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RTT Agentic Module: Fel’s Conjecture on Syzygies of Numerical Semigroups

Module ID: fel_syzygies_rtt
Source paper: https://arxiv.org/pdf/2602.03716

This module wraps the paper “Fel’s Conjecture on Syzygies of Numerical Semigroups” in RTT operator grammar.
It preserves the authors’ mathematics while exposing the structural regimes that govern syzygy power sums, gap power sums, and the universal symmetric polynomials Tn.

1. Purpose#

  • Make the paper agentic and machine-navigable.
  • Clarify the regime structure behind Fel’s conjecture.
  • Provide students with a clean conceptual map.
  • Support AI agents in reasoning over the paper without drift.

2. Core RTT view of the paper#

The paper proves Fel’s conjecture:

[ K_p(S) = \sum_{r=0}^p \binom{p}{r} T_{p-r}(\sigma), G_r(S) ;+; \frac{2^{p+1}}{p+1} T_{p+1}(\delta) ]

This identity links three interacting structures:

  • Hilbert series regime:
    Syzygy degrees appear in the Hilbert numerator (Q_S(z)).

  • Gap power regime:
    The gap set Δ produces power sums (G_r(S)).

  • Universal polynomial regime:
    The polynomials (T_n(\sigma)) and (T_n(\delta)) encode universal identities.

The proof uses exponential generating functions to unify these regimes.

3. RTT structures in this module#

Regimes#

  • hilbert_series_regime
  • gap_power_regime
  • universal_polynomial_regime
  • formalization_regime

Tensions#

  • combinatorial_vs_algebraic
  • explicit_formula_vs_universal_identity
  • natural_language_vs_formal_proof

Transitions#

  • hilbert_to_gap_transition
  • gap_to_universal_transition
  • informal_to_formal_transition

4. Operators#

  • syzygy_power_operator — computes alternating syzygy power sums.
  • gap_power_operator — computes gap power sums.
  • universal_T_operator — evaluates universal symmetric polynomials.
  • egf_operator — performs exponential generating function conversions.

5. How to use this module#

  • Students:
    Use this README alongside the PDF to understand how syzygies, gaps, and universal polynomials interact.

  • Researchers:
    Query the module’s regimes and operators to explore structural dependencies.

  • Agents:
    Treat module.json as the canonical structural map of the paper.

6. Provenance#

  • Module authoring: TriadicFrameworks (RTT / agentic mapping).
  • Original content: Authors of arXiv:2602.03716.
  • License: Open educational use permitted.

✅ diagram.txt#

(ASCII regime–tension–transition map)

           +---------------------------------------+
           | fel_syzygies_rtt                      |
           +---------------------------------------+
 
REGIMES
  [R1] hilbert_series_regime
  [R2] gap_power_regime
  [R3] universal_polynomial_regime
  [R4] formalization_regime
 
TENSIONS
  [T1] combinatorial_vs_algebraic      (R1 <--> R2)
  [T2] explicit_formula_vs_universal   (R2 <--> R3)
  [T3] natural_language_vs_formal      (R3 <--> R4)
 
TRANSITIONS
  [X1] hilbert_to_gap_transition       (R1 -> R2)
  [X2] gap_to_universal_transition     (R2 -> R3)
  [X3] informal_to_formal_transition   (R3 -> R4)
 
FLOW
  hilbert_series_regime (R1)
        |
        v
  gap_power_regime (R2)
        |
        v
  universal_polynomial_regime (R3)
        |
        v
  formalization_regime (R4)

Updated

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