Theory Center

Unified Mathematical Map

The cards below summarize shared mathematical notions across reports and map each notion to concrete report pages.

Goal: keep the mathematics continuous from notation to derivation, then to evidence and cross-report verification.

Reading Protocol

Start from Chapter 0/1 in Book for notation and core FPT objects. /book
Use concept cards below to locate every notion in concrete reports.
Inspect stage matrix and claim ledger to verify model/method/result/finding continuity.
Use duplication governance to distinguish shared foundations from harmful repetition.

Theory cards 6 Claims 130 Book chapters 8

Continuity Spine

Acts define the long-form argument structure, and chapter spine transitions show how one verified block feeds the next.

Act I: Foundations

Unify notation, FPT objects, and baseline assumptions before model-specific branches.

Chapters: 2

Act II: Family Mechanisms

Expand Grid2D and Ring families with derivation-backed interaction checkpoints.

Chapters: 3

Act III: Cross-Model Synthesis

Merge claims into a shared explanatory frame and validate reproducibility pathways.

Chapters: 2

Act IV: Outlook

Stabilize conclusions, isolate unresolved gaps, and define next-step research questions.

Chapters: 1

Chapter 7: Outlook & Open Questions

Open chapter transitions

Chapter 0: Reading Guide & Notation: Carry notation and verified claims from Chapter 0: Reading Guide & Notation into Chapter 1: Core FPT Concepts, then extend mechanism and evidence without resetting assumptions.
Chapter 1: Core FPT Concepts: Carry notation and verified claims from Chapter 1: Core FPT Concepts into Chapter 2: Grid2D Family, then extend mechanism and evidence without resetting assumptions.
Chapter 2: Grid2D Family: Bridge note: keep the same hazard/survival diagnostics from Grid2D, then switch geometry to ring so shortcut and lazy parameters can be isolated without boundary-shape confounders.
Chapter 3: Ring Family Baseline: Carry notation and verified claims from Chapter 3: Ring Family Baseline into Chapter 4: Shortcut Variants, then extend mechanism and evidence without resetting assumptions.
Chapter 4: Shortcut Variants: Carry notation and verified claims from Chapter 4: Shortcut Variants into Chapter 5: Cross-Model Synthesis, then extend mechanism and evidence without resetting assumptions.
Chapter 5: Cross-Model Synthesis: Before any new claims are added, move from synthesis to reproducibility gates and verify command-, schema-, and artifact-level closure.
Chapter 6: Reproducibility & Validation: Only unresolved items that pass reproducibility constraints should enter the outlook as auditable hypotheses.
Chapter 7: Outlook & Open Questions: No downstream chapter; consolidate assumptions, claims, and open questions.

Concept Cards

First-passage distribution

Core PMF/CDF/survival quantities used across the major report families.

Reports: 26

Show all linked reports

Survival and hazard

Links between f(t), S(t), and hazard-style diagnostics.

Reports: 14

Show all linked reports

Hazard interpretation

Peak/valley interpretation using hazard dynamics.

Reports: 12

Show all linked reports

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Reports: 13

Show all linked reports

Spectral decomposition

Eigenvalue / resolvent based derivations.

Reports: 3

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Reports: 11

Show all linked reports

Notation & Term Lock

These locked terms keep CN/EN semantics aligned and provide formula anchors for quick lookup.

AW Inversion

Discrete Cauchy/FFT-based inversion from generating functions to time-domain FPT quantities.

f_t \approx FFT(F(z_k))

method Reports 14

Beta Scan

Parameter sweep over shortcut strength β to identify phase shifts and regime boundaries.

β ∈ [0,1]

parameter Reports 14

Bimodality Criterion

Operational criterion to separate true two-peak structure from noisy shoulders.

peak-valley-peak consistency

diagnostic Reports 14

Claim Ledger

Structured mapping from statement to evidence paths and cross-report links.

claim -> evidence -> linked reports

workflow Reports 14

Equal4 Baseline

Four-way equalized baseline used to compare shortcut effects under symmetric local movement.

p(±1)=p(±2)

shortcut-variant Reports 14

First-Passage Time (FPT)

Random time needed for the trajectory to hit an absorbing target for the first time.

f(t)

core-metric Reports 14

Hazard Rate

Conditional probability of first passage at step t given survival up to t.

h(t)=f(t)/S(t-1)

core-metric Reports 14

Renormalize Shortcut Mode

Base transition weights are rescaled after shortcut injection to preserve normalization constraints.

p_i' = c · p_i

shortcut-variant Reports 14

Selfloop Shortcut Mode

Shortcut probability mass is taken from self-loop probability without renormalizing other moves.

p_stay -> p_stay-β

shortcut-variant Reports 14

Survival Function

Probability that first passage has not happened by step t.

S(t)=P(T>t)

core-metric Reports 14

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

theory-card Reports 11

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

theory-card Reports 13

Stage Coverage Matrix

Each row tracks whether one report closes the full chain from assumptions to findings.

Report	Model	Method	Result	Finding	Claims	Formulas	Book
Cross-Model Luca Regime Map	1	1	1	2	5	1	chapter-5-cross-model-synthesis
Reflecting Encounter Mean Validation	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D Bimodality Baseline	1	1	1	2	5	14	chapter-0-reading-guide
Grid2D Blackboard Endpoint Case	1	1	1	2	5	8	chapter-2-grid2d-family
Grid2D Membrane Near Target	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D One Target — Base	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D One Target — Window Measures	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D One vs Two Target — Gating	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D Rectangle Bimodality	1	1	1	2	5	1	chapter-2-grid2d-family
Grid2D Reflecting-Boundary Bimodality	1	1	1	2	5	14	chapter-2-grid2d-family
Grid2D Two-Target Bias-Radius Scaffold	1	1	1	2	5	1	chapter-6-repro-validation
Grid2D Two-Target Double-Peak	1	1	1	2	5	1	chapter-1-core-fpt
2D Two-Walker Encounter With Shortcut	1	1	1	2	5	1	chapter-6-repro-validation
Ring Derivation Backbone	1	1	1	2	5	14	chapter-0-reading-guide
Lazy Ring Flux Baseline	1	1	1	2	5	2	chapter-1-core-fpt
Lazy Ring Jump-Over Mechanism (K2 vs K4)	1	1	1	2	5	1	chapter-3-ring-baseline
Lazy Ring Shortcut Beta Scan	1	1	1	2	5	1	chapter-4-shortcut-variants
Lazy Ring Shortcut Figure-1 Revision	1	1	1	2	5	1	chapter-4-shortcut-variants
Two-Target Lazy Ring Mechanics	1	1	1	2	5	1	chapter-0-reading-guide
1D Ring Two-Walker Encounter With Shortcut	1	1	1	2	5	1	chapter-6-repro-validation
Ring Valley Regime Map	1	1	1	2	5	6	chapter-3-ring-baseline
Destination-Scan Valley Control	1	1	1	2	5	1	chapter-4-shortcut-variants
Reflecting Encounter Diagonal Decomposition	1	1	1	1	4	1	chapter-6-repro-validation
Exact Recursion — Method Guide	1	1	1	1	4	1	chapter-6-repro-validation
Final Multitimescale FPT and Encounter Report	1	1	1	1	4	10	chapter-6-repro-validation
Grid2D One Target — Exit Timing	1	1	1	1	4	1	chapter-6-repro-validation
Grid2D One Target — Valley/Peak Budget	1	1	1	1	4	1	chapter-6-repro-validation

Formula Depth Governance

Thresholds are applied by report family so lightweight notes are explicit exceptions instead of silent gaps.

Report	Group	Formulas	Min	Target	Status	Exception
Cross-Model Luca Regime Map	cross	1	2	4	FAIL	-
Reflecting Encounter Diagonal Decomposition	misc	1	2	4	FAIL	-
Reflecting Encounter Mean Validation	misc	1	2	4	FAIL	-
Exact Recursion — Method Guide	misc	1	2	4	FAIL	-
Final Multitimescale FPT and Encounter Report	misc	10	2	4	PASS	-
Grid2D Bimodality Baseline	grid2d	14	8	12	PASS	-
Grid2D Blackboard Endpoint Case	grid2d	8	8	12	PASS	-
Grid2D Membrane Near Target	grid2d	1	8	12	FAIL	-
Grid2D One Target — Base	grid2d	1	8	12	FAIL	-
Grid2D One Target — Exit Timing	grid2d	1	8	12	FAIL	-
Grid2D One Target — Valley/Peak Budget	grid2d	1	8	12	FAIL	-
Grid2D One Target — Window Measures	grid2d	1	8	12	FAIL	-
Grid2D One vs Two Target — Gating	grid2d	1	8	12	FAIL	-
Grid2D Rectangle Bimodality	grid2d	1	8	12	FAIL	-
Grid2D Reflecting-Boundary Bimodality	grid2d	14	8	12	PASS	-
Grid2D Two-Target Bias-Radius Scaffold	grid2d	1	8	12	FAIL	-
Grid2D Two-Target Double-Peak	grid2d	1	8	12	FAIL	-
2D Two-Walker Encounter With Shortcut	grid2d	1	8	12	FAIL	-
Ring Derivation Backbone	ring	14	4	8	PASS	-
Lazy Ring Flux Baseline	ring	2	4	8	FAIL	-
Lazy Ring Jump-Over Mechanism (K2 vs K4)	ring	1	4	8	FAIL	-
Lazy Ring Shortcut Beta Scan	ring	1	2	4	FAIL	extension_delta Extension report focuses on delta from baseline derivation.
Lazy Ring Shortcut Figure-1 Revision	ring	1	2	4	FAIL	extension_revision Revision emphasizes corrected claims over repeated derivations.
Two-Target Lazy Ring Mechanics	ring	1	4	8	FAIL	-
1D Ring Two-Walker Encounter With Shortcut	ring	1	4	8	FAIL	-
Ring Valley Regime Map	ring	6	4	6	PASS	-
Destination-Scan Valley Control	ring	1	4	8	FAIL	-

Notion → Claim → Evidence

First-passage distribution

Core PMF/CDF/survival quantities used across the major report families.

Model ring_valley (evidence 5)
The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut…
Model ring_deriv_k2 (evidence 4)
The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-…
Model grid2d_bimodality (evidence 4)
The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Model grid2d_blackboard_bimodality (evidence 4)
The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.

Survival and hazard

Links between f(t), S(t), and hazard-style diagnostics.

Model ring_valley (evidence 5)
The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Model ring_lazy_jump (evidence 3)
A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.
Model ring_lazy_jump_ext (evidence 3)
The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched param…
Model ring_valley_dst (evidence 3)
The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target;…

Hazard interpretation

Peak/valley interpretation using hazard dynamics.

Model ring_valley (evidence 5)
The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Model ring_lazy_jump (evidence 3)
A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.
Model ring_lazy_jump_ext (evidence 3)
The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched param…
Model ring_valley_dst (evidence 3)
The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target;…

Beta / shortcut scan

How shortcut strength changes bimodality and phase behavior.

Model ring_valley (evidence 5)
The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut…
Model ring_deriv_k2 (evidence 4)
The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Model grid2d_blackboard_bimodality (evidence 4)
The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.
Model ring_lazy_jump (evidence 3)
A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.

Spectral decomposition

Eigenvalue / resolvent based derivations.

Model ring_deriv_k2 (evidence 4)
The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-…
Model grid2d_bimodality (evidence 4)
The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Method ring_deriv_k2 (evidence 5)
The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be n…
Method grid2d_bimodality (evidence 5)
The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter sca…

AW inversion

Discrete Cauchy / FFT inversion from generating functions.

Model ring_valley (evidence 5)
The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut…
Model ring_deriv_k2 (evidence 4)
The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-…
Model grid2d_bimodality (evidence 4)
The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary…
Model grid2d_reflecting_bimodality (evidence 4)
The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).
Model final_multitimescale_fpt_encounter (evidence 4)
Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).

Duplication Governance

Shared Foundations

Count: 8

f(t)=\Pr[T=t],\quad S(t)=\Pr[T>t],\quad h(t)=\frac{f(t)}{S(t-1)} (0)
\begin{aligned} p_{\text{left}} &= \frac{q}{4}(1+g_x),\quad p_{\text{right}} = \frac{q}{4}(1-g_x),\\ p_{\text{down}} &=… (0)
\Delta=\delta p_{\text{stay}} (0)
q_{\text{site}}=\text{factor}\cdot q (0)
A_{v,u}=\Pr(u\to v) (0)
\mathbf{p}_{t+1}=A\mathbf{p}_t (0)

Redundant Duplicates

Count: 1

f(t)=\Pr[T=t] (0)

Consistency Checks

all_reports_have_formula: PASS

Details

ring_lazy_jump: 1
ring_lazy_jump_ext: 1
ring_lazy_jump_ext_rev2: 1
ring_lazy_flux: 2
ring_valley: 6
22 more fields in raw JSON.

Open raw check payload

{
  "ring_lazy_jump": 1,
  "ring_lazy_jump_ext": 1,
  "ring_lazy_jump_ext_rev2": 1,
  "ring_lazy_flux": 2,
  "ring_valley": 6,
  "ring_valley_dst": 1,
  "ring_deriv_k2": 14,
  "ring_two_target": 1,
  "ring_two_walker_encounter_shortcut": 1,
  "grid2d_bimodality": 14,
  "grid2d_reflecting_bimodality": 14,
  "grid2d_blackboard_bimodality": 8,
  "grid2d_two_target_double_peak": 1,
  "grid2d_two_walker_encounter_shortcut": 1,
  "grid2d_rect_bimodality": 1,
  "grid2d_membrane_near_target": 1,
  "cross_luca_regime_map": 1,
  "final_multitimescale_fpt_encounter": 10,
  "grid2d_one_target_base": 1,
  "grid2d_one_target_exit_timing": 1,
  "grid2d_one_target_valley_peak_budget": 1,
  "grid2d_one_target_window_measures": 1,
  "grid2d_one_two_target_gating": 1,
  "exact_recursion_method_guide": 1,
  "encounter_reflecting_diagonal_decomp": 1,
  "encounter_reflecting_mean_validation": 1,
  "grid2d_two_target_bias_radius": 1
}

Raw payload is available in /data/v1/theory_map.json

formula_depth_policy: FAIL

Details

rows: 27
failure_report_ids: 21
exception_rows: 2

Open raw check payload

{
  "rows": [
    {
      "report_id": "cross_luca_regime_map",
      "group": "cross",
      "formula_count": 1,
      "min_required": 2,
      "target": 4,
      "pass": false,
      "policy_note": "Cross synthesis can be concise if it cites upstream derivations.",
      "exception_tag": "",
      "exception_reason": ""
    },
    {
      "report_id": "grid2d_bimodality",
      "group": "grid2d",
      "formula_count": 14,
      "min_required": 8,
      "target": 12,
      "pass": true,
      "policy_note": "Grid2D reports should carry full derivation context.",
      "exception_tag": "",
      "exception_reason": ""
    },
    {
      "report_id": "grid2d_blackboard_bimodality",
      "group": "grid2d",
      "formula_count": 8,
      "min_required": 8,
      "target": 12,
      "pass": true,
      "policy_note": "Grid2D reports should carry full derivation context.",
      "exception_tag": "",
      "exception_reason": ""
    },
    {
      "report_id": "grid2d_membrane_near_target",
      "group": "grid2d",
      "formula_count": 1,
      "min_required": 8,
      "target": 12,
      "pass": false,
      "policy_note": "Grid2D reports should carry full derivation context.",
      "exception_tag": "",
      "exception_reason": ""
    },
    {
      "report_id": "grid2d_one_target_base",
      "group": "grid2d",
      "formula_count": 1,
      "min_required": 8,
      "target": 12,
      "pass": false,
      "policy_note": "Grid2D reports should carry full derivation context.",
      "exception_tag": "",
      "exception_reason": ""
    },
    {
      "report_id": "grid2d_one_target_exit_timing",
      "group": "grid2d",
      "formula_count": 1,
      "min_required": 8,
      "target": 12,
      "pass": false,

... truncated 8572 chars. Open /data/v1/theory_map.json for full payload.

Raw payload is available in /data/v1/theory_map.json

all_reports_mapped_in_theory_cards: PASS
Details
- unmapped_report_ids:
Open raw check payload
```
{
  "unmapped_report_ids": []
}
```
Raw payload is available in /data/v1/theory_map.json
duplicate_key_findings: PASS
Details
- No issues detected.
Open raw check payload
```
[]
```
Raw payload is available in /data/v1/theory_map.json

duplicate_math_signatures: PASS

Details

redundant_count: 1
shared_core_count: 8

Open raw check payload

{
  "redundant_signatures": [
    {
      "latex": "f(t)=\\Pr[T=t]",
      "report_count": 8,
      "occurrence_count": 8,
      "report_ids": [
        "encounter_reflecting_diagonal_decomp",
        "encounter_reflecting_mean_validation",
        "exact_recursion_method_guide",
        "grid2d_one_target_base",
        "grid2d_one_target_exit_timing",
        "grid2d_one_target_window_measures",
        "grid2d_one_two_target_gating",
        "grid2d_two_target_bias_radius"
      ]
    }
  ],
  "shared_core_signatures": [
    {
      "latex": "f(t)=\\Pr[T=t],\\quad S(t)=\\Pr[T>t],\\quad h(t)=\\frac{f(t)}{S(t-1)}",
      "report_count": 10,
      "occurrence_count": 10,
      "report_ids": [
        "cross_luca_regime_map",
        "grid2d_membrane_near_target",
        "grid2d_rect_bimodality",
        "grid2d_two_target_double_peak",
        "grid2d_two_walker_encounter_shortcut",
        "ring_lazy_jump",
        "ring_lazy_jump_ext",
        "ring_two_target",
        "ring_two_walker_encounter_shortcut",
        "ring_valley_dst"
      ]
    },
    {
      "latex": "\\begin{aligned} p_{\\text{left}} &= \\frac{q}{4}(1+g_x),\\quad p_{\\text{right}} = \\frac{q}{4}(1-g_x),\\\\ p_{\\text{down}} &= \\frac{q}{4}(1+g_y),\\quad p_{\\text{up}} = \\frac{q}{4}(1-g_y),\\\\ p_{\\text{stay}} &= 1-q. \\end{aligned}",
      "report_count": 3,
      "occurrence_count": 3,
      "report_ids": [
        "grid2d_bimodality",
        "grid2d_blackboard_bimodality",
        "grid2d_reflecting_bimodality"
      ]
    },
    {
      "latex": "\\Delta=\\delta p_{\\text{stay}}",
      "report_count": 3,
      "occurrence_count": 3,
      "report_ids": [
        "grid2d_bimodality",
        "grid2d_blackboard_bimodality",
        "grid2d_reflecting_bimodality"
      ]
    },
    {

... truncated 1255 chars. Open /data/v1/theory_map.json for full payload.

Raw payload is available in /data/v1/theory_map.json

asset_label_duplication: PASS
Details
- duplicate_count: 0
Open raw check payload
```
{
  "duplicate_count": 0,
  "reports_with_duplicate_labels": 0,
  "total_reports_scanned": 22,
  "examples": {}
}
```
Raw payload is available in /data/v1/theory_map.json

Cross-Report Routes

These routes stitch reports into continuous tracks instead of isolated pages.

ring

grid2d

cross

Cross-Model Luca Regime Map

misc

Content-Level Arcs

ring progression

ring track links 9 reports into one continuous argument.

Checkpoints: 9 Claims: 45

Open checkpoints

grid2d progression

grid2d track links 13 reports into one continuous argument.

Checkpoints: 13 Claims: 63

Open checkpoints

cross progression

cross track links 1 reports into one continuous argument.

Checkpoints: 1 Claims: 5

Open checkpoints

Cross-Model Luca Regime Map

misc progression

misc track links 4 reports into one continuous argument.

Checkpoints: 4 Claims: 17

Open checkpoints

Grid and ring mechanisms converge into cross-report synthesis.

Global storyline that connects all report families from mechanism to synthesis.

Checkpoints: 27 Claims: 130

Open checkpoints