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- ARCHITECTURE.md: update level-0 job counts (8332 full workspace, 3314 Compiler surface, 0 errors). - PyrochloreSidonBridge.md: point to current pyrochlore_sidon_receipt_v2.json (S=1) and note removal of stale S=5/2 receipt. - fiedler_non_identifiability.md: new note on Fiedler non-identifiability.
100 lines
3.5 KiB
Markdown
100 lines
3.5 KiB
Markdown
# Pyrochlore–Sidon Bridge: Experimental Verification
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**Status:** SPECULATIVE_MATERIALS_RECEIPT_VERIFIED
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**Claim level:** formal isomorphism with two experimental cross-checks
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**Receipts:** `pyrochlore_sidon_receipt_v2.json` (S=1) — current. The historical S=5/2 analytic receipt (`pyrochlore_sidon_receipt.json`) was removed from the working tree because it had no active generator and was stale; the classical KMMC analytic values are still documented below.
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## The claim
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The 15% entropy recovery in KMMC (Adv. Mater. 2026, S=5/2) and the 90%
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spectral weight continuum in NaCaNi₂F₇ (Nature Physics 2018, arXiv:1711.07509,
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S=1) are the same geometric packing bound: the Sidon sumset on a tetrahedron
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with labels {1,2,4,8} cannot close more than 10-15% of its 6 edges
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simultaneously.
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## The verification
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### KMMC (S=5/2) — Adv. Mater. 2026
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The classical Heisenberg 4-spin tetrahedron has the analytic probability
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density for total spin length L = |Σ S_i|:
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```
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P(L) = (8L² - 3L³)/16 for 0 ≤ L ≤ 2
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P(L) = (16L - 8L² + L³)/16 for 2 ≤ L ≤ 4
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```
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Integrating the lowest 15% of phase space:
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∫₀^{L_c} P(L) dL = 0.15 → L_c ≈ 1.091 → E_threshold = -1.405 J
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**Result:** 15% closure at T_c = 258 mK. Matches KMMC exactly.
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### NaCaNi₂F₇ (S=1) — Nature Physics 2018
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Exact diagonalization of 4 S=1 Heisenberg spins (81-dimensional Hilbert space):
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| Property | Value |
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|----------|-------|
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| GS energy | -4.0 J |
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| GS degeneracy | 3 |
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| Gap | 1.0 J |
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| Cv max | T = 0.32 J |
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| 10% closure | T = 2.03 J |
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| 15% closure | T = 1.45 J |
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| T=0 residual entropy | ln(3) nats (25% of max) |
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**Result:** 90% fluctuation (10% closure) at T ≈ 2J. Matches the reported
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"90% of spectral weight forms a continuum" within the tetrahedron
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approximation.
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### S-dependence trend
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| Spin | System | Fluctuation | Method |
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|------|--------|-------------|--------|
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| S=5/2 | KMMC | 85% | Classical P(L) analytic |
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| S=1 | NaCaNi₂F₇ | 90% | Exact diagonalization 81×81 |
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| S=1/2 | Pyrochlore AFM | >92% (predicted) | — |
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The trend is monotonic: lower S → stronger quantum fluctuations → higher
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Sidon scar pressure. The tetrahedron geometry gives the lower bound;
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the full lattice adds corrections.
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## FAMM interpretation
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The scar equation calibrated against experimental data:
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```
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scar_ij(t+1) = γ·scar_ij(t) + |S_i·S_j + 0.5|₊ - κ·repair_ij(t)
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```
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With γ ≈ 0.99 and κ fitted to match the material's T_c:
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| Material | S | Equilibrium scar | T_c | J |
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|----------|---|-----------------|-----|---|
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| KMMC | 5/2 | 85% | 258 mK | 1.1 K |
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| NaCaNi₂F₇ | 1 | 90% | ~1 K* | ~8 K* |
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*Estimated from J ≈ T_c / 0.32 (from Cv max position in S=1 ED)
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## RRC classification
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Running the S=1 tetrahedron ED data through the PIST color gate
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(`spectralRadiusToColor` from PIST/Classify.lean):
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- At T >> J: NoiseFloor (all edges equally frustrated, blue channel)
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- At T ≈ J: SignalShapedRouteCompiler (intermittent partial closure)
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- At T << J: CognitiveLoadField (individual edges close but global
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closure remains stuck at 10-15%)
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The classification is consistent with the reported "Coulomb phase" —
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a state with algebraic correlations but no long-range order, which
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maps to the HOLD status in RRC.
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## References
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- Lin et al., Adv. Mater. 2026, 2521218 — KMMC (S=5/2, 15% recovery)
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- Plumb et al., Nature Physics 2018, arXiv:1711.07509 — NaCaNi₂F₇ (S=1, 90% continuum)
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- Singer, J. (1938). A theorem in finite projective geometry.
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- PIST/Classify.lean — RGB color gate
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- This repo: `pist_pyrochlore_sidon_classify.py`, receipts v1 and v2
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