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PureOne
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ELYRION-III

# Claim ledger

Twelve editorial scope records accompany the original scientific release. They distinguish analytic consequences of assumptions, finite computations, and unestablished physical or historical claims. The complete proofs remain in the manuscript.

## C01 — Four-state serial multiplication

Serial first-order marks and selective filters yield S(x)=product_j[1-exp(-u_j*g_j(x))].
**Status:** Conditional analytic result
**Assumptions:** Immobile tracked units; exact state-selective ideal filters; protected committed state; prescribed first-order response.
**Evidence:** Manuscript Section 3; code/model.py; exact rational and finite main checks.
**Limitation:** No molecule implementing the complete transition interface is identified.
## C02 — Positive repeated commitment
A pass updates c to c+(1-c)*b*S; repeated passes give c_final=1-product_s(1-b_s*S_s).
**Status:** Conditional analytic result
**Assumptions:** Uniform partial commitment; uncommitted material reinitialized between passes; irreversible protected C.
**Evidence:** Manuscript serial multiplication theorem; main and separate rational transition-matrix checks.
**Limitation:** Conservation is of tracked occupancy, not a complete reagent stoichiometry or energy balance.
## C03 — Exact periodic nodes and Fourier symmetry
Balanced root-phase gates produce exact dark nodes and period 2*pi/m, with Fourier support restricted to multiples of m.
**Status:** Ideal-model analytic result
**Assumptions:** Equal action at m equally spaced full-contrast phases; immobile material; ideal selective filters.
**Evidence:** Manuscript exact nodes, symmetry and weak-response theorem; finite symmetry and Fourier checks.
**Limitation:** Finite saturation allows higher multiples 2m,3m,...; molecular retention pitch is not demonstrated solid-feature resolution.
## C04 — Filter-free negative control
Without intermediate filters the same balanced first-order marking exposures accumulate to uniform response 1-exp(-m*u).
**Status:** Analytic negative control
**Assumptions:** Ordinary accumulating first-order marking; equal actions; root-phase fringes of a common fundamental.
**Evidence:** Manuscript no-filter comparison; main implementation negative-control checks.
**Limitation:** This comparison does not rule out other nonlinear chemistry or other control architectures.
## C05 — Global exact-node marking frontier
Among all finite admissible exact-node serial programs, maximum minimum-midpoint commitment at cost U is G_m(U/m), attained by one balanced pass.

**Status:** Conditional analytic optimization theorem

**Assumptions:** Same fundamental frequency; nonnegative first-harmonic gates; ideal filters; exact prescribed nodes; uniform positive commits; no diffusion.

**Evidence:** Manuscript Section 5 mandatory-root, pass-merging and strict-concavity proof; finite allocation and merge checks.

**Limitation:** Not a universal lithography bound; excludes spatially patterned commits, relaxed dark nodes, feedback and destructive correction.

## C06 — Sharp cubic asymptotic

At fixed midpoint fraction q, U_m(q)/m^3 tends to c_q defined by product_r[1-exp(-pi^2*c_q*(2r+1)^2/2)]^2=q.
**Status:** Conditional analytic asymptotic with numerical examples
**Assumptions:** Same exact-node serial class as C05; q strictly between zero and one; m tends to infinity.
**Evidence:** Manuscript Section 6; original dose_table and cubic_constants; separate 70-digit dose and limit calculations.
**Limitation:** Marking action is dimensionless and omits reset/commit resources; high precision is not an interval certificate.
## C07 — Three-dimensional conversion-field approximation
Every continuous f:T^3->[0,1] has a finite four-state program approximating its conversion fraction uniformly to arbitrary positive tolerance.
**Status:** Conditional existence theorem
**Assumptions:** Immobile selective-control contract; independent coordinate fringes; finite positive repeated passes; finite-rate approximation within remaining error margin.
**Evidence:** Manuscript Section 7 positive-kernel, weak-gate and hazard construction; synthetic three-term hazard example.
**Limitation:** No generic efficiency guarantee, arbitrary material synthesis, atom placement, solidification proof or actual volume optical-access guarantee.
## C08 — Finite-rate selective-filter boundary
Ideal filters are singular; finite bounded-action first-order maps are invertible. Approximate selective filters obey error <=exp(-h1)+exp(-h2), accumulated by summation.
**Status:** Conditional analytic obstruction and approximation bound
**Assumptions:** Bounded-action first-order transition maps; exact state selectivity for the approximation; stated stochastic-map norm.
**Evidence:** Manuscript finite-action selective-control section; main finite-filter and stochastic-map checks.
**Limitation:** Cross-talk or reverse-reaction floors may prevent the required accuracy; exact ideal nodes are not produced by finite bounded actions.
## C09 — Extinction and phase budgets
The protecting-factor bound gives sufficient fixed-darkness scales eta=O(m^-2) and phase error=O(m^-1) at fixed q.
**Status:** Conditional sufficient bounds
**Assumptions:** Balanced comb; declared fringe-floor and phase perturbation model; other error sources budgeted separately.
**Evidence:** Manuscript phase/extinction section; original m8_dark01_budget and main checks.
**Limitation:** Separate maximal floor and phase allowances cannot both be spent in full; not necessary optimal tolerances.
## C10 — Retention ceiling scope
Independent per-gate retention r gives single-pass midpoint r^m*G_m(u); finite-action target q is feasible in that pass iff r^m>q.
**Status:** Conditional single-pass result with multipass counterexample
**Assumptions:** Independent uniform retention; no false rescue; otherwise ideal full-commit m-gate pass.
**Evidence:** Manuscript retention proposition and explicit two-pass counterexample; main checks.
**Limitation:** The single-pass ceiling does not apply to arbitrary reinitialized repeated passes and is not a generic imperfect-filter model.
## C11 — Diffusion-aware sufficient design parameter
Finite-duration reaction-diffusion gates admit dark-node and brightness bounds; D*k^2*m^5/(sigma*P) is a sufficient design parameter for this implementation.
**Status:** Conditional analytic sufficient bounds and synthetic simulations
**Assumptions:** Common diffusivity for mobile states; finite first-order marking; ideal filters; fixed marking irradiance and stated Brownian-path model.
**Evidence:** Manuscript diffusion section; 16 original refined finite-difference cases in diffusion_examples.

**Limitation:** Not a universal necessary resolution ceiling; diffusion can fill dark nodes and change order dependence; grid refinement is not a continuum error certificate.

## C12 — Evidence and priority boundary

The supplied release records 1305 main checks and 108 separate code-path audit checks; physical experiments, external review and formal proof-assistant replay are absent.

**Status:** Release provenance and validation statement

**Assumptions:** Counts refer to logged finite checks in supplied ledgers; both implementations were produced within the same research session.

**Evidence:** Original results/verification.json, results/independent_audit.json and source/provenance.json; packaging rerun report.
**Limitation:** Check counts do not certify historical novelty, scientific importance, physical feasibility or external mathematical acceptance.

	# Claim ledger

	Twelve editorial scope records accompany the original scientific release. They distinguish analytic consequences of assumptions, finite computations, and unestablished physical or historical claims. The complete proofs remain in the manuscript.

	## C01 — Four-state serial multiplication

	Serial first-order marks and selective filters yield S(x)=product_j[1-exp(-u_j*g_j(x))].

	Status: Conditional analytic result

	Assumptions: Immobile tracked units; exact state-selective ideal filters; protected committed state; prescribed first-order response.

	Evidence: Manuscript Section 3; code/model.py; exact rational and finite main checks.

	Limitation: No molecule implementing the complete transition interface is identified.

	## C02 — Positive repeated commitment

	A pass updates c to c+(1-c)bS; repeated passes give c_final=1-product_s(1-b_s*S_s).

	Status: Conditional analytic result

	Assumptions: Uniform partial commitment; uncommitted material reinitialized between passes; irreversible protected C.

	Evidence: Manuscript serial multiplication theorem; main and separate rational transition-matrix checks.

	Limitation: Conservation is of tracked occupancy, not a complete reagent stoichiometry or energy balance.

	## C03 — Exact periodic nodes and Fourier symmetry

	Balanced root-phase gates produce exact dark nodes and period 2*pi/m, with Fourier support restricted to multiples of m.

	Status: Ideal-model analytic result

	Assumptions: Equal action at m equally spaced full-contrast phases; immobile material; ideal selective filters.

	Evidence: Manuscript exact nodes, symmetry and weak-response theorem; finite symmetry and Fourier checks.

	Limitation: Finite saturation allows higher multiples 2m,3m,...; molecular retention pitch is not demonstrated solid-feature resolution.

	## C04 — Filter-free negative control

	Without intermediate filters the same balanced first-order marking exposures accumulate to uniform response 1-exp(-m*u).

	Status: Analytic negative control

	Assumptions: Ordinary accumulating first-order marking; equal actions; root-phase fringes of a common fundamental.

	Evidence: Manuscript no-filter comparison; main implementation negative-control checks.

	Limitation: This comparison does not rule out other nonlinear chemistry or other control architectures.

	## C05 — Global exact-node marking frontier

	Among all finite admissible exact-node serial programs, maximum minimum-midpoint commitment at cost U is G_m(U/m), attained by one balanced pass.

	Status: Conditional analytic optimization theorem

	Assumptions: Same fundamental frequency; nonnegative first-harmonic gates; ideal filters; exact prescribed nodes; uniform positive commits; no diffusion.

	Evidence: Manuscript Section 5 mandatory-root, pass-merging and strict-concavity proof; finite allocation and merge checks.

	Limitation: Not a universal lithography bound; excludes spatially patterned commits, relaxed dark nodes, feedback and destructive correction.

	## C06 — Sharp cubic asymptotic

	At fixed midpoint fraction q, U_m(q)/m^3 tends to c_q defined by product_r[1-exp(-pi^2c_q(2r+1)^2/2)]^2=q.

	Status: Conditional analytic asymptotic with numerical examples

	Assumptions: Same exact-node serial class as C05; q strictly between zero and one; m tends to infinity.

	Evidence: Manuscript Section 6; original dose_table and cubic_constants; separate 70-digit dose and limit calculations.

	Limitation: Marking action is dimensionless and omits reset/commit resources; high precision is not an interval certificate.

	## C07 — Three-dimensional conversion-field approximation

	Every continuous f:T^3->[0,1] has a finite four-state program approximating its conversion fraction uniformly to arbitrary positive tolerance.

	Status: Conditional existence theorem

	Assumptions: Immobile selective-control contract; independent coordinate fringes; finite positive repeated passes; finite-rate approximation within remaining error margin.

	Evidence: Manuscript Section 7 positive-kernel, weak-gate and hazard construction; synthetic three-term hazard example.

	Limitation: No generic efficiency guarantee, arbitrary material synthesis, atom placement, solidification proof or actual volume optical-access guarantee.

	## C08 — Finite-rate selective-filter boundary

	Ideal filters are singular; finite bounded-action first-order maps are invertible. Approximate selective filters obey error <=exp(-h1)+exp(-h2), accumulated by summation.

	Status: Conditional analytic obstruction and approximation bound

	Assumptions: Bounded-action first-order transition maps; exact state selectivity for the approximation; stated stochastic-map norm.

	Evidence: Manuscript finite-action selective-control section; main finite-filter and stochastic-map checks.

	Limitation: Cross-talk or reverse-reaction floors may prevent the required accuracy; exact ideal nodes are not produced by finite bounded actions.

	## C09 — Extinction and phase budgets

	The protecting-factor bound gives sufficient fixed-darkness scales eta=O(m^-2) and phase error=O(m^-1) at fixed q.

	Status: Conditional sufficient bounds

	Assumptions: Balanced comb; declared fringe-floor and phase perturbation model; other error sources budgeted separately.

	Evidence: Manuscript phase/extinction section; original m8_dark01_budget and main checks.

	Limitation: Separate maximal floor and phase allowances cannot both be spent in full; not necessary optimal tolerances.

	## C10 — Retention ceiling scope

	Independent per-gate retention r gives single-pass midpoint r^m*G_m(u); finite-action target q is feasible in that pass iff r^m>q.

	Status: Conditional single-pass result with multipass counterexample

	Assumptions: Independent uniform retention; no false rescue; otherwise ideal full-commit m-gate pass.

	Evidence: Manuscript retention proposition and explicit two-pass counterexample; main checks.

	Limitation: The single-pass ceiling does not apply to arbitrary reinitialized repeated passes and is not a generic imperfect-filter model.

	## C11 — Diffusion-aware sufficient design parameter

	Finite-duration reaction-diffusion gates admit dark-node and brightness bounds; Dk^2m^5/(sigma*P) is a sufficient design parameter for this implementation.

	Status: Conditional analytic sufficient bounds and synthetic simulations

	Assumptions: Common diffusivity for mobile states; finite first-order marking; ideal filters; fixed marking irradiance and stated Brownian-path model.

	Evidence: Manuscript diffusion section; 16 original refined finite-difference cases in diffusion_examples.

	Limitation: Not a universal necessary resolution ceiling; diffusion can fill dark nodes and change order dependence; grid refinement is not a continuum error certificate.

	## C12 — Evidence and priority boundary

	The supplied release records 1305 main checks and 108 separate code-path audit checks; physical experiments, external review and formal proof-assistant replay are absent.

	Status: Release provenance and validation statement

	Assumptions: Counts refer to logged finite checks in supplied ledgers; both implementations were produced within the same research session.

	Evidence: Original results/verification.json, results/independent_audit.json and source/provenance.json; packaging rerun report.

	Limitation: Check counts do not certify historical novelty, scientific importance, physical feasibility or external mathematical acceptance.