USR-2026-0057 · Quantum information
Entanglement counts a theory's degrees of freedom
Entanglement counts degrees of freedom: fitting the Calabrese–Cardy law to block entropies alone recovers the central charge c = 0.500062 for the critical Ising chain (CFT value ½, converging to 0.500013 on inner windows) and c → 1 for the free boson as the regulator mass goes to zero. Two universality classes, both counted correctly from entanglement with no other input.
PROVISIONAL
c = 0.500062 (Ising, CFT ½) and c → 1 (boson) — read off entanglement alone
reproduced ≠ discovered
re-run & confirmed 2026-07-26 (R0 — David Thompson's CI; independence pending)
Calabrese–Cardy 2004, recomputed. The entanglement ledger does not merely scale with boundaries — its universal coefficient counts the field content of the underlying theory.
Falsify-box — how to kill this claim
SURVIVED
fitted c deviates from the CFT value by more than ~5% under window shifts, or the two universality classes are not distinguished
Verification record — every quoted number, re-run 2026-07-26
| quantity | measured | verdict |
|---|---|---|
| c_Ising | 0.500062 ± 0.000006 (fit s.e. — fit quality, not evidence strength) | MATCH |
| c_boson | 0.9908 at m=1e-4, trending to 1 as m→0 | MATCH |
| exact-diagonalisation cross-check | agreement 1.4e-15 | MATCH |
Provenance
script: information_seam/central_charge.py
script status: PRESENT
datasets: information-seam-scripts (downloadable from /data — run it yourself)
re-run: 2026-07-26
script status: PRESENT
datasets: information-seam-scripts (downloadable from /data — run it yourself)
re-run: 2026-07-26
Status, honestly
verified by adversarial re-run (workflow, 2026-07-26); admitted from MCP intake INTAKE-814293f286. This status was assigned by hand from the public re-run audit above — it is a judgement made from that evidence, not a number this site computed for itself. What cannot move it is authority: no vote, endorsement, or say-so, from anyone including the maintainer. Independent reproduction would move it; nothing else will.
§ Reproduction
central_charge.py
David Thompson's own run of this script, captured verbatim. A match proves the result is reproducible; it is still R0 on this registry's independence rings — same code, so it cannot move a status. Only an outside run does that.
A · Recorded
David Thompson's machine · 2026-08-02
========================================================================
TEST 8: CENTRAL CHARGE FROM ENTANGLEMENT (Calabrese-Cardy 2004)
S(l) = (c/3) ln[(N/pi) sin(pi l/N)] + const, periodic chain
STATUS: REPRODUCED known result -- machinery check only
========================================================================
numpy 1.26.4 | scipy 1.17.1 available (not required; pure-numpy path used) | seed 42 (unused: exact linear algebra)
N = 200, blocks l = 2..100, fit windows [(4, 96), (8, 92), (16, 84)]
------------------------------------------------------------------------
PART A: transverse-field Ising at h = 1 (critical), periodic N = 200
Jordan-Wigner free fermions, even/antiperiodic (NS) sector
------------------------------------------------------------------------
[validation, N=8] ED ground energy = -10.251661790966 (gap 0.196983)
[validation, N=8] NS free-fermion E0 = -10.251661790966 |diff| = 5.329e-15
[validation, N=8] block entropies ED vs free-fermion, l=1..4:
l=1 S_ED = 0.470840432692 S_ff = 0.470840432692 |diff| = 6.106e-16
l=2 S_ED = 0.576923708075 S_ff = 0.576923708075 |diff| = 1.110e-15
l=3 S_ED = 0.621840572029 S_ff = 0.621840572029 |diff| = 2.109e-15
l=4 S_ED = 0.635095665499 S_ff = 0.635095665499 |diff| = 2.887e-15
max |diff| = 2.887e-15 (machine-precision CODE check, not evidence)
[validation, N=200] max |S(l) - S(N-l)| over l in [40, 60, 80] = 5.535e-12
sample S(l) [Ising]:
l = 2 x(l) = 0.230994 S = 0.593382758
l = 4 x(l) = 0.461879 S = 0.709363114
l = 8 x(l) = 0.692269 S = 0.824661392
l = 16 x(l) = 0.920680 S = 0.938891212
l = 32 x(l) = 1.141088 S = 1.049101701
l = 50 x(l) = 1.269005 S = 1.113061015
l = 64 x(l) = 1.328124 S = 1.142621200
l = 80 x(l) = 1.367802 S = 1.162460108
l = 100 x(l) = 1.384529 S = 1.170823794
Ising window l in [ 4, 96]: c = 0.500062 +/- 0.000006 (fit s.e.), const = +0.478479
Ising window l in [ 8, 92]: c = 0.500027 +/- 0.000002 (fit s.e.), const = +0.478524
Ising window l in [ 16, 84]: c = 0.500013 +/- 0.000001 (fit s.e.), const = +0.478542
Ising window-shift movement of c: 0.000049 (expected c = 1/2 (free Majorana))
------------------------------------------------------------------------
PART B: harmonic ring (free boson), N = 200, regulator masses [0.001, 0.0001]
exact Gaussian covariance method (as wave-1 area_law)
------------------------------------------------------------------------
mass m = 1e-03 (correlation length 1/m = 1000 sites >> N)
sample S(l) [boson, m=1e-03]:
l = 2 x(l) = 0.230994 S = 1.582083819
l = 4 x(l) = 0.461879 S = 1.795940885
l = 8 x(l) = 0.692269 S = 2.007800801
l = 16 x(l) = 0.920680 S = 2.216973002
l = 32 x(l) = 1.141088 S = 2.418091265
l = 50 x(l) = 1.269005 S = 2.534495373
l = 64 x(l) = 1.328124 S = 2.588217987
l = 80 x(l) = 1.367802 S = 2.624246725
l = 100 x(l) = 1.384529 S = 2.639429432
boson m=1e-03 window l in [ 4, 96]: c = 0.912511 +/- 0.000194 (fit s.e.), const = +1.376407
boson m=1e-03 window l in [ 8, 92]: c = 0.911443 +/- 0.000145 (fit s.e.), const = +1.377767
boson m=1e-03 window l in [ 16, 84]: c = 0.910370 +/- 0.000103 (fit s.e.), const = +1.379159
boson m=1e-03 window-shift movement of c: 0.002141 (expected c = 1 (free boson))
mass m = 1e-04 (correlation length 1/m = 10000 sites >> N)
sample S(l) [boson, m=1e-04]:
l = 2 x(l) = 0.230994 S = 2.637050502
l = 4 x(l) = 0.461879 S = 2.867466929
l = 8 x(l) = 0.692269 S = 3.096070221
l = 16 x(l) = 0.920680 S = 3.322419640
l = 32 x(l) = 1.141088 S = 3.540765432
l = 50 x(l) = 1.269005 S = 3.667471979
l = 64 x(l) = 1.328124 S = 3.726030880
l = 80 x(l) = 1.367802 S = 3.765331489
l = 100 x(l) = 1.384529 S = 3.781899751
boson m=1e-04 window l in [ 4, 96]: c = 0.990768 +/- 0.000023 (fit s.e.), const = +2.410181
boson m=1e-04 window l in [ 8, 92]: c = 0.990625 +/- 0.000008 (fit s.e.), const = +2.410361
boson m=1e-04 window l in [ 16, 84]: c = 0.990561 +/- 0.000003 (fit s.e.), const = +2.410444
boson m=1e-04 window-shift movement of c: 0.000207 (expected c = 1 (free boson))
mass sensitivity: c(m=1e-4) - c(m=1e-3) = +0.078256 in primary window
mass moves the CONSTANT by +1.033774 (non-universal, zero-mode inflated)
========================================================================
HEADLINE NUMBERS (all +/- are OLS fit s.e. -- fit quality, NOT evidence sigmas)
[H1] c_Ising (l in [4,96]) = 0.500062 +/- 0.000006 [CFT: 0.5]
[H2] c_Ising window-shift movement = 0.000049
[H3] c_boson (m=1e-4, same window) = 0.990768 +/- 0.000023 [CFT: 1.0]
[H4] c_boson (m=1e-3, same window) = 0.912511
[H5] c_boson window-shift movement = 0.000207 (m=1e-4), 0.002141 (m=1e-3)
[H6] c_boson mass-shift movement = 0.078256
[H7] validation: ED-vs-fermion max entropy diff (N=8) = 2.887e-15
[H8] validation: S(l)=S(N-l) max deviation (N=200) = 5.535e-12
========================================================================
INTERPRETATION (honesty rules apply)
At criticality the entanglement ledger does not merely scale with
the boundary: its universal log-coefficient counts the degrees of
freedom of the underlying field theory. Two universality classes
(free Majorana fermion, c=1/2; free boson, c=1) are both read off
correctly from entanglement alone, matching Calabrese-Cardy 2004.
STATUS: reproduced-known-result; validates machinery, not nature.
B · Yours
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