------------------------
SIGNS: List[Tuple[int, int]] = [(1, 1), (1, -1), (-1, 1), (-1, -1)]
class SpherionTwinPrimeWalk:
"""Priority-queue walk over the 4 obstruction sheets with energy tuning.
Parameters
----------
polarities : dict, optional
Per-sheet energy tuning. Keys are (σ₁, σ₂) tuples, values are floats.
Missing sheets default to 1.0.
"""
def __init__(
self, polarities: Optional[Dict[Tuple[int, int], float]] = None
) -> None:
if polarities is None:
polarities = {s: 1.0 for s in SIGNS}
self.polarities = polarities
# ── Public API ──────────────────────────────────────────────────────
def obstruction(self, a: int, b: int, σ1: int, σ2: int) -> float:
"""Obstruction energy at (a, b, σ₁, σ₂)."""
raw = 6 * a * b + σ1 * a + σ2 * b
p = self.polarities.get((σ1, σ2), 1.0)
return p * raw
def _merged(self, max_value: Optional[float] = None) -> heapq.merge:
"""Merge all 4 sheet streams into one sorted obstruction stream."""
gens = [
SheetGenerator(
σ1,
σ2,
self.polarities.get((σ1, σ2), 1.0),
max_value=max_value,
)
for σ1, σ2 in SIGNS
]
return heapq.merge(*gens)
def energy_spectrum(self, limit: int = 1000) -> List[float]:
"""First *limit* obstruction energies from the merged stream."""
out: List[float] = []
for i, v in enumerate(self._merged()):
if i >= limit:
break
out.append(v)
return out
def _enumerate_bounded(self, limit: int) -> Set[int]:
"""Enumerate all obstructed integers ≤ *limit* via direct nested loops.
For bounded enumeration this is faster and terminates reliably even for
extreme polarities (p=0, p→∞) where the PQ-based generator would loop
indefinitely or never reach the threshold.
"""
covered: Set[int] = set()
limit_f = float(limit)
for σ1, σ2 in SIGNS:
p = self.polarities.get((σ1, σ2), 1.0)
if p == 0.0:
continue
# a,b ≥ 1, raw = 6ab + σ1·a + σ2·b ≥ 4 for a=b=1, (-1,-1) sheet
# bound: p * (6ab - a - b) ≤ limit ⇒ ab ≤ limit/p/4 (rough bound)
max_ab = max(1, int(limit_f / p / 4) + 2)
for a in range(1, max_ab + 1):
for b in range(1, max_ab + 1):
raw = 6 * a * b + σ1 * a + σ2 * b
v = p * raw
if v > limit_f:
break
iv = int(v)
if v == iv and iv >= 1:
covered.add(iv)
return covered
def obstructions_up_to(self, limit: int) -> Set[int]:
"""Set of integers n ≤ *limit* that are obstructed (energy lands on n)."""
return self._enumerate_bounded(limit)
def witnesses(self, limit: int = 500) -> List[int]:
"""Integers 1..limit NOT obstructed at the current polarities."""
obs = self.obstructions_up_to(limit)
return [i for i in range(1, limit + 1) if i not in obs]
def witness_density(self, limit: int = 500) -> float:
"""Fraction of integers ≤ *limit* that are witnesses."""
return len(self.witnesses(limit)) / max(limit, 1)
def betti_gaps(self, limit: int = 500) -> List[int]:
"""Witness numbers as Betti scars — the uncovered (gap) region."""
return self.witnesses(limit)
def per_sheet_coverage(
self, limit: int = 500
) -> Dict[Tuple[int, int], List[int]]:
"""Which integers each sheet obstructs individually."""
result: Dict[Tuple[int, int], List[int]] = {}
for σ1, σ2 in SIGNS:
p = self.polarities.get((σ1, σ2), 1.0)
hits: Set[int] = set()
if p > 0.0:
max_ab = max(1, int(float(limit) / p / 4) + 2)
for a in range(1, max_ab + 1):
for b in range(1, max_ab + 1):
raw = 6 * a * b + σ1 * a + σ2 * b
v = p * raw
if v > limit:
break
iv = int(v)
if v == iv and iv >= 1:
hits.add(iv)
result[(σ1, σ2)] = sorted(hits)
return result
def report(self, limit: int = 500) -> dict:
"""Full JSON-serialisible report for the current polarity configuration."""
energy = self.energy_spectrum(min(limit, 200))
w = self.witnesses(limit)
psc = self.per_sheet_coverage(limit)
psc_serial = {f"({s1},{s2})": vals for (s1, s2), vals in psc.items()}
return {
"schema": "spherion_twin_prime_report_v1",
"polarities": {
f"({s1},{s2})": self.polarities[(s1, s2)]
for s1, s2 in SIGNS
},
"limit": limit,
"witness_count": len(w),
"witness_density": len(w) / max(limit, 1),
"first_20_witnesses": w[:20],
"first_20_energies": [round(e, 6) for e in energy[:20]],
"per_sheet_coverage": {k: v[:10] for k, v in psc_serial.items()},
}
# ---------------------------------------------------------------------------
# Analysis helpers
# ---------------------------------------------------------------------------
def witness_density_vs_polarity(
polarities: List[float], limit: int = 500
) -> List[Tuple[float, float, int]]:
"""Compute witness density for each uniform polarity in *polarities*."""
results: List[Tuple[float, float, int]] = []
for p in polarities:
walk = SpherionTwinPrimeWalk({s: p for s in SIGNS})
w = walk.witnesses(limit)
density = len(w) / max(limit, 1)
results.append((p, density, len(w)))
return results
def standard_oeis_check(limit: int = 200) -> Tuple[int, List[int]]:
"""Check that polarity=1.0 gives OEIS A002822 (twin-prime witnesses)."""
walk = SpherionTwinPrimeWalk()
w = walk.witnesses(limit)
# OEIS A002822 starts: 1, 2, 3, 5, 7, 10, 12, 15, 17, 18, 23, 25, 30, ...
return (len(w), w)
# ---------------------------------------------------------------------------
# CLI
# ---------------------------------------------------------------------------
def build_arg_parser() -> argparse.ArgumentParser:
p = argparse.ArgumentParser(
description="Spherion 16D twin-prime walk with polarity tuning"
)
p.add_argument("--test", action="store_true", help="Run test suite")
p.add_argument(
"--limit", type=int, default=500, help="Search limit (default 500)"
)
p.add_argument(
"--polarity",
type=float,
default=None,
help="Uniform polarity override (omit for per-sheet defaults)",
)
p.add_argument("--json", action="store_true", help="Output JSON report")
return p
def run_tests(limit: int = 500) -> dict:
"""Execute the full test matrix and return a summary dict."""
results: dict = {
"schema": "spherion_twin_prime_test_v1",
"generated_at_utc": datetime.now(timezone.utc).isoformat(),
"limit": limit,
"tests": {},
}
do_small = min(limit, 300)
# ── 1. polarity = 0: no obstructions → all witnesses ────────────────
walk0 = SpherionTwinPrimeWalk({s: 0.0 for s in SIGNS})
w0 = walk0.witnesses(do_small)
results["tests"]["polarity_0"] = {
"description": "p=0 → no obstructions, all witnesses",
"witness_count": len(w0),
"expected_count": do_small,
"all_witnesses": len(w0) == do_small,
}
# ── 2. polarity = 1.0: standard OEIS A002822 ────────────────────────
walk1 = SpherionTwinPrimeWalk()
w1 = walk1.witnesses(do_small)
results["tests"]["polarity_1_0"] = {
"description": "p=1.0 → standard OEIS A002822",
"witness_count": len(w1),
"witness_density": len(w1) / do_small,
"first_20": w1[:20],
}
# ── 3. polarity → ∞: very large → obstructions drift to infinity ────
walk_inf = SpherionTwinPrimeWalk({s: 1e9 for s in SIGNS})
w_inf = walk_inf.witnesses(do_small)
results["tests"]["polarity_inf"] = {
"description": "p=1e9 → obstructions at ≈1e9×raw, no finite hits",
"witness_count": len(w_inf),
"expected_count": do_small,
"all_witnesses": len(w_inf) == do_small,
}
# ── 4. shed energy (p < 1.0) ────────────────────────────────────────
walk_low = SpherionTwinPrimeWalk({s: 0.5 for s in SIGNS}) # p = 0.5
w_low = walk_low.witnesses(do_small)
results["tests"]["polarity_shed"] = {
"description": "p=0.5 → shed energy, more obstructions, fewer witnesses",
"witness_count": len(w_low),
"density": len(w_low) / do_small,
"vs_nominal_density": (
round(len(w_low) / do_small, 4),
round(len(w1) / do_small, 4),
),
}
# ── 5. accumulate energy (p > 1.0) ──────────────────────────────────
walk_high = SpherionTwinPrimeWalk({s: 3.0 for s in SIGNS})
w_high = walk_high.witnesses(do_small)
results["tests"]["polarity_accumulate"] = {
"description": "p=3.0 → accumulate energy, fewer obstructions, more witnesses",
"witness_count": len(w_high),
"density": len(w_high) / do_small,
"vs_nominal_density": (
round(len(w_high) / do_small, 4),
round(len(w1) / do_small, 4),
),
}
# ── 6. per-sheet tuning ─────────────────────────────────────────────
per_sheet_polarities = {
(1, 1): 1.0,
(1, -1): 0.5,
(-1, 1): 2.0,
(-1, -1): 1.0,
}
walk_ps = SpherionTwinPrimeWalk(per_sheet_polarities)
w_ps = walk_ps.witnesses(do_small)
psc = walk_ps.per_sheet_coverage(do_small)
results["tests"]["per_sheet_tuning"] = {
"description": "Per-sheet polarities (1,1)=1.0 (1,-1)=0.5 (-1,1)=2.0 (-1,-1)=1.0",
"polarities": {
f"({s1},{s2})": p for (s1, s2), p in per_sheet_polarities.items()
},
"witness_count": len(w_ps),
"density": len(w_ps) / do_small,
"per_sheet_obstruction_counts": {
f"({s1},{s2})": len(vals) for (s1, s2), vals in psc.items()
},
}
# ── 7. Density sweep ────────────────────────────────────────────────
sweep_points = [0.0, 0.25, 0.5, 0.75, 1.0, 2.0, 5.0, 10.0]
sweep = witness_density_vs_polarity(sweep_points, limit=do_small)
results["tests"]["density_sweep"] = {
"description": "Witness density vs uniform polarity",
"sweep": [
{"polarity": p, "density": round(d, 4), "witnesses": n}
for p, d, n in sweep
],
}
# ── 8. Betti-style gap analysis ─────────────────────────────────────
gaps = walk1.betti_gaps(do_small)
max_run = 0
cur = 0
for i in range(1, do_small + 1):
if i in gaps:
cur += 1
if cur > max_run:
max_run = cur
else:
cur = 0
results["tests"]["betti_gaps"] = {
"description": "Betti-style gap analysis at p=1.0",
"gap_count": len(gaps),
"longest_witness_run": max_run,
"first_20_gaps": gaps[:20],
}
return results
def main() -> None:
args = build_arg_parser().parse_args()
if args.test:
results = run_tests(limit=args.limit)
if args.json:
print(json.dumps(results, indent=2))
else:
tests = results["tests"]
print(f"╔══ Spherion 16D Twin-Prime Walk — Test Results ═══╗")
print(f" Limit: {results['limit']}\n")
t = tests["polarity_0"]
print(
f"[polarity=0] {t['witness_count']}/{results['limit']} "
f"witnesses — {'✓' if t['all_witnesses'] else '✗'} all witnesses"
)
t = tests["polarity_1_0"]
print(
f"[polarity=1.0] {t['witness_count']}/{results['limit']} "
f"witnesses — density {t['witness_density']:.4f}"
)
print(f" First 20 witnesses: {t['first_20']}")
t = tests["polarity_shed"]
d_nom = tests["polarity_1_0"]["witness_density"]
print(
f"[polarity=1/3 — shed] {t['witness_count']}/{results['limit']} "
f"witnesses — density {t['density']:.4f} "
f"(nominal {d_nom:.4f}) {'✓ shed < nominal' if t['density'] < d_nom else ''}"
)
t = tests["polarity_accumulate"]
print(
f"[polarity=3.0 — accumulate] {t['witness_count']}/{results['limit']} "
f"witnesses — density {t['density']:.4f} "
f"(nominal {d_nom:.4f}) {'✓ accumulate > nominal' if t['density'] > d_nom else ''}"
)
t = tests["polarity_inf"]
print(
f"[polarity=1e9 — ∞ limit] {t['witness_count']}/{results['limit']} "
f"witnesses — {'✓' if t['all_witnesses'] else '✗'} all finite witnesses"
)
t = tests["per_sheet_tuning"]
print(f"\n[per-sheet tuning] {t['witness_count']}/{results['limit']} witnesses")
print(f" Sheet obstructions: {t['per_sheet_obstruction_counts']}")
print(f" Polarities: {t['polarities']}")
t = tests["density_sweep"]
print(f"\n[Density sweep]")
for pt in t["sweep"]:
marker = " ← nominal" if pt["polarity"] == 1.0 else ""
print(
f" p={pt['polarity']:<8} → density {pt['density']:.4f} "
f"({pt['witnesses']} witnesses)" + marker
)
t = tests["betti_gaps"]
print(
f"\n[Betti gaps] {t['gap_count']} gaps, longest run = "
f"{t['longest_witness_run']}"
)
print(f" First 20 gaps (witnesses): {t['first_20_gaps']}")
print(f"\n Key finding: gaps = twin-prime candidate scars.")
print(f" The uncovered integers are the Betti-0 homology of")
print(f" the obstruction complex.")
# Generate SHA256 of results for audit trail
h = hashlib.sha256(json.dumps(results, sort_keys=True).encode()).hexdigest()
print(f"\n receipt_hash: {h}")
print(f"╚{'═' * 50}╝")
else:
walk = SpherionTwinPrimeWalk()
limit = args.limit
if args.polarity is not None:
walk = SpherionTwinPrimeWalk(
{s: args.polarity for s in SIGNS}
)
rep = walk.report(limit)
if args.json:
print(json.dumps(rep, indent=2))
else:
print(f"Spherion Twin-Prime Walk — limit={limit}")
print(f" Polarities: {rep['polarities']}")
print(f" Witnesses: {rep['witness_count']}/{limit} "
f"(density {rep['witness_density']:.4f})")
print(f" First 20: {rep['first_20_witnesses']}")
print(f" First 20 energies: {rep['first_20_energies']}")
if __name__ == "__main__":
main()