IndisputableMonolith.Gravity.SevenGaps.Gap2GluingDerivation
IndisputableMonolith/Gravity/SevenGaps/Gap2GluingDerivation.lean · 2070 lines · 181 declarations
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1import IndisputableMonolith.Gravity.SevenGaps.Gap2GaugeVolume
2
3/-!
4# Gap 2: deriving the gluing law instead of assuming it
5
6`Gap2GaugeVolume` §6d proves that a gluing law forces the inverse factorial and
7hence implies the gauge-counting principle. The gluing law was assumed there.
8This module attempts to derive it, following the route the 2026-07-28 adversarial
9panel's judge selected as the only candidate that closes rather than relocates.
10
11The route needs a disjoint union on the carrier, which `MeasureInvarianceNoGo`
12explicitly records as missing ("the existing `BoundedComplex` machinery carries no
13disjoint-union operation to state it against"). §1 supplies it.
14
15## The shape of the derivation
16
17Two premises, neither mentioning `mu`, `Aut`, factorials, or the counting
18principle:
19
20* **(i) size-blindness**: the labeled weight depends only on the three index
21 sizes. This, and not independence, is what excludes the known decoy
22 `1/orbitCard`, whose class mass is identically one and is therefore trivially
23 multiplicative.
24* **(ii) gluing multiplicativity**: class mass multiplies over disjoint unions.
25
26Given those, the *orbit counts* supply a binomial interleaving factor on their
27own, and cancelling them turns (ii) into the three-variable shuffle identity,
28whose solutions are `f(a,b,c) = x^a y^b z^c / (a! b! c!)`. So the premise is
29reduced from a normalization to a locality statement plus three couplings.
30-/
31
32namespace IndisputableMonolith
33namespace Gravity
34namespace SevenGaps
35namespace Gap2GluingDerivation
36
37open PathSumMeasure ExactShellGaugePreflight Gap2GaugeVolume
38
39/-! ## §1. Disjoint union on the carrier
40
41The union of a `BoundedComplex B` and a `BoundedComplex B'` lives at cap `B + B'`.
42Index sets add, and the incidence maps are the two originals pushed into the two
43halves of the summed index sets by `finSumFinEquiv`. -/
44
45/-- Push a vertex of the left summand into the union's vertex set. -/
46abbrev inlV {m n : ℕ} (i : Fin m) : Fin (m + n) := finSumFinEquiv (Sum.inl i)
47
48/-- Push a vertex of the right summand into the union's vertex set. -/
49abbrev inrV {m n : ℕ} (i : Fin n) : Fin (m + n) := finSumFinEquiv (Sum.inr i)
50
51/-- The **disjoint union** of two bounded complexes, at the summed cap. -/
52def dunion {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B') :
53 BoundedComplex (B + B') where
54 nV := K.nV + L.nV
55 nE := K.nE + L.nE
56 nT := K.nT + L.nT
57 hV := Nat.add_le_add K.hV L.hV
58 hE := Nat.add_le_add K.hE L.hE
59 hT := Nat.add_le_add K.hT L.hT
60 edgeVerts := fun e =>
61 Sum.elim
62 (fun e' : Fin K.nE => (inlV (K.edgeVerts e').1, inlV (K.edgeVerts e').2))
63 (fun e' : Fin L.nE => (inrV (L.edgeVerts e').1, inrV (L.edgeVerts e').2))
64 (finSumFinEquiv.symm e)
65 tetVerts := fun t i =>
66 Sum.elim
67 (fun t' : Fin K.nT => inlV (K.tetVerts t' i))
68 (fun t' : Fin L.nT => inrV (L.tetVerts t' i))
69 (finSumFinEquiv.symm t)
70
71@[simp] theorem dunion_nV {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B') :
72 (dunion K L).nV = K.nV + L.nV := rfl
73
74@[simp] theorem dunion_nE {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B') :
75 (dunion K L).nE = K.nE + L.nE := rfl
76
77@[simp] theorem dunion_nT {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B') :
78 (dunion K L).nT = K.nT + L.nT := rfl
79
80/-! ## §2. The gauge volume of a union, and where the binomial comes from
81
82This is pure factorial arithmetic and it is the reason the interleaving count is
83binomial: the gauge volume of the union exceeds the product of the parts' gauge
84volumes by exactly the three binomial coefficients counting how the label blocks
85can be interleaved. -/
86
87/-- The three-way interleaving count for a pair of size triples. -/
88def interleave (a b c a' b' c' : ℕ) : ℕ :=
89 Nat.choose (a + a') a * (Nat.choose (b + b') b * Nat.choose (c + c') c)
90
91/-- The gauge volume of a size triple. -/
92def gaugeVol (a b c : ℕ) : ℕ :=
93 Nat.factorial a * (Nat.factorial b * Nat.factorial c)
94
95/-- **THEOREM (the interleaving count is binomial).** The gauge volume of a sum
96of size triples is the interleaving count times the product of the parts' gauge
97volumes. Pure arithmetic, no complexes involved. -/
98theorem gaugeVol_add (a b c a' b' c' : ℕ) :
99 gaugeVol (a + a') (b + b') (c + c')
100 = interleave a b c a' b' c' * (gaugeVol a b c * gaugeVol a' b' c') := by
101 have key : ∀ m n : ℕ,
102 Nat.factorial (m + n)
103 = Nat.choose (m + n) m * (Nat.factorial m * Nat.factorial n) := by
104 intro m n
105 have h := Nat.choose_mul_factorial_mul_factorial (Nat.le_add_right m n)
106 have hsub : m + n - m = n := Nat.add_sub_cancel_left m n
107 rw [hsub] at h
108 rw [← h, mul_assoc]
109 unfold gaugeVol interleave
110 rw [key a a', key b b', key c c']
111 ring
112
113/-- The gauge volume of a union, in the module's own terms. -/
114theorem gaugeVol_dunion {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B') :
115 gaugeVol (dunion K L).nV (dunion K L).nE (dunion K L).nT
116 = interleave K.nV K.nE K.nT L.nV L.nE L.nT
117 * (gaugeVol K.nV K.nE K.nT * gaugeVol L.nV L.nE L.nT) := by
118 simp only [dunion_nV, dunion_nE, dunion_nT]
119 exact gaugeVol_add _ _ _ _ _ _
120
121/-- `pairCount` is the gauge volume of the index sizes, in the notation of this
122module. -/
123theorem pairCount_eq_gaugeVol {B : ℕ} (K : BoundedComplex B) :
124 pairCount K = gaugeVol K.nV K.nE K.nT :=
125 pairCount_eq_factorials K
126
127/-- The interleaving count is positive. -/
128theorem interleave_pos (a b c a' b' c' : ℕ) : 0 < interleave a b c a' b' c' := by
129 unfold interleave
130 have h1 : 0 < Nat.choose (a + a') a := Nat.choose_pos (Nat.le_add_right a a')
131 have h2 : 0 < Nat.choose (b + b') b := Nat.choose_pos (Nat.le_add_right b b')
132 have h3 : 0 < Nat.choose (c + c') c := Nat.choose_pos (Nat.le_add_right c c')
133 positivity
134
135/-- The gauge volume is positive. -/
136theorem gaugeVol_pos (a b c : ℕ) : 0 < gaugeVol a b c := by
137 unfold gaugeVol
138 have := Nat.factorial_pos a
139 have := Nat.factorial_pos b
140 have := Nat.factorial_pos c
141 positivity
142
143/-! ## §3. Dust, and the refutation of unrestricted gluing multiplicativity
144
145Before using premise (ii) it must be checked against the intended answer. It
146fails. `dust n` is `n` isolated vertices; its automorphism group is the full
147symmetric group, so `mu (dust n) = 1/n!`, while dust glues to dust. Unrestricted
148multiplicativity would demand `1/(a+b)! = 1/a! · 1/b!`, which is false as soon as
149both parts are nonempty. So premise (ii) is not a free lunch: it must carry a
150side condition excluding repeated isomorphic pieces, and that side condition is
151mandatory rather than a convenience. -/
152
153/-- `n` isolated vertices: no edges, no tetrahedra. -/
154def dust (n : ℕ) : BoundedComplex n where
155 nV := n
156 nE := 0
157 nT := 0
158 hV := le_refl n
159 hE := Nat.zero_le n
160 hT := Nat.zero_le n
161 edgeVerts := Fin.elim0
162 tetVerts := Fin.elim0
163
164@[simp] theorem dust_nV (n : ℕ) : (dust n).nV = n := rfl
165@[simp] theorem dust_nE (n : ℕ) : (dust n).nE = 0 := rfl
166@[simp] theorem dust_nT (n : ℕ) : (dust n).nT = 0 := rfl
167
168/-- **The automorphism group of dust is the full symmetric group.** With no
169incidence data, both commutation conditions are vacuous, so every vertex
170permutation is an automorphism. -/
171def autDustEquiv (n : ℕ) : Aut (dust n) ≃ Equiv.Perm (Fin n) where
172 toFun := fun r => r.vEquiv
173 invFun := fun v =>
174 { vEquiv := v
175 eEquiv := Equiv.refl _
176 tEquiv := Equiv.refl _
177 edge_comm := fun e => Fin.elim0 e
178 tet_comm := fun t _ => Fin.elim0 t }
179 left_inv := by
180 intro r
181 have he : r.eEquiv = Equiv.refl (Fin (dust n).nE) := Equiv.ext fun e => Fin.elim0 e
182 have ht : r.tEquiv = Equiv.refl (Fin (dust n).nT) := Equiv.ext fun t => Fin.elim0 t
183 cases r
184 simp_all
185 right_inv := by intro v; rfl
186
187/-- `|Aut (dust n)| = n!`. -/
188theorem autCard_dust (n : ℕ) : Nat.card (Aut (dust n)) = Nat.factorial n := by
189 rw [Nat.card_congr (autDustEquiv n), Nat.card_eq_fintype_card, Fintype.card_perm,
190 Fintype.card_fin]
191
192/-- The RS measure on dust is the reciprocal factorial. -/
193theorem mu_dust (n : ℕ) : mu (dust n) = 1 / (Nat.factorial n : ℝ) := by
194 unfold mu
195 rw [autCard_dust]
196
197/-- Dust glues to dust: the union of `a` and `b` isolated vertices is `a + b`
198isolated vertices. -/
199def dunionDustRelabel (a b : ℕ) : Relabel (dunion (dust a) (dust b)) (dust (a + b)) where
200 vEquiv := Equiv.refl _
201 eEquiv := Equiv.refl _
202 tEquiv := Equiv.refl _
203 edge_comm := fun e => Fin.elim0 e
204 tet_comm := fun t _ => Fin.elim0 t
205
206theorem dunion_dust_equivalent (a b : ℕ) :
207 Equivalent (dunion (dust a) (dust b)) (dust (a + b)) :=
208 ⟨dunionDustRelabel a b⟩
209
210/-- **THEOREM (unrestricted gluing multiplicativity is FALSE).** There is no
211version of premise (ii) that applies to every disjoint union, because the
212intended answer `mu` itself violates it. Witness: one vertex glued to one
213vertex. `mu` of the union is `1/2`; the product of the parts' `mu` is `1`. -/
214theorem unrestricted_gluing_multiplicativity_false :
215 ¬ (∀ (B B' : ℕ) (K : BoundedComplex B) (L : BoundedComplex B'),
216 mu (dunion K L) = mu K * mu L) := by
217 intro h
218 have hd := h 1 1 (dust 1) (dust 1)
219 rw [mu_congr (dunion_dust_equivalent 1 1)] at hd
220 rw [mu_dust, mu_dust] at hd
221 norm_num at hd
222
223/-- The same failure, quantified: `mu` of a dust union undershoots the product of
224the parts by exactly the interleaving count, which is the binomial coefficient.
225This is the extra symmetry that repeated isomorphic pieces create. -/
226theorem mu_dust_union_off_by_binomial (a b : ℕ) :
227 mu (dunion (dust a) (dust b)) * (Nat.choose (a + b) a : ℝ)
228 = mu (dust a) * mu (dust b) := by
229 rw [mu_congr (dunion_dust_equivalent a b), mu_dust, mu_dust, mu_dust]
230 have hfac : (Nat.factorial (a + b) : ℝ)
231 = (Nat.choose (a + b) a : ℝ) * ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ)) := by
232 have h := Nat.choose_mul_factorial_mul_factorial (Nat.le_add_right a b)
233 have hsub : a + b - a = b := Nat.add_sub_cancel_left a b
234 rw [hsub] at h
235 have : ((Nat.choose (a + b) a * Nat.factorial a * Nat.factorial b : ℕ) : ℝ)
236 = ((Nat.factorial (a + b) : ℕ) : ℝ) := by exact_mod_cast h
237 push_cast at this
238 linarith [this]
239 rw [hfac]
240 have hca : (0 : ℝ) < (Nat.choose (a + b) a : ℝ) := by
241 have := Nat.choose_pos (Nat.le_add_right a b)
242 exact_mod_cast this
243 have ha : (0 : ℝ) < (Nat.factorial a : ℝ) := by
244 have := Nat.factorial_pos a; exact_mod_cast this
245 have hb : (0 : ℝ) < (Nat.factorial b : ℝ) := by
246 have := Nat.factorial_pos b; exact_mod_cast this
247 field_simp
248
249/-! ## §4. What the interleaving count is, at the level of orbit counts
250
251Kill condition 1 of the pre-registered gate asks whether the interleaving count is
252binomial. §2 settles that for the gauge volume unconditionally. Transporting it
253to orbit counts costs exactly one thing: multiplicativity of the automorphism
254count. This section proves the transport, so the residue is isolated to a single
255identity about `Aut`. -/
256
257/-- **THEOREM (the interleaving count is binomial, given `Aut` multiplicativity).**
258If the automorphism count multiplies over a union, the orbit counts satisfy the
259binomial interleaving identity. No other hypothesis. -/
260theorem orbitCard_dunion_of_autMul {B B' : ℕ}
261 (K : BoundedComplex B) (L : BoundedComplex B')
262 (haut : Nat.card (Aut (dunion K L)) = Nat.card (Aut K) * Nat.card (Aut L)) :
263 gaugeOrbitCard (dunion K L)
264 = interleave K.nV K.nE K.nT L.nV L.nE L.nT
265 * (gaugeOrbitCard K * gaugeOrbitCard L) := by
266 have hU : pairCount (dunion K L)
267 = gaugeOrbitCard (dunion K L) * Nat.card (Aut (dunion K L)) :=
268 pairCount_eq_orbitCard_mul_autCard _
269 have hK : pairCount K = gaugeOrbitCard K * Nat.card (Aut K) :=
270 pairCount_eq_orbitCard_mul_autCard _
271 have hL : pairCount L = gaugeOrbitCard L * Nat.card (Aut L) :=
272 pairCount_eq_orbitCard_mul_autCard _
273 have hvol : pairCount (dunion K L)
274 = interleave K.nV K.nE K.nT L.nV L.nE L.nT * (pairCount K * pairCount L) := by
275 rw [pairCount_eq_gaugeVol, pairCount_eq_gaugeVol, pairCount_eq_gaugeVol]
276 exact gaugeVol_dunion K L
277 rw [hU, haut, hK, hL] at hvol
278 have hpos : 0 < Nat.card (Aut K) * Nat.card (Aut L) :=
279 Nat.mul_pos (autCard_pos K) (autCard_pos L)
280 have hcancel : gaugeOrbitCard (dunion K L) * (Nat.card (Aut K) * Nat.card (Aut L))
281 = (interleave K.nV K.nE K.nT L.nV L.nE L.nT
282 * (gaugeOrbitCard K * gaugeOrbitCard L))
283 * (Nat.card (Aut K) * Nat.card (Aut L)) := by
284 rw [hvol]; ring
285 exact Nat.eq_of_mul_eq_mul_right hpos hcancel
286
287/-! ## §5. Premise (i): size-blindness, and the transport to a shuffle identity -/
288
289/-- **Premise (i), size-blindness.** A single function of the three index sizes,
290used as the labeled weight at every cap. Nothing here mentions `mu`, `Aut`,
291factorials, or the counting principle. -/
292def sizeWeight (f : ℕ → ℕ → ℕ → ℝ) {B : ℕ} (K : BoundedComplex B) : ℝ :=
293 f K.nV K.nE K.nT
294
295/-- A size-blind weight is relabeling-invariant, since a relabeling preserves the
296three index sizes. -/
297theorem sizeWeight_invariant (f : ℕ → ℕ → ℕ → ℝ) {B : ℕ} {K K' : BoundedComplex B}
298 (h : Equivalent K K') : sizeWeight f K = sizeWeight f K' := by
299 obtain ⟨r⟩ := h
300 unfold sizeWeight
301 rw [size_v r, size_e r, size_t r]
302
303/-- The class mass of a size-blind weight is the orbit count times the size value.
304This is where the orbit counts enter, and they are the only thing that does. -/
305theorem classMass_sizeWeight (f : ℕ → ℕ → ℕ → ℝ) {B : ℕ} (K : BoundedComplex B) :
306 classMass (sizeWeight f) (Quotient.mk (relabelSetoid B) K)
307 = (gaugeOrbitCard K : ℝ) * f K.nV K.nE K.nT := by
308 rw [classMass_of_invariant _ (fun _ _ h => sizeWeight_invariant f h)]
309 have hout : sizeWeight f (Quotient.out (Quotient.mk (relabelSetoid B) K))
310 = sizeWeight f K :=
311 sizeWeight_invariant f (equivalent_out K)
312 rw [hout, orbitCardClass_mk]
313 rfl
314
315/-- **Premise (ii), gluing multiplicativity, at one pair.** The class mass of a
316union is the product of the parts' class masses. Stated with no reference to
317`mu`, `Aut`, factorials, the gauge volume, or the counting principle: only the
318class mass of the size-blind weight and the disjoint union. -/
319def GluesAt (f : ℕ → ℕ → ℕ → ℝ) {B B' : ℕ}
320 (K : BoundedComplex B) (L : BoundedComplex B') : Prop :=
321 classMass (sizeWeight f) (Quotient.mk (relabelSetoid (B + B')) (dunion K L))
322 = classMass (sizeWeight f) (Quotient.mk (relabelSetoid B) K)
323 * classMass (sizeWeight f) (Quotient.mk (relabelSetoid B') L)
324
325/-- **THEOREM (the transport).** At any pair where the automorphism count
326multiplies, premise (ii) for a size-blind weight is *exactly* the shuffle identity
327on the size function. The orbit counts cancel; their entire contribution is the
328binomial interleaving factor. This is the step that converts a locality premise
329into a normalization identity, and it is where the compiled arithmetic of §2 and
330§4 is spent. -/
331theorem shuffle_of_gluesAt (f : ℕ → ℕ → ℕ → ℝ) {B B' : ℕ}
332 (K : BoundedComplex B) (L : BoundedComplex B')
333 (haut : Nat.card (Aut (dunion K L)) = Nat.card (Aut K) * Nat.card (Aut L))
334 (hglue : GluesAt f K L) :
335 f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT)
336 * (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ)
337 = f K.nV K.nE K.nT * f L.nV L.nE L.nT := by
338 have hob := orbitCard_dunion_of_autMul K L haut
339 unfold GluesAt at hglue
340 rw [classMass_sizeWeight, classMass_sizeWeight, classMass_sizeWeight] at hglue
341 simp only [dunion_nV, dunion_nE, dunion_nT] at hglue
342 rw [hob] at hglue
343 push_cast at hglue
344 have hK : (0 : ℝ) < (gaugeOrbitCard K : ℝ) := by
345 exact_mod_cast gaugeOrbitCard_pos K
346 have hL : (0 : ℝ) < (gaugeOrbitCard L : ℝ) := by
347 exact_mod_cast gaugeOrbitCard_pos L
348 have hprod : (gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ) ≠ 0 :=
349 (mul_pos hK hL).ne'
350 refine mul_right_cancel₀ hprod ?_
351 calc f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT)
352 * (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ)
353 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ))
354 = (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ)
355 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ))
356 * f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT) := by ring
357 _ = (gaugeOrbitCard K : ℝ) * f K.nV K.nE K.nT
358 * ((gaugeOrbitCard L : ℝ) * f L.nV L.nE L.nT) := hglue
359 _ = f K.nV K.nE K.nT * f L.nV L.nE L.nT
360 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ)) := by ring
361
362/-! ## §6. What the two premises force
363
364The premise supplies the shuffle identity only at pairs the carrier can realize
365with non-mixing automorphisms. §3 proves this restriction is mandatory. Four
366families suffice, and each is a union of two complexes sharing no isomorphic
367component:
368
369* `dust a` glued to a **bouquet**: one vertex carrying `b` loops and `c`
370 degenerate tetrahedra, with at least one incidence. The bouquet vertex has an
371 incidence and the dust vertices do not, so no automorphism exchanges them.
372* `dust a` glued to a single **edge**.
373* a bouquet glued to a single **edge** (one vertex versus two).
374* a bouquet glued to a single **tetrahedron** (one vertex versus four).
375
376Nothing below mentions `mu`, `Aut`, factorials, or the counting principle: the
377input is four instances of premise (ii) transported through
378`shuffle_of_gluesAt`. -/
379
380/-- Two times a middle binomial, in closed form. Needed because the dust-edge
381instance carries `C(a+2, a)` rather than a linear factor. -/
382theorem two_mul_choose (a : ℕ) : 2 * Nat.choose (a + 2) a = (a + 2) * (a + 1) := by
383 have h := Nat.choose_mul_factorial_mul_factorial (show a ≤ a + 2 by omega)
384 have h2 : a + 2 - a = 2 := by omega
385 rw [h2] at h
386 have hf : Nat.factorial (a + 2) = (a + 2) * ((a + 1) * Nat.factorial a) := by
387 rw [Nat.factorial_succ, Nat.factorial_succ]
388 rw [hf] at h
389 have hfac : Nat.factorial 2 = 2 := rfl
390 rw [hfac] at h
391 have hcancel : (2 * Nat.choose (a + 2) a) * Nat.factorial a
392 = ((a + 2) * (a + 1)) * Nat.factorial a := by
393 calc (2 * Nat.choose (a + 2) a) * Nat.factorial a
394 = Nat.choose (a + 2) a * Nat.factorial a * 2 := by ring
395 _ = (a + 2) * ((a + 1) * Nat.factorial a) := h
396 _ = ((a + 2) * (a + 1)) * Nat.factorial a := by ring
397 exact Nat.eq_of_mul_eq_mul_right (Nat.factorial_pos a) hcancel
398
399@[simp] theorem interleave_pt (a b c : ℕ) : interleave a 0 0 1 b c = a + 1 := by
400 unfold interleave
401 simp
402
403@[simp] theorem interleave_edge (a : ℕ) : interleave a 0 0 2 1 0 = Nat.choose (a + 2) a := by
404 unfold interleave
405 simp
406
407@[simp] theorem interleave_bqEdge (b c : ℕ) : interleave 1 b c 2 1 0 = 3 * (b + 1) := by
408 unfold interleave
409 simp [Nat.choose_one_right]
410
411@[simp] theorem interleave_bqTet (b c : ℕ) : interleave 1 b c 4 0 1 = 5 * (c + 1) := by
412 unfold interleave
413 simp [Nat.choose_one_right]
414
415/-- **The four gluing instances the carrier supplies**, written purely as
416identities on the size function. Each is premise (ii) at one family of pairs,
417already transported through `shuffle_of_gluesAt`. -/
418structure CarrierShuffle (f : ℕ → ℕ → ℕ → ℝ) : Prop where
419 /-- A weight is a positive number. -/
420 pos : ∀ a b c, 0 < f a b c
421 /-- The empty complex has unit weight (the normalization of the sum). -/
422 unit : f 0 0 0 = 1
423 /-- `dust a ⊔ bouquet(b,c)`, the bouquet carrying at least one incidence. -/
424 dust_bouquet : ∀ a b c, 1 ≤ b + c →
425 f (a + 1) b c * (interleave a 0 0 1 b c : ℝ) = f a 0 0 * f 1 b c
426 /-- `dust a ⊔ edge`. -/
427 dust_edge : ∀ a,
428 f (a + 2) 1 0 * (interleave a 0 0 2 1 0 : ℝ) = f a 0 0 * f 2 1 0
429 /-- `bouquet(b,c) ⊔ edge`. -/
430 bouquet_edge : ∀ b c,
431 f 3 (b + 1) c * (interleave 1 b c 2 1 0 : ℝ) = f 1 b c * f 2 1 0
432 /-- `bouquet(b,c) ⊔ tetrahedron`. -/
433 bouquet_tet : ∀ b c,
434 f 5 b (c + 1) * (interleave 1 b c 4 0 1 : ℝ) = f 1 b c * f 4 0 1
435
436namespace CarrierShuffle
437
438variable {f : ℕ → ℕ → ℕ → ℝ}
439
440/-- The single-edge weight, from the loop-plus-point realization of the same size
441triple. This is the step where size-blindness does real work: the size triple
442`(2,1,0)` is realized both by the indecomposable edge and by the decomposable
443`loop ⊔ point`, and premise (i) identifies them. -/
444theorem edgeWeight (h : CarrierShuffle f) : f 2 1 0 * 2 = f 1 0 0 * f 1 1 0 := by
445 have hd := h.dust_bouquet 1 1 0 (by omega)
446 rw [interleave_pt] at hd
447 push_cast at hd
448 linarith [hd]
449
450/-- **The vertex recursion.** Adding one isolated vertex divides the weight by the
451new vertex count. Derived from the dust-bouquet and dust-edge instances; the
452binomial `C(a+2,a)` cancels against the linear factors. -/
453theorem vertexRec (h : CarrierShuffle f) (a : ℕ) :
454 f (a + 1) 0 0 * ((a : ℝ) + 1) = f a 0 0 * f 1 0 0 := by
455 have hP1 := h.dust_bouquet (a + 1) 1 0 (by omega)
456 rw [interleave_pt] at hP1
457 have hQ := h.dust_edge a
458 rw [interleave_edge] at hQ
459 have hR := edgeWeight h
460 have hC : (2 : ℝ) * (Nat.choose (a + 2) a : ℝ) = ((a : ℝ) + 2) * ((a : ℝ) + 1) := by
461 have := two_mul_choose a
462 have hcast : ((2 * Nat.choose (a + 2) a : ℕ) : ℝ) = (((a + 2) * (a + 1) : ℕ) : ℝ) := by
463 exact_mod_cast congrArg (fun n : ℕ => (n : ℝ)) this
464 push_cast at hcast
465 linarith [hcast]
466 push_cast at hP1
467 have hq : f 1 1 0 ≠ 0 := (h.pos 1 1 0).ne'
468 refine mul_right_cancel₀ hq ?_
469 linear_combination (-((a : ℝ) + 1)) * hP1 + (-(f (a + 2) 1 0)) * hC + 2 * hQ
470 + (f a 0 0) * hR
471
472/-- **The dust row.** Iterating the vertex recursion: `f(a,0,0) · a! = x^a`. -/
473theorem dustRow (h : CarrierShuffle f) (a : ℕ) :
474 f a 0 0 * (Nat.factorial a : ℝ) = (f 1 0 0) ^ a := by
475 induction a with
476 | zero => simpa using h.unit
477 | succ n ih =>
478 have hv := vertexRec h n
479 have hfac : (Nat.factorial (n + 1) : ℝ)
480 = ((n : ℝ) + 1) * (Nat.factorial n : ℝ) := by
481 rw [Nat.factorial_succ]; push_cast; ring
482 calc f (n + 1) 0 0 * (Nat.factorial (n + 1) : ℝ)
483 = (f (n + 1) 0 0 * ((n : ℝ) + 1)) * (Nat.factorial n : ℝ) := by
484 rw [hfac]; ring
485 _ = (f n 0 0 * f 1 0 0) * (Nat.factorial n : ℝ) := by rw [hv]
486 _ = (f n 0 0 * (Nat.factorial n : ℝ)) * f 1 0 0 := by ring
487 _ = (f 1 0 0) ^ n * f 1 0 0 := by rw [ih]
488 _ = (f 1 0 0) ^ (n + 1) := by ring
489
490/-- **The edge recursion.** Adding one loop to a bouquet divides by the new edge
491count, with the edge fugacity `f(1,1,0)/f(1,0,0)` as the ratio. -/
492theorem bRec (h : CarrierShuffle f) (b c : ℕ) :
493 f 1 (b + 1) c * ((b : ℝ) + 1) * f 1 0 0 = f 1 b c * f 1 1 0 := by
494 have hi := h.bouquet_edge b c
495 rw [interleave_bqEdge] at hi
496 have hii := h.dust_bouquet 2 (b + 1) c (by omega)
497 rw [interleave_pt] at hii
498 have hiii : f 2 0 0 * 2 = (f 1 0 0) ^ 2 := by
499 have := dustRow h 2
500 have hf2 : (Nat.factorial 2 : ℝ) = 2 := by norm_num [Nat.factorial]
501 rw [hf2] at this
502 linarith [this]
503 have hiv := edgeWeight h
504 push_cast at hi hii
505 have hx : f 1 0 0 ≠ 0 := (h.pos 1 0 0).ne'
506 refine mul_right_cancel₀ hx ?_
507 linear_combination (-(f 1 (b + 1) c * ((b : ℝ) + 1))) * hiii
508 + (-2 * ((b : ℝ) + 1)) * hii + 2 * hi + (f 1 b c) * hiv
509
510/-- The four-vertex tetrahedron weight, from the point-plus-tetrahedron
511realization. -/
512theorem tetWeight (h : CarrierShuffle f) : f 4 0 1 * 4 = f 3 0 0 * f 1 0 1 := by
513 have hd := h.dust_bouquet 3 0 1 (by omega)
514 rw [interleave_pt] at hd
515 push_cast at hd
516 linarith [hd]
517
518/-- **The tetrahedron recursion.** Adding one degenerate tetrahedron to a bouquet
519divides by the new tetrahedron count, with fugacity `f(1,0,1)/f(1,0,0)`. -/
520theorem cRec (h : CarrierShuffle f) (b c : ℕ) :
521 f 1 b (c + 1) * ((c : ℝ) + 1) * f 1 0 0 = f 1 b c * f 1 0 1 := by
522 have hi := h.bouquet_tet b c
523 rw [interleave_bqTet] at hi
524 have hii := h.dust_bouquet 4 b (c + 1) (by omega)
525 rw [interleave_pt] at hii
526 have hiii := tetWeight h
527 have hF3 : f 3 0 0 * 6 = (f 1 0 0) ^ 3 := by
528 have := dustRow h 3
529 have hf3 : (Nat.factorial 3 : ℝ) = 6 := by norm_num [Nat.factorial]
530 rw [hf3] at this
531 linarith [this]
532 have hF4 : f 4 0 0 * 24 = (f 1 0 0) ^ 4 := by
533 have := dustRow h 4
534 have hf4 : (Nat.factorial 4 : ℝ) = 24 := by norm_num [Nat.factorial]
535 rw [hf4] at this
536 linarith [this]
537 push_cast at hi hii
538 have hx3 : (f 1 0 0) ^ 3 ≠ 0 := pow_ne_zero _ (h.pos 1 0 0).ne'
539 refine mul_right_cancel₀ hx3 ?_
540 linear_combination (-(f 1 b (c + 1) * ((c : ℝ) + 1))) * hF4
541 + (-24 * ((c : ℝ) + 1)) * hii + 24 * hi + (6 * f 1 b c) * hiii
542 + (f 1 b c * f 1 0 1) * hF3
543
544/-- The bouquet column: iterating the tetrahedron recursion at `b = 0`. -/
545theorem bouquetTetCol (h : CarrierShuffle f) (c : ℕ) :
546 f 1 0 c * (Nat.factorial c : ℝ) * (f 1 0 0) ^ c
547 = f 1 0 0 * (f 1 0 1) ^ c := by
548 induction c with
549 | zero => simp
550 | succ n ih =>
551 have hc := cRec h 0 n
552 have hfac : (Nat.factorial (n + 1) : ℝ)
553 = ((n : ℝ) + 1) * (Nat.factorial n : ℝ) := by
554 rw [Nat.factorial_succ]; push_cast; ring
555 calc f 1 0 (n + 1) * (Nat.factorial (n + 1) : ℝ) * (f 1 0 0) ^ (n + 1)
556 = (f 1 0 (n + 1) * ((n : ℝ) + 1) * f 1 0 0)
557 * ((Nat.factorial n : ℝ) * (f 1 0 0) ^ n) := by
558 rw [hfac]; ring
559 _ = (f 1 0 n * f 1 0 1) * ((Nat.factorial n : ℝ) * (f 1 0 0) ^ n) := by rw [hc]
560 _ = (f 1 0 n * (Nat.factorial n : ℝ) * (f 1 0 0) ^ n) * f 1 0 1 := by ring
561 _ = (f 1 0 0 * (f 1 0 1) ^ n) * f 1 0 1 := by rw [ih]
562 _ = f 1 0 0 * (f 1 0 1) ^ (n + 1) := by ring
563
564/-- **The bouquet weight in closed form.** A one-vertex bouquet with `b` loops and
565`c` degenerate tetrahedra: the two fugacities appear as powers over factorials. -/
566theorem bouquetRow (h : CarrierShuffle f) (b c : ℕ) :
567 f 1 b c * ((Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
568 * (f 1 0 0) ^ (b + c)
569 = f 1 0 0 * (f 1 1 0) ^ b * (f 1 0 1) ^ c := by
570 induction b with
571 | zero =>
572 have hcol := bouquetTetCol h c
573 simp only [Nat.factorial_zero, Nat.cast_one, one_mul, zero_add, pow_zero, mul_one]
574 exact hcol
575 | succ n ih =>
576 have hb := bRec h n c
577 have hfac : (Nat.factorial (n + 1) : ℝ)
578 = ((n : ℝ) + 1) * (Nat.factorial n : ℝ) := by
579 rw [Nat.factorial_succ]; push_cast; ring
580 calc f 1 (n + 1) c * ((Nat.factorial (n + 1) : ℝ) * (Nat.factorial c : ℝ))
581 * (f 1 0 0) ^ (n + 1 + c)
582 = (f 1 (n + 1) c * ((n : ℝ) + 1) * f 1 0 0)
583 * ((Nat.factorial n : ℝ) * (Nat.factorial c : ℝ) * (f 1 0 0) ^ (n + c)) := by
584 rw [hfac]
585 have : (f 1 0 0) ^ (n + 1 + c) = (f 1 0 0) ^ (n + c) * f 1 0 0 := by
586 rw [show n + 1 + c = (n + c) + 1 by omega, pow_succ]
587 rw [this]; ring
588 _ = (f 1 n c * f 1 1 0)
589 * ((Nat.factorial n : ℝ) * (Nat.factorial c : ℝ) * (f 1 0 0) ^ (n + c)) := by
590 rw [hb]
591 _ = (f 1 n c * ((Nat.factorial n : ℝ) * (Nat.factorial c : ℝ))
592 * (f 1 0 0) ^ (n + c)) * f 1 1 0 := by ring
593 _ = (f 1 0 0 * (f 1 1 0) ^ n * (f 1 0 1) ^ c) * f 1 1 0 := by rw [ih]
594 _ = f 1 0 0 * (f 1 1 0) ^ (n + 1) * (f 1 0 1) ^ c := by ring
595
596/-- **THEOREM (the two premises determine the weight up to three constants).**
597On every size triple the carrier can realize, the size-blind weight is the inverse
598gauge volume times three fugacities: one per index type. The infinite-dimensional
599residue of `Gap2GaugeVolume` §6c (an arbitrary `a : ℕ³ → ℝ`) collapses to three
600real numbers. -/
601theorem closedForm (h : CarrierShuffle f) (a b c : ℕ) (ha : 1 ≤ a) :
602 f a b c * ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
603 * (f 1 0 0) ^ (b + c)
604 = (f 1 0 0) ^ a * (f 1 1 0) ^ b * (f 1 0 1) ^ c := by
605 obtain ⟨a', rfl⟩ : ∃ a', a = a' + 1 := ⟨a - 1, by omega⟩
606 rcases Nat.eq_zero_or_pos (b + c) with hbc | hbc
607 · -- No edges and no tetrahedra: the dust row already settles it.
608 have hb : b = 0 := by omega
609 have hc : c = 0 := by omega
610 subst hb; subst hc
611 have := dustRow h (a' + 1)
612 simpa using this
613 · -- At least one incidence: the dust-bouquet instance plus the bouquet row.
614 have hd := h.dust_bouquet a' b c hbc
615 rw [interleave_pt] at hd
616 push_cast at hd
617 have hrow := dustRow h a'
618 have hbq := bouquetRow h b c
619 have hfac : (Nat.factorial (a' + 1) : ℝ)
620 = ((a' : ℝ) + 1) * (Nat.factorial a' : ℝ) := by
621 rw [Nat.factorial_succ]; push_cast; ring
622 have hx : f 1 0 0 ≠ 0 := (h.pos 1 0 0).ne'
623 calc f (a' + 1) b c
624 * ((Nat.factorial (a' + 1) : ℝ) * (Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
625 * (f 1 0 0) ^ (b + c)
626 = (f (a' + 1) b c * ((a' : ℝ) + 1)) * (Nat.factorial a' : ℝ)
627 * (f 1 b c * ((Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
628 * (f 1 0 0) ^ (b + c)) / f 1 b c := by
629 rw [hfac]
630 field_simp [(h.pos 1 b c).ne']
631 _ = (f a' 0 0 * f 1 b c) * (Nat.factorial a' : ℝ)
632 * (f 1 0 0 * (f 1 1 0) ^ b * (f 1 0 1) ^ c) / f 1 b c := by
633 rw [hd, hbq]
634 _ = (f a' 0 0 * (Nat.factorial a' : ℝ))
635 * (f 1 0 0 * (f 1 1 0) ^ b * (f 1 0 1) ^ c) := by
636 field_simp [(h.pos 1 b c).ne']
637 _ = (f 1 0 0) ^ a' * (f 1 0 0 * (f 1 1 0) ^ b * (f 1 0 1) ^ c) := by rw [hrow]
638 _ = (f 1 0 0) ^ (a' + 1) * (f 1 1 0) ^ b * (f 1 0 1) ^ c := by ring
639
640/-- **THEOREM (the Gibbs weight, from the two premises plus three unit
641normalizations).** If a single labeled vertex, a single labeled loop, and a
642single labeled degenerate tetrahedron each carry unit weight, the size-blind
643weight is forced to be the inverse gauge volume: exactly `gibbsWeight`. By
644`Gap2GaugeVolume.gibbsWeight_gives_gaugeCounting` the counting principle then
645follows, so the measure is derived. -/
646theorem gibbs_of_unit_fugacities (h : CarrierShuffle f) (hx : f 1 0 0 = 1) (hy : f 1 1 0 = 1)
647 (hz : f 1 0 1 = 1) (a b c : ℕ) (ha : 1 ≤ a) :
648 f a b c
649 = 1 / ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ) * (Nat.factorial c : ℝ)) := by
650 have hcf := closedForm h a b c ha
651 rw [hx, hy, hz] at hcf
652 simp only [one_pow, mul_one] at hcf
653 have hpos : (0 : ℝ) < (Nat.factorial a : ℝ) * (Nat.factorial b : ℝ)
654 * (Nat.factorial c : ℝ) := by
655 have h1 : (0 : ℝ) < (Nat.factorial a : ℝ) := by
656 have := Nat.factorial_pos a; exact_mod_cast this
657 have h2 : (0 : ℝ) < (Nat.factorial b : ℝ) := by
658 have := Nat.factorial_pos b; exact_mod_cast this
659 have h3 : (0 : ℝ) < (Nat.factorial c : ℝ) := by
660 have := Nat.factorial_pos c; exact_mod_cast this
661 positivity
662 field_simp at hcf ⊢
663 linarith [hcf]
664
665end CarrierShuffle
666
667/-! ## §7. Availability: the gluing instances are not vacuous
668
669§6 assumes the four gluing instances. They are legitimate only if the pairs
670really do have multiplicative automorphism counts, since §4 shows that is what the
671binomial interleaving needs and §3 shows it genuinely fails for repeated
672isomorphic pieces. This section compiles the check on a nontrivial slice of
673family (a): `dust 1` glued to a bouquet with `b` loops and `c` degenerate
674tetrahedra. The automorphism groups involved have order `b! · c!`, so this is not
675a rigid or trivial-group witness; it is unbounded in both index directions.
676
677The general theorem (parts sharing no isomorphic component have multiplicative
678`Aut`) is not formalized here. That is the remaining formalization debt and it is
679recorded in the certificate rather than assumed away. -/
680
681/-- A **bouquet**: one vertex carrying `b` loops and `c` degenerate tetrahedra
682(every corner of every tetrahedron at the same vertex). -/
683def bouquet (b c : ℕ) : BoundedComplex (1 + b + c) where
684 nV := 1
685 nE := b
686 nT := c
687 hV := by omega
688 hE := by omega
689 hT := by omega
690 edgeVerts := fun _ => (0, 0)
691 tetVerts := fun _ _ => 0
692
693@[simp] theorem bouquet_nV (b c : ℕ) : (bouquet b c).nV = 1 := rfl
694@[simp] theorem bouquet_nE (b c : ℕ) : (bouquet b c).nE = b := rfl
695@[simp] theorem bouquet_nT (b c : ℕ) : (bouquet b c).nT = c := rfl
696
697/-- A **cone**: `a` isolated vertices plus one apex carrying `b` loops and `c`
698degenerate tetrahedra. The cap matches `dunion (dust a) (bouquet b c)` so the two
699are comparable. -/
700def cone (a b c : ℕ) : BoundedComplex (a + (1 + b + c)) where
701 nV := a + 1
702 nE := b
703 nT := c
704 hV := by omega
705 hE := by omega
706 hT := by omega
707 edgeVerts := fun _ => (Fin.last a, Fin.last a)
708 tetVerts := fun _ _ => Fin.last a
709
710@[simp] theorem cone_nV (a b c : ℕ) : (cone a b c).nV = a + 1 := rfl
711@[simp] theorem cone_nE (a b c : ℕ) : (cone a b c).nE = b := rfl
712@[simp] theorem cone_nT (a b c : ℕ) : (cone a b c).nT = c := rfl
713
714/-- The right-hand injection of the single bouquet vertex lands on the apex. -/
715theorem inrV_zero (a : ℕ) : (inrV (0 : Fin 1) : Fin (a + 1)) = Fin.last a := by
716 apply Fin.ext
717 simp [inrV]
718
719/-- In a union whose left part has no edges, every edge comes from the right part.
720Stated for a right part whose edges all sit on one vertex. -/
721theorem dunion_dust_edgeVerts {B' : ℕ} (a : ℕ) (L : BoundedComplex B') (v : Fin L.nV)
722 (hL : ∀ e, L.edgeVerts e = (v, v)) (e : Fin (dunion (dust a) L).nE) :
723 (dunion (dust a) L).edgeVerts e
724 = ((inrV v : Fin (a + L.nV)), (inrV v : Fin (a + L.nV))) := by
725 simp only [dunion]
726 cases hsum : finSumFinEquiv.symm e with
727 | inl i => exact Fin.elim0 i
728 | inr e' => simp [hL e']
729
730/-- The same for tetrahedra. -/
731theorem dunion_dust_tetVerts {B' : ℕ} (a : ℕ) (L : BoundedComplex B') (v : Fin L.nV)
732 (hL : ∀ t i, L.tetVerts t i = v) (t : Fin (dunion (dust a) L).nT) (i : Fin 4) :
733 (dunion (dust a) L).tetVerts t i = (inrV v : Fin (a + L.nV)) := by
734 simp only [dunion]
735 cases hsum : finSumFinEquiv.symm t with
736 | inl j => exact Fin.elim0 j
737 | inr t' => simp [hL t']
738
739/-- **The union of dust and a bouquet is the cone.** Every edge and tetrahedron of
740the union comes from the bouquet, since dust carries none, and the bouquet's single
741vertex sits at the apex. -/
742def dunionConeRelabel (a b c : ℕ) :
743 Relabel (dunion (dust a) (bouquet b c)) (cone a b c) where
744 vEquiv := Equiv.refl _
745 eEquiv := finCongr (Nat.zero_add b)
746 tEquiv := finCongr (Nat.zero_add c)
747 edge_comm := by
748 intro e
749 rw [dunion_dust_edgeVerts a (bouquet b c)
750 (show Fin (bouquet b c).nV from (0 : Fin 1)) (fun _ => rfl) e]
751 simp [cone]
752 apply Fin.ext
753 simp
754 tet_comm := by
755 intro t i
756 rw [dunion_dust_tetVerts a (bouquet b c)
757 (show Fin (bouquet b c).nV from (0 : Fin 1)) (fun _ _ => rfl) t i]
758 simp [cone]
759 apply Fin.ext
760 simp
761
762theorem dunion_cone_equivalent (a b c : ℕ) :
763 Equivalent (dunion (dust a) (bouquet b c)) (cone a b c) :=
764 ⟨dunionConeRelabel a b c⟩
765
766/-- **The automorphism group of a bouquet.** The single vertex has no choice, and
767both incidence maps are constant, so every permutation of loops and of
768tetrahedra is an automorphism and nothing else is required. -/
769def autBouquetEquiv (b c : ℕ) :
770 Aut (bouquet b c) ≃ Equiv.Perm (Fin b) × Equiv.Perm (Fin c) where
771 toFun := fun r => (r.eEquiv, r.tEquiv)
772 invFun := fun p =>
773 { vEquiv := Equiv.refl _
774 eEquiv := p.1
775 tEquiv := p.2
776 edge_comm := fun e => by simp [bouquet]
777 tet_comm := fun t i => by simp [bouquet] }
778 left_inv := by
779 intro r
780 have hv : r.vEquiv = Equiv.refl (Fin (bouquet b c).nV) := by
781 refine Equiv.ext fun i => ?_
782 simp only [Equiv.refl_apply]
783 apply Fin.ext
784 have h1 := (r.vEquiv i).isLt
785 have h2 := i.isLt
786 simp only [bouquet_nV] at h1 h2
787 omega
788 cases r
789 simp_all
790 right_inv := by intro p; rfl
791
792theorem autCard_bouquet (b c : ℕ) :
793 Nat.card (Aut (bouquet b c)) = Nat.factorial b * Nat.factorial c := by
794 rw [Nat.card_congr (autBouquetEquiv b c), Nat.card_eq_fintype_card,
795 Fintype.card_prod, Fintype.card_perm, Fintype.card_perm, Fintype.card_fin,
796 Fintype.card_fin]
797
798/-- A permutation of a two-element index set that fixes one point is the identity. -/
799theorem perm_fin_two_fixes (σ : Equiv.Perm (Fin 2)) (h : σ (Fin.last 1) = Fin.last 1) :
800 σ = Equiv.refl (Fin 2) := by
801 revert h
802 revert σ
803 decide
804
805/-- **The automorphism group of a one-dust cone.** The apex is the only vertex
806carrying an incidence, so it is fixed; with only two vertices the remaining vertex
807is fixed too, and the loops and tetrahedra permute freely. Hence
808`|Aut| = b! · c!`, which equals `|Aut (dust 1)| · |Aut (bouquet b c)|`. -/
809def autConeOneEquiv (b c : ℕ) (hbc : 1 ≤ b + c) :
810 Aut (cone 1 b c) ≃ Equiv.Perm (Fin b) × Equiv.Perm (Fin c) where
811 toFun := fun r => (r.eEquiv, r.tEquiv)
812 invFun := fun p =>
813 { vEquiv := Equiv.refl _
814 eEquiv := p.1
815 tEquiv := p.2
816 edge_comm := fun e => by simp [cone]
817 tet_comm := fun t i => by simp [cone] }
818 left_inv := by
819 intro r
820 have hfix : r.vEquiv (Fin.last 1) = Fin.last 1 := by
821 rcases Nat.eq_zero_or_pos b with hb | hb
822 · -- no loops, so there is at least one tetrahedron
823 have hc : 0 < c := by omega
824 have ht := r.tet_comm ⟨0, hc⟩ 0
825 simpa [cone] using ht.symm
826 · have he := r.edge_comm ⟨0, hb⟩
827 have := congrArg Prod.fst he
828 simpa [cone] using this.symm
829 have hv : r.vEquiv = Equiv.refl (Fin (cone 1 b c).nV) := by
830 have : r.vEquiv = Equiv.refl (Fin 2) := perm_fin_two_fixes r.vEquiv hfix
831 exact this
832 cases r
833 simp_all
834 right_inv := by intro p; rfl
835
836theorem autCard_cone_one (b c : ℕ) (hbc : 1 ≤ b + c) :
837 Nat.card (Aut (cone 1 b c)) = Nat.factorial b * Nat.factorial c := by
838 rw [Nat.card_congr (autConeOneEquiv b c hbc), Nat.card_eq_fintype_card,
839 Fintype.card_prod, Fintype.card_perm, Fintype.card_perm, Fintype.card_fin,
840 Fintype.card_fin]
841
842/-- The permutations of a vertex set fixing one distinguished vertex number the
843permutations of the rest. -/
844theorem stab_card (a : ℕ) :
845 Nat.card {σ : Equiv.Perm (Fin (a + 1)) // σ (Fin.last a) = Fin.last a}
846 = Nat.factorial a := by
847 classical
848 have hiff : ∀ f : Equiv.Perm (Fin (a + 1)),
849 f (Fin.last a) = Fin.last a ↔ ∀ x, ¬(x ≠ Fin.last a) → f x = x := by
850 intro f
851 constructor
852 · intro h x hx
853 have hxe : x = Fin.last a := by by_contra hc; exact hx hc
854 rw [hxe]; exact h
855 · intro h; exact h _ (by simp)
856 have e2 : {σ : Equiv.Perm (Fin (a + 1)) // σ (Fin.last a) = Fin.last a}
857 ≃ {f : Equiv.Perm (Fin (a + 1)) // ∀ x, ¬(x ≠ Fin.last a) → f x = x} :=
858 Equiv.subtypeEquivRight (fun f => hiff f)
859 have e1 : {f : Equiv.Perm (Fin (a + 1)) // ∀ x, ¬(x ≠ Fin.last a) → f x = x}
860 ≃ Equiv.Perm {y : Fin (a + 1) // y ≠ Fin.last a} :=
861 (Equiv.Perm.subtypeEquivSubtypePerm (fun y : Fin (a + 1) => y ≠ Fin.last a)).symm
862 have hc : Fintype.card {y : Fin (a + 1) // y ≠ Fin.last a} = a := by
863 have h := Fintype.card_subtype_compl (p := fun y : Fin (a + 1) => y = Fin.last a)
864 rw [Fintype.card_subtype_eq, Fintype.card_fin] at h
865 simpa using h
866 rw [Nat.card_congr (e2.trans e1), Nat.card_eq_fintype_card, Fintype.card_perm, hc]
867
868/-- **The automorphism group of a cone, for every amount of dust.** The apex is the
869only vertex carrying an incidence, so it is fixed; the remaining vertices permute
870freely, and so do the loops and the tetrahedra. -/
871def autConeEquiv (a b c : ℕ) (hbc : 1 ≤ b + c) :
872 Aut (cone a b c)
873 ≃ {σ : Equiv.Perm (Fin (a + 1)) // σ (Fin.last a) = Fin.last a}
874 × (Equiv.Perm (Fin b) × Equiv.Perm (Fin c)) where
875 toFun := fun r =>
876 (⟨r.vEquiv, by
877 rcases Nat.eq_zero_or_pos b with hb | hb
878 · have hc : 0 < c := by omega
879 have ht := r.tet_comm ⟨0, hc⟩ 0
880 simpa [cone] using ht.symm
881 · have he := r.edge_comm ⟨0, hb⟩
882 have h1 := congrArg Prod.fst he
883 simpa [cone] using h1.symm⟩,
884 (r.eEquiv, r.tEquiv))
885 invFun := fun p =>
886 { vEquiv := p.1.val
887 eEquiv := p.2.1
888 tEquiv := p.2.2
889 edge_comm := fun e => by simp [cone, p.1.property]
890 tet_comm := fun t i => by simp [cone, p.1.property] }
891 left_inv := by intro r; rfl
892 right_inv := by intro p; rfl
893
894theorem autCard_cone (a b c : ℕ) (hbc : 1 ≤ b + c) :
895 Nat.card (Aut (cone a b c))
896 = Nat.factorial a * (Nat.factorial b * Nat.factorial c) := by
897 classical
898 rw [Nat.card_congr (autConeEquiv a b c hbc), Nat.card_eq_fintype_card,
899 Fintype.card_prod, Fintype.card_prod, Fintype.card_perm, Fintype.card_perm,
900 Fintype.card_fin, Fintype.card_fin]
901 have hs : Fintype.card {σ : Equiv.Perm (Fin (a + 1)) // σ (Fin.last a) = Fin.last a}
902 = Nat.factorial a := by
903 have := stab_card a
904 rwa [Nat.card_eq_fintype_card] at this
905 rw [hs]
906
907/-- **THEOREM (family (a) is available in full).** For every amount of dust and every
908bouquet carrying at least one incidence, the automorphism count multiplies over the
909union. This is the family that carries the vertex, edge and tetrahedron recursions
910of §6, so the most-used gluing instance is verified rather than assumed. -/
911theorem autMul_dust_bouquet (a b c : ℕ) (hbc : 1 ≤ b + c) :
912 Nat.card (Aut (dunion (dust a) (bouquet b c)))
913 = Nat.card (Aut (dust a)) * Nat.card (Aut (bouquet b c)) := by
914 rw [autCard_congr (dunion_cone_equivalent a b c), autCard_cone a b c hbc,
915 autCard_dust, autCard_bouquet]
916
917/-- The orbit-count form: the interleaving factor is `a + 1`, and each part has a
918single labeled presentation. -/
919theorem orbitCard_dust_bouquet (a b c : ℕ) (hbc : 1 ≤ b + c) :
920 gaugeOrbitCard (dunion (dust a) (bouquet b c))
921 = interleave a 0 0 1 b c
922 * (gaugeOrbitCard (dust a) * gaugeOrbitCard (bouquet b c)) :=
923 orbitCard_dunion_of_autMul _ _ (autMul_dust_bouquet a b c hbc)
924
925/-- **THEOREM (family (a) is available at `a = 1`, for every `b` and `c`).** The
926automorphism count multiplies over `dust 1 ⊔ bouquet(b,c)`. Together with §4 this
927means the orbit counts there satisfy the binomial interleaving identity, so the
928gluing instance used in §6 is a real instance and not an assumption with no
929models. The groups have order `b! · c!`, unbounded in both directions. -/
930theorem autMul_dust_one_bouquet (b c : ℕ) (hbc : 1 ≤ b + c) :
931 Nat.card (Aut (dunion (dust 1) (bouquet b c)))
932 = Nat.card (Aut (dust 1)) * Nat.card (Aut (bouquet b c)) := by
933 rw [autCard_congr (dunion_cone_equivalent 1 b c), autCard_cone_one b c hbc,
934 autCard_dust, autCard_bouquet]
935 simp [Nat.factorial]
936
937/-- The corresponding orbit-count identity, spelled out: the interleaving factor is
938exactly `2`, which is `C(1+1, 1)`, and both parts have a single labeled
939presentation. -/
940theorem orbitCard_dust_one_bouquet (b c : ℕ) (hbc : 1 ≤ b + c) :
941 gaugeOrbitCard (dunion (dust 1) (bouquet b c))
942 = interleave 1 0 0 1 b c
943 * (gaugeOrbitCard (dust 1) * gaugeOrbitCard (bouquet b c)) :=
944 orbitCard_dunion_of_autMul _ _ (autMul_dust_one_bouquet b c hbc)
945
946/-! ### §7b. Families (b), (c) and (d): the rigid parts
947
948Family (a) is dust glued to a bouquet, and its availability is above. The other three
949instances glue something to a *rigid* part, meaning a part with no automorphisms: a
950single edge with distinct endpoints, or a single nondegenerate tetrahedron. Rigidity is
951what makes those parts contribute a factor of one, and it is a consequence of the
952incidence data being *ordered*: a relabeling must match the endpoint pair in order, so it
953cannot reverse an edge or rotate a tetrahedron.
954
955The no-mixing argument is the same in all three cases and rests on invariants of the
956incidence pattern rather than on any count. Whether an edge's two endpoints coincide is
957preserved by relabeling, so a loop can never map to a proper edge; whether a
958tetrahedron's four corners coincide is preserved likewise. So the parts cannot exchange
959cells, every vertex carrying an incidence is pinned, and the remaining vertices permute
960freely. -/
961
962/-- A permutation fixing every point but one fixes that one too, by injectivity. -/
963theorem perm_fix_of_fixes_others {α : Type*} (σ : Equiv.Perm α) (x : α)
964 (h : ∀ y, y ≠ x → σ y = y) : σ x = x := by
965 by_contra hx
966 exact hx (σ.injective (h (σ x) hx))
967
968/-- The permutations of an index set fixing one distinguished point number the
969permutations of the rest. The `Fin.last` case is `stab_card`; this is the general one,
970needed because the distinguished point of a union is an injection image. -/
971theorem stab1_card {m : ℕ} (p : Fin (m + 1)) :
972 Nat.card {σ : Equiv.Perm (Fin (m + 1)) // σ p = p} = Nat.factorial m := by
973 classical
974 have hiff : ∀ f : Equiv.Perm (Fin (m + 1)),
975 f p = p ↔ ∀ x, ¬(x ≠ p) → f x = x := by
976 intro f
977 constructor
978 · intro h x hx
979 rw [not_not.mp hx]; exact h
980 · intro h; exact h _ (by simp)
981 have e2 : {σ : Equiv.Perm (Fin (m + 1)) // σ p = p}
982 ≃ {f : Equiv.Perm (Fin (m + 1)) // ∀ x, ¬(x ≠ p) → f x = x} :=
983 Equiv.subtypeEquivRight (fun f => hiff f)
984 have e1 : {f : Equiv.Perm (Fin (m + 1)) // ∀ x, ¬(x ≠ p) → f x = x}
985 ≃ Equiv.Perm {y : Fin (m + 1) // y ≠ p} :=
986 (Equiv.Perm.subtypeEquivSubtypePerm (fun y : Fin (m + 1) => y ≠ p)).symm
987 have hc : Fintype.card {y : Fin (m + 1) // y ≠ p} = m := by
988 have h := Fintype.card_subtype_compl (p := fun y : Fin (m + 1) => y = p)
989 rw [Fintype.card_subtype_eq, Fintype.card_fin] at h
990 simpa using h
991 rw [Nat.card_congr (e2.trans e1), Nat.card_eq_fintype_card, Fintype.card_perm, hc]
992
993/-! The four accessors for a union's incidence maps. Every cell of the union is the
994image of a cell of one part, and these say what its vertices are. -/
995
996theorem dunion_edgeVerts_inl {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B')
997 (e : Fin K.nE) :
998 (dunion K L).edgeVerts (finSumFinEquiv (Sum.inl e))
999 = ((inlV (K.edgeVerts e).1 : Fin (K.nV + L.nV)), inlV (K.edgeVerts e).2) := by
1000 simp only [dunion, Equiv.symm_apply_apply, Sum.elim_inl]
1001
1002theorem dunion_edgeVerts_inr {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B')
1003 (e : Fin L.nE) :
1004 (dunion K L).edgeVerts (finSumFinEquiv (Sum.inr e))
1005 = ((inrV (L.edgeVerts e).1 : Fin (K.nV + L.nV)), inrV (L.edgeVerts e).2) := by
1006 simp only [dunion, Equiv.symm_apply_apply, Sum.elim_inr]
1007
1008theorem dunion_tetVerts_inl {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B')
1009 (t : Fin K.nT) (i : Fin 4) :
1010 (dunion K L).tetVerts (finSumFinEquiv (Sum.inl t)) i
1011 = (inlV (K.tetVerts t i) : Fin (K.nV + L.nV)) := by
1012 simp only [dunion, Equiv.symm_apply_apply, Sum.elim_inl]
1013
1014theorem dunion_tetVerts_inr {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B')
1015 (t : Fin L.nT) (i : Fin 4) :
1016 (dunion K L).tetVerts (finSumFinEquiv (Sum.inr t)) i
1017 = (inrV (L.tetVerts t i) : Fin (K.nV + L.nV)) := by
1018 simp only [dunion, Equiv.symm_apply_apply, Sum.elim_inr]
1019
1020/-- A **single edge** with distinct endpoints: two vertices, one edge joining them. -/
1021def edge : BoundedComplex 2 where
1022 nV := 2
1023 nE := 1
1024 nT := 0
1025 hV := le_refl 2
1026 hE := by omega
1027 hT := by omega
1028 edgeVerts := fun _ => (0, Fin.last 1)
1029 tetVerts := fun t _ => Fin.elim0 t
1030
1031@[simp] theorem edge_nV : edge.nV = 2 := rfl
1032@[simp] theorem edge_nE : edge.nE = 1 := rfl
1033@[simp] theorem edge_nT : edge.nT = 0 := rfl
1034
1035/-- A permutation of a one-element index set is the identity. -/
1036theorem perm_fin_one (σ : Equiv.Perm (Fin 1)) : σ = Equiv.refl (Fin 1) := by
1037 refine Equiv.ext fun i => ?_
1038 apply Fin.ext
1039 have h1 := (σ i).isLt
1040 have h2 := i.isLt
1041 omega
1042
1043/-- A permutation of an empty index set is the identity. -/
1044theorem perm_fin_zero (σ : Equiv.Perm (Fin 0)) : σ = Equiv.refl (Fin 0) :=
1045 Equiv.ext fun t => Fin.elim0 t
1046
1047/-- The permutations of a vertex set fixing two distinguished vertices number the
1048permutations of the rest. Used by all three rigid families. -/
1049theorem stab2_card {m : ℕ} (p q : Fin (m + 2)) (hpq : p ≠ q) :
1050 Nat.card {σ : Equiv.Perm (Fin (m + 2)) // σ p = p ∧ σ q = q} = Nat.factorial m := by
1051 classical
1052 have hiff : ∀ f : Equiv.Perm (Fin (m + 2)),
1053 (f p = p ∧ f q = q) ↔ ∀ x, ¬(x ≠ p ∧ x ≠ q) → f x = x := by
1054 intro f
1055 constructor
1056 · intro h x hx
1057 rcases not_and_or.mp hx with hxp | hxq
1058 · rw [not_not.mp hxp]; exact h.1
1059 · rw [not_not.mp hxq]; exact h.2
1060 · intro h
1061 exact ⟨h p (by simp), h q (by simp)⟩
1062 have e2 : {σ : Equiv.Perm (Fin (m + 2)) // σ p = p ∧ σ q = q}
1063 ≃ {f : Equiv.Perm (Fin (m + 2)) // ∀ x, ¬(x ≠ p ∧ x ≠ q) → f x = x} :=
1064 Equiv.subtypeEquivRight (fun f => hiff f)
1065 have e1 : {f : Equiv.Perm (Fin (m + 2)) // ∀ x, ¬(x ≠ p ∧ x ≠ q) → f x = x}
1066 ≃ Equiv.Perm {y : Fin (m + 2) // y ≠ p ∧ y ≠ q} :=
1067 (Equiv.Perm.subtypeEquivSubtypePerm (fun y : Fin (m + 2) => y ≠ p ∧ y ≠ q)).symm
1068 have hc : Fintype.card {y : Fin (m + 2) // y ≠ p ∧ y ≠ q} = m := by
1069 rw [Fintype.card_subtype]
1070 have hfil : (Finset.univ.filter (fun y : Fin (m + 2) => y ≠ p ∧ y ≠ q))
1071 = Finset.univ \ {p, q} := by
1072 ext y
1073 simp only [Finset.mem_filter, Finset.mem_univ, true_and, Finset.mem_sdiff,
1074 Finset.mem_insert, Finset.mem_singleton]
1075 tauto
1076 rw [hfil, Finset.card_sdiff, Finset.inter_univ, Finset.card_univ, Fintype.card_fin,
1077 Finset.card_insert_of_notMem (by simpa using hpq), Finset.card_singleton]
1078 omega
1079 rw [Nat.card_congr (e2.trans e1), Nat.card_eq_fintype_card, Fintype.card_perm, hc]
1080
1081/-- The single edge is **rigid**: ordered endpoints leave no automorphism. -/
1082def autEdgeEquiv : Aut edge ≃ Unit where
1083 toFun := fun _ => ()
1084 invFun := fun _ =>
1085 { vEquiv := Equiv.refl _
1086 eEquiv := Equiv.refl _
1087 tEquiv := Equiv.refl _
1088 edge_comm := fun e => by simp [edge]
1089 tet_comm := fun t _ => Fin.elim0 t }
1090 left_inv := by
1091 intro r
1092 have hsnd : r.vEquiv (Fin.last 1) = Fin.last 1 := by
1093 have he := r.edge_comm (show Fin edge.nE from (0 : Fin 1))
1094 have h2 := congrArg Prod.snd he
1095 simpa [edge] using h2.symm
1096 have hv : r.vEquiv = Equiv.refl (Fin edge.nV) := perm_fin_two_fixes r.vEquiv hsnd
1097 have he : r.eEquiv = Equiv.refl (Fin edge.nE) := perm_fin_one r.eEquiv
1098 have ht : r.tEquiv = Equiv.refl (Fin edge.nT) := perm_fin_zero r.tEquiv
1099 cases r
1100 simp_all
1101 right_inv := by intro _; rfl
1102
1103theorem autCard_edge : Nat.card (Aut edge) = 1 := by
1104 rw [Nat.card_congr autEdgeEquiv, Nat.card_eq_fintype_card, Fintype.card_unit]
1105
1106/-- In a union of dust with the single edge, the one edge is the pushed-in edge of the
1107right part, so its endpoints are the two pushed-in vertices. -/
1108theorem dunion_dust_edge_edgeVerts (a : ℕ) (e : Fin (dunion (dust a) edge).nE) :
1109 (dunion (dust a) edge).edgeVerts e
1110 = ((inrV (0 : Fin 2) : Fin (a + 2)), (inrV (Fin.last 1) : Fin (a + 2))) := by
1111 simp only [dunion]
1112 cases hsum : finSumFinEquiv.symm e with
1113 | inl i => exact Fin.elim0 i
1114 | inr e' => simp [edge]
1115
1116/-- The two pushed-in endpoints are distinct, since the injection is injective. -/
1117theorem inrV_edge_ne (a : ℕ) :
1118 (inrV (0 : Fin 2) : Fin (a + 2)) ≠ (inrV (Fin.last 1) : Fin (a + 2)) := by
1119 intro h
1120 have hv := congrArg Fin.val h
1121 simp [inrV] at hv
1122
1123/-- **The automorphism group of dust glued to an edge.** The two endpoints are the only
1124vertices carrying an incidence and the edge is ordered, so both are pinned; the dust
1125permutes freely. -/
1126def autDustEdgeEquiv (a : ℕ) :
1127 Aut (dunion (dust a) edge)
1128 ≃ {σ : Equiv.Perm (Fin (a + 2)) //
1129 σ (inrV (0 : Fin 2)) = inrV (0 : Fin 2)
1130 ∧ σ (inrV (Fin.last 1)) = inrV (Fin.last 1)} where
1131 toFun := fun r =>
1132 ⟨r.vEquiv, by
1133 have he := r.edge_comm (show Fin (dunion (dust a) edge).nE from (0 : Fin 1))
1134 rw [dunion_dust_edge_edgeVerts, dunion_dust_edge_edgeVerts] at he
1135 simp only [Prod.map, Prod.mk.injEq] at he
1136 exact ⟨he.1.symm, he.2.symm⟩⟩
1137 invFun := fun σ =>
1138 { vEquiv := σ.val
1139 eEquiv := Equiv.refl _
1140 tEquiv := Equiv.refl _
1141 edge_comm := fun e => by
1142 rw [dunion_dust_edge_edgeVerts, dunion_dust_edge_edgeVerts]
1143 simp only [Prod.map, Prod.mk.injEq]
1144 exact ⟨σ.property.1.symm, σ.property.2.symm⟩
1145 tet_comm := fun t _ => Fin.elim0 t }
1146 left_inv := by
1147 intro r
1148 have he : r.eEquiv = Equiv.refl (Fin (dunion (dust a) edge).nE) :=
1149 perm_fin_one r.eEquiv
1150 have ht : r.tEquiv = Equiv.refl (Fin (dunion (dust a) edge).nT) :=
1151 perm_fin_zero r.tEquiv
1152 cases r
1153 simp_all
1154 right_inv := by intro _; rfl
1155
1156theorem autCard_dust_edge (a : ℕ) :
1157 Nat.card (Aut (dunion (dust a) edge)) = Nat.factorial a := by
1158 rw [Nat.card_congr (autDustEdgeEquiv a)]
1159 exact stab2_card _ _ (inrV_edge_ne a)
1160
1161/-- **THEOREM (family (b) is available).** The automorphism count multiplies over dust
1162glued to a single edge, for every amount of dust. -/
1163theorem autMul_dust_edge (a : ℕ) :
1164 Nat.card (Aut (dunion (dust a) edge))
1165 = Nat.card (Aut (dust a)) * Nat.card (Aut edge) := by
1166 rw [autCard_dust_edge, autCard_dust, autCard_edge, mul_one]
1167
1168/-- The orbit-count form of family (b). -/
1169theorem orbitCard_dust_edge (a : ℕ) :
1170 gaugeOrbitCard (dunion (dust a) edge)
1171 = interleave a 0 0 2 1 0
1172 * (gaugeOrbitCard (dust a) * gaugeOrbitCard edge) :=
1173 orbitCard_dunion_of_autMul _ _ (autMul_dust_edge a)
1174
1175/-! ### Family (c): a bouquet glued to an edge
1176
1177Three vertices: the bouquet vertex, and the edge's two ends. The proper edge cannot map
1178to a loop, because a relabeling carries the two endpoints of an edge to the two endpoints
1179of its image *in order*, so it preserves whether those endpoints coincide. That pins both
1180ends of the proper edge, and then the bouquet vertex is pinned because a permutation
1181fixing all but one point fixes that one. Loops and tetrahedra then permute freely. -/
1182
1183/-- The bouquet vertex of the union. -/
1184def bqeV (b c : ℕ) : Fin (dunion (bouquet b c) edge).nV := inlV (0 : Fin 1)
1185/-- The first endpoint of the proper edge. -/
1186def bqeV0 (b c : ℕ) : Fin (dunion (bouquet b c) edge).nV := inrV (0 : Fin 2)
1187/-- The second endpoint of the proper edge. -/
1188def bqeV1 (b c : ℕ) : Fin (dunion (bouquet b c) edge).nV := inrV (Fin.last 1)
1189/-- The index of the proper edge. -/
1190def bqePE (b c : ℕ) : Fin (dunion (bouquet b c) edge).nE := finSumFinEquiv (Sum.inr (0 : Fin 1))
1191
1192theorem bqeV_val (b c : ℕ) : (bqeV b c).val = 0 := rfl
1193theorem bqeV0_val (b c : ℕ) : (bqeV0 b c).val = 1 := rfl
1194theorem bqeV1_val (b c : ℕ) : (bqeV1 b c).val = 2 := rfl
1195
1196/-- The proper edge joins the two pushed-in endpoints. -/
1197theorem bqe_edgeVerts_pE (b c : ℕ) :
1198 (dunion (bouquet b c) edge).edgeVerts (bqePE b c) = (bqeV0 b c, bqeV1 b c) := by
1199 rw [bqePE, dunion_edgeVerts_inr]
1200 rfl
1201
1202/-- Every other edge of the union is a loop at the bouquet vertex. -/
1203theorem bqe_edgeVerts_loop (b c : ℕ) (e : Fin (dunion (bouquet b c) edge).nE)
1204 (h : e ≠ bqePE b c) :
1205 (dunion (bouquet b c) edge).edgeVerts e = (bqeV b c, bqeV b c) := by
1206 cases hsum : finSumFinEquiv.symm e with
1207 | inl i =>
1208 have he : e = finSumFinEquiv (Sum.inl i) := by rw [← hsum, Equiv.apply_symm_apply]
1209 rw [he, dunion_edgeVerts_inl]
1210 rfl
1211 | inr j =>
1212 exfalso
1213 apply h
1214 apply finSumFinEquiv.symm.injective
1215 rw [hsum, bqePE, Equiv.symm_apply_apply]
1216 congr 1
1217 apply Fin.ext
1218 have h1 : edge.nE = 1 := rfl
1219 have h2 := j.isLt
1220 have h3 : ((0 : Fin 1) : ℕ) = 0 := rfl
1221 omega
1222
1223/-- Every tetrahedron of the union sits at the bouquet vertex, the edge having none. -/
1224theorem bqe_tetVerts (b c : ℕ) (t : Fin (dunion (bouquet b c) edge).nT) (i : Fin 4) :
1225 (dunion (bouquet b c) edge).tetVerts t i = bqeV b c := by
1226 cases hsum : finSumFinEquiv.symm t with
1227 | inl t' =>
1228 have ht : t = finSumFinEquiv (Sum.inl t') := by rw [← hsum, Equiv.apply_symm_apply]
1229 rw [ht, dunion_tetVerts_inl]
1230 rfl
1231 | inr t' => exact Fin.elim0 t'
1232
1233/-- **The proper edge is fixed.** If it mapped to a loop its two distinct endpoints
1234would have the same image. -/
1235theorem bqe_eEquiv_fixes (b c : ℕ) (r : Aut (dunion (bouquet b c) edge)) :
1236 r.eEquiv (bqePE b c) = bqePE b c := by
1237 by_contra hne
1238 have he := r.edge_comm (bqePE b c)
1239 rw [bqe_edgeVerts_loop b c _ hne, bqe_edgeVerts_pE] at he
1240 simp only [Prod.map, Prod.mk.injEq] at he
1241 have hcoll : bqeV0 b c = bqeV1 b c := r.vEquiv.injective (he.1.symm.trans he.2)
1242 have h0 := bqeV0_val b c
1243 rw [hcoll] at h0
1244 rw [bqeV1_val] at h0
1245 omega
1246
1247/-- **Every vertex is pinned.** -/
1248theorem bqe_vEquiv_refl (b c : ℕ) (r : Aut (dunion (bouquet b c) edge)) :
1249 r.vEquiv = Equiv.refl _ := by
1250 have he := r.edge_comm (bqePE b c)
1251 rw [bqe_eEquiv_fixes, bqe_edgeVerts_pE] at he
1252 simp only [Prod.map, Prod.mk.injEq] at he
1253 have h0 : r.vEquiv (bqeV0 b c) = bqeV0 b c := he.1.symm
1254 have h1 : r.vEquiv (bqeV1 b c) = bqeV1 b c := he.2.symm
1255 have hother : ∀ y, y ≠ bqeV b c → r.vEquiv y = y := by
1256 intro y hy
1257 have hlt : y.val < 3 := by
1258 have := y.isLt
1259 simpa using this
1260 have hne0 : y.val ≠ 0 := fun hc => hy (Fin.ext (by rw [hc, bqeV_val]))
1261 rcases (show y.val = 1 ∨ y.val = 2 by omega) with h | h
1262 · have hy0 : y = bqeV0 b c := Fin.ext (by rw [h, bqeV0_val])
1263 rw [hy0]; exact h0
1264 · have hy1 : y = bqeV1 b c := Fin.ext (by rw [h, bqeV1_val])
1265 rw [hy1]; exact h1
1266 have hb := perm_fix_of_fixes_others r.vEquiv (bqeV b c) hother
1267 refine Equiv.ext fun y => ?_
1268 by_cases hy : y = bqeV b c
1269 · rw [hy]; exact hb
1270 · exact hother y hy
1271
1272/-- **The automorphism group of a bouquet glued to an edge.** The edge contributes
1273nothing; the loops and the tetrahedra permute freely. -/
1274def autBqEdgeEquiv (b c : ℕ) :
1275 Aut (dunion (bouquet b c) edge)
1276 ≃ {σ : Equiv.Perm (Fin (dunion (bouquet b c) edge).nE) // σ (bqePE b c) = bqePE b c}
1277 × Equiv.Perm (Fin (dunion (bouquet b c) edge).nT) where
1278 toFun := fun r => (⟨r.eEquiv, bqe_eEquiv_fixes b c r⟩, r.tEquiv)
1279 invFun := fun p =>
1280 { vEquiv := Equiv.refl _
1281 eEquiv := p.1.val
1282 tEquiv := p.2
1283 edge_comm := fun e => by
1284 by_cases he : e = bqePE b c
1285 · rw [he, p.1.property, bqe_edgeVerts_pE]; rfl
1286 · have hne : p.1.val e ≠ bqePE b c := fun hc =>
1287 he (p.1.val.injective (hc.trans p.1.property.symm))
1288 rw [bqe_edgeVerts_loop b c _ hne, bqe_edgeVerts_loop b c e he]; rfl
1289 tet_comm := fun t i => by rw [bqe_tetVerts, bqe_tetVerts]; rfl }
1290 left_inv := by
1291 intro r
1292 have hv := bqe_vEquiv_refl b c r
1293 cases r
1294 simp_all
1295 right_inv := by intro _; rfl
1296
1297theorem autCard_bouquet_edge (b c : ℕ) :
1298 Nat.card (Aut (dunion (bouquet b c) edge))
1299 = Nat.factorial b * Nat.factorial c := by
1300 classical
1301 rw [Nat.card_congr (autBqEdgeEquiv b c), Nat.card_eq_fintype_card, Fintype.card_prod,
1302 Fintype.card_perm, Fintype.card_fin]
1303 have hs : Fintype.card {σ : Equiv.Perm (Fin (dunion (bouquet b c) edge).nE) //
1304 σ (bqePE b c) = bqePE b c} = Nat.factorial b := by
1305 have := stab1_card (bqePE b c)
1306 rwa [Nat.card_eq_fintype_card] at this
1307 rw [hs]
1308 simp
1309
1310/-- **THEOREM (family (c) is available).** The automorphism count multiplies over a
1311bouquet glued to a single edge, for every number of loops and tetrahedra. -/
1312theorem autMul_bouquet_edge (b c : ℕ) :
1313 Nat.card (Aut (dunion (bouquet b c) edge))
1314 = Nat.card (Aut (bouquet b c)) * Nat.card (Aut edge) := by
1315 rw [autCard_bouquet_edge, autCard_bouquet, autCard_edge, mul_one]
1316
1317/-- The orbit-count form of family (c). -/
1318theorem orbitCard_bouquet_edge (b c : ℕ) :
1319 gaugeOrbitCard (dunion (bouquet b c) edge)
1320 = interleave 1 b c 2 1 0
1321 * (gaugeOrbitCard (bouquet b c) * gaugeOrbitCard edge) :=
1322 orbitCard_dunion_of_autMul _ _ (autMul_bouquet_edge b c)
1323
1324/-! ### Family (d): a bouquet glued to a tetrahedron
1325
1326Five vertices. A nondegenerate tetrahedron cannot map to a degenerate one, because a
1327relabeling carries the four corners of a tetrahedron to the four corners of its image in
1328order, so it preserves whether they coincide. That pins all four corners, and the
1329bouquet vertex follows. -/
1330
1331/-- A **single nondegenerate tetrahedron**: four vertices, one tetrahedron on them. -/
1332def tetra : BoundedComplex 4 where
1333 nV := 4
1334 nE := 0
1335 nT := 1
1336 hV := le_refl 4
1337 hE := by omega
1338 hT := by omega
1339 edgeVerts := Fin.elim0
1340 tetVerts := fun _ i => i
1341
1342@[simp] theorem tetra_nV : tetra.nV = 4 := rfl
1343@[simp] theorem tetra_nE : tetra.nE = 0 := rfl
1344@[simp] theorem tetra_nT : tetra.nT = 1 := rfl
1345
1346/-- The tetrahedron is **rigid**: ordered corners leave no automorphism. -/
1347def autTetraEquiv : Aut tetra ≃ Unit where
1348 toFun := fun _ => ()
1349 invFun := fun _ =>
1350 { vEquiv := Equiv.refl _
1351 eEquiv := Equiv.refl _
1352 tEquiv := Equiv.refl _
1353 edge_comm := fun e => Fin.elim0 e
1354 tet_comm := fun t i => by simp [tetra] }
1355 left_inv := by
1356 intro r
1357 have hv : r.vEquiv = Equiv.refl (Fin tetra.nV) := by
1358 refine Equiv.ext fun i => ?_
1359 have ht := r.tet_comm (show Fin tetra.nT from (0 : Fin 1)) i
1360 simpa [tetra] using ht.symm
1361 have he : r.eEquiv = Equiv.refl (Fin tetra.nE) := perm_fin_zero r.eEquiv
1362 have ht : r.tEquiv = Equiv.refl (Fin tetra.nT) := perm_fin_one r.tEquiv
1363 cases r
1364 simp_all
1365 right_inv := by intro _; rfl
1366
1367theorem autCard_tetra : Nat.card (Aut tetra) = 1 := by
1368 rw [Nat.card_congr autTetraEquiv, Nat.card_eq_fintype_card, Fintype.card_unit]
1369
1370/-- The bouquet vertex of the union. -/
1371def bqtV (b c : ℕ) : Fin (dunion (bouquet b c) tetra).nV := inlV (0 : Fin 1)
1372/-- The `i`-th corner of the tetrahedron, in the union. -/
1373def bqtVR (b c : ℕ) (i : Fin 4) : Fin (dunion (bouquet b c) tetra).nV := inrV i
1374/-- The index of the nondegenerate tetrahedron. -/
1375def bqtPT (b c : ℕ) : Fin (dunion (bouquet b c) tetra).nT := finSumFinEquiv (Sum.inr (0 : Fin 1))
1376
1377theorem bqtV_val (b c : ℕ) : (bqtV b c).val = 0 := rfl
1378theorem bqtVR_val (b c : ℕ) (i : Fin 4) : (bqtVR b c i).val = 1 + i.val := rfl
1379
1380/-- The nondegenerate tetrahedron has the four pushed-in corners. -/
1381theorem bqt_tetVerts_pT (b c : ℕ) (i : Fin 4) :
1382 (dunion (bouquet b c) tetra).tetVerts (bqtPT b c) i = bqtVR b c i := by
1383 rw [bqtPT, dunion_tetVerts_inr]
1384 rfl
1385
1386/-- Every other tetrahedron sits at the bouquet vertex. -/
1387theorem bqt_tetVerts_deg (b c : ℕ) (t : Fin (dunion (bouquet b c) tetra).nT)
1388 (h : t ≠ bqtPT b c) (i : Fin 4) :
1389 (dunion (bouquet b c) tetra).tetVerts t i = bqtV b c := by
1390 cases hsum : finSumFinEquiv.symm t with
1391 | inl t' =>
1392 have ht : t = finSumFinEquiv (Sum.inl t') := by rw [← hsum, Equiv.apply_symm_apply]
1393 rw [ht, dunion_tetVerts_inl]
1394 rfl
1395 | inr t' =>
1396 exfalso
1397 apply h
1398 apply finSumFinEquiv.symm.injective
1399 rw [hsum, bqtPT, Equiv.symm_apply_apply]
1400 congr 1
1401 apply Fin.ext
1402 have h1 : tetra.nT = 1 := rfl
1403 have h2 := t'.isLt
1404 have h3 : ((0 : Fin 1) : ℕ) = 0 := rfl
1405 omega
1406
1407/-- Every edge of the union is a loop at the bouquet vertex, the tetrahedron having none. -/
1408theorem bqt_edgeVerts (b c : ℕ) (e : Fin (dunion (bouquet b c) tetra).nE) :
1409 (dunion (bouquet b c) tetra).edgeVerts e = (bqtV b c, bqtV b c) := by
1410 cases hsum : finSumFinEquiv.symm e with
1411 | inl i =>
1412 have he : e = finSumFinEquiv (Sum.inl i) := by rw [← hsum, Equiv.apply_symm_apply]
1413 rw [he, dunion_edgeVerts_inl]
1414 rfl
1415 | inr j => exact Fin.elim0 j
1416
1417/-- **The nondegenerate tetrahedron is fixed.** If it mapped to a degenerate one, its
1418four distinct corners would all have the same image. -/
1419theorem bqt_tEquiv_fixes (b c : ℕ) (r : Aut (dunion (bouquet b c) tetra)) :
1420 r.tEquiv (bqtPT b c) = bqtPT b c := by
1421 by_contra hne
1422 have h0 := r.tet_comm (bqtPT b c) 0
1423 have h1 := r.tet_comm (bqtPT b c) 1
1424 rw [bqt_tetVerts_deg b c _ hne, bqt_tetVerts_pT] at h0
1425 rw [bqt_tetVerts_deg b c _ hne, bqt_tetVerts_pT] at h1
1426 have hcoll : bqtVR b c 0 = bqtVR b c 1 := r.vEquiv.injective (h0.symm.trans h1)
1427 have hv := bqtVR_val b c 0
1428 rw [hcoll, bqtVR_val] at hv
1429 simp at hv
1430
1431/-- **Every vertex is pinned.** -/
1432theorem bqt_vEquiv_refl (b c : ℕ) (r : Aut (dunion (bouquet b c) tetra)) :
1433 r.vEquiv = Equiv.refl _ := by
1434 have hcorner : ∀ i : Fin 4, r.vEquiv (bqtVR b c i) = bqtVR b c i := by
1435 intro i
1436 have ht := r.tet_comm (bqtPT b c) i
1437 rw [bqt_tEquiv_fixes, bqt_tetVerts_pT] at ht
1438 exact ht.symm
1439 have hother : ∀ y, y ≠ bqtV b c → r.vEquiv y = y := by
1440 intro y hy
1441 have hlt : y.val < 5 := by
1442 have := y.isLt
1443 simpa using this
1444 have hne0 : y.val ≠ 0 := fun hc => hy (Fin.ext (by rw [hc, bqtV_val]))
1445 have hyi : y = bqtVR b c ⟨y.val - 1, by omega⟩ := by
1446 apply Fin.ext
1447 rw [bqtVR_val]
1448 show y.val = 1 + (y.val - 1)
1449 omega
1450 rw [hyi]
1451 exact hcorner _
1452 have hb := perm_fix_of_fixes_others r.vEquiv (bqtV b c) hother
1453 refine Equiv.ext fun y => ?_
1454 by_cases hy : y = bqtV b c
1455 · rw [hy]; exact hb
1456 · exact hother y hy
1457
1458/-- **The automorphism group of a bouquet glued to a tetrahedron.** -/
1459def autBqTetEquiv (b c : ℕ) :
1460 Aut (dunion (bouquet b c) tetra)
1461 ≃ Equiv.Perm (Fin (dunion (bouquet b c) tetra).nE)
1462 × {τ : Equiv.Perm (Fin (dunion (bouquet b c) tetra).nT) //
1463 τ (bqtPT b c) = bqtPT b c} where
1464 toFun := fun r => (r.eEquiv, ⟨r.tEquiv, bqt_tEquiv_fixes b c r⟩)
1465 invFun := fun p =>
1466 { vEquiv := Equiv.refl _
1467 eEquiv := p.1
1468 tEquiv := p.2.val
1469 edge_comm := fun e => by rw [bqt_edgeVerts, bqt_edgeVerts]; rfl
1470 tet_comm := fun t i => by
1471 by_cases ht : t = bqtPT b c
1472 · rw [ht, p.2.property, bqt_tetVerts_pT]; rfl
1473 · have hne : p.2.val t ≠ bqtPT b c := fun hc =>
1474 ht (p.2.val.injective (hc.trans p.2.property.symm))
1475 rw [bqt_tetVerts_deg b c _ hne, bqt_tetVerts_deg b c t ht]; rfl }
1476 left_inv := by
1477 intro r
1478 have hv := bqt_vEquiv_refl b c r
1479 cases r
1480 simp_all
1481 right_inv := by intro _; rfl
1482
1483theorem autCard_bouquet_tetra (b c : ℕ) :
1484 Nat.card (Aut (dunion (bouquet b c) tetra))
1485 = Nat.factorial b * Nat.factorial c := by
1486 classical
1487 rw [Nat.card_congr (autBqTetEquiv b c), Nat.card_eq_fintype_card, Fintype.card_prod,
1488 Fintype.card_perm, Fintype.card_fin]
1489 have hs : Fintype.card {τ : Equiv.Perm (Fin (dunion (bouquet b c) tetra).nT) //
1490 τ (bqtPT b c) = bqtPT b c} = Nat.factorial c := by
1491 have := stab1_card (bqtPT b c)
1492 rwa [Nat.card_eq_fintype_card] at this
1493 rw [hs]
1494 simp
1495
1496/-- **THEOREM (family (d) is available).** The automorphism count multiplies over a
1497bouquet glued to a single nondegenerate tetrahedron. -/
1498theorem autMul_bouquet_tetra (b c : ℕ) :
1499 Nat.card (Aut (dunion (bouquet b c) tetra))
1500 = Nat.card (Aut (bouquet b c)) * Nat.card (Aut tetra) := by
1501 rw [autCard_bouquet_tetra, autCard_bouquet, autCard_tetra, mul_one]
1502
1503/-- The orbit-count form of family (d). -/
1504theorem orbitCard_bouquet_tetra (b c : ℕ) :
1505 gaugeOrbitCard (dunion (bouquet b c) tetra)
1506 = interleave 1 b c 4 0 1
1507 * (gaugeOrbitCard (bouquet b c) * gaugeOrbitCard tetra) :=
1508 orbitCard_dunion_of_autMul _ _ (autMul_bouquet_tetra b c)
1509
1510/-- **THEOREM (all four gluing instances of the premise set are available).** Each of
1511the four unions used by `CarrierShuffle` has automorphism counts that multiply, so the
1512premise set is not an assumption about instances that might fail to exist: it is a
1513statement about four verified families, each unbounded in the sizes it ranges over. -/
1514theorem all_four_families_available :
1515 (∀ a b c : ℕ, 1 ≤ b + c →
1516 Nat.card (Aut (dunion (dust a) (bouquet b c)))
1517 = Nat.card (Aut (dust a)) * Nat.card (Aut (bouquet b c)))
1518 ∧ (∀ a : ℕ, Nat.card (Aut (dunion (dust a) edge))
1519 = Nat.card (Aut (dust a)) * Nat.card (Aut edge))
1520 ∧ (∀ b c : ℕ, Nat.card (Aut (dunion (bouquet b c) edge))
1521 = Nat.card (Aut (bouquet b c)) * Nat.card (Aut edge))
1522 ∧ (∀ b c : ℕ, Nat.card (Aut (dunion (bouquet b c) tetra))
1523 = Nat.card (Aut (bouquet b c)) * Nat.card (Aut tetra)) :=
1524 ⟨fun a b c h => autMul_dust_bouquet a b c h, autMul_dust_edge,
1525 autMul_bouquet_edge, autMul_bouquet_tetra⟩
1526
1527/-! ## §8. How much each premise does: the strength measurement
1528
1529A hostile panel run on §1-§7 converged on one correction: the headline "two premises
1530force the measure" hides the fact that the two premises are not equal partners. This
1531section measures them, because a premise set is only worth what its weakest member
1532excludes.
1533
1534The first theorem is a decoy: a relabeling-invariant weight that satisfies premise (ii)
1535at **every** pair, with no side condition whatsoever, and is not the RS measure. So
1536premise (ii) alone excludes nothing at all, and every bit of the derivation's force
1537comes from premise (i).
1538
1539The second is the converse receipt. Premise (ii) is *silent* on `dust a ⊔ dust b`,
1540because the parts share a component; C3 shows why it must be. The weight the two
1541premises force nonetheless satisfies the binomial-corrected identity there, as a
1542theorem rather than a premise. A premise fitted to produce a wanted answer would
1543have its bodies buried exactly in the region it declines to speak about; this one
1544predicts that region correctly. That is the non-circularity receipt. -/
1545
1546/-- The **uniform weight**: unit mass per class, spread evenly over the labeled
1547complexes presenting it. This is the honest "labels are physical, every class counts
1548once" alternative to the RS measure. -/
1549noncomputable def uniformWeight {B : ℕ} (K : BoundedComplex B) : ℝ :=
1550 1 / (gaugeOrbitCard K : ℝ)
1551
1552/-- The gauge orbit count is relabeling-invariant: it is determined by the index sizes
1553and the automorphism count, and a relabeling preserves both. -/
1554theorem gaugeOrbitCard_congr {B : ℕ} {K K' : BoundedComplex B} (h : Equivalent K K') :
1555 gaugeOrbitCard K = gaugeOrbitCard K' := by
1556 have hK := orbitCard_mul_autCard K
1557 have hK' := orbitCard_mul_autCard K'
1558 obtain ⟨r⟩ := h
1559 rw [size_v r, size_e r, size_t r] at hK
1560 rw [autCard_congr ⟨r⟩] at hK
1561 exact Nat.eq_of_mul_eq_mul_right (autCard_pos K') (hK.trans hK'.symm)
1562
1563theorem uniformWeight_invariant {B : ℕ} {K K' : BoundedComplex B} (h : Equivalent K K') :
1564 uniformWeight K = uniformWeight K' := by
1565 unfold uniformWeight
1566 rw [gaugeOrbitCard_congr h]
1567
1568/-- **Every class carries unit mass under the uniform weight.** -/
1569theorem classMass_uniform {B : ℕ} (K : BoundedComplex B) :
1570 classMass uniformWeight (Quotient.mk (relabelSetoid B) K) = 1 := by
1571 rw [classMass_of_invariant _ (fun _ _ h => uniformWeight_invariant h) _,
1572 uniformWeight_invariant (equivalent_out K), orbitCardClass_mk]
1573 unfold uniformWeight
1574 have hpos : (0 : ℝ) < (gaugeOrbitCard K : ℝ) := by
1575 exact_mod_cast gaugeOrbitCard_pos K
1576 field_simp
1577
1578/-- **Premise (ii), stated for a family of labeled weights**, one at each size cap:
1579the class mass of a union is the product of the class masses of the parts. This is
1580the unrestricted form, with no side condition on the parts. -/
1581def GluesGenerally (W : ∀ B : ℕ, BoundedComplex B → ℝ) : Prop :=
1582 ∀ {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B'),
1583 classMass (W (B + B')) (Quotient.mk (relabelSetoid (B + B')) (dunion K L))
1584 = classMass (W B) (Quotient.mk (relabelSetoid B) K)
1585 * classMass (W B') (Quotient.mk (relabelSetoid B') L)
1586
1587/-- The uniform weight satisfies unrestricted gluing, at every pair, trivially. -/
1588theorem uniform_gluesGenerally :
1589 GluesGenerally (fun B => (uniformWeight : BoundedComplex B → ℝ)) := by
1590 intro B B' K L
1591 rw [classMass_uniform, classMass_uniform, classMass_uniform]
1592 norm_num
1593
1594/-- The uniform weight's class mass is not the RS measure: a bouquet with two loops
1595has automorphism count two, so `mu` there is one half, while uniform mass is one. -/
1596theorem uniform_classMass_ne_mu :
1597 classMass uniformWeight (Quotient.mk (relabelSetoid (1 + 2 + 0)) (bouquet 2 0))
1598 ≠ mu (bouquet 2 0) := by
1599 rw [classMass_uniform]
1600 unfold mu
1601 rw [autCard_bouquet]
1602 norm_num [Nat.factorial]
1603
1604/-- **THEOREM (premise (ii) alone excludes nothing).** There is a relabeling-invariant
1605labeled weight whose class mass multiplies over *every* disjoint union, with no side
1606condition, and which is not the RS measure. So the gluing premise carries no
1607discriminating power by itself: all of it is contributed by size-blindness, premise (i).
1608
1609This is the strength measurement the honest-tag discipline requires, and it corrects
1610the framing of §5-§6: the two premises are not equal partners. -/
1611theorem gluing_alone_does_not_force_mu :
1612 GluesGenerally (fun B => (uniformWeight : BoundedComplex B → ℝ))
1613 ∧ classMass uniformWeight (Quotient.mk (relabelSetoid (1 + 2 + 0)) (bouquet 2 0))
1614 ≠ mu (bouquet 2 0) :=
1615 ⟨uniform_gluesGenerally, uniform_classMass_ne_mu⟩
1616
1617/-- Dust has a single labeled presentation per class: with no incidence data to move,
1618every relabeling returns the same complex. -/
1619theorem orbitCard_dust (n : ℕ) : gaugeOrbitCard (dust n) = 1 := by
1620 have h := orbitCard_mul_autCard (dust n)
1621 rw [autCard_dust] at h
1622 have hd : (dust n).nV = n ∧ (dust n).nE = 0 ∧ (dust n).nT = 0 := ⟨rfl, rfl, rfl⟩
1623 rw [hd.1, hd.2.1, hd.2.2] at h
1624 simp only [Nat.factorial_zero, mul_one, one_mul] at h
1625 exact Nat.eq_of_mul_eq_mul_right (Nat.factorial_pos n) (by simpa using h)
1626
1627/-- The class mass of a size-blind weight on dust is just its value at that size. -/
1628theorem classMass_sizeWeight_dust (f : ℕ → ℕ → ℕ → ℝ) (n : ℕ) :
1629 classMass (sizeWeight f) (Quotient.mk (relabelSetoid n) (dust n)) = f n 0 0 := by
1630 rw [classMass_of_invariant _ (fun _ _ h => sizeWeight_invariant f h) _,
1631 sizeWeight_invariant f (equivalent_out (dust n)), orbitCardClass_mk, orbitCard_dust]
1632 unfold sizeWeight
1633 norm_num
1634
1635namespace CarrierShuffle
1636
1637variable {f : ℕ → ℕ → ℕ → ℝ}
1638
1639/-- **THEOREM (the derived weight predicts the region the premise excludes).** Premise
1640(ii) says nothing about `dust a ⊔ dust b`, since the two parts share a component. The
1641weight forced by the two premises satisfies the binomial-corrected gluing identity
1642there anyway, matching `mu`'s own behaviour from C3 exactly.
1643
1644This is the non-circularity receipt. Had the side condition been reverse-engineered to
1645carve out the cases where the wanted answer fails, the excluded region would be where
1646the derivation breaks. Instead it is where the derivation is confirmed. -/
1647theorem excludedRegion_predicted (h : CarrierShuffle f) (a b : ℕ) :
1648 classMass (sizeWeight f) (Quotient.mk (relabelSetoid (a + b)) (dust (a + b)))
1649 * (Nat.choose (a + b) a : ℝ)
1650 = classMass (sizeWeight f) (Quotient.mk (relabelSetoid a) (dust a))
1651 * classMass (sizeWeight f) (Quotient.mk (relabelSetoid b) (dust b)) := by
1652 rw [classMass_sizeWeight_dust, classMass_sizeWeight_dust, classMass_sizeWeight_dust]
1653 have hA := dustRow h a
1654 have hB := dustRow h b
1655 have hAB := dustRow h (a + b)
1656 have hchoose : (Nat.choose (a + b) a : ℝ) * (Nat.factorial a : ℝ) * (Nat.factorial b : ℝ)
1657 = (Nat.factorial (a + b) : ℝ) := by
1658 have hn : a ≤ a + b := Nat.le_add_right a b
1659 have hsub : a + b - a = b := by omega
1660 have := Nat.choose_mul_factorial_mul_factorial hn
1661 rw [hsub] at this
1662 exact_mod_cast congrArg (fun n : ℕ => (n : ℝ)) this
1663 have hfa : (Nat.factorial a : ℝ) ≠ 0 := by
1664 exact_mod_cast (Nat.factorial_pos a).ne'
1665 have hfb : (Nat.factorial b : ℝ) ≠ 0 := by
1666 exact_mod_cast (Nat.factorial_pos b).ne'
1667 refine mul_right_cancel₀ (mul_ne_zero hfa hfb) ?_
1668 calc f (a + b) 0 0 * (Nat.choose (a + b) a : ℝ)
1669 * ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ))
1670 = f (a + b) 0 0 * ((Nat.choose (a + b) a : ℝ) * (Nat.factorial a : ℝ)
1671 * (Nat.factorial b : ℝ)) := by ring
1672 _ = f (a + b) 0 0 * (Nat.factorial (a + b) : ℝ) := by rw [hchoose]
1673 _ = (f 1 0 0) ^ (a + b) := hAB
1674 _ = (f 1 0 0) ^ a * (f 1 0 0) ^ b := by rw [pow_add]
1675 _ = (f a 0 0 * (Nat.factorial a : ℝ)) * (f b 0 0 * (Nat.factorial b : ℝ)) := by
1676 rw [hA, hB]
1677 _ = f a 0 0 * f b 0 0 * ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ)) := by ring
1678
1679end CarrierShuffle
1680
1681/-! ## §9. The side condition is forced, not chosen
1682
1683The sharpest charge against §5 is that the side condition on premise (ii) was
1684reverse-engineered: keep the pairs where the wanted answer survives, drop the rest.
1685Two compiled facts answer it, in place of an argument.
1686
1687First, the premise set is **satisfiable**, and satisfied at the intended point: the
1688inverse gauge volume, which is the size function of `gibbsWeight`, meets all four
1689gluing instances with all three constants equal to one. So §6 is not vacuously true.
1690
1691Second, the **unrestricted** premise is *inconsistent* with size-blindness and
1692positivity. Not "inconsistent with `mu`", which is §3 and invites the reply that the
1693cases where the answer fails were deleted. Inconsistent full stop, on a statement
1694mentioning no measure, no automorphism count, and no factorial. The four instances
1695the carrier supplies force the vertex recursion, which divides by the new vertex
1696count; unrestricted gluing applied to two piles of dust does not divide.
1697
1698Together: the restriction is the boundary of consistency of the two premises, fixed
1699before any measure enters. It could not have been chosen otherwise, so it cannot have
1700been fitted to an answer. -/
1701
1702/-- The size function of the Gibbs weight: the inverse gauge volume. -/
1703noncomputable def gibbsSize (a b c : ℕ) : ℝ :=
1704 1 / ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
1705
1706theorem factorial_cast_pos (n : ℕ) : (0 : ℝ) < (Nat.factorial n : ℝ) := by
1707 exact_mod_cast Nat.factorial_pos n
1708
1709theorem gibbsSize_pos (a b c : ℕ) : 0 < gibbsSize a b c := by
1710 unfold gibbsSize
1711 have := factorial_cast_pos a
1712 have := factorial_cast_pos b
1713 have := factorial_cast_pos c
1714 positivity
1715
1716theorem gibbsSize_eq_inv_gaugeVol (a b c : ℕ) :
1717 gibbsSize a b c = 1 / (gaugeVol a b c : ℝ) := by
1718 unfold gibbsSize gaugeVol
1719 push_cast
1720 ring
1721
1722theorem gaugeVol_cast_pos (a b c : ℕ) : (0 : ℝ) < (gaugeVol a b c : ℝ) := by
1723 exact_mod_cast gaugeVol_pos a b c
1724
1725/-- **THEOREM (the inverse gauge volume satisfies the shuffle identity at every
1726pair).** This is `gaugeVol_add` read as a statement about weights: the gauge volume
1727of a sum of size triples exceeds the product by exactly the interleaving count, so its
1728reciprocal shuffles. No side condition and no hypothesis. -/
1729theorem gibbsSize_shuffle (a b c a' b' c' : ℕ) :
1730 gibbsSize (a + a') (b + b') (c + c') * (interleave a b c a' b' c' : ℝ)
1731 = gibbsSize a b c * gibbsSize a' b' c' := by
1732 rw [gibbsSize_eq_inv_gaugeVol, gibbsSize_eq_inv_gaugeVol, gibbsSize_eq_inv_gaugeVol,
1733 gaugeVol_add a b c a' b' c']
1734 have h1 := gaugeVol_cast_pos a b c
1735 have h2 := gaugeVol_cast_pos a' b' c'
1736 have h3 : (0 : ℝ) < (interleave a b c a' b' c' : ℝ) := by
1737 exact_mod_cast interleave_pos a b c a' b' c'
1738 push_cast
1739 field_simp
1740
1741/-- **THEOREM (the premise set is satisfiable, at the intended point).** The inverse
1742gauge volume satisfies all four gluing instances. Since `gibbsSize 1 0 0`,
1743`gibbsSize 1 1 0` and `gibbsSize 1 0 1` are all one, §6's three constants are attained
1744at unity, so `gibbs_of_unit_fugacities` has a witness and the derivation is not
1745vacuously true. -/
1746theorem gibbsSize_carrierShuffle : CarrierShuffle gibbsSize where
1747 pos := gibbsSize_pos
1748 unit := by norm_num [gibbsSize]
1749 dust_bouquet := by
1750 intro a b c _
1751 simpa [Nat.zero_add] using gibbsSize_shuffle a 0 0 1 b c
1752 dust_edge := by
1753 intro a
1754 simpa [Nat.zero_add] using gibbsSize_shuffle a 0 0 2 1 0
1755 bouquet_edge := by
1756 intro b c
1757 simpa [Nat.add_zero] using gibbsSize_shuffle 1 b c 2 1 0
1758 bouquet_tet := by
1759 intro b c
1760 simpa [Nat.add_zero] using gibbsSize_shuffle 1 b c 4 0 1
1761
1762/-- **Premise (ii) with no side condition**: class mass multiplies over every disjoint
1763union, whatever the parts. -/
1764def GluesEverywhere (f : ℕ → ℕ → ℕ → ℝ) : Prop :=
1765 ∀ {B B' : ℕ} (K : BoundedComplex B) (L : BoundedComplex B'), GluesAt f K L
1766
1767/-- Under the unrestricted premise, dust glues with no binomial correction at all. -/
1768theorem dust_glues_plainly (f : ℕ → ℕ → ℕ → ℝ) (hplain : GluesEverywhere f) (a b : ℕ) :
1769 f (a + b) 0 0 = f a 0 0 * f b 0 0 := by
1770 have h := hplain (dust a) (dust b)
1771 unfold GluesAt at h
1772 rw [show Quotient.mk (relabelSetoid (a + b)) (dunion (dust a) (dust b))
1773 = Quotient.mk (relabelSetoid (a + b)) (dust (a + b)) from
1774 Quotient.sound (dunion_dust_equivalent a b)] at h
1775 rw [classMass_sizeWeight_dust, classMass_sizeWeight_dust,
1776 classMass_sizeWeight_dust] at h
1777 exact h
1778
1779/-- **THEOREM (the unrestricted premise is inconsistent, and the statement never
1780mentions the measure).** No positive size-blind weight satisfies the four gluing
1781instances the carrier supplies *and* gluing at every pair. The first forces
1782`f(a+1,0,0)·(a+1) = f(a,0,0)·f(1,0,0)`; the second forces
1783`f(a+b,0,0) = f(a,0,0)·f(b,0,0)`; at `a = b = 1` these give `2x² = x²`, so `x = 0`,
1784against positivity.
1785
1786This is the receipt that the side condition is forced rather than fitted. §3 showed
1787that `mu` fails unrestricted gluing, which leaves open the reply that the failing cases
1788were simply excluded. This shows the *premises* fail unrestricted gluing, with no
1789measure named, so the exclusion is prior to any answer. -/
1790theorem unrestricted_gluing_inconsistent (f : ℕ → ℕ → ℕ → ℝ)
1791 (h : CarrierShuffle f) (hplain : GluesEverywhere f) : False := by
1792 have h1 := CarrierShuffle.vertexRec h 1
1793 have h2 := dust_glues_plainly f hplain 1 1
1794 have hpos := h.pos 1 0 0
1795 norm_num at h1 h2
1796 nlinarith [h1, h2, hpos]
1797
1798/-- **THEOREM (some restriction is mandatory).** The restricted premise is
1799satisfiable, by the inverse gauge volume; the unrestricted premise is satisfied by
1800nothing. So a restriction on premise (ii) is not a convenience: dropping it entirely
1801makes the premise set empty.
1802
1803Read the scope exactly. This says a restriction is *necessary*. It does not say the
1804particular restriction used here is the *only* one that would restore consistency, and
1805several others plainly would. What removes the remaining worry is not this theorem but
1806`all_four_families_available`: the derivation consumes four named instances, each of
1807them proved, so it never depends on where the general boundary is drawn. -/
1808theorem restriction_is_mandatory :
1809 CarrierShuffle gibbsSize ∧ (∀ f : ℕ → ℕ → ℕ → ℝ, CarrierShuffle f → ¬ GluesEverywhere f) :=
1810 ⟨gibbsSize_carrierShuffle, fun f h hp => unrestricted_gluing_inconsistent f h hp⟩
1811
1812/-! ## §9a. The transport runs both ways, and which size triples exist
1813
1814Two gaps a second hostile read found in the write-up, both closed here rather than
1815hedged in prose. The first: `shuffle_of_gluesAt` proves the premise implies the
1816shuffle identity, so `CarrierShuffle` was only known to be *implied by* premise (ii),
1817and a reader was entitled to say the structure is four algebraic equations rather than
1818the premise. At an eligible pair the transport is an equivalence, so the converse is
1819the same algebra run backwards. The second: `closedForm` says nothing at triples with
1820no vertex, which is harmless only because the carrier cannot realize them, and that
1821fact was assumed rather than proved. -/
1822
1823/-- **THEOREM (the transport is an equivalence at an eligible pair).** Converse of
1824`shuffle_of_gluesAt`. Together they say that at a pair where automorphism counts
1825multiply, the shuffle identity on the size function and premise (ii) on the class mass
1826are the same statement, so `CarrierShuffle` is the premise at those four families and
1827not merely a consequence of it. -/
1828theorem gluesAt_of_shuffle (f : ℕ → ℕ → ℕ → ℝ) {B B' : ℕ}
1829 (K : BoundedComplex B) (L : BoundedComplex B')
1830 (haut : Nat.card (Aut (dunion K L)) = Nat.card (Aut K) * Nat.card (Aut L))
1831 (hsh : f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT)
1832 * (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ)
1833 = f K.nV K.nE K.nT * f L.nV L.nE L.nT) :
1834 GluesAt f K L := by
1835 have hob := orbitCard_dunion_of_autMul K L haut
1836 unfold GluesAt
1837 rw [classMass_sizeWeight, classMass_sizeWeight, classMass_sizeWeight]
1838 simp only [dunion_nV, dunion_nE, dunion_nT]
1839 rw [hob]
1840 push_cast
1841 calc (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ)
1842 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ))
1843 * f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT)
1844 = (f (K.nV + L.nV) (K.nE + L.nE) (K.nT + L.nT)
1845 * (interleave K.nV K.nE K.nT L.nV L.nE L.nT : ℝ))
1846 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ)) := by ring
1847 _ = (f K.nV K.nE K.nT * f L.nV L.nE L.nT)
1848 * ((gaugeOrbitCard K : ℝ) * (gaugeOrbitCard L : ℝ)) := by rw [hsh]
1849 _ = (gaugeOrbitCard K : ℝ) * f K.nV K.nE K.nT
1850 * ((gaugeOrbitCard L : ℝ) * f L.nV L.nE L.nT) := by ring
1851
1852/-- **THEOREM (any incidence needs a vertex to carry it).** A complex with an edge or
1853a tetrahedron has at least one vertex, since both incidence maps land in `Fin nV` and
1854that type is empty when `nV = 0`. So the size triples the carrier realizes all have
1855`1 ≤ nV`, which is exactly the hypothesis `closedForm` carries: the closed form pins
1856the size function at every triple that any complex actually has, and its silence at
1857`(0, b, c)` with `b + c ≥ 1` is silence about sizes nothing can have. -/
1858theorem vertex_of_incidence {B : ℕ} (K : BoundedComplex B) (h : 1 ≤ K.nE + K.nT) :
1859 1 ≤ K.nV := by
1860 by_contra hv
1861 push_neg at hv
1862 have hv0 : K.nV = 0 := by omega
1863 rcases Nat.lt_or_ge 0 K.nE with he | he
1864 · have hx := (K.edgeVerts ⟨0, he⟩).1.isLt
1865 omega
1866 · have ht : 0 < K.nT := by omega
1867 have hx := (K.tetVerts ⟨0, ht⟩ 0).isLt
1868 omega
1869
1870/-- The closed form applies to every complex the carrier contains: either it has a
1871vertex, or it is the empty complex, where `unit` already gives the value. -/
1872theorem closedForm_or_empty {f : ℕ → ℕ → ℕ → ℝ} (h : CarrierShuffle f) {B : ℕ}
1873 (K : BoundedComplex B) :
1874 (1 ≤ K.nV) ∨ (K.nV = 0 ∧ K.nE = 0 ∧ K.nT = 0) := by
1875 rcases Nat.lt_or_ge 0 (K.nE + K.nT) with hi | hi
1876 · exact Or.inl (vertex_of_incidence K hi)
1877 · rcases Nat.lt_or_ge 0 K.nV with hv | hv
1878 · exact Or.inl hv
1879 · exact Or.inr ⟨by omega, by omega, by omega⟩
1880
1881/-! ## §9b. Where the symmetry factor actually comes from
1882
1883Added 2026-07-28 after a hostile read of the write-up located a misattribution in
1884the headline, not in any proof. The natural summary of this module, "two premises
1885force the class measure to `1/|Aut|` up to three constants", credits the wrong
1886premise with the `1/|Aut|`. The two theorems below measure it instead of asserting
1887it, and the first of them deflates the headline. Keep them next to the derivation
1888so the next reader cannot restate it the old way. -/
1889
1890/-- The **sector fugacity** of a size-blind weight: the gauge volume of a size triple
1891times the weight there. It is everything about a size-blind weight that
1892orbit-stabilizer has not already fixed. -/
1893noncomputable def fugacityOf (f : ℕ → ℕ → ℕ → ℝ) (a b c : ℕ) : ℝ :=
1894 (gaugeVol a b c : ℝ) * f a b c
1895
1896/-- **THEOREM (premise (i) alone already produces the symmetry factor).** For *any*
1897size-blind weight whatever, positive or not, normalized or not, the class mass of a
1898class is its sector fugacity divided by the automorphism count. The only inputs are
1899premise (i) and `orbitCard_mul_autCard`, which the carrier had before this module
1900existed. No gluing premise is used, and none is even available at this point.
1901
1902The consequence for the derivation is worth stating flatly, because it is easy to
1903claim the opposite. The `1/|Aut K|` dependence is a consequence of size-blindness.
1904Premise (ii) does not produce it and could not, since it is already here before
1905premise (ii) is stated. What premise (ii) does is force `fugacityOf f` to be a
1906character rather than an arbitrary function of the three sizes, which is the content
1907of `closedForm`; and the unit character is what makes the constant one. -/
1908theorem classMass_sizeWeight_eq_fugacity_div_autCard
1909 (f : ℕ → ℕ → ℕ → ℝ) {B : ℕ} (K : BoundedComplex B) :
1910 classMass (sizeWeight f) (Quotient.mk (relabelSetoid B) K)
1911 = fugacityOf f K.nV K.nE K.nT / (Nat.card (Aut K) : ℝ) := by
1912 have hmul : (gaugeOrbitCard K : ℝ) * (Nat.card (Aut K) : ℝ)
1913 = (gaugeVol K.nV K.nE K.nT : ℝ) := by
1914 have h := orbitCard_mul_autCard K
1915 unfold gaugeVol
1916 exact_mod_cast h
1917 have hA : (0 : ℝ) < (Nat.card (Aut K) : ℝ) := by
1918 exact_mod_cast autCard_pos K
1919 rw [classMass_sizeWeight, eq_div_iff hA.ne', fugacityOf]
1920 calc (gaugeOrbitCard K : ℝ) * f K.nV K.nE K.nT * (Nat.card (Aut K) : ℝ)
1921 = ((gaugeOrbitCard K : ℝ) * (Nat.card (Aut K) : ℝ)) * f K.nV K.nE K.nT := by ring
1922 _ = (gaugeVol K.nV K.nE K.nT : ℝ) * f K.nV K.nE K.nT := by rw [hmul]
1923
1924/-- The **character weights**: one positive number per index type, spread over the
1925gauge volume. `gibbsSize` is the member with all three equal to one. -/
1926noncomputable def characterSize (u v w : ℝ) (a b c : ℕ) : ℝ :=
1927 (u ^ a * v ^ b * w ^ c) / (gaugeVol a b c : ℝ)
1928
1929theorem characterSize_shuffle (u v w : ℝ) (a b c a' b' c' : ℕ) :
1930 characterSize u v w (a + a') (b + b') (c + c') * (interleave a b c a' b' c' : ℝ)
1931 = characterSize u v w a b c * characterSize u v w a' b' c' := by
1932 unfold characterSize
1933 rw [gaugeVol_add a b c a' b' c']
1934 have h1 := gaugeVol_cast_pos a b c
1935 have h2 := gaugeVol_cast_pos a' b' c'
1936 have h3 : (0 : ℝ) < (interleave a b c a' b' c' : ℝ) := by
1937 exact_mod_cast interleave_pos a b c a' b' c'
1938 push_cast
1939 field_simp
1940 ring
1941
1942theorem characterSize_carrierShuffle {u v w : ℝ} (hu : 0 < u) (hv : 0 < v) (hw : 0 < w) :
1943 CarrierShuffle (characterSize u v w) where
1944 pos := by
1945 intro a b c
1946 unfold characterSize
1947 have := gaugeVol_cast_pos a b c
1948 positivity
1949 unit := by norm_num [characterSize, gaugeVol]
1950 dust_bouquet := by
1951 intro a b c _
1952 simpa [Nat.zero_add] using characterSize_shuffle u v w a 0 0 1 b c
1953 dust_edge := by
1954 intro a
1955 simpa [Nat.zero_add] using characterSize_shuffle u v w a 0 0 2 1 0
1956 bouquet_edge := by
1957 intro b c
1958 simpa [Nat.add_zero] using characterSize_shuffle u v w 1 b c 2 1 0
1959 bouquet_tet := by
1960 intro b c
1961 simpa [Nat.add_zero] using characterSize_shuffle u v w 1 b c 4 0 1
1962
1963theorem gibbsSize_eq_characterSize_one : gibbsSize = characterSize 1 1 1 := by
1964 funext a b c
1965 rw [gibbsSize_eq_inv_gaugeVol]
1966 unfold characterSize
1967 norm_num
1968
1969/-- **THEOREM (the residue is exactly three positive constants, not at most three).**
1970Forward, `closedForm`: every solution of the premise set has the three-constant form.
1971Backward: every positive triple is realized by an actual solution. Both directions
1972are needed before the residue can be called a three-parameter family, and the
1973backward one also settles a question about strategy: no further gluing family can
1974ever shrink the residue, because the whole three-parameter family satisfies the
1975shuffle identity at every pair unconditionally, by `characterSize_shuffle`. -/
1976theorem residue_is_exactly_three_positive_constants :
1977 (∀ f : ℕ → ℕ → ℕ → ℝ, CarrierShuffle f → ∀ a b c : ℕ, 1 ≤ a →
1978 f a b c * ((Nat.factorial a : ℝ) * (Nat.factorial b : ℝ) * (Nat.factorial c : ℝ))
1979 * (f 1 0 0) ^ (b + c)
1980 = (f 1 0 0) ^ a * (f 1 1 0) ^ b * (f 1 0 1) ^ c)
1981 ∧ (∀ u v w : ℝ, 0 < u → 0 < v → 0 < w → CarrierShuffle (characterSize u v w)) :=
1982 ⟨fun _ h a b c ha => h.closedForm a b c ha,
1983 fun _ _ _ hu hv hw => characterSize_carrierShuffle hu hv hw⟩
1984
1985/-! ## §10. Certificate: what is derived, what is assumed, what remains -/
1986
1987/-- The state of the gluing route after this module. -/
1988structure GluingStatus where
1989 /-- The carrier now carries a disjoint union (previously recorded as missing). -/
1990 dunion_exists : Bool
1991 /-- The interleaving count is binomial, as pure arithmetic on gauge volumes. -/
1992 interleaving_binomial : Bool
1993 /-- Unrestricted gluing multiplicativity is refuted against `mu` itself. -/
1994 unrestricted_gluing_refuted : Bool
1995 /-- Automorphism multiplicativity is the exact residue for transporting the
1996 binomial identity to orbit counts. -/
1997 autmul_is_the_residue : Bool
1998 /-- Size-blindness plus gluing is exactly the shuffle identity where `Aut`
1999 multiplies. -/
2000 premises_give_shuffle : Bool
2001 /-- The two premises force the weight up to three constants. -/
2002 three_constants : Bool
2003 /-- Unit values for those three constants give exactly `gibbsWeight`. -/
2004 units_give_gibbs : Bool
2005 /-- One nontrivial family of gluing instances is verified available. -/
2006 availability_witnessed : Bool
2007 /-- All four gluing instances the premise set uses are verified available, each over
2008 an unbounded family. -/
2009 all_four_families_available : Bool
2010 /-- Measured: premise (ii) alone excludes nothing, so premise (i) carries the force. -/
2011 gluing_alone_is_empty : Bool
2012 /-- Measured: the derived weight predicts the region premise (ii) declines to
2013 speak about, which is the non-circularity receipt. -/
2014 excluded_region_predicted : Bool
2015 /-- The premise set is satisfiable, and satisfied at the intended unit point. -/
2016 premises_satisfiable : Bool
2017 /-- The unrestricted premise is satisfied by nothing, so some restriction on
2018 premise (ii) is mandatory rather than a carve-out. -/
2019 restriction_is_mandatory : Bool
2020 /-- NOT proved: the general no-mixing theorem for parts sharing no isomorphic
2021 component. -/
2022 general_autmul_formalized : Bool
2023 /-- NOT proved: that the three constants must equal one. -/
2024 three_constants_forced : Bool
2025
2026/-- Status after this module. -/
2027def gluingStatus : GluingStatus where
2028 dunion_exists := true
2029 interleaving_binomial := true
2030 unrestricted_gluing_refuted := true
2031 autmul_is_the_residue := true
2032 premises_give_shuffle := true
2033 three_constants := true
2034 units_give_gibbs := true
2035 availability_witnessed := true
2036 all_four_families_available := true
2037 gluing_alone_is_empty := true
2038 excluded_region_predicted := true
2039 premises_satisfiable := true
2040 restriction_is_mandatory := true
2041 general_autmul_formalized := false
2042 three_constants_forced := false
2043
2044theorem status_dunion : gluingStatus.dunion_exists = true := rfl
2045theorem status_binomial : gluingStatus.interleaving_binomial = true := rfl
2046theorem status_refuted : gluingStatus.unrestricted_gluing_refuted = true := rfl
2047theorem status_residue : gluingStatus.autmul_is_the_residue = true := rfl
2048theorem status_shuffle : gluingStatus.premises_give_shuffle = true := rfl
2049theorem status_three : gluingStatus.three_constants = true := rfl
2050theorem status_units : gluingStatus.units_give_gibbs = true := rfl
2051theorem status_witness : gluingStatus.availability_witnessed = true := rfl
2052theorem status_all_four : gluingStatus.all_four_families_available = true := rfl
2053theorem status_gluing_empty : gluingStatus.gluing_alone_is_empty = true := rfl
2054theorem status_excluded : gluingStatus.excluded_region_predicted = true := rfl
2055theorem status_satisfiable : gluingStatus.premises_satisfiable = true := rfl
2056theorem status_forced : gluingStatus.restriction_is_mandatory = true := rfl
2057
2058/-- **The two honest negatives.** The general no-mixing theorem is not formalized,
2059and nothing here forces the three constants to equal one. Both flags are `false`
2060by construction, so the certificate cannot drift into claiming them. -/
2061theorem status_open :
2062 gluingStatus.general_autmul_formalized = false ∧
2063 gluingStatus.three_constants_forced = false :=
2064 ⟨rfl, rfl⟩
2065
2066end Gap2GluingDerivation
2067end SevenGaps
2068end Gravity
2069end IndisputableMonolith
2070