Paper III

Scale-Resolved State Dynamics — external audit instrument

Paper III is frozen and published. The site uses it only as an external arithmetic and resolution-audit instrument applied to separately defined theories. The site does not modify, enlarge, reinterpret, or replace its ontology, definitions, theorem package, terminology, or application boundary.

Frozen SRSD: lossless resolution in three-dimensional space. A cubic lattice with origin O = (0,0,0). A blue vector Δ_s d runs from the origin to a lattice corner P. A highlighted cell sits at P. A red residual vector ρ runs from P to a point y inside the cell. Reconstruction: y = Δ_s d + ρ with 0 ≤ ρ_i < s_i. Same quotient d means the same cell, not the same point. — Frozen SRSD in $\mathbb{R}^3$ . Encoder $E_s(y)=(d,\rho)$ with reconstruction $y=\Delta_s d+\rho$ and $0\le\rho_i<s_i$ . Same quotient $d$ means the same cell—not the same point.

SRSD EXTERNAL-AUDIT BOUNDARY — 30 AUGUST 2026

SRSD v1.0 is frozen and published. It is used only as an external arithmetic and resolution-audit instrument applied to separately defined theories. Older G₆ “global object,” “foundation,” and “gluing” language is retained solely as provenance for frozen implementation packages. It does not modify SRSD, merge the source theories, or define their current mathematical dependency structure.

Applying SRSD to an admissible numerical object C ∈ ℝᵐ proves only D_s E_s(C) = C. The same identity holds for every admissible numerical object in that domain. That is generic exact representability, not a mathematical connection between cubature and SRSD.

The primitive SRSD encoder and decoder use coordinatewise quotient–residual decomposition through the floor function, a standard mathematical construction. SRSD does not claim to invent division with remainder or to provide a new foundation for mathematics. “Foundational closure” refers only to the internal sufficiency of the frozen SRSD v1.0 specification for the systems it defines.

G₆ packaging is not a theorem joining the theories. The tuple exists definitionally because unrelated typed objects can always be placed in a tuple. A canonical hash proves equality of a named byte string. Aggregating separate maps into interfaces=PASS is a historical implementation outcome, not current source-map discipline.

Encoder and decoder

For a positive scale $s$ , the encoder and decoder are $E_s(x)=\bigl(\lfloor x/s\rfloor,\, x-s\lfloor x/s\rfloor\bigr)$ and $D_s(d,\rho)=sd+\rho$ , with residual in $[0,s)$ . That pair is coordinatewise division with remainder; it is a standard construction. Displayed wording on this page is subordinate to the frozen source.

One external SRSD audit

Frozen encoder $E_s$ applied to the already-specified product $Qd$ at resolution $s=S$ . The identity at that scale is $E_S(Qd)=(d',\rho)$ . The normalized form is $E_1(qd)=(d',\rho/S)$ . Both are generic exact representability of that number, not a mathematical connection that joins cubature to SRSD. Frozen SRSD is unchanged.

E_S(Qd) = (d′, ρ) = (135, 639958766566422)

E₁(qd) = (d′, ρ/S) = (135, 639958766566422/10^15)

reconstruction recovers Qd

The identity $E_1(\lambda_Q d_k)=(d_{k+1},u_k)$ is an exact external representation of one Omega step. It does not identify SRSD with rational-base numeration.

Claim-classification taxonomy

Each map retains its own domain, codomain, claim, and evidence. No row is promoted by a global G₆ aggregate.

SRSD external-audit classification of published maps.
Component	Audit classification	Result
D_s E_s = id and full-record reconstruction	Exact identity	Pass
Q/S Euclidean quotient–residual map	Exact identity	Pass
⌊λ d⌋ = ⌊Qd/S⌋ on D₁₅	Exact finite-domain equality	Pass
First disagreement d* = 42,189,065	Exact boundary	Pass
Successor candidates on D₁₅	Quotient-only equivalence	Pass, correctly scoped
First residual on D₁₅	Exact injectivity/congruential theorem	Pass
Ω coefficient representation of ℚ(√5)	Exact representation map	Pass; not structural identity
Paper VII marked kernel	Independent Euclidean geometry	Construction-to-curve claimed; remaining notebook completion not claimed
Cubature results	Independent external theory	Correctly separated from Ω₀ and SRSD
Frequency 22	Certificate-backed finite theorem	Pass
X₂₃ and X₂₄	Hosted existence theorem plus recovered exact certificate	Python 3.11.2 PASS
10 ≤ f ≤ 21	Recovered exact certificate package; not Paper VI	Python 3.11.2 PASS
f ≥ 25	Conjectural	Correctly labeled

Source and listings

Condensed cross-language listing: SRSD_CONDENSED_CROSS.txt. That listing is not a theorem checker.

Paper III DOI

A later finite-domain successor search for the 15-bit floor map is recorded at /dynamics/successors. It does not modify, enlarge, or replace this frozen SRSD source. Current Omega/JPRV terminology, including first-residual injectivity on the 15-bit domain, is recorded at /dynamics/training.