CP-H1-FORM-MOSCO-001: Analytical Theorem Audit and Closure of Gate A3 — Strong Common-Space Consistency of Restricted-Voronoi Reconstruction and Averaging v0.1.0

This record provides the theorem-level audit and closure of gate A3 for CP-H1-FORM-MOSCO-001.

 

The scope is deliberately narrow:

 

to establish strong common-space consistency of the restricted-Voronoi reconstruction and averaging operators used in the source-free native-graph quadratic-form continuum programme.

 

Current theorem status

----------------------

 

A1:

PROVED

 

A2:

PROVED

 

A3:

PROVED

 

A4–A8:

OPEN

 

M1–M2:

OPEN

 

Therefore:

 

NO Mosco convergence theorem is claimed.

 

NO overall FORM-MOSCO theorem-level PASS is claimed.

 

The global standing remains a partially closed theorem audit with A4–A8 and M1–M2 unresolved.

 

Provenance

----------

 

The controlling Mathematical Core is:

 

Krüger, M. (2026).

A Rigorous Mathematical Architecture for a 6D-to-3D Cut-and-Project and Incidence-Spectral Research Programme — FORM–Mosco Finite Audit, Threshold Provenance, and Explicit Finest-Level Quadrature Residual — Rigorous Consolidated Core v2.7.1.

Zenodo.

DOI: 10.5281/zenodo.22685169

 

The controlling CP-H1-FORM-MOSCO-001 records are:

 

Analytical architecture:

DOI: 10.5281/zenodo.22657352

 

Pre-execution protocol:

DOI: 10.5281/zenodo.22658157

 

Authoritative corrected implementation v0.1.3:

DOI: 10.5281/zenodo.22664013

 

Locked finite diagnostic result:

DOI: 10.5281/zenodo.22680099

 

A1 theorem audit / closure:

DOI: 10.5281/zenodo.22703339

 

A2 theorem audit / closure:

DOI: 10.5281/zenodo.22709453

 

No frozen target diagnostic is rerun in this record.

 

No form, source, numerical threshold, boundary condition, theorem condition, or previous verdict is altered.

 

A3 theorem obligation

---------------------

 

Gate A3 concerns strong consistency of the exact common-space reconstruction/averaging pair.

 

Let

 

J_j : H_j^0 → L²(Ω)

 

be the exact restricted-Voronoi reconstruction operator and let

 

Π_j : L²(Ω) → H_j^0

 

be the exact cell-averaging operator.

 

Define

 

P_j = J_j Π_j.

 

The A3 obligation is to establish

 

||P_j f - f||_{L²(Ω)} → 0

 

for every

 

f ∈ L²(Ω).

 

Exact projection structure

--------------------------

 

The restricted-Voronoi cells form the declared measurable partition of the fixed physical domain Ω.

 

For each level j, P_j = J_j Π_j is therefore the orthogonal projection in L²(Ω) onto the finite-dimensional subspace of functions that are constant on each restricted-Voronoi cell.

 

Consequently,

 

||P_j||_{L²→L²} ≤ 1.

 

A1 provides the load-bearing geometric input:

 

h_j^V

=

maximal restricted-Voronoi cell diameter

→ 0.

 

Continuous-function consistency

-------------------------------

 

For any

 

g ∈ C(cl Ω),

 

let ω_g denote its modulus of continuity.

 

Because every point in a restricted-Voronoi cell lies within cell diameter h_j^V of every other point in that cell, the difference between g and its cell average satisfies the uniform estimate

 

|P_j g(x) - g(x)|

≤

ω_g(h_j^V).

 

Therefore,

 

||P_j g - g||_{L²(Ω)}

≤

|Ω|^{1/2} ω_g(h_j^V).

 

Since

 

h_j^V → 0,

 

uniform continuity on the compact domain gives

 

ω_g(h_j^V) → 0.

 

Hence

 

||P_j g - g||_{L²(Ω)} → 0

 

for every continuous g.

 

Extension to all L² functions

-----------------------------

 

Continuous functions on the compact closure of Ω are dense in L²(Ω).

 

For arbitrary

 

f ∈ L²(Ω)

 

and ε > 0, choose

 

g ∈ C(cl Ω)

 

such that

 

||f - g||_{L²(Ω)} < ε.

 

Using the contraction property of P_j,

 

||P_j f - f||

≤

||P_j(f-g)||

+

||P_j g-g||

+

||g-f||

 

≤

2 ε

+

||P_j g-g||.

 

For sufficiently large j,

 

||P_j g-g|| < ε.

 

Thus

 

limsup_{j→∞} ||P_j f-f|| ≤ 3ε.

 

Because ε is arbitrary,

 

||J_j Π_j f - f||_{L²(Ω)} → 0

 

for every f ∈ L²(Ω).

 

A3 conclusion

-------------

 

Gate A3 is therefore classified as:

 

A3 = PROVED.

 

The result is theorem-level and does not depend on a numerical percentage threshold.

 

It establishes strong common-space consistency of the exact restricted-Voronoi averaging/reconstruction architecture.

 

Finite-versus-exact boundary

----------------------------

 

The proof applies to the exact mathematical restricted-Voronoi cells, exact Lebesgue cell measures, and exact averaging operators.

 

It does NOT promote the finite Sobol/QMC quadrature realization to exact arithmetic.

 

The previously reported finest-level restricted-Voronoi weight-stability residual

 

TV_{2^21→2^22}(j=6)

=

0.02520942687988281

 

remains preserved.

 

That value exceeds the historical 0.02 CP-H1-CONT-BRIDGE-001 TP6 comparator.

 

The 0.02 value was not a frozen FORM-MOSCO theorem or I0–I9 verdict gate, so the residual does not invalidate the locked finite FORM-MOSCO execution.

 

It remains, however, unfavorable finite numerical evidence regarding the precision of the finest-level quadrature realization.

 

Likewise, the finite projection errors reported in the locked diagnostic run remain finite numerical diagnostics and are not erased or reinterpreted by the A3 theorem.

 

Residual preservation

---------------------

 

This record does not modify or erase:

 

- the j = 6 restricted-Voronoi weight-stability residual;

 

- the finite projection-error values;

 

- the narrow continuum-reference quality PASS under the frozen 0.06 TP6 numerical tolerance;

 

- any earlier negative, open, or inconclusive result;

 

- the A2 distinction between asymptotic exclusion of persistent spurious zero modes and exact eventual connectivity.

 

A4–A8 remain OPEN.

 

M1–M2 remain OPEN.

 

No Mosco theorem is claimed.

 

Claim boundary

--------------

 

This A3 result does not establish:

 

- bulk pair-frequency convergence;

 

- isotropic continuum-coefficient identification;

 

- discrete compactness;

 

- liminf identification;

 

- recovery-sequence existence;

 

- exclusion of hidden anisotropic, defect, boundary, or surface terms;

 

- Neumann boundary identification;

 

- Mosco convergence;

 

- strong-resolvent convergence;

 

- continuum Hodge theory;

 

- physical spacetime;

 

- gauge dynamics;

 

- particle physics;

 

- gravity;

 

- cosmology;

 

- empirical validation of HLV.

 

Evidence status

---------------

 

Under STG-CHAT Governance v0.2.2:

 

Frozen A3 obligation:

E2 — source verified.

 

Analytical projection/contraction argument and dense-subspace extension:

E3 — directly checked within the originating workflow.

 

No E4 independent external validation is claimed.

 

Programme status after A3

-------------------------

 

The theorem ledger is now:

 

A1 = PROVED

 

A2 = PROVED

 

A3 = PROVED

 

A4–A8 = OPEN

 

M1–M2 = OPEN

 

The next load-bearing theorem-audit target is A4:

 

bulk native pair-frequency / directional-moment control and identification of the declared isotropic coefficient η.

 

Keywords:

cut-and-project geometry; Mosco convergence; theorem audit; restricted Voronoi cells; common-space reconstruction; averaging operator; orthogonal projection; strong L2 convergence; continuum limit; quadratic forms; exact mathematics; HLV testbed theory; reproducibility