billiards-triangles 005: death law completed - parametric necessity for all (a,b), death = gamma_d exactly for 20 members

mechanisms.json

@@ -124,6 +124,10 @@
       "field": "logic-methodology",
       "description": "Cross-field ideation sweep: apply untried field lenses to one problem in parallel to generate candidate attack routes; output is a MAP of triaged leads and recorded no-purchase verdicts (docs/IDEATE.md)."
     },
+    "formal-parameter-specialization": {
+      "field": "algebra",
+      "description": "Prove an infinite family of identities at one stroke by moving the family parameters into the exponent lattice: adjoin e^{i a alpha}, e^{i b beta} (or analogous parameter-indexed units) as independent formal Laurent variables, verify the identity once by exact polynomial subtraction in the enlarged ring, and specialize by ring homomorphism to every integer parameter value. Turns a hand geometric-series computation into a finite machine check; needs a short hand derivation that the geometric quantities equal the formal closed forms."
+    },
     "girth-restriction": {
       "field": "graph-theory",
       "description": "Restrict the search to high-girth instances, where short-cycle escapes are unavailable and structure is forced."
@@ -212,6 +216,10 @@
       "field": "analysis",
       "description": "Examine the set of achieved objective values for gaps and accumulation points, not just the extremum."
     },
+    "structure-class-design-search": {
+      "field": "dynamics",
+      "description": "Invert a proven structural mechanism into a design tool: parametrize the structure class the mechanism applies to, enumerate the class, derive each member's governing invariant (symbolically where the mechanism factors, numerically where it does not), and hunt for members whose invariant lands in a target range."
+    },
     "symbolic-trig-closed-form": {
       "field": "algebra",
       "description": "Recast an iterated geometric construction in an exact division-free symbolic domain (here Laurent polynomials in half-angle exponentials over Q(i)) so that the quantities of interest become closed-form trigonometric polynomials whose factorizations expose the law being sought."

problems/billiards-triangles/data/design_bind_N1_death.json

@@ -0,0 +1,32 @@
+{
+ "word": "0121212020202121202020201212120202021212020202",
+ "gamma": 136.8,
+ "argmax_alpha_deg": 26.4705,
+ "width": 0.0025074725538049014,
+ "bindings": [
+  {
+   "gate": 1,
+   "end": "v",
+   "value": -0.0012537362769024507,
+   "factorization": "sin(2b)/2 * sin(2a)/2 * [-1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 - 1/4*cos(2a+6b)/2 - 1/4*cos(2a+10b)/2 - 1/4*cos(2a+14b)/2 - 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]"
+  },
+  {
+   "gate": 8,
+   "end": "v",
+   "value": 0.005027838592129741,
+   "factorization": "sin(2b)/2 * sin(2a)/2 * [1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 - 1/4*cos(2a+6b)/2 - 1/4*cos(2a+10b)/2 + 1/4*cos(2a+14b)/2 + 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]"
+  },
+  {
+   "gate": 12,
+   "end": "v",
+   "value": 0.03407876093434886,
+   "factorization": "sin(2b)/2 * sin(2a)/2 * [1/4*cos(2a-14b)/2 - 1/4*cos(2a+2b)/2 + 1/4*cos(2a+6b)/2 + 1/4*cos(2a+10b)/2 + 1/4*cos(2a+14b)/2 + 1/4*cos(6a-14b)/2 - 1/4*cos(6a+2b)/2 - 1/4*cos(10a-10b)/2 - 1/4*cos(10a-6b)/2 - 1/4*cos(10a-2b)/2 - 1/4*cos(10a+2b)/2]"
+  },
+  {
+   "gate": 2,
+   "end": "v",
+   "value": 0.05477806685912051,
+   "factorization": "sin(2a+2b)/2 * sin(2b)/2 * cos(2a)/2 * [-1/8 - 1/4*cos(4b)/2 - 1/4*cos(8b)/2 - 1/4*cos(12b)/2 + 1/4*cos(4a-12b)/2 + 1/4*cos(4a-8b)/2 + 1/4*cos(4a-4b)/2 + 1/4*cos(4a)/2 - 1/4*cos(8a-12b)/2 - 1/4*cos(8a-8b)/2 - 1/4*cos(8a-4b)/2 - 1/4*cos(8a)/2]"
+  }
+ ]
+}

problems/billiards-triangles/data/design_bind_W43_birth.json

@@ -0,0 +1,14 @@
+{
+ "word": "012121212020202012121212020202",
+ "gamma": 135.005,
+ "argmax_alpha_deg": 22.498046875,
+ "width": 0.00019281662726977622,
+ "bindings": [
+  {
+   "gate": 2,
+   "end": "v",
+   "value": 9.640831363488811e-05,
+   "factorization": "sin(8b)/2 * cos(8a)/2 * sin(2a+2b)/2 * (-1/8+0i)"
+  }
+ ]
+}

problems/billiards-triangles/data/design_cert_W43_gapdeep.json

@@ -0,0 +1,23 @@
+{
+ "word": "012121212020202012121212020202",
+ "len": 30,
+ "apex": [
+  "2251801040365531/4503599627370496",
+  "932723720360161/4503599627370496"
+ ],
+ "float_alpha_beta_deg": [
+  22.49993896484375,
+  22.499961035156247
+ ],
+ "exact_corridor_width": "131852611212664972320325170954221175946544606520120959968945332768641429496875547107980738970737674186988783948148884366495552831013630505861283434096662671882642779615726129483669834740517511090741177713840137594385499390468880290970112000/6058825575870784362575239525877639160407355821405085040618645712553868948462080266021314321336795173294904804542675498664055451078355898392936773952590538706996082238687924758547289133217492095791195251348368239508083929774402377623685359316001",
+ "exact_corridor_width_float": 2.1762074111815788e-05,
+ "orbit_verified_by_independent_simulation": true,
+ "orbit_msg": "exact periodic orbit confirmed, 30 bounces",
+ "certified_gamma_bracket_deg": [
+  "1350000001/10000000",
+  "135001/1000"
+ ],
+ "gamma_gt_glo_certified": true,
+ "gamma_lt_ghi_certified": true,
+ "conclusion": "word alive (exact positive corridor + independent orbit simulation) at a triangle with gamma in (1350000001/10000000, 135001/1000) deg, certified"
+}

problems/billiards-triangles/data/design_cert_W43_pinchgap.json

@@ -0,0 +1,23 @@
+{
+ "word": "012121212020202012121212020202",
+ "len": 30,
+ "apex": [
+  "35188172861353/70368744177664",
+  "7283325023095/35184372088832"
+ ],
+ "float_alpha_beta_deg": [
+  22.48781249999999,
+  22.4921875
+ ],
+ "exact_corridor_width": "85322988579010084987906284814713283794810913771430798553461834331342257290981055849092546339049269857248444195217552641209061417471520793636538451548132940689781195948278132114889760374622659879215554296566825472000/19554606598896063695130231153430179046962225987906151309754278982475070264737536593659297498782382008314463754120577971246126026823440817327321628009704888615202566734079786295044907626150021471044687875495608035526241",
+ "exact_corridor_width_float": 0.004363319105781802,
+ "orbit_verified_by_independent_simulation": true,
+ "orbit_msg": "exact periodic orbit confirmed, 30 bounces",
+ "certified_gamma_bracket_deg": [
+  "135001/1000",
+  "16881/125"
+ ],
+ "gamma_gt_glo_certified": true,
+ "gamma_lt_ghi_certified": true,
+ "conclusion": "word alive (exact positive corridor + independent orbit simulation) at a triangle with gamma in (135001/1000, 16881/125) deg, certified"
+}

problems/billiards-triangles/data/design_cert_W54_touch144.json

@@ -0,0 +1,23 @@
+{
+ "word": "01212121212020202020121212121202020202",
+ "len": 38,
+ "apex": [
+  "9007193351942221/18014398509481984",
+  "5853215511333387/36028797018963968"
+ ],
+ "float_alpha_beta_deg": [
+  17.999961035156247,
+  17.99993896484375
+ ],
+ "exact_corridor_width": "16280363379130760593874671049280704734196942703937674958531194646429730405019837345275579247533894557264138632858168431770032028004641388745643249555306647526246960193947232563573045806191864148828975201635113579002338688855614983989974649205478061754238152045054206936081341465575577167649693288778472173601205184432/598485725031739533988988558976876226449986620588172898991679462219379426327748997081939588610260834247861638869225328200258866656356728147827085206879755161293360731481967365996631617645210143368515575994521807314222155819998524881672417840120201493541680164534640337059740617165052951659420530159866566474175916580290625",
+ "exact_corridor_width_float": 2.72025926403965e-05,
+ "orbit_verified_by_independent_simulation": true,
+ "orbit_msg": "exact periodic orbit confirmed, 38 bounces",
+ "certified_gamma_bracket_deg": [
+  "1440000001/10000000",
+  "144001/1000"
+ ],
+ "gamma_gt_glo_certified": true,
+ "gamma_lt_ghi_certified": true,
+ "conclusion": "word alive (exact positive corridor + independent orbit simulation) at a triangle with gamma in (1440000001/10000000, 144001/1000) deg, certified"
+}