Research · Research Note
Section 9 — One-page interface contract \(Paper 2 → Paper 1\)
This file is the single copy-paste certificate sheet Paper 2 should output once the reduction is complete.
It is designed so Paper 1 can be invoked with no interpretation.
Machine-parsable payload conventions for the same objects (graphs, words, logs) are fixed in 09D_CERTIFICATE_SCHEMA.md.
For the dependency arrows and the theorem-versus-certification separation (so no symbol drift occurs across sections), see 13_LOGICAL_ARROWS_AND_CERTIFICATION_SEPARATION.md.
If you run the end-to-end runner paper2_reduction/tools/run_pipeline.py on a payload directory, it writes a machine-produced contract_generated.md alongside the JSON payloads. That file is a concrete instantiation of the fields in this contract sheet, referencing the exact artifact filenames.
The same runner also writes verification.json and verification_report.md (see Section 11) unless verification is skipped. These are finite regression checks for the PA bridge constants and the bounded closing constant.
The fields below are the only quantities Paper 1 uses.
9.1 Certificate sheet (fill-in values)
| Item | Symbol / Value | How to obtain (finite / checkable) | Used by Paper 1 |
|---|---|---|---|
| Reduced base system | \((\Sigma_{\mathcal D},\sigma)\), presentation graph \(G\) | finite presentation from descriptor reduction | scope + locality |
| Descriptor size | \(d=|\mathcal D|\) | count states | constants |
| Template set | \(\mathcal T=\{T_1,\dots,T_m\}\) | explicit finite list | FTG |
| Template count | \(m\) | count templates | FTG |
| Positive entry bounds | \(\underline a,\overline a\) | min/max over positive entries | all constants |
| Nucleus periodic word | \(w_*\) (length \(L:=|w_*|\)) | chosen candidate cycle | nucleus |
| Nucleus product | \(P_*:=P(w_*)\) | multiply templates along \(w_*\) | Doeblin block |
| Core support graph | \(G(P_*)\) | edge \(i\to j\) iff \((P_*)_{j,i}>0\) | primitivity |
| Primitivity exponent | \(k_0\) | power test / Wielandt bound on \(G(P_*)\) | Doeblin block |
| Doeblin block word | \(u_*:=w_*^{k_0}\) (length \(|u_*|=Lk_0\)) | definition | Doeblin return block |
| Doeblin block matrix | \(B_*:=P(u_*)=P_*^{k_0}>0\) | numeric power | contraction |
| Entry bounds on \(B_*\) | \(\varepsilon_*:=\min(B_*)_{ij}\), \(\overline M_*:=\max(B_*)_{ij}\) | direct min/max or coarse FTG bound | Hilbert diameter |
| Hilbert diameter bound | \(\Delta_*\) or coarse bound via \(\varepsilon_*,\overline M_*\) | standard formula | Birkhoff |
| Contraction factor | \(\tau_*:=\tanh(\Delta_*/4)\) | standard | Birkhoff |
9.2 Breakthrough certificate (PA)
If Paper 2 certifies periodic approximation at the minimum (PA) via Theorem 8.3, record:
| Item | Symbol / Value | How to obtain (finite / checkable) | Used by Paper 1 |
|---|---|---|---|
| Positive bridge word | \(v\) | take \(v:=u_*\) so \(P(v)=B_*>0\) | PA |
| Bridge matrix | \(B:=P(v)>0\) | multiply | PA |
| Doeblin constants | \(\beta_{\min}:=\min B_{ij}\), \(\beta_{\max}:=\max B_{ij}\) | direct min/max or coarse bound | quasi-multiplicativity |
| Mixing/closing bound | \(L_0\) | compute mixing time on \(G\) (boolean powers) | bounded closing |
| Statement | “(PA) holds by Theorem 8.3 with \((v,B,\beta_{\min},\beta_{\max},L_0)\)” | finite verification | PA ⇒ PO route |
The machine checklist for \(L_0\), \(k_0\), and FTG-compatible bounds is in 08A_PA_CERTIFICATE_CHECKLIST.md.
9.3 Uniqueness certificate on periodic orbits (SCG)
Paper 1 treats SCG as a certificate input.
SCG must be packaged as a machine-verifiable object per 09A_SCG_CERTIFICATE_CHECKLIST.md.
Before claiming SCG, run the obstruction test and regime classification in:
09C_SCG_OBSTRUCTION_AND_REGIMES.md.
Record the following fields:
| Item | Symbol / Value | How to obtain (finite / checkable) | Used by Paper 1 |
|---|---|---|---|
| Nucleus periodic word | \(w_*\) | same \(w_*\) as above | identity |
| Claimed strict gap | \(\delta>0\) | certificate value | strict uniqueness |
| Numeric tolerance | \(\varepsilon_{\rm num}>0\) | certificate value | audit |
| Route tag | one of {FINITE_RADIUS, QUOTIENT_CLASS, INEQUALITY_SYSTEM, OTHER} | choose route | audit |
| Route payload location | file / appendix name | finite data used to close the universal quantifier | audit |
| SCG regime + obstruction test | P/A + pass/fail | run 09C sanity check | audit |
| SCG statement | \(\lambda(w)\ge \lambda(w_*)+\delta\) for all other primitive periodic \(w\not\sim w_*\) | verified from payload | strict uniqueness |
A practical guide to producing such a certificate is in 09B_SCG_CERTIFICATE_GUIDE.md.
9.4 Consequences to record (automatic once inputs are present)
If both (PA) and SCG are certified, then Paper 1 supplies:
- (PO) periodic realization at \(\mathcal O_*\)
- positive return density to \([\mathcal U_*]\) for every ergodic minimizing measure \(\mu\), hence exponential projective synchronization along minimizing dynamics
In the purely periodic case (\(\mathcal M_* = \{\mu_{w_*}\}\)), Paper 1 also gives explicit periodic return constants:
- \(\eta_* = 1/|u_*| = 1/(|w_*|k_0)\)
- return times to \([\mathcal U_*]\) form an arithmetic progression with step \(|u_*|\) (up to phase)
9.5 Projective regularity certificate (PR) on the minimizing support (Regime A route)
Use this section when the SCG obstruction triggers (Regime A) and a global cycle-gap statement is not being claimed.
A PR certificate replaces the SCG→PO→return-density route by an explicit return certificate on a specified minimizing subshift.
Record the following fields:
| Item | Symbol / Value | How to obtain (finite / checkable) | Used by Paper 1 | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Regime tag | A or P | run 09C obstruction test | audit | ||||||
| Minimizing support subgraph | G_min ⊂ G (payload location) | explicit vertex/edge lists or attached file | defines support Y | ||||||
| Selection route tag | EDGE_PRUNING_LB | per 10B | audit | ||||||
| Selection parameters | δprune, εnum, L_max (if used) | record from pruning payload | audit | ||||||
| Selection payload log | path / hash | attach payload list for deleted edges | audit | ||||||
| Doeblin return word | u_* | same as Section 9.1 | contraction block | ||||||
| Return check route tag | AUTONOMOUS_RETURN_AUTOMATON | per 10A | audit | ||||||
| Return check result | pass/fail | cycle test on the avoidance automaton graph H | validity | ||||||
| Bounded-gap constant | W_* | safe bound W_* = | V_min | · | u_* | + | u_* | or exact longest-path bound | return density |
| Uniform return density lower bound | η* := 1/W* | derived | contraction rate |
Once PR is present \(Doeblin block + τ* from Section 9.1, plus W* here\), Paper 1 can be invoked on the minimizing support without SCG (see Section 10).
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