Why, and why it is separate from #637
#637 asks about critical polynomials and transfer-matrix machinery. This asks about the
observable itself, and the two searches have almost no overlap.
Our defining quantity is a difference of wrapping indicators on a torus:
D(C) = 1{black NN wraps} - 1{white NN+NNN wraps}
M(p) = sum_k a_k p^k (1-p)^(N-k), threshold condition M(p_c) = 0
Wrapping probabilities on a torus at criticality are one of the few things in 2D percolation
that are known exactly, and this project has never systematically read that literature
against its own observable. That is a gap we can close cheaply.
Q1 — the exact wrapping results, at first hand
Pinson (1994) computed crossing/wrapping probabilities for critical percolation on a torus
exactly, resolved by wrapping topology (neither / horizontal / vertical / both), in terms of
theta functions of the aspect ratio. Later work (Ziff, Newman, Langlands and others) extended
and checked these.
Read the primary statements and report:
- The exact expressions, resolved by wrapping type, and the modular / aspect-ratio
dependence.
- What universality class and what lattice they hold for, and precisely how they
transfer (or do not) to site percolation on the square lattice.
- Whether the results are for the primal lattice only, or whether the matching lattice's
wrapping is also given.
Item 3 is the one we most need.
Q2 — the primal/matching relation on a torus
On an infinite plane, primal and matching lattice connectivity are complementary in a clean
way. On a torus they are famously not: both can wrap, or neither.
What is known exactly about the joint law of primal and matching wrapping on a torus?
Is there a published identity, an exact complementarity defect, or an exact expression for
the difference of the two wrapping probabilities?
This is the continuum shadow of the object we measure. #628 established the finite-L fact
that M(p) + M(1-p) != 0 exactly and that the site duality constraint does not transport to
the square-bond lab (r_b + r_w = 2 fails in 118133 of 262144 configurations). We want the
published version of that story.
If the difference of the two wrapping probabilities is a known exact function at
criticality, that is potentially decisive — it would say what M converges to and at what
rate, which is exactly the input #618/#622 report as missing.
Q3 — the Mertens-Ziff matching function specifically
The construction we use has a specific provenance.
Where is the matching function D(C) = 1{black wraps} - 1{white wraps} introduced, what
is it used for there, and what is already proved about it?
Specifically: is M(p_N) = 0 known to converge to p_c, and if so at what rate and under
what hypotheses? #613/#276 have a "critical-point bridge" and #618 reports that
Q_N(u) -> p_c gives no polynomial rate from the named inputs H1-H3. If the rate is in the
literature, that unblocks several tickets at once. If it is only conjectural there too, say
that — it is equally useful.
Q4 — finite-size corrections to wrapping probabilities
What is the known finite-size correction structure for torus wrapping probabilities — the
leading exponent, whether it is universal or lattice-dependent, and whether the corrections
for the difference of two wrapping probabilities are smaller than for either separately?
We measure a shape flow whose amplitude goes roughly as N^-0.97 (L^-1.94) after
projecting out location and scale (omega = 0.970 ± 0.031, unity not excluded). Whether
that is a known correction exponent for this class of observable, an analytic background, or
a coincidence, is a question the literature may simply answer.
Q5 — aspect ratio
Pinson's results depend on the torus aspect ratio through modular functions. This project has
an aspect-ladder line (#567, #573, #575, #577) that measured an amplitude ratio across
aspect ratios and reported it underpowered with three surviving hypotheses.
Does the exact aspect-ratio dependence of wrapping probabilities predict that ratio?
If it does, the ladder can be scored against a theory number instead of against three
hypotheses, which converts an underpowered result into a test.
What a useful answer looks like
Per question: cite or gap, with exact statements, theorem numbers, and enough
bibliographic detail to find the source. Quoted statements are the payload. Mark conjectures
as conjectures — we have inherited one as a theorem before.
A clean "Q2 has no published answer" is a real result and tells us the finite-L census in
#640 is the frontier rather than a rediscovery.
Boundaries
Related: #640, #635, #618, #622, #613, #276, #577.
Why, and why it is separate from #637
#637 asks about critical polynomials and transfer-matrix machinery. This asks about the
observable itself, and the two searches have almost no overlap.
Our defining quantity is a difference of wrapping indicators on a torus:
Wrapping probabilities on a torus at criticality are one of the few things in 2D percolation
that are known exactly, and this project has never systematically read that literature
against its own observable. That is a gap we can close cheaply.
Q1 — the exact wrapping results, at first hand
Pinson (1994) computed crossing/wrapping probabilities for critical percolation on a torus
exactly, resolved by wrapping topology (neither / horizontal / vertical / both), in terms of
theta functions of the aspect ratio. Later work (Ziff, Newman, Langlands and others) extended
and checked these.
Read the primary statements and report:
dependence.
transfer (or do not) to site percolation on the square lattice.
wrapping is also given.
Item 3 is the one we most need.
Q2 — the primal/matching relation on a torus
On an infinite plane, primal and matching lattice connectivity are complementary in a clean
way. On a torus they are famously not: both can wrap, or neither.
This is the continuum shadow of the object we measure. #628 established the finite-
Lfactthat
M(p) + M(1-p) != 0exactly and that the site duality constraint does not transport tothe square-bond lab (
r_b + r_w = 2fails in 118133 of 262144 configurations). We want thepublished version of that story.
If the difference of the two wrapping probabilities is a known exact function at
criticality, that is potentially decisive — it would say what
Mconverges to and at whatrate, which is exactly the input #618/#622 report as missing.
Q3 — the Mertens-Ziff matching function specifically
The construction we use has a specific provenance.
Specifically: is
M(p_N) = 0known to converge top_c, and if so at what rate and underwhat hypotheses? #613/#276 have a "critical-point bridge" and #618 reports that
Q_N(u) -> p_cgives no polynomial rate from the named inputsH1-H3. If the rate is in theliterature, that unblocks several tickets at once. If it is only conjectural there too, say
that — it is equally useful.
Q4 — finite-size corrections to wrapping probabilities
We measure a shape flow whose amplitude goes roughly as
N^-0.97(L^-1.94) afterprojecting out location and scale (
omega = 0.970 ± 0.031, unity not excluded). Whetherthat is a known correction exponent for this class of observable, an analytic background, or
a coincidence, is a question the literature may simply answer.
Q5 — aspect ratio
Pinson's results depend on the torus aspect ratio through modular functions. This project has
an aspect-ladder line (#567, #573, #575, #577) that measured an amplitude ratio across
aspect ratios and reported it underpowered with three surviving hypotheses.
If it does, the ladder can be scored against a theory number instead of against three
hypotheses, which converts an underpowered result into a test.
What a useful answer looks like
Per question: cite or gap, with exact statements, theorem numbers, and enough
bibliographic detail to find the source. Quoted statements are the payload. Mark conjectures
as conjectures — we have inherited one as a theorem before.
A clean "Q2 has no published answer" is a real result and tells us the finite-
Lcensus in#640 is the frontier rather than a rediscovery.
Boundaries
docs/STATUS.md, does not close a ticket.[Grok / deep retrieval] Symmetry quotients, selection rules and Fieller design: five things #598 may be rediscovering #601 (symmetry quotients and selection rules). If an answer belongs to one of those, say so
and put it there.
scripts/threshold_claim_intake.pyfirst.Related: #640, #635, #618, #622, #613, #276, #577.