Trigger
The recent modulus program has done something useful by failing.
The next theoretical response should not be to add a longer list of isolated functions of tau.
Recent loop-model work gives a more appropriate object: a modular-covariant solution space labelled not only by fields but by connectivity/combinatorial-map data.
External input
Roux--Ribault--Jacobsen, arXiv:2604.24491, construct torus one-point functions in critical loop models using sphere four-point data. For the simplest primary insertions they find multiple modular-covariance solutions rather than one universal scalar shape; the torus functions are infinite non-chiral conformal-block combinations and the correlation function includes a combinatorial-map/connectivity pattern as part of its definition.
This continues the program of Grans-Samuelsson et al., arXiv:2302.08168, where ribbon/combinatorial maps organize loop-model correlation functions and bootstrap solution spaces.
Ang et al., arXiv:2604.05503, now supplies exact generic-loop three-point structure constants for many charged leg fields, so surviving map/charge sectors can eventually be tied to OPE data rather than only fitted as arbitrary torus functions.
The practical lesson is:
The correct torus hypothesis may be a low-dimensional subspace of map-resolved modular-covariant solutions, not one ray such as E4, r, or a single logarithmic block.
Primary question
For the exact lattice observable being measured, what is the smallest symmetry-allowed torus solution space that survives existing modulus data?
Write schematically
y(tau) = sum_a c_a F_a(tau),
where each F_a is a declared modular-covariant/map sector and the coefficient vector is constrained by exact lattice symmetries/selection rules.
The first result should be the dimension and identity of the allowed solution subspace, not a named continuum field.
Phase A — no new Monte Carlo: build the candidate sector dictionary
Start from already-measured torus/modulus assets only.
For each candidate sector record:
field/representation label when known,
combinatorial-map/connectivity label,
spin and C4/D4 transformation,
primal/matching parity when defined,
deck/homology character compatibility,
modular covariance law,
known logarithmic/non-chiral block content,
which existing lattice observable could realize it.
Use exact repository constraints aggressively:
If no credible lattice-to-map dictionary exists for a candidate solution, do not include it merely because its numerical shape is convenient.
Phase B — score subspaces, not ratios
For every existing modulus block retain the raw correlated response vector and full covariance. Compare a candidate sector space V by
D = min_a (y - V a)^T S^+ (y - V a),
using #579's projective/subspace machinery.
This has three advantages:
- no amplitude denominator is nominated;
- a legitimate two-sector continuum hypothesis is tested as a two-dimensional space rather than as an arbitrary fitted ratio;
- a known systematic direction can be carried as a nuisance column instead of deleting the coordinate that could falsify it.
Report identifiable rank and degrees of freedom explicitly.
Phase C — held-out modulus tomography
A multi-dimensional solution space only becomes informative if it predicts something not used to choose its coefficients.
Use the existing modulus ladder in a leave-one-modulus-out design where possible:
fit amplitudes/map mixture on source moduli,
freeze the solution subspace,
predict the full held-out response vector.
If current data have too few independent moduli, identify the one next modulus that maximizes principal-angle separation between surviving solution spaces. Do not default to another larger aspect ratio.
The held-out target can also be a different exact symmetry projector, e.g. a C3 homology character, if it is genuinely tied to the same continuum sector.
Phase D — exploit the sphere--torus relation
A particularly strong outcome of arXiv:2604.24491 is that torus modular covariance is related to sphere four-point crossing in a corresponding channel/map description.
For any torus sector that survives:
- identify the associated sphere four-point solution/map;
- construct a lattice sphere/plane four-point observable with the same connectivity semantics if feasible;
- compare a normalization-free cross-ratio vector or OPE ratio.
This creates an independent geometric test of the same sector. A torus fit that has no consistent sphere crossing realization should be downgraded.
Phase E — interface to exact three-point/OPE data
Do not rerun the old broad charged tables merely because exact 2026 OPE constants exist.
For a surviving map/charge sector, build an insertion faithful to the generic-loop field V_(r,s) and compare normalization-independent structure constants from arXiv:2604.05503. The target is
map/representation tensor
+
scaling dimension/spin
+
normalized OPE coefficient,
not another one-point amplitude.
This is the appropriate point to connect to #250.
What this changes about the current modulus program
It does not rescue a rejected ray automatically
A failed E4/weight-4 ray remains failed. Enlarging to a solution space is justified only by a concrete map/representation sector, not by post-hoc basis expansion.
It changes the meaning of high rank
Several torus solutions or combinatorial-map sectors can contribute to one scalar lattice projection. A multi-dimensional modulus response need not mean many unrelated irrelevant exponents.
It makes angular calibration prior to field identification mandatory
#583 should first establish what angular Fourier content is actually being measured. A torus solution-space fit should consume calibrated angular coordinates rather than ask continuum amplitudes to absorb an incorrect A4 extraction.
Decision table
One map-resolved ray survives held-out moduli
Promote a specific torus sector and demand its sphere/OPE fingerprint before operator naming.
A small map-resolved subspace survives, one-dimensional rays fail
Interpret the observable as a mixture/solution-space projection. Use exact symmetry/charge readouts to separate the components rather than adding more radial sizes.
The required subspace dimension grows with every modulus
Downgrade low-dimensional torus closure. Move to annulus/context-Hankel/process descriptions rather than a larger list of modular functions.
No candidate has a defensible lattice-to-map dictionary
Stop scalar modulus production. Design a marked/charged/connectivity-resolved observable whose combinatorial-map semantics are explicit.
Claim boundary
- A modular-covariant solution is not by itself a field identification.
- A combinatorial-map label is part of the correlation-function definition, not evidence of extra microscopic degrees of freedom.
- Fitting a larger subspace after reveal is exploratory until it predicts a held-out modulus/readout.
- Existing N=290/N=580 views remain the same evidence blocks under reanalysis.
Deliverable
1. map/representation/symmetry candidate dictionary;
2. numerical basis for the smallest relevant torus solution spaces;
3. covariance-weighted projective/subspace scores on existing moduli;
4. leave-one-modulus/readout-out predictions;
5. ranked next modulus or marked-observable design;
6. sphere/OPE cross-check for any surviving sector.
Literature anchors:
- Roux, Ribault, Jacobsen, arXiv:2604.24491.
- Grans-Samuelsson et al., arXiv:2302.08168.
- Ang et al., arXiv:2604.05503.
- Jacobsen, Nivesvivat, Ribault, Roux, arXiv:2510.04701.
Related: #114, #156, #220, #244, #249, #250, #333, #337, #576, #579, #583.
Trigger
The recent modulus program has done something useful by failing.
A8/A4.The next theoretical response should not be to add a longer list of isolated functions of
tau.Recent loop-model work gives a more appropriate object: a modular-covariant solution space labelled not only by fields but by connectivity/combinatorial-map data.
External input
Roux--Ribault--Jacobsen, arXiv:2604.24491, construct torus one-point functions in critical loop models using sphere four-point data. For the simplest primary insertions they find multiple modular-covariance solutions rather than one universal scalar shape; the torus functions are infinite non-chiral conformal-block combinations and the correlation function includes a combinatorial-map/connectivity pattern as part of its definition.
This continues the program of Grans-Samuelsson et al., arXiv:2302.08168, where ribbon/combinatorial maps organize loop-model correlation functions and bootstrap solution spaces.
Ang et al., arXiv:2604.05503, now supplies exact generic-loop three-point structure constants for many charged leg fields, so surviving map/charge sectors can eventually be tied to OPE data rather than only fitted as arbitrary torus functions.
The practical lesson is:
Primary question
For the exact lattice observable being measured, what is the smallest symmetry-allowed torus solution space that survives existing modulus data?
Write schematically
where each
F_ais a declared modular-covariant/map sector and the coefficient vector is constrained by exact lattice symmetries/selection rules.The first result should be the dimension and identity of the allowed solution subspace, not a named continuum field.
Phase A — no new Monte Carlo: build the candidate sector dictionary
Start from already-measured torus/modulus assets only.
For each candidate sector record:
Use exact repository constraints aggressively:
pi({1,0});X=r-1is a typed topological coordinate;If no credible lattice-to-map dictionary exists for a candidate solution, do not include it merely because its numerical shape is convenient.
Phase B — score subspaces, not ratios
For every existing modulus block retain the raw correlated response vector and full covariance. Compare a candidate sector space
Vbyusing #579's projective/subspace machinery.
This has three advantages:
Report identifiable rank and degrees of freedom explicitly.
Phase C — held-out modulus tomography
A multi-dimensional solution space only becomes informative if it predicts something not used to choose its coefficients.
Use the existing modulus ladder in a leave-one-modulus-out design where possible:
If current data have too few independent moduli, identify the one next modulus that maximizes principal-angle separation between surviving solution spaces. Do not default to another larger aspect ratio.
The held-out target can also be a different exact symmetry projector, e.g. a C3 homology character, if it is genuinely tied to the same continuum sector.
Phase D — exploit the sphere--torus relation
A particularly strong outcome of arXiv:2604.24491 is that torus modular covariance is related to sphere four-point crossing in a corresponding channel/map description.
For any torus sector that survives:
This creates an independent geometric test of the same sector. A torus fit that has no consistent sphere crossing realization should be downgraded.
Phase E — interface to exact three-point/OPE data
Do not rerun the old broad charged tables merely because exact 2026 OPE constants exist.
For a surviving map/charge sector, build an insertion faithful to the generic-loop field
V_(r,s)and compare normalization-independent structure constants from arXiv:2604.05503. The target isnot another one-point amplitude.
This is the appropriate point to connect to #250.
What this changes about the current modulus program
It does not rescue a rejected ray automatically
A failed
E4/weight-4 ray remains failed. Enlarging to a solution space is justified only by a concrete map/representation sector, not by post-hoc basis expansion.It changes the meaning of high rank
Several torus solutions or combinatorial-map sectors can contribute to one scalar lattice projection. A multi-dimensional modulus response need not mean many unrelated irrelevant exponents.
It makes angular calibration prior to field identification mandatory
#583 should first establish what angular Fourier content is actually being measured. A torus solution-space fit should consume calibrated angular coordinates rather than ask continuum amplitudes to absorb an incorrect
A4extraction.Decision table
One map-resolved ray survives held-out moduli
Promote a specific torus sector and demand its sphere/OPE fingerprint before operator naming.
A small map-resolved subspace survives, one-dimensional rays fail
Interpret the observable as a mixture/solution-space projection. Use exact symmetry/charge readouts to separate the components rather than adding more radial sizes.
The required subspace dimension grows with every modulus
Downgrade low-dimensional torus closure. Move to annulus/context-Hankel/process descriptions rather than a larger list of modular functions.
No candidate has a defensible lattice-to-map dictionary
Stop scalar modulus production. Design a marked/charged/connectivity-resolved observable whose combinatorial-map semantics are explicit.
Claim boundary
Deliverable
Literature anchors:
Related: #114, #156, #220, #244, #249, #250, #333, #337, #576, #579, #583.