IndisputableMonolith.NumberTheory.HilbertPolyaCandidate
IndisputableMonolith/NumberTheory/HilbertPolyaCandidate.lean · 307 lines · 29 declarations
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1import Mathlib
2import IndisputableMonolith.Cost
3
4/-!
5# A Recognition-Cost Candidate for the Hilbert--Pólya Operator
6
7The Hilbert--Pólya conjecture (Hilbert, Pólya, c. 1914) proposes that the
8imaginary parts `γ_n` of the non-trivial zeros `ρ_n = 1/2 + i γ_n` of the
9Riemann zeta function are eigenvalues of some self-adjoint operator on
10a Hilbert space. Such an operator's existence would prove RH.
11
12This module constructs the algebraic skeleton of a candidate operator
13on the free `ℝ`-module on the multiplicative group `ℚ_{>0}^×` (which is
14the free abelian group on the primes), built from the Recognition Science
15cost function `J`. The reciprocal symmetry `J(x) = J(1/x)` translates
16into an operator-level intertwining with the multiplicative involution
17`q ↦ 1/q`, mirroring the zeta functional equation `ξ(s) = ξ(1-s)`.
18
19We do NOT prove that the spectrum is the imaginary parts of zeta zeros.
20That is the Hilbert--Pólya conjecture. We construct the candidate
21operator, prove its structural symmetries, and identify the open
22spectral question.
23
24## Main definitions
25
26* `MultIndex` : `Nat.Primes →₀ ℤ`, the multiplicative index space
27 (free abelian group on primes, isomorphic to
28 `ℚ_{>0}^×`).
29* `toRat` : `MultIndex → ℝ`, the rational `∏ p^(v p)`.
30* `costAt` : `MultIndex → ℝ`, the J-cost at a rational.
31* `StateSpace` : `MultIndex →₀ ℝ`, the free `ℝ`-module that
32 serves as our pre-Hilbert space.
33* `diagOp` : the diagonal cost operator `D`.
34* `shiftOp p` : the prime-shift operator `V_p`.
35* `involutionOp` : the reciprocal-involution operator `U`.
36
37## Main theorems (all 0 sorry)
38
39* `costAt_neg_eq` : `J(1/q) = J(q)` at the index level.
40* `involutionOp_involutive` : `U^2 = id`.
41* `involutionOp_diagOp_comm`: `U ∘ D = D ∘ U`.
42* `involutionOp_shiftOp` : `U ∘ V_p = V_p^{-1} ∘ U`.
43* `shiftOp_invertible` : `V_p ∘ V_p^{-1} = id` (formal unitarity).
44
45## Lean status: 0 sorry
46-/
47
48namespace IndisputableMonolith
49namespace NumberTheory
50namespace HilbertPolyaCandidate
51
52open Cost Finsupp
53
54noncomputable section
55
56/-! ## The multiplicative index space -/
57
58/-- Index for the multiplicative group `ℚ_{>0}^×`: a finitely-supported
59 function from primes to integers. -/
60abbrev MultIndex : Type := Nat.Primes →₀ ℤ
61
62/-- The positive rational corresponding to a multiplicative index,
63 interpreted as a real number: `toRat v = ∏ p^(v p)`. -/
64def toRat (v : MultIndex) : ℝ :=
65 v.prod (fun p k => (p.val : ℝ) ^ (k : ℤ))
66
67/-- The cost evaluated at the rational represented by `v`. -/
68def costAt (v : MultIndex) : ℝ := Jcost (toRat v)
69
70@[simp] theorem toRat_zero : toRat (0 : MultIndex) = 1 := by
71 simp [toRat]
72
73theorem toRat_pos (v : MultIndex) : 0 < toRat v := by
74 unfold toRat
75 rw [Finsupp.prod]
76 apply Finset.prod_pos
77 intro p _
78 apply zpow_pos
79 exact_mod_cast p.prop.pos
80
81theorem toRat_add (v w : MultIndex) :
82 toRat (v + w) = toRat v * toRat w := by
83 unfold toRat
84 rw [Finsupp.prod_add_index]
85 · intro p _
86 simp
87 · intro p _ k₁ k₂
88 rw [zpow_add₀ (by
89 have hp : p.val ≠ 0 := Nat.Prime.ne_zero p.prop
90 exact_mod_cast hp)]
91
92theorem toRat_neg (v : MultIndex) : toRat (-v) = (toRat v)⁻¹ := by
93 have h_sum : toRat ((-v) + v) = toRat (-v) * toRat v := toRat_add (-v) v
94 have h_zero : ((-v) + v) = (0 : MultIndex) := by simp
95 rw [h_zero, toRat_zero] at h_sum
96 have hv_pos : 0 < toRat v := toRat_pos v
97 have hv_ne : toRat v ≠ 0 := ne_of_gt hv_pos
98 field_simp [hv_ne]
99 linarith [h_sum]
100
101/-- Reciprocal symmetry of `J` at the index level: `J(1/q) = J(q)`. -/
102theorem costAt_neg_eq (v : MultIndex) : costAt (-v) = costAt v := by
103 unfold costAt
104 rw [toRat_neg]
105 exact (Jcost_symm (toRat_pos v)).symm
106
107/-! ## The state space: free ℝ-module on `MultIndex` -/
108
109/-- The pre-Hilbert space: free `ℝ`-module on `MultIndex`. -/
110abbrev StateSpace : Type := MultIndex →₀ ℝ
111
112/-! ## The three operators
113
114We use `Finsupp.lsum` and similar mathlib constructions to define
115linear endomorphisms of `StateSpace`. The "basis vector" `e_v` is
116`Finsupp.single v 1`. -/
117
118/-- The diagonal cost operator `D`: maps `e_v` to `J(toRat v) · e_v`.
119
120 Defined as the linear map sending each basis element `e_v` to
121 `costAt v • e_v`. -/
122def diagOp : StateSpace →ₗ[ℝ] StateSpace :=
123 Finsupp.lsum ℝ (fun v => costAt v • Finsupp.lsingle v)
124
125/-- Action of `diagOp` on a basis element: `D(e_v) = costAt v · e_v`. -/
126@[simp] theorem diagOp_single (v : MultIndex) (c : ℝ) :
127 diagOp (Finsupp.single v c) = Finsupp.single v (costAt v * c) := by
128 simp [diagOp, mul_comm]
129
130/-- The prime-shift operator `V_p`: maps `e_v` to `e_{v + δ_p}`,
131 i.e., multiplication of the underlying rational by `p`.
132
133 Defined via `Finsupp.lmapDomain` shifting the index by `δ_p`. -/
134def shiftOp (p : Nat.Primes) : StateSpace →ₗ[ℝ] StateSpace :=
135 Finsupp.lmapDomain ℝ ℝ (fun v => v + Finsupp.single p 1)
136
137/-- Action of `shiftOp p` on a basis element. -/
138@[simp] theorem shiftOp_single (p : Nat.Primes) (v : MultIndex) (c : ℝ) :
139 shiftOp p (Finsupp.single v c)
140 = Finsupp.single (v + Finsupp.single p 1) c := by
141 simp [shiftOp, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]
142
143/-- The inverse prime-shift operator `V_p^{-1}`: maps `e_v` to
144 `e_{v - δ_p}` (division of the underlying rational by `p`). -/
145def shiftInvOp (p : Nat.Primes) : StateSpace →ₗ[ℝ] StateSpace :=
146 Finsupp.lmapDomain ℝ ℝ (fun v => v - Finsupp.single p 1)
147
148@[simp] theorem shiftInvOp_single (p : Nat.Primes) (v : MultIndex) (c : ℝ) :
149 shiftInvOp p (Finsupp.single v c)
150 = Finsupp.single (v - Finsupp.single p 1) c := by
151 simp [shiftInvOp, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]
152
153/-- The reciprocal involution operator `U`: maps `e_v` to `e_{-v}`,
154 corresponding to the multiplicative inversion `q ↦ 1/q`. -/
155def involutionOp : StateSpace →ₗ[ℝ] StateSpace :=
156 Finsupp.lmapDomain ℝ ℝ (fun v => -v)
157
158@[simp] theorem involutionOp_single (v : MultIndex) (c : ℝ) :
159 involutionOp (Finsupp.single v c) = Finsupp.single (-v) c := by
160 simp [involutionOp, Finsupp.lmapDomain_apply, Finsupp.mapDomain_single]
161
162/-! ## Structural theorems -/
163
164/-- The reciprocal involution is involutive: `U ∘ U = id`. -/
165theorem involutionOp_involutive : involutionOp ∘ₗ involutionOp = LinearMap.id := by
166 ext v
167 simp
168
169/-- The reciprocal involution commutes with the diagonal cost operator
170 (consequence of `J(1/q) = J(q)`). -/
171theorem involutionOp_diagOp_comm :
172 involutionOp ∘ₗ diagOp = diagOp ∘ₗ involutionOp := by
173 ext v
174 simp [costAt_neg_eq]
175
176/-- The reciprocal involution intertwines the prime-shift with its
177 inverse: `U ∘ V_p = V_p^{-1} ∘ U`.
178
179 This is the operator-level analog of the zeta functional equation's
180 involution `s ↔ 1-s`. -/
181theorem involutionOp_shiftOp (p : Nat.Primes) :
182 involutionOp ∘ₗ shiftOp p = shiftInvOp p ∘ₗ involutionOp := by
183 ext v
184 simp only [LinearMap.coe_comp, Function.comp_apply,
185 shiftOp_single, involutionOp_single, shiftInvOp_single,
186 Finsupp.lsingle_apply]
187 congr 1
188 abel
189
190/-- Symmetric form of the previous: `U ∘ V_p^{-1} = V_p ∘ U`. -/
191theorem involutionOp_shiftInvOp (p : Nat.Primes) :
192 involutionOp ∘ₗ shiftInvOp p = shiftOp p ∘ₗ involutionOp := by
193 ext v
194 simp only [LinearMap.coe_comp, Function.comp_apply,
195 shiftInvOp_single, involutionOp_single, shiftOp_single,
196 Finsupp.lsingle_apply]
197 congr 1
198 abel
199
200/-- The shift and inverse-shift compose to the identity (formal unitarity
201 of `V_p`). -/
202theorem shiftOp_shiftInvOp (p : Nat.Primes) :
203 shiftOp p ∘ₗ shiftInvOp p = LinearMap.id := by
204 ext v
205 simp only [LinearMap.coe_comp, Function.comp_apply,
206 shiftInvOp_single, shiftOp_single, Finsupp.lsingle_apply,
207 LinearMap.id_apply]
208 congr 1
209 abel
210
211theorem shiftInvOp_shiftOp (p : Nat.Primes) :
212 shiftInvOp p ∘ₗ shiftOp p = LinearMap.id := by
213 ext v
214 simp only [LinearMap.coe_comp, Function.comp_apply,
215 shiftOp_single, shiftInvOp_single, Finsupp.lsingle_apply,
216 LinearMap.id_apply]
217 congr 1
218 abel
219
220/-! ## The candidate Hilbert--Pólya operator (algebraic part) -/
221
222/-- The off-diagonal piece for a single prime: `V_p + V_p^{-1}`.
223 This is the "hopping" term in the multiplicative direction `p`. -/
224def primeHop (p : Nat.Primes) : StateSpace →ₗ[ℝ] StateSpace :=
225 shiftOp p + shiftInvOp p
226
227/-- The reciprocal involution maps `V_p + V_p^{-1}` to itself
228 (consequence of intertwining shift and inverse shift). -/
229theorem involutionOp_primeHop (p : Nat.Primes) :
230 involutionOp ∘ₗ primeHop p = primeHop p ∘ₗ involutionOp := by
231 unfold primeHop
232 rw [LinearMap.comp_add, LinearMap.add_comp]
233 rw [involutionOp_shiftOp, involutionOp_shiftInvOp]
234 rw [add_comm (shiftInvOp p ∘ₗ involutionOp) (shiftOp p ∘ₗ involutionOp)]
235
236/-- The candidate Hilbert--Pólya operator with weights `λ : Nat.Primes → ℝ`,
237 defined on a finite set `S ⊆ Nat.Primes`:
238 `T_S(λ) := D + ∑_{p ∈ S} λ p · (V_p + V_p^{-1})`.
239
240 The full operator (sum over all primes) requires choosing a Hilbert space
241 closure and analyzing convergence; we work here at the algebraic level
242 with a finite truncation. -/
243def candidateOp (S : Finset Nat.Primes) (lam : Nat.Primes → ℝ) :
244 StateSpace →ₗ[ℝ] StateSpace :=
245 diagOp + S.sum (fun p => lam p • primeHop p)
246
247/-- Auxiliary: involution commutes with weighted sum of `primeHop` over a finset.
248 Proved by induction on the finset. -/
249private theorem involutionOp_sum_primeHop
250 (S : Finset Nat.Primes) (lam : Nat.Primes → ℝ) :
251 involutionOp ∘ₗ S.sum (fun p => lam p • primeHop p)
252 = S.sum (fun p => lam p • primeHop p) ∘ₗ involutionOp := by
253 classical
254 refine Finset.induction_on S ?_ ?_
255 · simp
256 · intro p S hp ih
257 rw [Finset.sum_insert hp]
258 rw [LinearMap.comp_add, LinearMap.add_comp, ih]
259 congr 1
260 rw [LinearMap.comp_smul, LinearMap.smul_comp]
261 congr 1
262 exact involutionOp_primeHop p
263
264/-- The candidate operator commutes with the reciprocal involution.
265 This is the Hilbert--Pólya-style structural symmetry: any spectrum
266 of the (closure of the) operator decomposes into eigenspaces of
267 the involution, mirroring `s ↔ 1-s`. -/
268theorem involutionOp_candidateOp (S : Finset Nat.Primes) (lam : Nat.Primes → ℝ) :
269 involutionOp ∘ₗ candidateOp S lam = candidateOp S lam ∘ₗ involutionOp := by
270 unfold candidateOp
271 rw [LinearMap.comp_add, LinearMap.add_comp]
272 rw [involutionOp_diagOp_comm]
273 rw [involutionOp_sum_primeHop]
274
275/-! ## Master certificate -/
276
277/-- Master certificate: the structural properties of the candidate
278 Hilbert--Pólya operator that this module establishes. -/
279theorem hilbert_polya_candidate_certificate :
280 -- (1) Reciprocal symmetry of the cost at the index level.
281 (∀ (v : MultIndex), costAt (-v) = costAt v) ∧
282 -- (2) The reciprocal involution is involutive.
283 (involutionOp ∘ₗ involutionOp = LinearMap.id) ∧
284 -- (3) The diagonal cost operator commutes with the involution.
285 (involutionOp ∘ₗ diagOp = diagOp ∘ₗ involutionOp) ∧
286 -- (4) Each prime shift inverts under the involution.
287 (∀ (p : Nat.Primes),
288 involutionOp ∘ₗ shiftOp p = shiftInvOp p ∘ₗ involutionOp) ∧
289 -- (5) Shifts are formally unitary.
290 (∀ (p : Nat.Primes),
291 shiftOp p ∘ₗ shiftInvOp p = LinearMap.id) ∧
292 -- (6) The full candidate operator commutes with the involution.
293 (∀ (S : Finset Nat.Primes) (lam : Nat.Primes → ℝ),
294 involutionOp ∘ₗ candidateOp S lam = candidateOp S lam ∘ₗ involutionOp) :=
295 ⟨costAt_neg_eq,
296 involutionOp_involutive,
297 involutionOp_diagOp_comm,
298 involutionOp_shiftOp,
299 shiftOp_shiftInvOp,
300 involutionOp_candidateOp⟩
301
302end
303
304end HilbertPolyaCandidate
305end NumberTheory
306end IndisputableMonolith
307