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Results index

One row per numbered statement of the revision. Generated by tools/build_reference.py from build/free_fermion_cft_v5.aux, the manuscript as last committed on 2026-09-22; do not edit by hand, the numbers come from the compiled document and change when it is recompiled.

The new column marks the 28 statements that do not appear in free_fermion_cft_v4.tex, derived by comparing the two auxiliary files rather than asserted.

Statuses: proved in the revision, numerical where the evidence is a computation, definition and convention where nothing is claimed. The last column links to the widget on the interactive page that lets you vary the parameters.

Section 3. Scaling limits of lattice fermions

statement	page	what it says	status	new	explore
Definition 3.1	p. 24	Convergent sequences of observables between scales	definition
Definition 3.2	p. 25	The wavelet renormalization group, built from a Daubechies scaling function	definition		shat
Proposition 3.3	p. 26	It is an inductive system; item (4) is the asymptotic compatibility that re-centring would break	proved
Definition 3.4	p. 27	The same maps for currents, quadratic in the fermions	definition
Remark 3.5	p. 27	Re-centring by the centre of mass, offered as a comparison convention and deliberately not built in	convention	yes	support
Definition 3.6	p. 28	Local one-particle spaces, discarding the last 2K-1 sites of an interval	definition
Proposition 3.7	p. 28	The wavelet group preserves localisation	proved
Remark 3.8	p. 28	A strict one-sided light cone of width eps_N(2K-1), and the linear trade-off of locality against regularity	proved	yes	support
Lemma 3.9	p. 29	Decay of the finite products of the low-pass symbol	proved
Corollary 3.10	p. 30	What that decay gives inside the Brillouin zone, and only there	proved	yes
Lemma 3.11	p. 30	Pointwise decay of the transform of the scaling function	proved		shat
Definition 3.12	p. 31	rho-regular means sigma_K > rho: the hypothesis the published version left as 'sufficiently regular'	definition	yes	ladder
Remark 3.13	p. 31	The table of sigma_K and rho, and the four thresholds that occur	numerical	yes	ladder
Lemma 3.14	p. 32	The uniform majorant: the two decays interlock across the Brillouin zone	proved	yes	majorant
Corollary 3.15	p. 33	The majorant in the form the dominated-convergence argument needs	proved	yes	majorant
Definition 3.16	p. 33	The momentum-cutoff renormalization group	definition
Proposition 3.17	p. 33	It is an inductive system, and the Hardy projections leave its core invariant	proved
Lemma 3.18	p. 35	The lattice vacua converge with no regularity hypothesis: the alias weights are probability weights	proved		alias
Corollary 3.19	p. 37	The scaling limit of the lattice vacua, with the rate	proved		alias
Remark 3.20	p. 38	The massless Ramond zero mode, where the limit is convention-dependent	proved	yes
Remark 3.21	p. 38	One argument covers both renormalization groups	proved
Remark 3.22	p. 39	Where the rate comes from, and why a supremum over the aliasing class would not be small	proved	yes	alias

Section 4. Approximation of conformal symmetries

statement	page	what it says	status	new	explore
Definition 4.1	p. 40	The Koo-Saleur approximants	definition
Remark 4.2	p. 46	The modified approximants' finite-scale algebra: the discrepancy vanishes for same-sign modes	proved	yes	discrepancy
Definition 4.3	p. 50	The one-particle Virasoro generators, renormalised: the published definition was off by L/pi	definition
Remark 4.4	p. 50	The central charge is 1 (resp. 1/2) for every L under this normalisation	proved	yes
Corollary 4.5	p. 51	Analytic vectors, with radius 1/	k	uniform in the mode	proved	radius
Remark 4.6	p. 51	The Majorana implementability condition	proved
Lemma 4.7	p. 52	Wavelet route, one particle: convergence on the whole domain, not just a core	proved		rates
Remark 4.8	p. 55	The sharp rate in the Sobolev scale, with a matching lower bound at the Brillouin-zone edge	proved	yes	rates
Theorem 4.9	p. 55	Convergence of the approximants in the Fock representation (c = 0)	proved		rates
Corollary 4.10	p. 57	The one-particle Bogoliubov transformations, made quantitative	proved		duhamel
Remark 4.11	p. 57	The estimate for multinomials in the creation and annihilation operators	proved
Lemma 4.12	p. 57	One-particle energy growth and Duhamel: linear in t at k = 0, exponential otherwise	proved	yes	duhamel
Remark 4.13	p. 58	The same for smeared generators; the constant is sufficient but not sharp	proved	yes	duhamel
Corollary 4.14	p. 59	The derivations, with no t-dependence at all	proved		duhamel
Lemma 4.15	p. 60	Momentum-cutoff route, one particle: the sharp rate, with no regularity of the scaling function	proved		rates
Theorem 4.16	p. 61	Convergence in the positive-energy representations (c = 1/2, 1). Theorem A of the introduction	proved		rates
Lemma 4.17	p. 62	The energy bound at fixed mode, needing no regularity	proved	yes
Corollary 4.18	p. 62	Essential self-adjointness on every core for the conformal Hamiltonian, by Nelson's commutator theorem	proved	yes
Lemma 4.19	p. 63	Energy growth and Duhamel in Fock space	proved	yes	duhamel
Corollary 4.20	p. 64	The unitary groups, with a rate	proved		duhamel
Corollary 4.21	p. 64	The Bogoliubov transformations on the momentum-cutoff route	proved		duhamel
Remark 4.22	p. 64	The Moebius generators in the Neveu-Schwarz sector	proved
Lemma 4.23	p. 65	The smeared approximants: uniform bound, domain and rate	proved		rates
Remark 4.24	p. 67	The smearing error is a pure phase, and re-centring makes it third order	proved	yes	phase
Theorem 4.25	p. 68	Convergence of the smeared approximants. Its rate needs K >= 7, not the K >= 6 printed	proved		phase
Corollary 4.26	p. 68	The smeared corollary	proved
Remark 4.27	p. 68	Lifting the restriction to basic sequences	proved		jackson
Lemma 4.28	p. 69	A Jackson estimate for the loop renormalization group, driven by the total aliasing mass	proved	yes	jackson
Theorem 4.29	p. 70	Reconstruction for every smooth loop, not only basic sequences; K >= 9, and K >= 10 for the rate	proved	yes	jackson
Remark 4.30	p. 71	The two-scale rate	proved	yes	jackson
Corollary 4.31	p. 71	The smeared Bogoliubov transformations	proved		duhamel
Remark 4.32	p. 73	The rate at nonzero mode is not an artefact of the estimate	numerical	yes	rates
Remark 4.33	p. 74	Why the supremum must be taken with care: the step Finding 1 shows to be false	proved	yes	shat
Remark 4.34	p. 74	Simulating the theory: what the scale and the resolution mean in qubits	proved		budget
Remark 4.35	p. 75	The error of the time evolution, linear in t at zero mode	proved		duhamel
Remark 4.36	p. 75	The error for nonzero modes, on analytic vectors	proved		radius

Section 5. Approximation of Wess-Zumino-Witten currents

statement	page	what it says	status	new	explore
Lemma 5.1	p. 80	The modified current is exact on the bulk and its off-diagonal blocks vanish identically	proved	yes	current
Theorem 5.2	p. 80	Convergence of the Wess-Zumino-Witten currents, with no regularity for the qualitative statement	proved		current
Remark 5.3	p. 81	Four ways the current is better behaved than the Virasoro generators	proved	yes	current

Section 6. Approximation of correlation functions

statement	page	what it says	status	new	explore
Theorem 6.1	p. 82	Convergence of the fermion correlation functions. Theorem B of the introduction	proved		budget
Corollary 6.2	p. 84	The same for a wider class of observables	proved
Proposition 6.3	p. 85	The explicit error bound for the chiral time evolution	proved	yes	budget
Remark 6.4	p. 86	The end-to-end qubit budget; locality costs a square root unless one re-centres	proved	yes	budget
Theorem 6.5	p. 87	Convergence of the Virasoro correlation functions. Theorem C, now unconditional	proved		budget
Proposition 6.6	p. 88	The scale-uniform energy bound that makes Theorem C unconditional	proved	yes
Remark 6.7	p. 89	Mixed correlation functions	proved