Formal reconstruction · Low-energy nuclear scattering
Theory of the Effective Range in Nuclear Scattering
The logical anatomy of exact radial comparison, controlled low-energy reduction, observable cross sections, bound-state extraction, Coulomb modification, and charge dependence
Definitions, structures, and assumptions
Structures, classes, and inductive alternatives
- structureThe controlled phase behavior obtained from the selected effective-range law. This proof-free type contains no phase-law, correction, or shape-resolution proposition.
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- structureProof-free constraint selected by the common bound-state extraction route. It retains the exact technical model but no proof that the route applies and no extracted-parameter output.
- structureShared semantics for extracting a qualified low-energy parameter estimate from a bound-state relation and empirical scattering data. `Experiment` is an opaque paper-local evidence package, so one selected value may aggregate binding, cross-section, and amplitude measurements.
- structureExplicit bridges from the shared calculation to theory and world claims.
- structureProof-free data selected by the shared bound-state extraction. The exact parent reduction is retained as one component, together with the three new physical indices and all eight extraction-specific relation families used by this route. `Experiment` denotes a paper-local evidence package and may aggregate several measurements rather than one apparatus or run.
- structureProof-free source constraint for one paper-local bound-state extraction. The typed token and its proposition-valued interpretation distinguish routes with the same effective-range estimate without storing evidence that either the source or the extraction output obtains.
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- structureProof-free data selected from one effective-range reduction. Besides the five physical indices, the model retains exactly the eleven proposition-valued relation families used by the shared route.
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- structureSource-sensitive structural-explanation constraint for one selected low-energy scattering problem. It retains the proof-free short-range description, one typed paper derivation, and its proposition-valued realization relation, but no reduced law, controlled behavior, or endpoint.
- structureExplicit bridges from a selected calculation to theory and world claims.
- structureProof-free source description of one paper-local effective-range derivation. It retains the exact selected model and a typed derivation token, but no calculation proof, reduced output, or controlled-behavior conclusion.
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- structureShared semantics for the repeated low-energy reduction. The meanings of the parameters and the derivation supplying each calculation remain paper-local.
- structureThese are asymptotic radial modes, not the regular interior solution.
- structureThe channel-indexed parameter estimate supported by one completed extraction. The evidence projection below keeps this estimate distinct from the full technical calculation model and from any claim that the estimate is exact.
- structureThe selected short-range scattering data before effective reduction.
- structureThe selected two-parameter low-energy description.
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- structureA selected claim must depend on exactly these source-map steps.
- structureA selected claim must retain exactly these epistemic statuses.
- structureA selected route must name exactly these immediate predecessor routes.
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- inductiveBethe's distinct source route from the comparison identity specialized to the deuteron ground state through the measured triplet-range extraction.
- inductiveBethe's comparison-wave-to-deuteron physical joining calculation.
- inductiveThe paper-local derivation whose source is Bethe's exact radial comparison identity followed by the controlled zero-energy interior replacement.
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- structureThe phase shift is relative to pure Coulomb scattering. The squared partial-wave amplitude below excludes the Coulomb phase, which has unit modulus. It is not the total proton cross section, with its spin weights and Coulomb interference.
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- structurePaper-local physical vocabulary. Numerical quantities are data; physical, calculation, approximation, empirical, and interpretive assertions are proposition-valued relations.
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- structureReduced selected-energy readout. The sine and cosine are retained as physical values constrained by the trigonometric identity; this allows exact rational countermodels without obscuring the source-level phase semantics.
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- structureOne fixed potential and one fixed zero-energy reference underlie the positive-momentum family. The phase readout remains an actual derivative.
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- structureAccessible obtained evidence supporting a model in a declared regime.
- structureRecord access within one specified physical description. Configurations may be classical runs or supported quantum alternatives; the context does not declare them observer-independent worlds or supply a probability law.
- structurePredictions made by descriptions in regimes. `predicts` is relational so the signature can represent deterministic values, sets of allowed values, or probabilistic outputs without choosing among them.
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Definitions and domain choices
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- definitionExact child semantics requires the supplied model to be the context's unique composed model, not one reconstructed from its own parent or child indices.
- definitionThe exact pre-output child setup used for theory/model/world grounding. It requires the context-selected composed model, an established parent two-parameter law, the selected bound state, the uncoupled S-wave scope, and both measured inputs. It assumes neither the bound-state relation calculation nor the extracted and qualified parameter estimate.
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- definitionSource-sensitive deuteron extraction retains the exact selected composed model, its inhabited effective-range/bound-state result, the independently sourced parent reduction, the independently sourced extraction, and a third paper-local derivation connecting the low-energy law to the measured bound state. In particular, extraction provenance cannot fabricate the separate higher-order shape-control assumptions of a reduction provenance.
- definitionSource-sensitive explanation by one typed paper-local derivation.
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- definitionThe unique child model fixed by the context's exact parent reduction, three new physical selections, and eight extraction-specific relation families.
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- definitionDirect evidence support for the selected qualified parameter estimate.
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- definitionExact proof-free route constraint selected by this extraction model.
- definitionExact proof-free source constraint selected by a model and derivation.
- definitionThe channel and parameter selected by this exact extraction model.
- definitionInterpret the aggregate binding-and-scattering package as evidence for one qualified effective-range parameter estimate. Accessibility fixes the selected evidence package and regime. Obtaining the evidence requires both measured inputs. The support relation additionally retains the nonvacuous parent law, selected bound state and uncoupled scope, calculated and controlled bound-state relation, extraction, and qualification. It asserts neither exactness of the estimate nor truth of a larger theory.
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- definitionA positive control radius makes the momentum sector inhabited. The actual coefficient limits follow from this bound; they are not a second independently supplied clause. The constants are existential, not a chosen error budget at an arbitrary prediction momentum.
- definitionThe phase function is k cot(delta), not the phase angle itself.
- definitionOne actual source/reduced pair, its derivation bridge, source support, and controlled reduced behavior. The endpoint deliberately retains failure to resolve detailed interaction shape.
- definitionSource-sensitive derivational reduction of the selected scattering model.
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- definitionThe direct order-`k^2` law available before either paper's later shape-term analysis. It retains the inhabited low-energy setup, controlled zero-energy reference, both parameter definitions, and the two-parameter phase law, but assumes no calculated or small higher-order correction and no shape conclusion.
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- definitionThe source-local reduction consumes the controlled reference and prior scattering-length definition, then returns the effective-range definition and phase law together. The shared order follows both original papers without inventing a dependency between those two sibling outputs.
- definitionThe source-local zero-energy calculation consumes the selected low-energy setup and returns control of the reference together with the scattering-length definition. Bethe identifies the scattering length in equation (11), and Blatt-Jackson identify it from the asymptote of their zero-energy wave before the effective-range integral is introduced.
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- definitionExact controlled phase behavior selected by the model.
- definitionSource-sensitive structural constraint selected by this reduction model.
- definitionThe exact model and typed paper-local derivation selected as source.
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- definitionExact source-side data selected by the low-energy reduction.
- definitionExact reduced two-parameter description selected by the model.
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- definitionInterpret the selected two-parameter law and its calculated higher-order correction in the generic truncation-assessment vocabulary. Calculation completeness retains the inhabited order-`k^2` law and the actual higher-order calculation. Remainder control is the source-backed smallness of that correction. The shared route makes no independent empirical-adequacy claim.
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- definitionSource-sensitive explanation of one controlled effective-range route.
- definitionOrdinary structural explanation of one controlled effective-range route.
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- definitionExact reduction semantics requires the supplied model to be the unique model selected by the context, rather than one reconstructed from its own indices.
- definitionThe exact pre-output setup used for theory/model/world grounding. It fixes the context-selected model and requires validity, a neutral S-wave channel, a short-range interaction, and the selected low-energy regime, but assumes none of the controlled-reference, two-parameter, or higher-order outputs.
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- definitionThe unique reduction model fixed by the context's five selected physical values and eleven route-defining relation families.
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- definitionSource-sensitive structural semantics for one controlled effective-range route. The source constraint adds one typed paper derivation without storing the reduced law, controlled behavior, established endpoint, or derivational reduction.
- definitionOrdinary structural semantics for one selected controlled effective-range reduction. Satisfaction requires the complete raw source-ordered route from the proof-free short-range description; the explained phenomenon is the controlled phase behavior at the same selected model.
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- definitionImmediate predecessor routes. Claims established by their transitive predecessors are inherited by the dependency closure.
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- definitionTemporary reductio-introduction steps discharged by this inference. Discharge removes only those exact origins from the conclusion's epistemic basis; it does not assert the temporary hypotheses globally.
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- definitionA source-mapped claim holds when its declared formula evaluates under the given paper-local semantic interpretation.
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- definitionEvery claim in the supplied identifier list holds.
- definitionA source-mapped step is semantically sound when the interpreted premises imply the interpreted conclusion. Physical principles and imported calculations therefore remain explicit hypotheses of paper-specific certificates rather than acquiring truth from their source-map labels. A reductio hypothesis is different: it introduces a temporary assumption into a subargument and therefore is not required to hold globally. Its later uses remain ordinary premise-sensitive steps, and discharging it into a negated conclusion remains an ordinary semantic obligation.
- definitionEvery step in the supplied identifier list is semantically sound.
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- definitionThe formula declared for a claim identifier, when that claim is present.
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- definitionEvery source-mapped step whose justification contributes to the claim through declared premise-to-conclusion steps.
- definitionAll justification statuses contributed by the exact source-mapped origins of a claim. This records epistemic provenance; it does not assert any paper claim or discharge any physical assumption.
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- definitionThe declared argument step with the requested identifier, when present.
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- definitionThe premise list and conclusion attached to a declared step identifier.
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- definitionThe complete Lean-side fidelity certificate. The first conjunct is computed from source-map data; the second is supplied by checked route semantics; the third requires each declared independence boundary to have a countermodel.
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- definitionA typed countermodel matrix avoids string-keyed audit rows. Each index selects one proposition-valued separation obligation.
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- definitionEvery route actually declared in the source map has a semantic certificate under the paper-local interpretation.
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- definitionOne countermodel row retains a weaker proposition while rejecting a stronger upgrade. This is a proposition, not a structure carrying proof fields.
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- definitionIndependent dependency and epistemic ledgers must review the same actual source claims, including every public conclusion and the selected paper conclusion. Empty, duplicated, nonexistent, or one-sided entries are invalid.
- definitionEvery actually declared source route must have exactly one independently reviewed predecessor row. Correct rows for a selected subset are insufficient.
- definitionThe declared main conclusion must be exposed by at least one public route output, rather than existing only as an unexported source-map claim.
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- definitionTwo finite lists describe the same dependency set. Duplicate expected entries are rejected so a ledger cannot pass by repeating a label.
- definitionEvery paper-level, claim-level, and step-level source reference is present and has a nonempty work and locator. PaperArgument.auditable separately checks that every claim and step carries at least one source.
- definitionA source reference is usable for review only when both the work and its locator are nonempty.
- definitionStatic fidelity checks that can be evaluated from a paper's source map alone. They do not assert the truth of any physical premise.
- definitionEvery declared output of a named source route holds under the semantic interpretation. This is the proof obligation discharged by paper-specific Lean theorems.
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- definitionA route is fully certified only when both its individual steps and its declared outputs are semantically checked.
- definitionEvery source-mapped inference step in a named route is semantically sound.
- definitionEvery route named in the supplied identifier list is semantically certified.
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- definitionThe source-mapped route with the requested identifier, when present.
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- definitionThe public semantic interface inherited from predecessor routes. Descendants receive only declared route outputs, not every internal intermediate claim. Outputs still carrying a temporary reductio origin are excluded.
- definitionAll direct and inherited route dependencies. The route-order check separately ensures that the declared graph is acyclic and topologically ordered.
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- definitionPositive radius from (2) and (20a), with the mass and hbar held fixed.
- definitionDividing the energy-radius relation by the baseline binding energy.
- definitionLiteral first-order profile printed in (25), including its plus sign.
- definitionLiteral affine readout in (27), in units of 10^-13 cm.
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- definitionThe well-posed normalized zero-energy scattering problem is an explicit mathematical input, quantified over actual radial solutions.
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- definitionThese normalized smooth curves diagnose inversion, not a selected Coulomb or nuclear radial equation.
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- definitionThe finite-radius comparison (44) and its boundary limits, summarized at the actual derivative difference before any phase-function identification.
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- definitionActual regular and irregular Coulomb solutions, with their origin and large-radius normalizations. Special-function realization is an imported calculation, not a stored phase-shift conclusion.
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- definitionOnly the common overlap and L1-error interface is used. The neutral comparison-wave equation at the origin is not asserted for Coulomb waves.
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- definitionThe common reference phase is fixed by the supplied Coulomb basis. The separate Gamma-function phase convention in (40) is not inferred here.
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- definitionScalar error controls at an actual Coulomb phase. These do not claim that the selected ranges arise from a solved nuclear potential.
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- definitionPositive lower radius avoids the Coulomb singularity. Integral radial dynamics permit finite steps in the nuclear potential.
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- definitionThe special-function limit and finite source intercept are explicit inputs. Their noncancellation is not inferred from barrier penetration.
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- definitionThe displayed leading small-radius expression, not an exact solution of the Coulomb radial equation. Its derivatives test the boundary inference.
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- definitionRaw paper premises interpreting Bethe's bound-state extraction token. They retain the comparison-wave construction, controlled order-`k^2` reduction, deuteron/data setup, relation calculation, and empirical extraction as separate contracts rather than storing an established shared endpoint.
- definitionRaw paper premises that realize Bethe's comparison-identity derivation. The token records a derivation route, not merely its two-parameter output.
- definitionEquations (10)-(11), specialized to the negative bound-state wave-number square and the actual derivative of the equation-(15) comparison function. The Wronskian/overlap evaluation remains the explicit source calculation.
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- definitionThe cited depth calculation is separate from scattering-length separation. Its common-range-and-shape premise remains conditional, not negated.
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- definitionThese quantitative bounds are explicit sufficient reconstruction inputs. The paper does not itself certify all of their numerical budgets.
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- definitionThe two unevaluated profiles are exponential and Gaussian. The original bracketing expectation precedes the finite-scattering-length corrections.
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- definitionPhysical Review p. 43: the sign of a resolved deviation has a directional meaning. Upward curvature would favor a concentrated potential; downward curvature would favor a long-tailed potential. Bethe then says that the available data do not make either deviation significant.
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- definitionPhysical Review p. 39, equation (11): k cot(delta) at zero energy is -alpha.
- definitionThe exact definition within equation (14): half of r0 is the actual zero-energy comparison-minus-exact square integral. The approximate replacement of rho(0,E) by r0 is a separate, later controlled input.
- definitionPhysical Review p. 39, equation (7): the asymptotic comparison wave is normalized to one at the origin by dividing sin(kr + delta) by sin(delta). The wave-number square relation and the stated origin-derivative readout k cot(delta) are retained so the k and delta used here are the same ones later used in equations (10)-(17).
- definitionPhysical Review p. 40, equation (17): the neutral S-wave cross section is `4 pi / (k^2 + (k cot delta)^2)`. The formula uses the paper's already typed phase-function value and does not silently transport the selected low-energy effective-range law to a different energy index.
- definitionBethe's first displayed equation-(17) form reads the physical selected phase shift directly: `sigma = 4 pi sin(delta)^2 / k^2`.
- definitionPhysical Review p. 39, equation (10): subtracting the two exact and comparison radial identities leaves the phase-function difference equal to the wave-number-square difference times the comparison-minus-exact overlap.
- definitionPhysical Review p. 39, equation (13): half of rho(0,E) is the comparison-minus exact overlap integral. This is the same integral appearing in equation (10), not an independently named range datum.
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- definitionDearnley's upward departure and Wilson's error-compatible comparison are different measurements, both above the selected low-energy sample. The potential-direction interpretation and excluded P-wave tensor explanation are distinct physical inputs, not consequences of the residual algebra.
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- definitionEquation (16) is the approximation obtained from (14), not an exact identity of the physical phase function. The tolerance is an explicit quantitative reconstruction, not a numerical bound printed by Bethe.
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- definitionThe exact finite-radius identities and boundary limit are imported analytic outputs. The desired infinite-radius comparison (10) is derived.
- definitionThe literal early radial readout used by Bethe's effective-range reduction. The last conjunct is the k1 = 0 specialization used between equations (10) and (11).
- definitionThe numerical S-wave unitarity claim on the selected physical energy: the observable cross section is positive and at most `4 pi / k^2`.
- definitionBethe's stated maximum is an equality of physical cross-section values, not merely a proposition named for a maximum.
- definitionThe permitted S-wave maximum is attained exactly when the selected numerical phase function is zero.
- definitionEquation (7)'s `k` and `delta` must be read at the actual selected cross-section energy; the earlier low-energy comparison does not transport those value identities to a distinct energy index.
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- definitionThe numerical fit evaluation is input; conversion to a range estimate is derived using (4) and (51). No exact integral-defined range is supplied here.
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- definitionBethe's pre-output setup for the shared bound-state extraction. The parent order-`k^2` law is already established, and the selected deuteron bound state, deferred tensor-force scope, plus binding and scattering data are available. No equation-(19) relation calculation, control output, extracted range, or qualification of that estimate is assumed.
- definitionBethe's paper-local pre-calculation setup for the shared reduction. It fixes a neutral S-wave channel with a finite-range nuclear potential at the selected low energy, but assumes no comparison identity, zero-energy replacement, effective-range law, higher-order estimate, or shape conclusion.
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- definitionThe zero-energy comparison function is linear with the inverse scattering length as minus its origin slope. It is not the regular interior wave.
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- definitionThe joining source keeps Bethe's two independently controlled histories, the printed comparison-wave/range formulas, deuteron ground-state specialization, both measured evidence inputs, and the explicitly qualified triplet estimate.
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- definitionEquation (15) is the normalized comparison mode, not the interior solution.
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- definitionThe neutron-proton singlet is not the triplet/deuteron channel used in the earlier neutral calculation.
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- definitionPaper-local realization of the repeated bound-state extraction inference. Bethe's comparison-wave derivation of the bound-state relation remains local; the shared component begins with its controlled output and the measured binding and scattering inputs on pp. 40-41, especially the relation in equation (19) and the extracted value in equation (27).
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- definitionPaper-local realization of the repeated low-energy effective-range inference. Bethe's exact comparison identity and its wavefunction integrals remain local; only their controlled low-energy output is shared.
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- definitionPhysical Review p. 40, Eq. (19), first equality: after selecting the deuteron bound state, the reciprocal triplet scattering length `alpha` equals `gamma - (1 / 2) gamma^2 rho(0, -epsilon)`. The later replacement of the energy-dependent range by `r0` remains a separate controlled approximation.
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- definitionThe model's two-parameter phase prediction is distinct from k cot(delta).
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- definitionThe coherent sign is a restriction on the admitted alternatives.
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- definitionA rate assumption, not just continuity at zero energy. The positive band and a single coefficient retain the uniform quantifiers of an order estimate.
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- definitionThe boundary and integral limits used between (8) and (9). Exact exterior matching with locally integrable overlap is sufficient, but a finite matching radius is not imposed on all short-range potentials.
- definitionA sufficient quantitative version of the interior-wave replacement in (14). It controls an actual integral, not a predicate named "small error".
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- definitionExact finite-radius integrated identities (6) and (8). Their analytic derivation is an explicit input, before subtraction and the infinite limit.
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- definitionOne actual neutral S-wave phase-function prediction at positive k squared.
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- definitionIntegral radial dynamics permit finite potential steps. The derivative is the actual one-sided radial derivative, not an independent readout.
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- definitionA bridge principle derives a macro behavior from any realizing microdescription that supports it.
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- definitionThe target's activity is the shared prediction-sector condition, not a second approximation-specific notion of having a prediction.
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- definitionA realized record constrains a physical value when every compatible configuration satisfies the specified property. This need not identify an exact value, assign a probability, or identify the actual configuration.
- definitionA realized record determines a value only if every compatible alternative agrees.
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- definitionDeterministic processing of the recorded data, not a change of the admitted physical configurations, their values, or the measurement dynamics.
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- definitionA selected observable sector.
- definitionA description makes at least one prediction in the selected sector.
- definitionOne description approximates another when every target prediction in the selected sector has an agreeing prediction in the approximating description. The direction is intentional.
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- definitionA model-to-world interpretation law. It does not assert that the model has produced a conclusion; use `GroundedModelConclusionAt` for that.
- definitionA model-to-theory interpretation law. It does not assert that the model has produced a conclusion; use `GroundedModelConclusionAt` for that.
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- definitionBuild the epistemic dependency basis in paper order. A conclusion inherits the justification origin of its own step and every origin already attached to its premises.
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- definitionExpand a set of route identifiers by one dependency edge, iterated with enough fuel to traverse any acyclic graph on the declared routes.
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Selected instances and coercions
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Source mappings and interpretation choices
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- checked statement**Alias** of `PhysicsLogic.Papers.Bethe1949.SemanticContext.all_routes_are_semantically_realizable`.
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Checked statements: parameters, hypotheses, and interpretation
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- checked statementThis conclusion uses the local fixed-mass relation, not a chosen formula for the radius. It therefore tests the first-order reading of (25).
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- checked statementThe source's positive binding energy is minus the shared bound-state energy. Here the reduced mass is half the nucleon mass in (2).
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- checked statementThe nominal source values R=4.332 and a_t=5.28 produce opposite signs under the two radius responses. No equality to the rounded 2.7 is claimed.
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- checked statementCoordinates are in the paper's unit 10^12 inverse centimeters. These are reported rounded estimates, not exact physical scattering lengths.
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- checked statementLean's totalized zero division makes the displayed equation true at `k^2 = 0` without giving a physical positive cross section or domain.
- checked statementA positive cross section satisfying the exact printed equation does not reach the permitted maximum unless the actual phase function vanishes.
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- checked statementEven an exact denominator formula with a physical `k cot delta` readout does not recover Bethe's sine-squared formula if the purported `k^2` is not the square of the same selected wave number.
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- checked statementGround the named extraction model from the established parent two-parameter law and Bethe's selected deuteron, deferred tensor-force scope, binding datum, and low-energy scattering datum on pp. 40-41. This step uses neither the equation (19) relation calculation and control output nor the parameter extraction and qualification in equation (27).
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- checked statementBethe's raw comparison/ground-state/data route realizes the source-sensitive shared explanation. No established shared extraction, nonvacuous endpoint, ordinary explanation, or evidence-support endpoint is a premise.
- checked statementBethe's complete comparison-to-deuteron composition requires the separate higher-order reduction source in addition to the distinct empirical extraction source; neither source manufactures the other's missing premises.
- checked statementGround the named low-energy model in Bethe's neutral S-wave, finite-range setting from Sections I-II, pp. 38-40. This step uses neither the comparison identity nor the controlled two-parameter and higher-order calculations.
- checked statementBethe's typed comparison-and-replacement source reconstructs the raw shared route rather than storing an established reduction or ordinary explanation.
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- checked statementThe raw exact-comparison route realizes a source-sensitive shared reduction. No established shared reduction is a premise.
- checked statementBethe's typed comparison-and-replacement source realizes the stricter source-sensitive structural explanation without identifying it with Blatt-Jackson's stationary-functional route.
- checked statementBethe's raw comparison-identity inputs structurally explain the selected controlled phase behavior without consuming an established endpoint or derivational-reduction wrapper.
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- checked statementPhysical Review p. 40, equations (17)-(18): inserting the phase law into the S-wave cross section retains the unitarity bound and its maximum. Bethe's warning that the next term is needed near that maximum is kept separately from the two-parameter phase law itself.
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- checked statementThe literal squared records are joined to the paper's equation-(19) calculation through the actual reciprocal triplet length. The coherent sign and the bound-energy-to-zero-energy replacement budget remain explicit. This does not turn the empirical central records into exact measurements.
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- checked statementBethe's second form of Eq. (19) follows algebraically only after an explicit replacement of the bound-energy range function by the effective range. This theorem does not establish that approximation; it exposes exactly what it would license.
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- checked statementEquation (7) really performs the advertised normalization: the selected comparison wave evaluates to one at r = 0 when sin(delta) is nonzero.
- checked statementThe derivative readout following equation (7) is not hidden in a field name: the selected phase function is explicitly the same k cot(delta) built from the comparison-wave k and delta.
- checked statementThe permitted S-wave maximum is attained only after the named vanishing phase is tied to the actual zero value in Bethe's displayed formula.
- checked statementWith positive selected `k^2`, the printed equation (17) itself yields a positive cross section and the S-wave unitarity bound.
- checked statementThe two forms printed together in Bethe's equation (17) agree only when the squared wave number and `k cot delta` are tied to the same selected physical `k` and phase shift.
- checked statementEvery genuinely nonzero selected phase suppresses the S-wave cross section strictly below the permitted unitarity maximum.
- checked statementIn Bethe's actual phase-shift variables, saturation of the S-wave bound is equivalent to `cos delta = 0`, not merely to a freely assigned scalar named for the phase function.
- checked statementAt physical positive `k^2`, equation (17) reaches its permitted maximum if and only if the actual phase function vanishes.
- checked statementEquations (10), (11), and (13), together with k1 = 0, algebraically give equation (12). The effective-range replacement in equation (14) is not used yet; it remains a later controlled approximation.
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- checked statementThe established maximum is characterized by the selected physical phase shift itself rather than only by an independently named scalar.
- checked statementThe original observable conclusion retains the full numerical characterization: saturation is equivalent to vanishing physical phase.
- checked statementThe selected observable route retains both equalities actually printed in equation (17), with the same physical wave number and phase shift.
- checked statementThe assessed observable branch retains Bethe's literal equation-(17) cross-section formula instead of only a named calculation predicate.
- checked statementThe established observable consequence contains the derived positive unitarity bound and its actual vanishing-phase saturation.
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- checked statementBethe's extracted triplet result retains the bound-state-specific control of equation (19), including the qualified replacement of the range function at the deuteron energy. It is not inferred from positive-energy scattering accuracy. Primary-source anchor: Physical Review pp. 40-41, equations (19) and (27), including the stated shape-dependent correction to the extracted range.
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- checked statementBethe's established order-`k^2` law supplies the shared parent needed by the deuteron extraction before Section IV's later shape-correction analysis.
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- checked statementThe shared setup recovers the neutral S-wave, finite-range, and low-energy premises that Bethe's reduction genuinely shares with Blatt-Jackson. The radial-wave and comparison-wave facts remain in the paper-local neutral result, because the shared component does not claim to reconstruct them.
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The paper reconstruction's physical
statements, dependencies, equations, qualifiers, and source locators
come from Bethe1949.lean,
Bethe1949/SourceMap.lean, and
Bethe1949/Coverage.lean. Numerical values reported in the
historical paper are not invented here when the Lean formalization
stores only their named data objects. The marked counterexamples also
use Bethe1949/Fidelity.lean; the independently constructed
elastic-amplitude diagnostic comes from
scripts/CheckIndependentReuse.lean, not from Bethe's paper.
The separately marked shared-error consumer comes from
tests/consumer/PhysicsLogicConsumer/NuclearScattering.lean
and tests the reusable framework rather than adding historical claims.
Two theoretical trunks support distinct extraction and comparison routes
The neutral trunk begins with an exact finite-range comparison and then introduces a controlled low-energy replacement. It branches into an observable cross-section assessment, an accuracy/shape assessment, and a deuteron bound-state extraction. A separate singlet-range route records the limited neutron–proton data and the distinct force-equality and borrowed-range assumptions. The charged trunk independently derives the exact source function and controlled charged law, then uses proton data, an independent exact weighted comparison, and controlled inverse readout to assess charge dependence. The singlet qualification and cited depth interpretation are subsequent inputs. Separate Coulomb-removal and threshold routes retain their own assumptions. The final route assembles both trunks with two explicitly deferred extensions.
| Transition | Status | Retained qualification |
|---|---|---|
| Neutral radial comparison | exact paper derivation | Does not itself justify the low-energy wave replacement. |
| Two-parameter neutral law | controlled approximation | Restricted to energy low relative to the force range. |
| Observable cross section | separate calculation and interpretation | The phase law does not itself supply the observable, and the next term matters near the unitarity maximum. |
| Shape assessment | calculation, interpretation, controlled accuracy | Curvature and cross-channel intrinsic-range arguments distinguish possibilities without selecting a shape from one channel. |
| Triplet extraction | imported data and paper calculation | Bound-state-specific control is not borrowed from positive-energy accuracy. |
| Singlet range qualification | imported limitation and interpretive borrowing | Limited neutron–proton data do not become a determined singlet range. |
| Coulomb comparison | exact conditional source formula | Actual basis, derivative expansion, and integrated comparison; no range replacement. |
| Charged range law | controlled reduction | Finite-energy overlap error is explicit. |
| Coulomb removal | separate controlled approximation | A second same-potential error budget, not section VII's weighted integral. |
| Coulomb threshold | physical and imported regularity inputs | Nuclear strength vanishes only under a noncancelling intercept condition. |
| Proton result | imported empirical data and fit | Direction, square-well excess, and failed tensor explanation are retained without closing the discrepancy. |
| Weighted charge comparison | exact conditional calculation | Positive cutoff, actual boundary limit, and weighted balance precede the empirical estimate. |
| Neutral-length estimate | controlled approximation | Cutoff, fit, wave, denominator margin, and truncation remainder are explicit. |
| Charge comparison | error separation and conditional depth inference | Unequal potentials follow from normalized-scattering uniqueness; pointwise closeness additionally requires common range and shape. |
| Additional potential shapes | author expectation and conditional calibration | The uncorrected bracket is not the corrected bracket or a computed exponential/Gaussian result. |
| Paper scope | deferred calculations | Tensor-force and deuteron photoeffect extensions are not completed. |
Subtracting the radial equations gives an exact finite-range comparison identity
The neutral setup selects the neutron–proton triplet S wave in a finite-range nuclear potential. Actual solutions are supplied at zero and selected low energy, with comparison waves matching them outside the force range. Equation (7) fixes the selected comparison wave by dividing sin(kr+δ) by sin δ, and Lean checks that this really gives value one at the origin. Its actual one-sided derivative is proved to be the same k cot δ phase function used below. The regular interior waves instead vanish at the origin. The common radial dynamics have an integral formulation that permits finite potential steps.
k cot δ − k0 cot δ0 = (k2−k02) ∫[ψ0(r)ψ(r)−u0(r)u(r)]dr. equations (7) and (10): normalized comparison wave, explicit phase-function readout, and exact comparison-minus-exact overlap identity (1)
u_i(0)=0, psi_i(0)=1; radial derivatives are actual one-sided derivatives.
W[f](R)=f_1(R)*f_0'(R)-f_0(R)*f_1'(R); B(R)=W[psi](R)-W[u](R).
F_1-F_0=(k_1^2-k_0^2)*integral_0^R (psi_0*psi_1-u_0*u_1) dr-B(R).
An integrable overlap and B(R)->0 give the exact infinite-radius comparison.
Control: V=0, k_0^2=0, k_1^2=1, u_0=r, u_1=sin(r), psi_0=1, psi_1=cos(r).
Both radial and finite Wronskian identities hold, but B(R)=R*cos(R) does not tend to zero. The free-wave control deliberately lacks exterior matching; it is not Bethe's selected scattering solution. Exact compact support is not imposed on every short-range potential.
Specializing one solution to zero energy gives an exact relation for the neutral phase function. Its range quantity is still energy-dependent; no two-parameter approximation has yet been made.
½ρ(0,E) = ∫[ψ0(r)ψ(r)−u0(r)u(r)]dr,
k cot δ = −α + ½k2ρ(0,E). equations (11)-(13): Lean derives the final relation from the shared overlap integral and k02=0 (2)
The radial setup, finite identities (6) and (8), convergent overlap, boundary limit, and exact range definitions derive the equations-(7)/(10)/(11)/(13)/(14) readout. The selected scalar readouts must represent those same phase differences and integrals. Equation (10) is a conclusion of this route, not an input named as its own calculation. Finite-radius integration identities retain their explicit analytic obligations.
equation_seven_normalizes_comparison_wave_at_origin · equation_seven_retains_k_cotangent_phase_function · equations_ten_eleven_and_thirteen_yield_zero_energy_relation · radial_comparison_and_formula_readout_yield_exact_neutral_identity · radial_comparison_yields_exact_neutral_identity · Physical Review pp. 38–40, equations (6)–(14)
The effective-range law requires a controlled zero-energy interior replacement
At energy low relative to the nuclear-force range, the finite-energy interior wave may be replaced by its zero-energy counterpart under an explicit integral error bound. Equation (14)'s exact square-integral definition of r0 is separate from replacing ρ(0,E) by that constant. The physical phase function and the two-parameter prediction remain different quantities.
F_ERE(k)=-alpha+k^2*r0/2.
F(k)-F_ERE(k)=k^2*(rho(0,E)/2-r0/2).
|F(k)-F_ERE(k)|<=k^2*epsilon, with epsilon>=0.
At k!=0, exact phase agreement is equivalent to rho(0,E)=r0. Equations (12)-(16): an explicit sufficient error budget, not a numerical bound supplied by the paper or an exact all-energy phase law. (3)
The paper-local route calculation first produces the controlled interior replacement integral bound. Lean derives the range error and its phase error from that bound and the exact comparison; no additional implication is assumed to supply the desired phase law. A separate identification bridge says that this phase-function relation is the effective-range law in the selected nuclear regime. The formal result retains the inhabited setup, controlled replacement, law, and identification output, but not the implication-valued route calculation or identification bridge. Source-sensitive adapters keep those raw contracts explicit instead of storing them inside the terminal result.
Exact comparison, low-energy validity, controlled zero-energy replacement integral control, and the identification bridge establish the two-parameter neutral law without converting the replacement into an exact identity.
controlled_zero_energy_replacement_yields_effective_range_law · Physical Review pp. 39–40, equations (14)–(16)
rho(0,E)/2-r0/2=integral_0^infinity J(r) dr.
J is integrable, epsilon>=0, and integral_0^infinity |J(r)| dr<=epsilon.
Therefore |rho(0,E)/2-r0/2|<=epsilon and |F(k)-F_ERE(k)|<=k^2*epsilon. A sufficient quantitative reconstruction of (14); cancellation may allow weaker controls, but smallness is not an arbitrary predicate or an assigned integral value.
|rho(0,E)/2-r0/2|<=C*k^2 implies |F(k)-F_ERE(k)|<=C*k^4.
Control: rho(k)=2+2*k is continuous, rho(0)=2, and F(k)=-1+k^2+k^3.
Its phase error k^3 has no uniform C*k^4 bound near k=0.
A nonzero quartic remainder, F(k)=-1+k^2+k^4, can satisfy the controlled approximation without exact equality. These rate controls are not asserted to arise from a fixed nuclear potential. Continuity alone is weaker than the required uniform quadratic range estimate.
S(k)=(F(k)+i*k)/(F(k)-i*k) has |S(k)|^2=1 for k>0.
At k=1/2 the actual phase residual is 1/8, not zero.
The shared approximation rule combines a 1/16 readout error with the 1/8 phase budget to give 3/16. An independently constructed elastic unitary amplitude, not a proved fixed-potential inverse-scattering realization or empirical error estimate.
The uniform radial extension fixes one potential and one zero-energy actual/comparison pair across momenta. Finite Wronskian identities, actual boundary limits, the zero-overlap integral, and a uniform interior L1 error bound derive the shared expansion. The theorem does not take its desired phase bound or coefficient limits as premises.
RadialComparison.Family.uniform_radial_inputs_realize_shared_effective_range_expansion · equations (12)-(16), with explicit sufficient analytic error inputs
Blatt-Jackson reaches this same error/limit contract by a different stationary-functional argument. Uniqueness applies to the coefficients of one phase function, not to the full phase function or a potential. Positive-momentum control supplies no bound-state continuation. The independent shared-API consumer makes the distinction observable:
The raw radial-comparison setup, exact comparison calculation,
finite-radius analytic inputs and boundary limit,
controlled interior replacement, identification bridge, and
higher-order assessments realize the typed
comparisonIdentityAndInteriorReplacement source. The
resulting shared reduction therefore records Bethe's derivation,
not merely an effective-range output with matching values.
bethe_source_inputs_make_shared_reduction_applicable · bethe_source_inputs_realize_source_effective_range_derivational_reduction · Physical Review pp. 39–43, equations (10)–(16)
The raw comparison setup and calculation, controlled interior replacement, identification bridge, and higher-order assessment explain the selected controlled phase behavior without consuming an established shared endpoint or derivational-reduction wrapper. The typed comparison-and-replacement source first reconstructs that raw route, then yields the stricter source-sensitive explanation. Matching low-energy output does not erase Bethe's derivational provenance.
bethe_source_inputs_realize_structural_effective_range_reduction_explanation · bethe_source_derivation_makes_shared_reduction_applicable · bethe_source_inputs_realize_source_structural_effective_range_reduction_explanation · Physical Review pp. 39–43, equations (10)–(16)
Shared low-energy reduction
toLowEnergyEffectiveRangeContext retains the S-wave,
short-range, low-energy, controlled-reference, two-parameter-law,
higher-order-correction, and unresolved-shape semantics. Separate
adapter reconstruction keeps Bethe's radial-wave facts local while
replaying the retained replacement, reduction, and higher-order
contracts from the shared setup and law. The paper-local grounding
setup supplies only exact selection, the S-wave and finite-range
conditions, and the selected low-energy condition. Grounding uses
none of the controlled-reference or law outputs before
grounded_bethe_reduction_transfers_every_shared_claim
transfers every shared claim.
The phase law reaches the observable cross section through a separate unitarity-scoped calculation
Bethe inserts the established neutral phase function into the S-wave
cross-section formula. The formalization keeps this as a separate
calculation rather than treating the word “phase” as already denoting
an observable. NeutralWaveNumberAndPhaseShiftReadoutAt
identifies the selected k² = k · k, requires nonzero
sin δ, and identifies the phase function with the
same-energy k cot δ. From that physical readout and
EquationSeventeenCrossSectionFormulaAt, Lean derives
EquationSeventeenSineSquaredCrossSectionFormulaAt:
both printed 4π sin²(δ) / k² and
4π / (k² + (k cot δ)²) forms hold at the actual
cross-section energy. The selected classical-energy point retains
the S-wave unitarity bound and its maximum exactly when the physical
phase shift satisfies cos δ = 0.
⟹ σ = 4π sin²(δ) / k² = 4π / (k² + (k cot δ)²);
σ satisfies its unitarity bound; σ is maximal ⟺ cos δ = 0. the observable calculation is distinct from the low-energy phase-law reduction (3a)
Near that maximum, the low-order cross-section expansion cannot be
used without qualification. Bethe explicitly notes that the next
term is required to keep the result below the permitted maximum.
NeutralCrossSectionTruncationInterpretation preserves
that warning independently of the calculation contract.
named_neutral_cross_section_does_not_supply_equation_seventeen_formula
separately blocks a name-only cross-section calculation from
supplying the displayed value relation.
The selected phase-law premise, same-energy physical wave/phase readout, cross-section calculation, and truncation interpretation establish both printed observable formulas, the phase-shift characterization of the bound's maximum, and the near-maximum warning together.
effective_range_phase_law_yields_unitarity_scoped_cross_section · established_neutral_cross_section_retains_both_equation_seventeen_forms · established_neutral_cross_section_characterizes_physical_phase_shift_maximum · established_neutral_cross_section_assessment_exposes_complete_route · Physical Review p. 40, equations (17)–(18)
A small higher-order correction supports accuracy but not reconstruction of the force shape
The higher-order calculation expands the exact and comparison waves, evaluates the first shape-dependent correction, finds its coefficient small, and judges the two-parameter approximation accurate in the selected energy range.
The interpretation distinguishes sensitivity from determination. Upward curvature of the phase function would indicate a concentrated potential; downward curvature would indicate a long-tailed one. Bethe's selected neutron–proton phase data do not make either deviation significant, so they do not resolve the detailed potential shape.
The same route retains a second, cross-channel distinction. A square well would require different intrinsic ranges for triplet and singlet scattering, whereas a long-tailed potential can accommodate their different effective ranges with a common intrinsic range. This favors a mode of explanation but does not let either channel’s energy dependence select the potential shape by itself.
The correction calculation, curvature diagnostic, and shape interpretation establish the expansion, small coefficient, accurate approximation, potential-tail discriminating power in principle, absent significant deviation, and failure of present shape determination.
small_shape_correction_supports_accuracy_not_shape_determination · Physical Review pp. 42–44, equations (29)–(36)
The completed accuracy scope retains both directional diagnostics and the absence of a significant measured deviation. It therefore does not promote in-principle tail sensitivity into an observed potential-tail classification.
accuracy_scope_retains_sign_sensitive_but_unresolved_curvature · Physical Review p. 43
The completed scope preserves the square-well mismatch, long-tail compatibility, and the failure of single-channel energy dependence to determine the interaction shape.
accuracy_scope_retains_cross_channel_intrinsic_range_distinctions · Physical Review pp. 41–44
Deuteron binding and scattering data extract the triplet effective range
The evidence setup supplies a deuteron bound state in the triplet
channel, explicitly retains the deferred tensor-force scope, and
supplies its measured binding datum and measured low-energy
scattering input. Combined with the established order-k2
neutral law, before the later shape analysis, an
explicit bound-energy comparison calculation and a separate explicit
parameter-extraction calculation produce the named, qualified
reported triplet effective range. The completed result stores the
controlled relation, extraction, and qualification outputs; another
theorem reconstructs the
paired calculation route. The relation control is indexed at the deuteron
energy and is not inferred from accuracy of the scattering law at a
selected positive energy. Its calculated companion is no longer a
label: BoundStateRadialComparisonAt retains the negative
wave-number square, zero-energy phase readout, and exact comparison
identity. The actual origin derivative of the normalized decaying
comparison derives tripletBoundStateRelationCalculatedAt,
rather than assuming equation (19) as the calculation's output.
k_bound^2=-gamma^2, k_zero^2=0, K_zero=-alpha.
psi_g'(0)-K_zero=(k_bound^2-k_zero^2)*rho/2.
Therefore alpha=gamma-gamma^2*rho/2. The exact overlap identity is an explicit input; the comparison is not the regular interior wave u with u(0)=0. (5a)
bound_comparison_retains_normalization_and_actual_origin_derivative
differentiates the comparison function itself.
bound_comparison_identity_derives_equation_nineteen then
executes the bound-energy substitution. Positive gamma also gives
an actual nonvacuous decaying comparison mode; it does not prove
a full radial matching or tensor-force calculation.
+ explicit range-replacement hypothesis
+ {deuteron binding datum, low-energy triplet scattering datum}
⟹ controlled effective-range form and qualified estimate of re,t. equation (19)'s first equality is explicit; its range replacement, shape-correction scope, deferred tensor force, and abstract reported value remain separate (5)
The exact comparison does not set rho=r0. Positive-energy scattering accuracy does not itself control this bound-energy range replacement. (5b)
equation_nineteen_exposes_the_range_replacement_residual
keeps the error from replacing the bound-energy range by r0.
triplet_extraction_retains_actual_bound_state_asymptotics
retains the decaying comparison in the qualified extraction result.
The data setup, order-k2 law, paired relation/extraction contracts, and bound-energy-specific control yield the qualified triplet-range estimate. equation_nineteen_range_replacement_yields_effective_range_form reaches the second equation-(19) form only after an explicit replacement hypothesis. The established result retains the terminal outputs; a checked theorem reconstructs the contracts for shared replay.
equation_nineteen_range_replacement_yields_effective_range_form · deuteron_and_scattering_data_extract_triplet_effective_range · established_triplet_extraction_exposes_complete_route · Physical Review pp. 40–41, equations (19)–(27)
established_triplet_extraction_retains_bound_state_relation_control directly exposes the distinct control proposition used by the shared adapter, while the adapter reconstructs the parent two-parameter law and replays the recovered contracts from the shared bound state, uncoupled S-wave scope, relation, control, and data.
The shared endpoint also preserves derivational provenance. bethe_bound_state_source_inputs_make_shared_extraction_applicable constructs the route from the raw comparison identity, controlled interior replacement, deuteron setup, equation-(19) relation calculation, and empirical extraction. bethe_bound_state_source_inputs_realize_source_structural_extraction_explanation records that exact route with a Bethe-specific typed token; it does not identify the derivation with Blatt and Jackson's scattering-matrix continuation merely because both support a qualified triplet-range estimate.
The singlet statement is deliberately not folded into this theorem. Existing neutron–proton singlet data do not determine the range accurately; a separate bridge retains both approximate equality with the proton–proton singlet force and the distinct borrowing of the proton–proton effective range. This qualification is later consumed by the charge comparison without making the deuteron extraction one of its premises.
The imported singlet-data limitation and the separate borrowing bridge establish the qualified singlet-range assessment without upgrading it to a measured range. singlet_range_assessment_retains_force_equality_and_borrowed_range exposes the two retained assumptions separately.
limited_neutron_proton_singlet_data_retain_borrowed_range_assumption · singlet_range_assessment_retains_force_equality_and_borrowed_range · Physical Review p. 41
Shared bound-state extraction
The paper also realizes a shared effective-range bound-state model: one binding datum and one scattering datum form an aggregate evidence package that feeds the selected two-parameter law and extracts a qualified range estimate. Its theory/model/world grounding requires the established shared parent two-parameter law, selected deuteron, deferred tensor-force scope, and measured inputs, but neither the bound-state relation calculation nor the extracted and qualified estimate. It remains separate from the paper-local singlet assessment.
Coulomb comparison, two range approximations, and threshold suppression are distinct arguments
The proton-proton singlet setup uses reduced mass M/2, giving the Bohr radius D=2 hbar2/(M e2), and eta=1/(kD). The positive scattering point is distinct from the zero-energy reference. Integral radial equations on strictly positive intervals admit finite nuclear-potential steps without treating the Coulomb origin as regular.
Actual Coulomb waves and their relative phase
The source basis has actual radial equations, origin limits, and sine/cosine asymptotes with one common reference phase. Its combination derives phi's unit origin limit and the relative nuclear phase in both comparison and exterior-matched physical waves. The common phase offset is a basis input; the Gamma-function convention printed in (40) is not silently replaced or independently derived.
f = g + 2 pi cot delta/[exp(2 pi eta)-1],
f/D = g/D + C02 k cot delta. the source series converges; f is dimensionless, f/D is not k cot delta
Each comparison derivative contains (2/D)[log(2r/D)+Euler's constant]. It diverges as r approaches zero. The finite derivative difference is obtained by cancelling this common logarithm at the same positive radius before taking the limit. Finite-radius integration and its boundary limits remain explicit analytical inputs, not an assumption of the final phase law.
phi'E(r)-phi'0(r) → [f(E)-f0]/D.
Each derivative diverges; only the same-radius difference has a finite limit. f0 is the zero-energy intercept, not the finite-momentum formula evaluated at k=0.
f = D[-alpha' + k2 rho(0,E)/2]. exact charged comparison and dimensionless source formula, before range replacement (6)
The actual basis, common-log expansion, integrated derivative comparison, and scalar representations establish the exact source function. This route does not use either range approximation.
the_coulomb_source_formula_is_established_before_range_replacement · the_actual_coulomb_basis_identifies_the_relative_nuclear_phase · Physical Review pp. 43-45, equations (37)-(51)
Two independent overlap errors
Replacing the energy-dependent charged overlap by its charged zero-energy value needs an L1 error control epsilonE. Removing Coulomb distortion from the zero-energy waves while keeping the same nuclear potential needs another control epsilonN. These sufficient quantitative reconstructions are explicit inputs; the paper does not supply the selected numerical tolerances.
|rC/2-rN/2| ≤ epsilonN,
|f/D - [-alpha' + k2 rN/2]| ≤ k2(epsilonE+epsilonN). the dimensionless f error has an additional factor D; no exact replacement is inferred (7)
These unweighted overlap bounds do not supply section VII's logarithmically weighted charge-comparison integrals.
The energy-replacement theorem derives the first phase error. The neutralization route requires its own same-potential calculation before combining the two budgets.
coulomb_comparison_yields_modified_effective_range_law · the_neutralized_range_requires_its_own_coulomb_removal_calculation · Physical Review p. 45, equation (52)
Threshold suppression needs noncancellation
The independent threshold route keeps k=1/(D eta) and actual scattering realizations for one nuclear potential. It assumes a finite limit f0, the imported special-function limit g → 2 Euler's constant, and f0 different from that limit. Their gap bounds the isolated nuclear partial-wave strength while C02 tends to zero. Neither a finite-energy fit nor the range replacement is a predecessor.
|fN|2 = 1/[k2+(k cot delta)2] = sin2 delta/k2 → 0. conditional nuclear partial-wave conclusion, not a vanishing total charged cross section
A separate resonant phase family has delta=pi/2, zero f-g, and increasing nuclear strength despite vanishing penetration. It is a diagnostic on-shell family, not a claimed realization by a fixed nuclear potential. Thus barrier suppression alone cannot replace the threshold regularity.
the_nonresonant_threshold_route_suppresses_only_the_nuclear_partial_wave · an_exponentially_small_penetration_factor_does_not_by_itself_suppress_nuclear_scattering · Physical Review p. 45, equation (51a)
Independent public-library use
The nuclear consumer reconstructs the conversion scale D C02 without importing this paper. The public elastic-scattering API gives unitary nuclear amplitudes and an error bound epsilon/(D C02 k2). A checked example has plotted-function difference at most 1/4 but squared amplitude distance 8. This tests observable error transfer, not a nuclear potential reconstruction or the total charged cross section.
|delta f| ≤ epsilon implies |delta aN| ≤ epsilon/(D C02 k2).
At D=1, k=1/4, K1=0, K2=1/4: |delta f| ≤ 1/4 but |delta aN|2=8. The shared g cancels; these unitary on-shell data do not reconstruct a nuclear potential.
Finite proton data support a calibrated range estimate, not an exact continuum law
The plotted quantity is the dimensionless function f in equation (51), computed from each measured nuclear phase shift. The fit uses laboratory energy, not relative kinetic energy. The finite selected data do not establish a continuum error bound or an exact potential range.
f_j=g(eta_j)+2*pi*cot(delta_j)/(exp(2*pi*eta_j)-1).
L(E_lab)=A+B*E_lab; alpha_hat=-A/D; r_hat=4*hbar^2*B/(D*M).
L(E_lab)=D*(-alpha_hat+k^2*r_hat/2).
E_lab>0, k>0, D>0, sin(delta_j)!=0; f_j is dimensionless. The source energy convention supplies the factor four in the slope-to-range conversion. (8)
Two distinct calibration energies constrain the affine model. A disjoint, nonempty finite validation sample tests the same line. Bethe draws it using selected Ragan-Kanne-Taschek and Blair et al. points and checks Herb et al. against it. He separately discusses the off-line Hafstad-Heydenburg-Tuve measurements; the selected-sample claim is not universal agreement of all historical measurements.
Two points in S_cal have different laboratory energies.
|f_j-L(E_j)|<=epsilon_j, epsilon_j>0, for j in S_cal union S_val.
Therefore |f_j-D*(-alpha_hat+k_j^2*r_hat/2)|<=epsilon_j on those same samples. Calibration and validation are distinct data selections, not a claim of statistical independence.
The source reports slope 0.942/MeV with accuracy five percent or better, and range (2.71 +/- 0.13) times 10^-13 cm. The fit errors, slope accuracy, and rounding allowance are explicit inputs, not newly certified experimental uncertainties. Lean derives the calibrated residual and propagation of these bounds; it does not identify the estimate with the exact integral-defined range or recompute the historical measurements.
0<delta_B<=B/20 implies 0<delta_r<=r_hat/20.
|B_star-B|<=delta_B and |r_report-r_hat|<=epsilon_round imply
|C_fit*B_star-r_report|<=delta_r+epsilon_round. A candidate slope and reporting error do not assert an exact physical range.
calibrated_finite_samples_give_range_estimates_and_actual_phase_residuals · established_proton_result_propagates_slope_and_rounding_uncertainties · Physical Review pp. 39, 45-46, equations (4), (51), (53)
Two different higher-energy outcomes
Wilson's comparison is within its reported error; Dearnley's selected point lies above the line beyond its error. Both lie above the selected low-energy samples, but they are distinct experiments. Dearnley's positive departure exceeds the positive square-well expectation. The potential-direction interpretation, exclusion of the proposed P-wave tensor explanation, and unresolved explanation remain separate physical inputs. The code does not infer a nuclear potential shape from a residual sign or certify the source's approximate fourfold ratio.
|R_W|<=epsilon_W; epsilon_D<R_D; 0<R_square<R_D.
E_W and E_D exceed every selected low-energy sample energy.
W and D are distinct experiments; the D discrepancy remains unexplained.
established_proton_result_distinguishes_the_two_high_energy_measurements · established_proton_result_retains_detailed_high_energy_diagnostics · Physical Review pp. 46-47, Figure 4
What the counterexamples actually test
The constructed phase families are on-shell diagnostics, not fixed-potential realizations or historical measurements. At positive energy the actual Coulomb formula reproduces each chosen curve, with nonzero phase sine and a unitary relative S matrix. The continuous control window is 0<E<=1; no zero-energy threshold value is substituted into the singular source formula.
For 0<E<=1, |f_plus-L| and |f_minus-L| are at most 1/10.
Actual second derivatives: f_plus''=1/5 and f_minus''=-1/5.
f_plus(4)-L(4)=8/5>1/10.
P(E)=(E-1/4)^2*(E-1/2)^2*(E-1)^2; f_sample(E)=L(E)+1024*P(E).
f_sample=L at calibration energies 1/4 and 1 and validation energy 1/2,
but f_sample(3/4)-L(3/4)=1.
At M=2, hbar=1, e^2=1, B=1: the laboratory range is 2, not the relative-energy value 1. Actual phase/source-function controls replace independently assigned curvature and low/high-energy residual flags.
finite_fitting_and_validation_do_not_control_an_unsampled_low_energy · low_energy_fit_tolerance_does_not_select_the_actual_curvature_sign · a_controlled_low_energy_line_does_not_supply_higher_energy_agreement · the_measured_laboratory_slope_does_not_have_the_relative_energy_range
The charge readout uses this fitted intercept but requires a separate bound on its difference from the true zero-energy intercept.
The result retains the Coulomb theory, finite empirical setup, derived range-fit residuals, qualified reported estimate, and both high-energy outcomes. The source map's theory predecessor is not silently promoted to an exact fit at unmeasured energies.
proton_data_support_low_energy_line_but_leave_high_energy_discrepancy · Physical Review pp. 46-47
Weighted comparison, controlled readout, and force interpretation are separate deductions
The comparison asks what neutral scattering would result if the proton-proton singlet nuclear potential were left unchanged while Coulomb repulsion were removed. The charged wave w and counterfactual neutral wave u solve actual zero-energy radial equations for that same nuclear potential. Their comparison waves are phi and psi(r)=1-alpha_P*r. The neutron-proton singlet solution is a distinct channel, not the triplet/deuteron solution from the earlier route.
W[u,w](R)-W[u,w](0)=(2/D)*integral_0^R u(r)*w(r)/r dr.
W[psi,phi](R)-W[psi,phi](r1)=(2/D)*integral_r1^R psi(r)*phi(r)/r dr.
integral_r1^infinity (psi*phi-u*w)/r dr=integral_0^r1 u*w/r dr.
B(R)=W[psi,phi](R)-W[u,w](R) tends to zero.
Therefore W[psi,phi](r1)=0. Finite identities, integrability, weighted balance, and the actual boundary limit are distinct inputs. (57-59)
At nonzero psi(r1) and phi(r1), psi'(r1)/psi(r1)=phi'(r1)/phi(r1).
F(r)=-phi'(r)/(phi(r)-r*phi'(r)); alpha_P=F(r1). The weighted Wronskian relation and phi(r1)!=0 imply the readout denominator is nonzero. (59-60)
the_weighted_charge_comparison_precedes_the_empirical_readout · the_linear_comparison_determines_the_equivalent_neutral_length · Original pp. 47-48
The cutoff estimate has its own model and error
r_star=b*exp(-integral_0^b u0(r)^2/r dr).
Yukawa trial reference: u0(r)=1-exp(-beta*r); r_star=2/(beta*exp(gamma_E)).
r_hat=r_star*(1+c), with c>-1 and the source-series remainder retained.
c_square=0.114*alpha_P*b+0.471*alpha_P^2*b^2+R_square.
c_tail=-0.245*s/beta-0.051*s^2/beta^2+R_tail.
|log(r1)-log(r_hat)|<=e_r; |F(r)-F(t)|<=C*|log(r)-log(t)| on a common positive-radius band.
Therefore |F(r1)-F(r_hat)|<=C*e_r. b is the actual square-well intrinsic range, not an interchangeable finite-length effective range. The tail is a supplied trial profile, not an exact Yukawa solution. (62-72a)
The original printed tail ansatz and neutral exterior require care about parameter signs. A global matching claim has the following checked obligation; a restricted-domain approximation would instead require its own domain and error.
u_trial(r)-psi(r)=(s+alpha_P)*r-exp(-beta*r).
For beta>0, u_trial-psi tends to zero at infinity if and only if s=-alpha_P. This preserves the literal plus sign in (72) and minus sign in (60). It does not declare an author-approved erratum or identify one nonzero parameter in both formulas. (60,72)
asymptotic_matching_requires_the_parameter_sign_identification · Original pp. 48-49
Fit, wave, and truncation errors do not disappear on inversion
v=1+r_hat*(q-2/D); s_phi=q+(2*r_hat/D)*(q-4/D).
alpha_first=-(q+(4*r_hat/D)*(q-2/D)).
-s_phi/(v-r_hat*s_phi)=alpha_first+R.
R=r_hat^2*(16/D^3-2*q^2/D-r_hat*(8*q^2/D^2-48*q/D^3+64/D^4))/(v-r_hat*s_phi).
eV=e_v+|r_hat/D|*e_f; eS=e_s+|(1+2*r_hat/D)/D|*e_f.
m>0; m+eV+|r_hat|*eS<=|v-r_hat*s_phi|.
E=C*e_r+(|s_phi|*eV+|v|*eS)/(m*|v-r_hat*s_phi|)+|R|.
|alpha_P-alpha_first|<=E. The true intercept is the same zero-energy Coulomb source. Wave errors are controlled against its expansion; e_f separately controls the fitted intercept. These are reconstructed sufficient bounds, not certified numerical error budgets from the paper. (55-61)
the_charge_estimate_retains_radius_fit_wave_and_truncation_errors · the_printed_first_order_formula_has_an_explicit_rational_remainder
Therefore alpha_P!=alpha_N.
|alpha_P-alpha_N|<=|alpha_first-alpha_N_obs|+E+e_N<=q_small/b_P, 0<q_small<=1/20. The reference inverse range is 1/b_P for the actual selected proton nuclear potential. A non-strict separation margin can still permit equal true lengths. (gap)
Normalized radial admissibility and same-potential readout uniqueness:
alpha_P!=alpha_N implies the nuclear potentials are not identical on all positive radii.
b_P=b_N, P_P=P_N, and |g_P-g_N|<=epsilon_V*g_N imply
|V_P(r)-V_N(r)|<=epsilon_V*|V_N(r)| for every r>0, 0<=epsilon_V<=3/100. The depth bound is a separate cited calculation. Small inverse-length response is not itself a pointwise potential bound. Common range and shape are unestablished inputs, not negated physical facts. (scope)
The controlled estimate, strict calibrated gap, separate singlet assessment, and normalized-scattering uniqueness yield unequal inverse lengths and nonidentical potentials. The cited depth input gives a conditional three-percent pointwise potential bound only with common range and shape. No opaque force-difference or nonestablishment flag supplies this conclusion.
proton_neutron_comparison_finds_small_but_definite_force_difference · established_charge_comparison_retains_robustness_and_conditions · Original pp. 49-50
Reported calculations and author expectation remain distinct
square uncorrected=-0.585; tail uncorrected=-0.517; neutron singlet=-0.421.
square corrected=-0.582; tail corrected=-0.529; corrected gaps=0.161 and 0.108.
The exponential/Gaussian expectation is the uncorrected bracket [-0.585,-0.517].
A bracketed estimate has a gap in [0.096,0.164] from the reported singlet value.
Additional model calibration and total errors below 0.096 times the inverse-length unit preserve a true gap. The extra shapes are actual exponential/Gaussian potential profiles, but their bracket is an author expectation, not a computed or corrected result. The normalized-model conclusion remains conditional on calibration. (reported)
extrapolating_to_other_shapes_retains_the_author_expectation_and_error_budget · the_source_shape_bracket_has_a_gap_before_finite_length_corrections · Original p. 49
Independent use of the public scattering library
|delta_alpha|<=e_alpha and |delta_r0|<=e_range imply
|delta_f|<=(e_alpha+(k^2/2)*e_range)/k^2.
At k=1, r0=0, alpha=1 and alpha=-1 give different amplitudes (-1+i)/2 and (1+i)/2.
Both have the same S-wave cross section 4*pi*|f|^2=2*pi. The source-only consumer imports public frameworks, not Bethe or production countermodels. These are dimensionless on-shell tests of the two-parameter response, not fixed-potential reconstructions or empirical fits. (reuse)
The complete result preserves 38 logical boundaries and two deferred extensions
| Available result | Not thereby established | Formal witness |
|---|---|---|
| Actual common-potential radial equations and both finite-radius Wronskian identities. | The asymptotic boundary limit required for the infinite comparison. | actual_free_radial_equations_do_not_supply_the_boundary_limit |
| An exact energy-dependent comparison and controlled range replacement. | Exact agreement of the physical and two-parameter phase functions. | controlled_replacement_does_not_erase_the_actual_phase_remainder |
| A continuous range function agreeing with its reference at zero energy. | A uniform quartic phase-error bound. | continuity_and_zero_energy_agreement_do_not_supply_quartic_accuracy |
| A named neutral cross-section calculation with positive wave-number and nonnegative cross-section data. | Bethe's printed equation-(17) value formula. | named_neutral_cross_section_does_not_supply_equation_seventeen_formula |
| The exact equation-(17) equality at zero wave number under totalized division. | A physical positive-wave-number and positive-cross-section scattering domain. | equation_seventeen_without_positive_wave_number_is_not_physical |
| A positive cross section satisfying equation (17) with a nonzero phase function. | Saturation of the numerical 4 pi / k squared unitarity maximum. | equation_seventeen_without_zero_phase_does_not_reach_maximum |
| The exact positive-domain denominator formula with the physical k cot delta. | The sine-squared equation when the retained k squared is not k times k. | exact_phase_function_without_wave_square_does_not_supply_sine_squared_formula |
| The exact positive-domain denominator formula with genuinely squared k. | The sine-squared equation when the phase function is not the selected k cot delta. | exact_wave_square_without_phase_readout_does_not_supply_sine_squared_formula |
| A bare deuteron bound-state label. | Bethe's exact equation-(19) relation. | bound_state_label_does_not_supply_equation_nineteen |
| A nonzero normalized comparison that decays at infinity. | Regularity at the origin of that extrapolated comparison. | normalized_decaying_comparison_is_not_a_regular_interior_wave |
| The actual positive Coulomb function at a pi/2 nuclear phase. | Equality with a penetration-only cotangent term. | omitting_the_additive_coulomb_function_changes_the_phase |
| The dimensionless source plot at D=2. | Equality with its inverse-length reduction f/D. | the_dimensionless_plot_is_not_the_inverse_length_phase |
| Unit leading comparisons with a finite same-radius derivative difference. | A finite individual comparison derivative at the origin. | each_actual_leading_derivative_diverges_but_their_difference_is_finite |
| A controlled error at an actual Coulomb phase readout. | Exact equality with its two-parameter approximation. | a_controlled_charged_phase_error_need_not_be_zero |
| Exact energy replacement and separately controlled neutralization. | A zero error budget after Coulomb removal. | exact_energy_replacement_does_not_remove_the_neutralization_error |
| Vanishing Coulomb penetration and admissible resonant phase data. | Vanishing nuclear partial-wave strength without noncancellation. | an_exponentially_small_penetration_factor_does_not_by_itself_suppress_nuclear_scattering |
| An exact radial comparison identity. | The controlled low-energy interior-wave replacement. | exact_comparison_identity_does_not_supply_low_energy_replacement |
| An accurate two-parameter phase law. | The detailed potential shape. | accurate_two_parameter_law_does_not_determine_potential_shape |
| Physical phases fit finite calibration and validation data. | A uniform error bound between those samples. | finite_fitting_and_validation_do_not_control_an_unsampled_low_energy |
| Actual Coulomb curves satisfy the same low-energy tolerance. | A definite positive or negative curvature sign. | low_energy_fit_tolerance_does_not_select_the_actual_curvature_sign |
| An actual source curve fits throughout a positive low-energy interval. | Agreement at a higher energy. | a_controlled_low_energy_line_does_not_supply_higher_energy_agreement |
| Finite measured phases and the laboratory-energy slope. | The relative-energy slope-to-range conversion. | the_measured_laboratory_slope_does_not_have_the_relative_energy_range |
| A completed proton-side charge calculation. | The separate neutron–proton singlet-range qualification. | proton_charge_calculation_does_not_supply_singlet_range_qualification |
| Approximate equality of the two singlet forces. | The separate borrowing of the proton–proton effective range. | approximate_singlet_force_equality_does_not_supply_borrowed_range |
| Charge independence as a good approximation. | Exact equality of the proton–proton and neutron–proton interactions. | approximate_charge_independence_does_not_establish_exact_equality |
| A controlled two-parameter phase law at the selected energy. | The observable cross-section calculation. | a_phase_law_does_not_supply_the_observable_cross_section_calculation |
| The S-wave unitarity bound at the maximum. | Control of the same low-order cross-section truncation there. | a_unitarity_bound_does_not_license_the_low_order_truncation_at_its_maximum |
| Long-tail compatibility with a common intrinsic range. | Selection of that shape from one channel. | cross_channel_range_compatibility_does_not_select_a_shape_from_one_channel |
| Direction, square-well excess, and tensor-force exclusion. | An explanation of the high-energy discrepancy. | detailed_high_energy_diagnostics_can_leave_the_discrepancy_unexplained |
| A continuous origin-vanishing overlap error with a small global unweighted integral. | The same bound for its inverse-radius weighted integral. | a_global_unweighted_error_bound_does_not_supply_the_same_coulomb_weighted_bound |
| Normalized smooth curves with small value/slope differences and nonzero readout denominators. | A small reciprocal-length error without a sufficient denominator margin. | small_wave_errors_need_not_give_a_small_inverse_length_error_near_a_pole |
| The source's first-order expression and its actual rational readout. | Zero truncation remainder. | the_source_first_order_truncation_can_have_a_nonzero_remainder |
| A sign-reparameterized tail with the required neutral asymptote. | Global matching with the same nonzero parameter in both literal sign conventions. | reparameterizing_the_tail_sign_matches_but_literal_parameter_reuse_does_not |
| A nonzero reported gap and calibrated errors satisfying a non-strict margin. | Unequal true inverse lengths. | a_nonzero_reported_gap_with_only_nonstrict_error_separation_can_hide_equal_lengths |
| A value inside the original uncorrected shape bracket. | Membership in the corrected bracket. | the_uncorrected_shape_bracket_does_not_supply_the_corrected_bracket |
| Physical square wells with equal depth and shape. | A pointwise relative potential bound when their ranges differ. | equal_depth_and_shape_do_not_control_potentials_with_different_ranges |
| Physical potentials with equal depth and range. | A pointwise relative potential bound when a square profile is replaced by a Gaussian. | equal_depth_and_range_do_not_control_potentials_with_different_shapes |
The 38 boundaries retain physical domains and source-shaped distinctions. Radial controls use actual solutions, derivatives, and integrals. Charged controls distinguish dimensionless and reduced functions, additive Coulomb correction, divergent origin derivatives, separate approximation budgets, and threshold regularity. The weighted-error and readout diagnostics are not asserted solutions of one fixed nuclear potential. Actual square and Gaussian profiles replace independently assigned range/shape-mismatch flags. The cross-section and bound-state witnesses retain their actual domains and normalization. Empirical controls use actual Coulomb phases, derivatives, finite samples, and computed low/high-energy residuals. No constant proposition, enum truth table, or assumed countermodel conclusion is used.
For a=1/120 the unweighted 1/120 bound holds and the weighted one fails.
phi_e(r)=1+(1-e)*r^2: at r=1, e=1/100 versus 1/50 changes value by 1/100 and slope by 1/50, but F by 100.
At D=2, r_hat=1/10, q=0: the rational readout is 5/23 while alpha_first=1/5. Dimensionless, explicitly constructed norm and inversion diagnostics. They test missing controls without claiming a fixed-potential scattering realization. (test)
The cross-section witnesses keep the positive equation-(17) denominator, distinguish exact nonzero-phase scattering from unitarity saturation, and require both the physical wave-number square and same-energy phase readout for the sine-squared formula.
Established
Exact neutral and Coulomb comparison identities, controlled low-energy laws, the observable unitarity-scoped cross section, accuracy assessment, triplet and proton range extraction, and the qualified charge comparison.
Explicitly not established
Potential-shape reconstruction, accurate neutron–proton singlet range from the selected data, high-energy agreement, exact charge independence, or equality of intrinsic range and shape.
Deferred
The tensor-force extension and the deuteron photoelectric and photomagnetic extension are reserved for subsequent work.
The final theorem conjoins twelve physical results, including the weighted comparison and estimate inside the charge result, with both deferred extensions. It neither drops the open high-energy discrepancy nor upgrades controlled, imported, interpretive, or author-expectation inputs.
independent_routes_establish_complete_scoped_result · Physical Review pp. 38–50
Linked concepts and notation
Linked terms lead to their controlling paper-local definition, theorem, or equation. Exact, approximate, empirical, interpretive, open, and deferred status are retained.
Exact neutral comparison
The boundary–overlap identity obtained by subtracting finite-range radial equations and their matched comparison equations.
ExactNeutralComparisonIdentityAt
Energy-dependent range
The exact range function appearing in the zero-energy comparison relation before low-energy replacement.
NeutralZeroEnergyRelationAt
Neutral effective-range law
The two-parameter scattering-length and constant-effective-range phase law obtained under controlled replacement.
NeutralEffectiveRangeLawAt
Zero-energy interior replacement
The controlled low-energy approximation replacing finite-energy wavefunctions inside the force range by zero-energy waves.
EffectiveRangeReferenceCalculation
Unitarity-scoped neutral cross section
The separate observable calculation, unitarity bound, vanishing-phase maximum, and near-maximum truncation warning.
EstablishedNeutralCrossSectionAssessment
Higher-order shape correction
The first calculated potential-shape-dependent term, whose small coefficient supports two-parameter accuracy.
HigherOrderCorrectionCalculation
Potential-shape scope
Curvature signs and triplet/singlet intrinsic-range compatibility distinguish concentrated and long-tailed possibilities, but no single channel resolves the detailed shape.
CurvatureDiagnosticInterpretation · ShapeSensitivityInterpretation
Triplet bound-state extraction
The use of deuteron binding and low-energy triplet scattering data, under deferred tensor-force scope, to extract the named qualified effective-range estimate.
TripletExtractionCalculation
Singlet range qualification
The failure of neutron–proton singlet data to determine the range accurately, followed by distinct assumptions of approximate force equality and a borrowed proton–proton effective range.
SingletRangeAssessment
Coulomb-modified effective-range law
The charged-channel two-parameter law derived from Coulomb comparison waves and controlled interior replacement.
CoulombEffectiveRangeLawAt
Low-energy proton fit
The empirical straight line of the Coulomb-modified phase function and its extracted proton–proton effective range.
ProtonProtonFitCalculation
Open high-energy discrepancy
The selected data depart in the concentrated-potential direction beyond the square-well expectation; the tensor-force account fails and the discrepancy remains unexplained.
HighEnergyAssessment
Same-potential counterfactual
The neutral scattering comparison obtained by keeping the proton nuclear potential while removing Coulomb effects.
ChargeComparisonSetup
Qualified charge dependence
A calibrated nonzero inverse-length gap excludes identical potentials only with normalized radial admissibility and uniqueness. Pointwise closeness requires a separate depth calculation and common range and shape. The additional-shape bracket remains an author expectation.
ChargeDependenceInterpretation
Deferred extensions
Tensor-force and deuteron photoelectric/photomagnetic calculations explicitly left to subsequent work.
PaperScopeQualifications
Theory/model/world grounding
Shared low-energy reduction and bound-state extraction models grounded separately in Bethe's nuclear regime and data-bearing system; the reduction grounding first requires the paper-local S-wave, finite-range, selected-low-energy setup.
shared_grounding_setup_selects_low_energy_effective_range_setup · toEffectiveRangeBoundStateExtractionInterpretation
Route inventory and formal provenance
The source map declares ten routes, thirty-eight semantic atoms, and
thirty-eight sourced steps.
Coverage interprets every atom and certifies each route from its
declared dependencies. For every nontrivial step it recovers the
exact selected atoms from ClaimsHold before applying the
paper-local theorem; route certificates witness dependency
availability only.
| Route | Dependencies | Principal output | Profile |
|---|---|---|---|
| Neutral comparison | — | Equations (7), (10), (11), (13)-(14), exact finite-range identity, and zero-energy relation | physical setup and exact derivation |
| Effective-range reduction | Neutral comparison | Neutral two-parameter law | controlled low-energy reduction |
| Neutral cross section | Effective-range reduction | Observable formula, unitarity maximum, truncation warning | calculation and interpretation |
| Accuracy and shape | Effective-range reduction | Accurate approximation with unresolved shape | calculation, interpretation, controlled assessment |
| Triplet extraction | Effective-range reduction | Reported triplet effective range | imported evidence and bound-state calculation |
| Singlet range qualification | — | Limited data plus distinct force-equality and borrowed-range assumptions | imported limitation and interpretive borrowing |
| Coulomb comparison | None | Actual basis, relative phase, and dimensionless source function | exact conditional comparison |
| Coulomb theory | Coulomb comparison | Modified law with charged energy-replacement error | controlled result |
| Coulomb neutralization | Coulomb comparison; Coulomb theory | Separate same-potential range error and combined phase budget | independent Coulomb-removal control |
| Coulomb threshold | None | Conditional zero-momentum nuclear partial-wave suppression | physical and imported regularity inputs |
| Proton experiment | Coulomb theory | Low-energy range fit with high-energy discrepancy | imported evidence, fit, open assessment |
| Charge comparison | Proton experiment; singlet range qualification | Small but definite qualified force difference | controlled counterfactual and interpretation |
| Complete scoped result | All twelve preceding routes | Eleven physical results and deferred extensions | logical synthesis retaining deferred scope |
Neutral route · neutral_comparison_route_is_semantically_realized
Reduction route · effective_range_reduction_route_is_semantically_realized
Cross-section route · neutral_cross_section_route_is_semantically_realized
Accuracy route · accuracy_and_shape_route_is_semantically_realized
Triplet route · triplet_extraction_route_is_semantically_realized
Singlet route · singlet_range_qualification_route_is_semantically_realized
Coulomb comparison · coulomb_comparison_route_is_semantically_realized
Coulomb range · coulomb_theory_route_is_semantically_realized
Coulomb neutralization · coulomb_neutralization_route_is_semantically_realized
Coulomb threshold · coulomb_threshold_route_is_semantically_realized
Proton route · proton_experiment_route_is_semantically_realized
Charge route · charge_comparison_route_is_semantically_realized
Complete route · complete_scoped_result_route_is_semantically_realized
Exhaustive coverage and inherited status
route_semantic_realization_is_exhaustive covers all thirteen
identifiers. Exact graph, independent-contract, result-application,
local-step, inherited-step, and premise-consumption guards
preserve both physical trunks. The cross-section, accuracy, and
triplet routes are checked as sibling descendants of the reduction,
while
singlet_borrowing_route_keeps_limitation_and_borrowing_distinct
prevents the p. 41 assumptions from collapsing. Dedicated
source-map audits retain the identification bridge and controlled
status in the neutral law,
neutral_result_depends_on_equations_7_10_11_13_and_14
keeps the early radial formula readout on the exact neutral route,
the cross-section calculation and
truncation warning, shape interpretation, imported empirical
ancestry in the proton result, open high-energy status, controlled
and interpretive ancestry in the charge result, and deferred scope
in the complete conclusion.
Policy rejects certificate-output replay, atom-output shortcuts,
and premise-ignoring step certification in this coverage file, and checks that
every result route invokes its paper-local bridge after recovering
every selected claim from ClaimsHold.
Lean-encoded bibliographic record
H. A. Bethe, Theory of the Effective Range in Nuclear
Scattering, Physical Review 76 (1949), 38–50. All page, equation,
figure, and provenance locators above are stored in
SourceMap.lean.
TraceComplete Lean semantic identifier ledger
Checked paper-local theorems: the_integral_radial_equation_fixes_its_origin_value, the_regular_actual_waves_and_normalized_comparison_waves_have_distinct_origins, the_finite_radius_comparison_retains_the_boundary_remainder, the_actual_boundary_limit_yields_bethes_comparison_identity, the_replacement_integrand_is_the_actual_overlap_difference, the_actual_phase_residual_is_the_range_replacement_error, controlled_range_replacement_gives_a_phase_error_bound, exact_phase_agreement_at_nonzero_energy_requires_exact_range_agreement, the_actual_integral_error_bounds_the_range_replacement, a_quadratic_range_error_gives_a_quartic_phase_error, uniform_quadratic_range_control_derives_quartic_accuracy, uniform_radial_inputs_realize_shared_effective_range_expansion, the_phase_error_is_a_controlled_observable_approximation, the_weighted_sine_force_has_the_actual_integral, the_weighted_cosine_force_has_the_actual_integral, the_unmatched_free_waves_satisfy_actual_radial_equations, the_four_free_derivatives_are_actual, the_unmatched_boundary_is_not_an_independent_scalar, the_regular_wave_product_has_the_actual_integral, the_unmatched_free_waves_satisfy_both_finite_wronskian_identities, actual_free_radial_equations_do_not_supply_the_boundary_limit, the_nonzero_shape_remainder_obeys_the_exact_range_comparison, the_nonzero_shape_remainder_has_controlled_range_replacement, controlled_replacement_does_not_erase_the_actual_phase_remainder, a_continuous_range_can_approach_its_reference_only_linearly, continuity_and_zero_energy_agreement_do_not_supply_quartic_accuracy, equation_thirty_eight_retains_the_two_proton_reduced_mass, positive_proton_parameters_give_a_positive_bohr_radius, the_scaled_coulomb_potential_is_the_printed_radial_term, the_leading_comparison_has_the_actual_logarithmic_derivative, the_common_logarithmic_derivative_has_no_finite_origin_limit, each_actual_leading_derivative_diverges_but_their_difference_is_finite, the_source_series_is_summable, the_repulsive_coulomb_factor_is_positive_and_below_one, equation_forty_three_has_the_actual_positive_normalization, the_dimensionless_source_function_has_the_printed_coulomb_coefficient, omitting_the_additive_coulomb_function_changes_the_phase, the_dimensionless_plot_is_not_the_inverse_length_phase, a_controlled_charged_phase_error_need_not_be_zero, exact_energy_replacement_does_not_remove_the_neutralization_error, the_repulsive_penetration_factor_vanishes_at_large_eta, the_positive_radius_integral_equation_uses_the_actual_wave_derivative, the_regular_irregular_combination_has_an_actual_unit_origin_limit, the_actual_coulomb_basis_identifies_the_relative_nuclear_phase, the_same_radius_logarithm_cancels_before_the_origin_limit, actual_derivative_comparison_derives_the_charged_range_identity, the_actual_charged_overlap_control_bounds_the_effective_range_replacement, the_charged_range_replacement_controls_the_phase_instead_of_making_it_exact, the_dimensionless_plot_retains_the_bohr_radius_in_its_error_budget, neutralizing_the_same_nuclear_potential_has_its_own_range_error, energy_replacement_and_coulomb_removal_have_separate_error_budgets, a_realized_threshold_point_has_the_actual_relative_coulomb_phase, large_coulomb_parameter_is_an_actual_zero_momentum_limit, a_noncancelling_source_intercept_supplies_the_uniform_gap, the_charged_readout_recovers_the_actual_nuclear_phase_function, the_nuclear_partial_wave_strength_is_sine_squared_over_momentum_squared, a_uniform_gap_bounds_the_nuclear_partial_wave_by_the_penetration_factor, the_nonresonant_source_gap_makes_the_nuclear_contribution_vanish, the_resonant_diagnostic_has_actual_charged_phase_and_partial_wave_strength, an_exponentially_small_penetration_factor_does_not_by_itself_suppress_nuclear_scattering, the_same_potential_regular_waves_have_zero_origin_wronskian, the_finite_weighted_comparison_retains_its_boundary_difference, weighted_balance_and_the_actual_boundary_limit_give_zero_comparison_wronskian, the_neutral_comparison_derivative_is_the_negative_reciprocal_length, a_zero_wronskian_gives_logarithmic_slope_agreement_only_at_nonzero_values, the_linear_comparison_determines_the_equivalent_neutral_length, the_source_weighted_comparison_determines_the_equivalent_neutral_length, actual_wave_errors_control_the_readout_denominator, the_reciprocal_length_difference_uses_both_actual_denominators, controlled_wave_and_slope_errors_bound_the_reciprocal_length, the_printed_first_order_formula_has_an_explicit_rational_remainder, changing_only_the_intercept_has_calculated_value_and_slope_effects, the_controlled_wave_readout_retains_the_first_order_remainder, intercept_calibration_has_distinct_value_and_slope_error_factors, expansion_and_intercept_errors_are_added_before_the_inverse_readout, the_printed_plus_profile_and_minus_exterior_have_an_actual_linear_difference, asymptotic_matching_requires_the_parameter_sign_identification, one_nonzero_parameter_cannot_be_used_in_both_literal_asymptotic_conventions, a_depth_comparison_controls_the_actual_potential_only_with_common_range_and_shape, equal_depth_and_shape_do_not_control_potentials_with_different_ranges, equal_depth_and_range_do_not_control_potentials_with_different_shapes, a_common_shape_realizes_nonzero_but_controlled_potential_difference, different_actual_scattering_lengths_exclude_equal_nuclear_potentials, estimate_and_measurement_errors_require_a_strict_separation_margin, an_observed_length_difference_and_both_errors_bound_the_true_difference, both_reported_corrected_estimates_have_a_small_nonzero_inverse_length_gap, the_source_shape_bracket_has_a_gap_before_finite_length_corrections, any_bracketed_estimate_retains_the_reported_gap, the_source_radius_estimate_is_positive, Data.a_controlled_log_radius_error_has_its_own_reciprocal_length_error, the_diagnostic_slope_is_the_derivative_of_the_normalized_curve, small_wave_errors_need_not_give_a_small_inverse_length_error_near_a_pole, the_source_first_order_truncation_can_have_a_nonzero_remainder, reparameterizing_the_tail_sign_matches_but_literal_parameter_reuse_does_not, a_nonzero_reported_gap_with_only_nonstrict_error_separation_can_hide_equal_lengths, the_uncorrected_shape_bracket_does_not_supply_the_corrected_bracket, the_weight_is_the_actual_inverse_radius, the_overlap_error_is_smooth_and_nonnegative, overlap_primitive_derivative, weighted_primitive_derivative, the_actual_unweighted_integral_is_linear_in_the_radius, the_actual_weighted_integral_is_independent_of_the_radius, arbitrarily_small_unweighted_errors_do_not_give_small_origin_weighted_errors, the_same_unweighted_tolerance_fails_for_a_smooth_origin_regular_profile, the_zero_extension_has_the_actual_positive_axis_integral, the_extended_weight_is_the_actual_inverse_radius, the_zero_extension_is_continuous_and_vanishes_at_its_endpoints, the_zero_extension_and_its_weight_are_nonnegative, the_compact_profile_has_distinct_global_unweighted_and_weighted_norms, a_global_unweighted_error_bound_does_not_supply_the_same_coulomb_weighted_bound, the_laboratory_energy_range_conversion_is_positive, laboratory_energy_calibration_derives_the_effective_range_fit, calibrated_finite_samples_give_range_estimates_and_actual_phase_residuals, slope_uncertainty_propagates_through_the_actual_range_conversion, a_reported_range_retains_separate_slope_and_rounding_errors, positive_laboratory_energy_gives_positive_momentum, the_momentum_has_the_source_laboratory_energy, the_constructed_relative_phase_has_nonzero_sine, the_constructed_phase_realizes_the_actual_coulomb_plot, the_source_curve_is_realized_at_each_positive_energy, the_constructed_phase_matrix_has_unit_norm, positive_energy_realization_retains_the_physical_phase_and_unitarity, the_curved_plot_has_its_actual_first_derivative, the_actual_source_curve_has_the_fitted_derivative, the_actual_source_curve_has_the_quadratic_curvature, each_curvature_sign_is_compatible_with_a_controlled_low_energy_line, a_controlled_low_energy_line_does_not_supply_higher_energy_agreement, low_energy_fit_tolerance_does_not_select_the_actual_curvature_sign, the_control_has_positive_parameters_and_an_actual_laboratory_energy, each_control_curve_has_positive_energy_phase_calibration, each_positive_energy_sample_has_an_actual_phase_and_laboratory_calibration, the_control_sample_is_finite_physical_and_has_held_out_validation, matching_the_three_sample_values_gives_the_finite_fit, finite_fitting_and_validation_do_not_control_an_unsampled_low_energy, the_measured_laboratory_slope_does_not_have_the_relative_energy_range, equation_seven_normalizes_comparison_wave_at_origin, equation_seven_retains_k_cotangent_phase_function, equation_seven_gives_the_actual_comparison_origin_derivative, the_zero_energy_comparison_has_the_actual_scattering_length_slope, lower_radial_inputs_derive_the_infinite_comparison_identity, lower_radial_inputs_derive_the_source_formula_readout, equations_ten_eleven_and_thirteen_yield_zero_energy_relation, radial_comparison_and_formula_readout_yield_exact_neutral_identity, radial_comparison_yields_exact_neutral_identity, the_actual_interior_integral_control_derives_the_range_replacement, the_exact_neutral_comparison_and_controlled_replacement_bound_the_phase_error, controlled_zero_energy_replacement_yields_effective_range_law, the_effective_range_output_is_a_controlled_observable_approximation, the_selected_phase_error_is_quartic_under_quadratic_replacement_control, equation_seventeen_coherent_phase_readout_yields_sine_squared_formula, coherent_phase_function_vanishes_iff_phase_shift_cosine_vanishes, equation_seventeen_at_positive_wave_number_implies_unitarity_bound, equation_seventeen_and_zero_phase_saturate_unitarity_maximum, equation_seventeen_saturates_unitarity_maximum_iff_phase_vanishes, equation_seventeen_saturates_unitarity_maximum_iff_phase_shift_cosine_vanishes, equation_seventeen_nonzero_phase_is_strictly_below_unitarity_maximum, effective_range_phase_law_yields_unitarity_scoped_cross_section, established_neutral_cross_section_retains_equation_seventeen_formula, established_neutral_cross_section_retains_numerical_bound_and_maximum, established_neutral_cross_section_characterizes_saturation_by_zero_phase, established_neutral_cross_section_retains_both_equation_seventeen_forms, established_neutral_cross_section_characterizes_physical_phase_shift_maximum, established_neutral_cross_section_assessment_exposes_complete_route, small_shape_correction_supports_accuracy_not_shape_determination, accuracy_scope_retains_sign_sensitive_but_unresolved_curvature, accuracy_scope_retains_cross_channel_intrinsic_range_distinctions, bound_radial_comparison_has_nonvacuous_decaying_mode, bound_comparison_retains_normalization_and_actual_origin_derivative, bound_radial_comparison_has_negative_wave_number_square, bound_comparison_identity_derives_equation_nineteen, equation_nineteen_exposes_the_range_replacement_residual, equation_nineteen_range_replacement_yields_effective_range_form, limited_neutron_proton_singlet_data_retain_borrowed_range_assumption, singlet_range_assessment_retains_force_equality_and_borrowed_range, deuteron_and_scattering_data_extract_triplet_effective_range, triplet_extraction_retains_actual_bound_state_asymptotics, established_triplet_extraction_exposes_complete_route, established_triplet_extraction_retains_bound_state_relation_control, the_charged_scattering_point_is_distinct_from_its_zero_energy_reference, the_charged_data_retain_the_zero_energy_intercept_definition, actual_coulomb_comparison_inputs_derive_the_exact_boundary_identity, actual_coulomb_comparison_inputs_recover_the_printed_source_function, the_coulomb_source_formula_is_established_before_range_replacement, coulomb_comparison_yields_modified_effective_range_law, established_coulomb_comparison_and_range_control_yield_the_modified_law, the_neutralized_range_requires_its_own_coulomb_removal_calculation, the_charged_law_and_independent_neutralization_control_give_the_combined_error, the_nonresonant_threshold_route_suppresses_only_the_nuclear_partial_wave, proton_data_support_low_energy_line_but_leave_high_energy_discrepancy, established_proton_result_retains_finite_data_and_the_derived_range_fit, established_proton_result_distinguishes_the_two_high_energy_measurements, established_proton_result_propagates_slope_and_rounding_uncertainties, established_proton_result_retains_detailed_high_energy_diagnostics, the_weighted_charge_comparison_precedes_the_empirical_readout, the_charge_readout_uses_the_same_zero_energy_coulomb_source, the_charge_estimate_retains_radius_fit_wave_and_truncation_errors, proton_neutron_comparison_finds_small_but_definite_force_difference, established_charge_comparison_retains_robustness_and_conditions, extrapolating_to_other_shapes_retains_the_author_expectation_and_error_budget, effective_range_reduction_model_retains_paper_semantics, shared_grounding_setup_selects_low_energy_effective_range_setup, shared_low_energy_setup_recovers_neutral_reduction_inputs, shared_established_two_parameter_law_recovers_neutral_effective_range_law, neutral_effective_range_law_establishes_shared_two_parameter_law, neutral_effective_range_routes_make_shared_route_applicable, bethe_source_inputs_make_shared_reduction_applicable, bethe_source_inputs_realize_structural_effective_range_reduction_explanation, bethe_source_derivation_makes_shared_reduction_applicable, bethe_source_inputs_realize_source_structural_effective_range_reduction_explanation, bethe_source_inputs_realize_source_effective_range_derivational_reduction, neutral_effective_range_routes_realize_derivational_reduction, neutral_effective_range_routes_establish_shared_reduction, neutral_effective_range_routes_exhibit_nonvacuous_shared_reduction, neutral_effective_range_routes_realize_controlled_truncation, established_bethe_reduction_realizes_every_shared_claim, bethe_effective_range_model_inputs_establish_shared_grounding, neutral_effective_range_routes_establish_grounded_shared_reduction, grounded_bethe_reduction_transfers_every_shared_claim, effective_range_bound_state_extraction_model_retains_paper_semantics, shared_bound_state_grounding_setup_selects_extraction_setup, triplet_routes_make_shared_bound_state_extraction_applicable, bethe_bound_state_source_inputs_make_shared_extraction_applicable, bethe_bound_state_source_inputs_realize_source_structural_extraction_explanation, bethe_comparison_ground_state_and_triplet_evidence_realize_source_sensitive_bound_state_extraction, triplet_routes_establish_shared_bound_state_extraction, triplet_routes_exhibit_nonvacuous_shared_bound_state_extraction, triplet_routes_realize_qualified_parameter_evidence_support, established_bethe_bound_state_extraction_realizes_every_shared_claim, bethe_bound_state_model_inputs_establish_shared_grounding, triplet_routes_establish_grounded_shared_bound_state_extraction, grounded_bethe_bound_state_extraction_transfers_every_shared_claim, independent_routes_establish_complete_scoped_result, exact_phase_function_without_wave_square_does_not_supply_sine_squared_formula, exact_wave_square_without_phase_readout_does_not_supply_sine_squared_formula, equation_seventeen_without_positive_wave_number_is_not_physical, equation_seventeen_without_zero_phase_does_not_reach_maximum, named_neutral_cross_section_does_not_supply_equation_seventeen_formula, bound_state_label_does_not_supply_equation_nineteen, exact_comparison_identity_does_not_supply_low_energy_replacement, accurate_two_parameter_law_does_not_determine_potential_shape, proton_charge_calculation_does_not_supply_singlet_range_qualification, approximate_singlet_force_equality_does_not_supply_borrowed_range, approximate_charge_independence_does_not_establish_exact_equality, a_phase_law_does_not_supply_the_observable_cross_section_calculation, a_unitarity_bound_does_not_license_the_low_order_truncation_at_its_maximum, cross_channel_range_compatibility_does_not_select_a_shape_from_one_channel, detailed_high_energy_diagnostics_can_leave_the_discrepancy_unexplained.
Checked route-realization theorems: every_source_claim_has_semantics, neutral_comparison_route_is_semantically_realized, effective_range_reduction_route_is_semantically_realized, neutral_cross_section_route_is_semantically_realized, accuracy_and_shape_route_is_semantically_realized, triplet_extraction_route_is_semantically_realized, singlet_range_qualification_route_is_semantically_realized, coulomb_comparison_route_is_semantically_realized, coulomb_theory_route_is_semantically_realized, coulomb_neutralization_route_is_semantically_realized, coulomb_threshold_route_is_semantically_realized, proton_experiment_route_is_semantically_realized, weighted_charge_route_is_semantically_realized, charge_estimate_route_is_semantically_realized, charge_comparison_route_is_semantically_realized, conditional_charge_shape_route_is_semantically_realized, complete_scoped_result_route_is_semantically_realized, all_routes_are_semantically_realizable, source_inputs_derive_the_actual_radial_comparison, source_interior_control_derives_the_physical_phase_error.
Checked source-map theorems: argument_wellFormed, argument_auditable, routes_wellFormed, routes_have_exact_predecessor_graph, routes_have_exact_dependency_closure, routes_expose_declared_dependency_outputs, local_setup_steps_do_not_claim_unearned_predecessors, dependent_steps_name_inherited_outputs, calculation_and_interpretation_contracts_are_independent_hypotheses, result_steps_apply_selected_setups_and_contracts, neutral_result_depends_on_equations_7_10_11_13_and_14, cross_section_accuracy_and_triplet_are_siblings_after_reduction, singlet_borrowing_route_keeps_limitation_and_borrowing_distinct, accuracy_route_keeps_curvature_diagnostic_distinct, dependent_routes_consume_certified_predecessor_outputs, effective_range_result_retains_approximate_status, effective_range_result_retains_identification_bridge, cross_section_result_retains_calculation_and_truncation_warning, cross_section_route_exposes_equation_seventeen_formula, accuracy_result_retains_shape_interpretation, the_exact_coulomb_formula_does_not_use_either_range_approximation, neutralization_retains_two_independent_approximation_inputs, threshold_regularity_is_not_an_empirical_fit_or_range_replacement, proton_result_retains_empirical_status, high_energy_assessment_retains_open_status, charge_result_retains_approximate_and_interpretive_status, charge_route_uses_singlet_qualification_without_triplet_extraction, weighted_charge_comparison_is_independent_of_fits_and_unweighted_range_control, charge_estimation_and_force_interpretation_retain_separate_inputs, shape_extrapolation_is_conditional_on_an_author_expectation_not_a_computed_model, complete_result_retains_deferred_scope, complete_result_relies_on_every_substantive_route.
Semantic atoms: neutralComparisonSetup, neutralComparisonCalculation, neutralRadialFormulaCalculation, neutralRadialFormulaReadout, exactNeutralComparison, effectiveRangeApproximationSetup, effectiveRangeReductionCalculation, effectiveRangeIdentificationBridge, neutralEffectiveRangeLaw, neutralCrossSectionSetup, neutralCrossSectionCalculation, neutralCrossSectionTruncationInterpretation, neutralCrossSectionAssessment, equationSeventeenCrossSectionFormula, higherOrderCalculation, curvatureDiagnosticInterpretation, shapeSensitivityInterpretation, accuracyAndShapeScope, tripletExtractionSetup, tripletExtractionCalculation, singletRangeDataLimitation, singletRangeBorrowingBridge, singletRangeAssessment, tripletExtractionResult, coulombComparisonSetup, coulombComparisonCalculation, coulombSourceFormula, coulombRangeReplacementCalculation, coulombIdentificationBridge, coulombEffectiveRangeLaw, coulombNeutralizationCalculation, coulombNeutralizationResult, coulombThresholdSetup, coulombThresholdRegularity, coulombThresholdResult, protonEmpiricalSetup, protonFitCalculation, highEnergyAssessment, protonEmpiricalResult, chargeComparisonSetup, chargeWeightedCalculation, chargeWeightedResult, chargeReadoutCalculation, chargeEstimateResult, chargeComparisonCalculation, chargeDependenceInterpretation, chargeComparisonResult, chargeShapeExpectation, chargeShapeErrorBudget, chargeShapeResult, paperScopeQualifications, completeScopedResult.
Source claims: neutralSetup, neutralCalculation, neutralFormulaCalculation, neutralFormulaReadout, neutralResult, approximationSetup, reductionCalculation, effectiveRangeIdentification, effectiveRangeResult, crossSectionSetup, crossSectionCalculation, crossSectionInterpretation, crossSectionResult, crossSectionFormula, correctionCalculation, curvatureDiagnostic, shapeInterpretation, accuracyResult, tripletSetup, tripletCalculation, singletLimitation, singletBorrowing, singletAssessment, tripletResult, coulombSetup, coulombCalculation, coulombSourceFormula, coulombRangeReplacement, coulombIdentification, coulombResult, coulombNeutralizationCalculation, coulombNeutralizationResult, coulombThresholdSetup, coulombThresholdRegularity, coulombThresholdResult, protonSetup, protonFit, highEnergyAssessment, protonResult, chargeSetup, chargeWeightedCalculation, chargeWeightedResult, chargeReadoutCalculation, chargeEstimateResult, chargeCalculation, chargeInterpretation, chargeResult, chargeShapeExpectation, chargeShapeErrorBudget, chargeShapeResult, scopeQualifications, completeResult.
Source-ordered inference steps: stateNeutralSetup, supplyNeutralCalculation, supplyNeutralRadialFormulaCalculation, deriveNeutralRadialFormulaReadout, deriveNeutralResult, stateApproximationSetup, supplyReductionCalculation, supplyEffectiveRangeIdentification, deriveEffectiveRangeResult, stateCrossSectionSetup, supplyCrossSectionCalculation, supplyCrossSectionInterpretation, deriveCrossSectionAssessment, deriveCrossSectionFormula, supplyCorrectionCalculation, interpretCurvatureDiagnostic, interpretShapeSensitivity, deriveAccuracyResult, stateTripletSetup, supplyTripletCalculation, deriveTripletResult, stateSingletLimitation, supplySingletBorrowing, deriveSingletAssessment, stateCoulombSetup, supplyCoulombCalculation, deriveCoulombSourceFormula, supplyCoulombRangeReplacement, supplyCoulombIdentification, deriveCoulombResult, supplyCoulombNeutralization, deriveCoulombNeutralization, stateCoulombThresholdSetup, supplyCoulombThresholdRegularity, deriveCoulombThreshold, stateProtonSetup, supplyProtonFit, assessHighEnergyData, deriveProtonResult, stateChargeSetup, supplyChargeWeightedCalculation, deriveChargeWeightedResult, supplyChargeReadoutCalculation, deriveChargeEstimate, supplyChargeCalculation, interpretChargeDependence, deriveChargeResult, stateChargeShapeExpectation, supplyChargeShapeErrorBudget, deriveChargeShapeAssessment, stateScopeQualifications, assembleCompleteResult.
Independent argument routes: neutralComparison, effectiveRangeReduction, neutralCrossSection, accuracyAndShape, tripletExtraction, singletRangeQualification, coulombComparison, coulombTheory, coulombNeutralization, coulombThreshold, protonExperiment, chargeWeightedComparison, chargeEstimate, chargeComparison, chargeShapeAssessment, completeScopedResult.
Argument-route classifications: exactFiniteRangeComparison, controlledLowEnergyReduction, sWaveCrossSectionUnitarity, higherOrderErrorAndShapeScope, boundStateDataExtraction, singletRangeDataQualification, exactChargedWaveComparison, coulombModifiedComparison, separateCoulombRemovalApproximation, nonresonantCoulombThreshold, empiricalLowAndHighEnergyAssessment, exactWeightedChargeComparison, controlledChargeReadout, counterfactualInteractionComparison, conditionalShapeExtrapolation, completePaperAssessment.
Current checked semantic identifiers
These complete lists follow current Lean declaration order. Historical and selected explanatory lists elsewhere retain their separate scope.
Primary semantic theorems (248)
the_integral_radial_equation_fixes_its_origin_value, the_regular_actual_waves_and_normalized_comparison_waves_have_distinct_origins, the_finite_radius_comparison_retains_the_boundary_remainder, the_actual_boundary_limit_yields_bethes_comparison_identity, the_replacement_integrand_is_the_actual_overlap_difference, the_actual_phase_residual_is_the_range_replacement_error, controlled_range_replacement_gives_a_phase_error_bound, exact_phase_agreement_at_nonzero_energy_requires_exact_range_agreement, the_actual_integral_error_bounds_the_range_replacement, a_quadratic_range_error_gives_a_quartic_phase_error, uniform_quadratic_range_control_derives_quartic_accuracy, uniform_radial_inputs_realize_shared_effective_range_expansion, the_phase_error_is_a_controlled_observable_approximation, the_weighted_sine_force_has_the_actual_integral, the_weighted_cosine_force_has_the_actual_integral, the_unmatched_free_waves_satisfy_actual_radial_equations, the_four_free_derivatives_are_actual, the_unmatched_boundary_is_not_an_independent_scalar, the_regular_wave_product_has_the_actual_integral, the_unmatched_free_waves_satisfy_both_finite_wronskian_identities, actual_free_radial_equations_do_not_supply_the_boundary_limit, the_nonzero_shape_remainder_obeys_the_exact_range_comparison, the_nonzero_shape_remainder_has_controlled_range_replacement, controlled_replacement_does_not_erase_the_actual_phase_remainder, a_continuous_range_can_approach_its_reference_only_linearly, continuity_and_zero_energy_agreement_do_not_supply_quartic_accuracy, equation_thirty_eight_retains_the_two_proton_reduced_mass, positive_proton_parameters_give_a_positive_bohr_radius, the_scaled_coulomb_potential_is_the_printed_radial_term, the_leading_comparison_has_the_actual_logarithmic_derivative, the_common_logarithmic_derivative_has_no_finite_origin_limit, each_actual_leading_derivative_diverges_but_their_difference_is_finite, the_source_series_is_summable, the_repulsive_coulomb_factor_is_positive_and_below_one, equation_forty_three_has_the_actual_positive_normalization, the_dimensionless_source_function_has_the_printed_coulomb_coefficient, omitting_the_additive_coulomb_function_changes_the_phase, the_dimensionless_plot_is_not_the_inverse_length_phase, a_controlled_charged_phase_error_need_not_be_zero, exact_energy_replacement_does_not_remove_the_neutralization_error, the_repulsive_penetration_factor_vanishes_at_large_eta, the_positive_radius_integral_equation_uses_the_actual_wave_derivative, the_regular_irregular_combination_has_an_actual_unit_origin_limit, the_actual_coulomb_basis_identifies_the_relative_nuclear_phase, the_same_radius_logarithm_cancels_before_the_origin_limit, actual_derivative_comparison_derives_the_charged_range_identity, the_actual_charged_overlap_control_bounds_the_effective_range_replacement, the_charged_range_replacement_controls_the_phase_instead_of_making_it_exact, the_dimensionless_plot_retains_the_bohr_radius_in_its_error_budget, neutralizing_the_same_nuclear_potential_has_its_own_range_error, energy_replacement_and_coulomb_removal_have_separate_error_budgets, a_realized_threshold_point_has_the_actual_relative_coulomb_phase, large_coulomb_parameter_is_an_actual_zero_momentum_limit, a_noncancelling_source_intercept_supplies_the_uniform_gap, the_charged_readout_recovers_the_actual_nuclear_phase_function, the_nuclear_partial_wave_strength_is_sine_squared_over_momentum_squared, a_uniform_gap_bounds_the_nuclear_partial_wave_by_the_penetration_factor, the_nonresonant_source_gap_makes_the_nuclear_contribution_vanish, the_resonant_diagnostic_has_actual_charged_phase_and_partial_wave_strength, an_exponentially_small_penetration_factor_does_not_by_itself_suppress_nuclear_scattering, the_same_potential_regular_waves_have_zero_origin_wronskian, the_finite_weighted_comparison_retains_its_boundary_difference, weighted_balance_and_the_actual_boundary_limit_give_zero_comparison_wronskian, the_neutral_comparison_derivative_is_the_negative_reciprocal_length, a_zero_wronskian_gives_logarithmic_slope_agreement_only_at_nonzero_values, the_linear_comparison_determines_the_equivalent_neutral_length, the_source_weighted_comparison_determines_the_equivalent_neutral_length, actual_wave_errors_control_the_readout_denominator, the_reciprocal_length_difference_uses_both_actual_denominators, controlled_wave_and_slope_errors_bound_the_reciprocal_length, the_printed_first_order_formula_has_an_explicit_rational_remainder, changing_only_the_intercept_has_calculated_value_and_slope_effects, the_controlled_wave_readout_retains_the_first_order_remainder, intercept_calibration_has_distinct_value_and_slope_error_factors, expansion_and_intercept_errors_are_added_before_the_inverse_readout, the_printed_plus_profile_and_minus_exterior_have_an_actual_linear_difference, asymptotic_matching_requires_the_parameter_sign_identification, one_nonzero_parameter_cannot_be_used_in_both_literal_asymptotic_conventions, a_depth_comparison_controls_the_actual_potential_only_with_common_range_and_shape, equal_depth_and_shape_do_not_control_potentials_with_different_ranges, equal_depth_and_range_do_not_control_potentials_with_different_shapes, a_common_shape_realizes_nonzero_but_controlled_potential_difference, different_actual_scattering_lengths_exclude_equal_nuclear_potentials, estimate_and_measurement_errors_require_a_strict_separation_margin, an_observed_length_difference_and_both_errors_bound_the_true_difference, both_reported_corrected_estimates_have_a_small_nonzero_inverse_length_gap, the_source_shape_bracket_has_a_gap_before_finite_length_corrections, any_bracketed_estimate_retains_the_reported_gap, the_source_radius_estimate_is_positive, Data.a_controlled_log_radius_error_has_its_own_reciprocal_length_error, the_diagnostic_slope_is_the_derivative_of_the_normalized_curve, small_wave_errors_need_not_give_a_small_inverse_length_error_near_a_pole, the_source_first_order_truncation_can_have_a_nonzero_remainder, reparameterizing_the_tail_sign_matches_but_literal_parameter_reuse_does_not, a_nonzero_reported_gap_with_only_nonstrict_error_separation_can_hide_equal_lengths, the_uncorrected_shape_bracket_does_not_supply_the_corrected_bracket, the_weight_is_the_actual_inverse_radius, the_overlap_error_is_smooth_and_nonnegative, overlap_primitive_derivative, weighted_primitive_derivative, the_actual_unweighted_integral_is_linear_in_the_radius, the_actual_weighted_integral_is_independent_of_the_radius, arbitrarily_small_unweighted_errors_do_not_give_small_origin_weighted_errors, the_same_unweighted_tolerance_fails_for_a_smooth_origin_regular_profile, the_zero_extension_has_the_actual_positive_axis_integral, the_extended_weight_is_the_actual_inverse_radius, the_zero_extension_is_continuous_and_vanishes_at_its_endpoints, the_zero_extension_and_its_weight_are_nonnegative, the_compact_profile_has_distinct_global_unweighted_and_weighted_norms, a_global_unweighted_error_bound_does_not_supply_the_same_coulomb_weighted_bound, the_laboratory_energy_range_conversion_is_positive, laboratory_energy_calibration_derives_the_effective_range_fit, calibrated_finite_samples_give_range_estimates_and_actual_phase_residuals, slope_uncertainty_propagates_through_the_actual_range_conversion, a_reported_range_retains_separate_slope_and_rounding_errors, positive_laboratory_energy_gives_positive_momentum, the_momentum_has_the_source_laboratory_energy, the_constructed_relative_phase_has_nonzero_sine, the_constructed_phase_realizes_the_actual_coulomb_plot, the_source_curve_is_realized_at_each_positive_energy, the_constructed_phase_matrix_has_unit_norm, positive_energy_realization_retains_the_physical_phase_and_unitarity, the_curved_plot_has_its_actual_first_derivative, the_actual_source_curve_has_the_fitted_derivative, the_actual_source_curve_has_the_quadratic_curvature, each_curvature_sign_is_compatible_with_a_controlled_low_energy_line, a_controlled_low_energy_line_does_not_supply_higher_energy_agreement, low_energy_fit_tolerance_does_not_select_the_actual_curvature_sign, the_control_has_positive_parameters_and_an_actual_laboratory_energy, each_control_curve_has_positive_energy_phase_calibration, each_positive_energy_sample_has_an_actual_phase_and_laboratory_calibration, the_control_sample_is_finite_physical_and_has_held_out_validation, matching_the_three_sample_values_gives_the_finite_fit, finite_fitting_and_validation_do_not_control_an_unsampled_low_energy, the_measured_laboratory_slope_does_not_have_the_relative_energy_range, equation_seven_normalizes_comparison_wave_at_origin, equation_seven_retains_k_cotangent_phase_function, equation_seven_gives_the_actual_comparison_origin_derivative, the_zero_energy_comparison_has_the_actual_scattering_length_slope, lower_radial_inputs_derive_the_infinite_comparison_identity, lower_radial_inputs_derive_the_source_formula_readout, equations_ten_eleven_and_thirteen_yield_zero_energy_relation, radial_comparison_and_formula_readout_yield_exact_neutral_identity, radial_comparison_yields_exact_neutral_identity, the_actual_interior_integral_control_derives_the_range_replacement, the_exact_neutral_comparison_and_controlled_replacement_bound_the_phase_error, controlled_zero_energy_replacement_yields_effective_range_law, the_effective_range_output_is_a_controlled_observable_approximation, the_selected_phase_error_is_quartic_under_quadratic_replacement_control, equation_seventeen_coherent_phase_readout_yields_sine_squared_formula, coherent_phase_function_vanishes_iff_phase_shift_cosine_vanishes, equation_seventeen_at_positive_wave_number_implies_unitarity_bound, equation_seventeen_and_zero_phase_saturate_unitarity_maximum, equation_seventeen_saturates_unitarity_maximum_iff_phase_vanishes, equation_seventeen_saturates_unitarity_maximum_iff_phase_shift_cosine_vanishes, equation_seventeen_nonzero_phase_is_strictly_below_unitarity_maximum, effective_range_phase_law_yields_unitarity_scoped_cross_section, established_neutral_cross_section_retains_equation_seventeen_formula, established_neutral_cross_section_retains_numerical_bound_and_maximum, established_neutral_cross_section_characterizes_saturation_by_zero_phase, established_neutral_cross_section_retains_both_equation_seventeen_forms, established_neutral_cross_section_characterizes_physical_phase_shift_maximum, established_neutral_cross_section_assessment_exposes_complete_route, small_shape_correction_supports_accuracy_not_shape_determination, accuracy_scope_retains_sign_sensitive_but_unresolved_curvature, accuracy_scope_retains_cross_channel_intrinsic_range_distinctions, bound_radial_comparison_has_nonvacuous_decaying_mode, bound_comparison_retains_normalization_and_actual_origin_derivative, bound_radial_comparison_has_negative_wave_number_square, bound_comparison_identity_derives_equation_nineteen, equation_nineteen_exposes_the_range_replacement_residual, equation_nineteen_range_replacement_yields_effective_range_form, equation_nineteen_and_signed_intensity_bound_the_triplet_range_error, limited_neutron_proton_singlet_data_retain_borrowed_range_assumption, singlet_range_assessment_retains_force_equality_and_borrowed_range, deuteron_and_scattering_data_extract_triplet_effective_range, triplet_extraction_retains_actual_bound_state_asymptotics, established_triplet_extraction_exposes_complete_route, established_triplet_extraction_retains_bound_state_relation_control, the_charged_scattering_point_is_distinct_from_its_zero_energy_reference, the_charged_data_retain_the_zero_energy_intercept_definition, actual_coulomb_comparison_inputs_derive_the_exact_boundary_identity, actual_coulomb_comparison_inputs_recover_the_printed_source_function, the_coulomb_source_formula_is_established_before_range_replacement, coulomb_comparison_yields_modified_effective_range_law, established_coulomb_comparison_and_range_control_yield_the_modified_law, the_neutralized_range_requires_its_own_coulomb_removal_calculation, the_charged_law_and_independent_neutralization_control_give_the_combined_error, the_nonresonant_threshold_route_suppresses_only_the_nuclear_partial_wave, proton_data_support_low_energy_line_but_leave_high_energy_discrepancy, established_proton_result_retains_finite_data_and_the_derived_range_fit, established_proton_result_distinguishes_the_two_high_energy_measurements, established_proton_result_propagates_slope_and_rounding_uncertainties, established_proton_result_retains_detailed_high_energy_diagnostics, the_weighted_charge_comparison_precedes_the_empirical_readout, the_charge_readout_uses_the_same_zero_energy_coulomb_source, the_charge_estimate_retains_radius_fit_wave_and_truncation_errors, proton_neutron_comparison_finds_small_but_definite_force_difference, established_charge_comparison_retains_robustness_and_conditions, extrapolating_to_other_shapes_retains_the_author_expectation_and_error_budget, effective_range_reduction_model_retains_paper_semantics, shared_grounding_setup_selects_low_energy_effective_range_setup, shared_low_energy_setup_recovers_neutral_reduction_inputs, shared_established_two_parameter_law_recovers_neutral_effective_range_law, neutral_effective_range_law_establishes_shared_two_parameter_law, neutral_effective_range_routes_make_shared_route_applicable, bethe_source_inputs_make_shared_reduction_applicable, bethe_source_inputs_realize_structural_effective_range_reduction_explanation, bethe_source_derivation_makes_shared_reduction_applicable, bethe_source_inputs_realize_source_structural_effective_range_reduction_explanation, bethe_source_inputs_realize_source_effective_range_derivational_reduction, neutral_effective_range_routes_realize_derivational_reduction, neutral_effective_range_routes_establish_shared_reduction, neutral_effective_range_routes_exhibit_nonvacuous_shared_reduction, neutral_effective_range_routes_realize_controlled_truncation, established_bethe_reduction_realizes_every_shared_claim, bethe_effective_range_model_inputs_establish_shared_grounding, neutral_effective_range_routes_establish_grounded_shared_reduction, grounded_bethe_reduction_transfers_every_shared_claim, effective_range_bound_state_extraction_model_retains_paper_semantics, shared_bound_state_grounding_setup_selects_extraction_setup, triplet_routes_make_shared_bound_state_extraction_applicable, bethe_bound_state_source_inputs_make_shared_extraction_applicable, bethe_bound_state_source_inputs_realize_source_structural_extraction_explanation, bethe_comparison_ground_state_and_triplet_evidence_realize_source_sensitive_bound_state_extraction, triplet_routes_establish_shared_bound_state_extraction, triplet_routes_exhibit_nonvacuous_shared_bound_state_extraction, triplet_routes_realize_qualified_parameter_evidence_support, established_bethe_bound_state_extraction_realizes_every_shared_claim, bethe_bound_state_model_inputs_establish_shared_grounding, triplet_routes_establish_grounded_shared_bound_state_extraction, grounded_bethe_bound_state_extraction_transfers_every_shared_claim, independent_routes_establish_complete_scoped_result, exact_phase_function_without_wave_square_does_not_supply_sine_squared_formula, exact_wave_square_without_phase_readout_does_not_supply_sine_squared_formula, equation_seventeen_without_positive_wave_number_is_not_physical, equation_seventeen_without_zero_phase_does_not_reach_maximum, named_neutral_cross_section_does_not_supply_equation_seventeen_formula, bound_state_label_does_not_supply_equation_nineteen, exact_comparison_identity_does_not_supply_low_energy_replacement, accurate_two_parameter_law_does_not_determine_potential_shape, proton_charge_calculation_does_not_supply_singlet_range_qualification, approximate_singlet_force_equality_does_not_supply_borrowed_range, approximate_charge_independence_does_not_establish_exact_equality, a_phase_law_does_not_supply_the_observable_cross_section_calculation, a_unitarity_bound_does_not_license_the_low_order_truncation_at_its_maximum, cross_channel_range_compatibility_does_not_select_a_shape_from_one_channel, detailed_high_energy_diagnostics_can_leave_the_discrepancy_unexplained
Coverage theorems (20)
every_source_claim_has_semantics, neutral_comparison_route_is_semantically_realized, effective_range_reduction_route_is_semantically_realized, neutral_cross_section_route_is_semantically_realized, accuracy_and_shape_route_is_semantically_realized, triplet_extraction_route_is_semantically_realized, singlet_range_qualification_route_is_semantically_realized, coulomb_comparison_route_is_semantically_realized, coulomb_theory_route_is_semantically_realized, coulomb_neutralization_route_is_semantically_realized, coulomb_threshold_route_is_semantically_realized, proton_experiment_route_is_semantically_realized, weighted_charge_route_is_semantically_realized, charge_estimate_route_is_semantically_realized, charge_comparison_route_is_semantically_realized, conditional_charge_shape_route_is_semantically_realized, complete_scoped_result_route_is_semantically_realized, all_routes_are_semantically_realizable, source_inputs_derive_the_actual_radial_comparison, source_interior_control_derives_the_physical_phase_error
Source-map audit theorems (32)
argument_wellFormed, argument_auditable, routes_wellFormed, routes_have_exact_predecessor_graph, routes_have_exact_dependency_closure, routes_expose_declared_dependency_outputs, local_setup_steps_do_not_claim_unearned_predecessors, dependent_steps_name_inherited_outputs, calculation_and_interpretation_contracts_are_independent_hypotheses, result_steps_apply_selected_setups_and_contracts, neutral_result_depends_on_equations_7_10_11_13_and_14, cross_section_accuracy_and_triplet_are_siblings_after_reduction, singlet_borrowing_route_keeps_limitation_and_borrowing_distinct, accuracy_route_keeps_curvature_diagnostic_distinct, dependent_routes_consume_certified_predecessor_outputs, effective_range_result_retains_approximate_status, effective_range_result_retains_identification_bridge, cross_section_result_retains_calculation_and_truncation_warning, cross_section_route_exposes_equation_seventeen_formula, accuracy_result_retains_shape_interpretation, the_exact_coulomb_formula_does_not_use_either_range_approximation, neutralization_retains_two_independent_approximation_inputs, threshold_regularity_is_not_an_empirical_fit_or_range_replacement, proton_result_retains_empirical_status, high_energy_assessment_retains_open_status, charge_result_retains_approximate_and_interpretive_status, charge_route_uses_singlet_qualification_without_triplet_extraction, weighted_charge_comparison_is_independent_of_fits_and_unweighted_range_control, charge_estimation_and_force_interpretation_retain_separate_inputs, shape_extrapolation_is_conditional_on_an_author_expectation_not_a_computed_model, complete_result_retains_deferred_scope, complete_result_relies_on_every_substantive_route
Semantic atoms (52)
neutralComparisonSetup, neutralComparisonCalculation, neutralRadialFormulaCalculation, neutralRadialFormulaReadout, exactNeutralComparison, effectiveRangeApproximationSetup, effectiveRangeReductionCalculation, effectiveRangeIdentificationBridge, neutralEffectiveRangeLaw, neutralCrossSectionSetup, neutralCrossSectionCalculation, neutralCrossSectionTruncationInterpretation, neutralCrossSectionAssessment, equationSeventeenCrossSectionFormula, higherOrderCalculation, curvatureDiagnosticInterpretation, shapeSensitivityInterpretation, accuracyAndShapeScope, tripletExtractionSetup, tripletExtractionCalculation, singletRangeDataLimitation, singletRangeBorrowingBridge, singletRangeAssessment, tripletExtractionResult, coulombComparisonSetup, coulombComparisonCalculation, coulombSourceFormula, coulombRangeReplacementCalculation, coulombIdentificationBridge, coulombEffectiveRangeLaw, coulombNeutralizationCalculation, coulombNeutralizationResult, coulombThresholdSetup, coulombThresholdRegularity, coulombThresholdResult, protonEmpiricalSetup, protonFitCalculation, highEnergyAssessment, protonEmpiricalResult, chargeComparisonSetup, chargeWeightedCalculation, chargeWeightedResult, chargeReadoutCalculation, chargeEstimateResult, chargeComparisonCalculation, chargeDependenceInterpretation, chargeComparisonResult, chargeShapeExpectation, chargeShapeErrorBudget, chargeShapeResult, paperScopeQualifications, completeScopedResult
Claim identifiers (52)
neutralSetup, neutralCalculation, neutralFormulaCalculation, neutralFormulaReadout, neutralResult, approximationSetup, reductionCalculation, effectiveRangeIdentification, effectiveRangeResult, crossSectionSetup, crossSectionCalculation, crossSectionInterpretation, crossSectionResult, crossSectionFormula, correctionCalculation, curvatureDiagnostic, shapeInterpretation, accuracyResult, tripletSetup, tripletCalculation, singletLimitation, singletBorrowing, singletAssessment, tripletResult, coulombSetup, coulombCalculation, coulombSourceFormula, coulombRangeReplacement, coulombIdentification, coulombResult, coulombNeutralizationCalculation, coulombNeutralizationResult, coulombThresholdSetup, coulombThresholdRegularity, coulombThresholdResult, protonSetup, protonFit, highEnergyAssessment, protonResult, chargeSetup, chargeWeightedCalculation, chargeWeightedResult, chargeReadoutCalculation, chargeEstimateResult, chargeCalculation, chargeInterpretation, chargeResult, chargeShapeExpectation, chargeShapeErrorBudget, chargeShapeResult, scopeQualifications, completeResult
Step identifiers (52)
stateNeutralSetup, supplyNeutralCalculation, supplyNeutralRadialFormulaCalculation, deriveNeutralRadialFormulaReadout, deriveNeutralResult, stateApproximationSetup, supplyReductionCalculation, supplyEffectiveRangeIdentification, deriveEffectiveRangeResult, stateCrossSectionSetup, supplyCrossSectionCalculation, supplyCrossSectionInterpretation, deriveCrossSectionAssessment, deriveCrossSectionFormula, supplyCorrectionCalculation, interpretCurvatureDiagnostic, interpretShapeSensitivity, deriveAccuracyResult, stateTripletSetup, supplyTripletCalculation, deriveTripletResult, stateSingletLimitation, supplySingletBorrowing, deriveSingletAssessment, stateCoulombSetup, supplyCoulombCalculation, deriveCoulombSourceFormula, supplyCoulombRangeReplacement, supplyCoulombIdentification, deriveCoulombResult, supplyCoulombNeutralization, deriveCoulombNeutralization, stateCoulombThresholdSetup, supplyCoulombThresholdRegularity, deriveCoulombThreshold, stateProtonSetup, supplyProtonFit, assessHighEnergyData, deriveProtonResult, stateChargeSetup, supplyChargeWeightedCalculation, deriveChargeWeightedResult, supplyChargeReadoutCalculation, deriveChargeEstimate, supplyChargeCalculation, interpretChargeDependence, deriveChargeResult, stateChargeShapeExpectation, supplyChargeShapeErrorBudget, deriveChargeShapeAssessment, stateScopeQualifications, assembleCompleteResult
Route identifiers (16)
neutralComparison, effectiveRangeReduction, neutralCrossSection, accuracyAndShape, tripletExtraction, singletRangeQualification, coulombComparison, coulombTheory, coulombNeutralization, coulombThreshold, protonExperiment, chargeWeightedComparison, chargeEstimate, chargeComparison, chargeShapeAssessment, completeScopedResult
Route kinds (16)
exactFiniteRangeComparison, controlledLowEnergyReduction, sWaveCrossSectionUnitarity, higherOrderErrorAndShapeScope, boundStateDataExtraction, singletRangeDataQualification, exactChargedWaveComparison, coulombModifiedComparison, separateCoulombRemovalApproximation, nonresonantCoulombThreshold, empiricalLowAndHighEnergyAssessment, exactWeightedChargeComparison, controlledChargeReadout, counterfactualInteractionComparison, conditionalShapeExtrapolation, completePaperAssessment
Checked source-route details
This ledger is generated from the elaborated Lean route records. It lists direct predecessors, ordered inference steps, and exported claims. The independent fidelity audit checks every field; this ledger does not replace the physical semantics or certify every explanatory diagram.
neutralComparison
Logical kind: exactFiniteRangeComparison.
Required predecessor routes: none.
Ordered inference steps: stateNeutralSetup, supplyNeutralCalculation, supplyNeutralRadialFormulaCalculation, deriveNeutralRadialFormulaReadout, deriveNeutralResult.
Exported conclusions: neutralFormulaReadout, neutralResult.
effectiveRangeReduction
Logical kind: controlledLowEnergyReduction.
Required predecessor routes: neutralComparison.
Ordered inference steps: stateApproximationSetup, supplyReductionCalculation, supplyEffectiveRangeIdentification, deriveEffectiveRangeResult.
Exported conclusions: effectiveRangeResult.
neutralCrossSection
Logical kind: sWaveCrossSectionUnitarity.
Required predecessor routes: effectiveRangeReduction.
Ordered inference steps: stateCrossSectionSetup, supplyCrossSectionCalculation, supplyCrossSectionInterpretation, deriveCrossSectionAssessment, deriveCrossSectionFormula.
Exported conclusions: crossSectionResult, crossSectionFormula.
accuracyAndShape
Logical kind: higherOrderErrorAndShapeScope.
Required predecessor routes: effectiveRangeReduction.
Ordered inference steps: supplyCorrectionCalculation, interpretCurvatureDiagnostic, interpretShapeSensitivity, deriveAccuracyResult.
Exported conclusions: accuracyResult.
tripletExtraction
Logical kind: boundStateDataExtraction.
Required predecessor routes: effectiveRangeReduction.
Ordered inference steps: stateTripletSetup, supplyTripletCalculation, deriveTripletResult.
Exported conclusions: tripletResult.
singletRangeQualification
Logical kind: singletRangeDataQualification.
Required predecessor routes: none.
Ordered inference steps: stateSingletLimitation, supplySingletBorrowing, deriveSingletAssessment.
Exported conclusions: singletAssessment.
coulombComparison
Logical kind: exactChargedWaveComparison.
Required predecessor routes: none.
Ordered inference steps: stateCoulombSetup, supplyCoulombCalculation, deriveCoulombSourceFormula.
Exported conclusions: coulombSourceFormula.
coulombTheory
Logical kind: coulombModifiedComparison.
Required predecessor routes: coulombComparison.
Ordered inference steps: supplyCoulombRangeReplacement, supplyCoulombIdentification, deriveCoulombResult.
Exported conclusions: coulombResult.
coulombNeutralization
Logical kind: separateCoulombRemovalApproximation.
Required predecessor routes: coulombComparison, coulombTheory.
Ordered inference steps: supplyCoulombNeutralization, deriveCoulombNeutralization.
Exported conclusions: coulombNeutralizationResult.
coulombThreshold
Logical kind: nonresonantCoulombThreshold.
Required predecessor routes: none.
Ordered inference steps: stateCoulombThresholdSetup, supplyCoulombThresholdRegularity, deriveCoulombThreshold.
Exported conclusions: coulombThresholdResult.
protonExperiment
Logical kind: empiricalLowAndHighEnergyAssessment.
Required predecessor routes: coulombTheory.
Ordered inference steps: stateProtonSetup, supplyProtonFit, assessHighEnergyData, deriveProtonResult.
Exported conclusions: protonResult.
chargeWeightedComparison
Logical kind: exactWeightedChargeComparison.
Required predecessor routes: none.
Ordered inference steps: stateChargeSetup, supplyChargeWeightedCalculation, deriveChargeWeightedResult.
Exported conclusions: chargeWeightedResult.
chargeEstimate
Logical kind: controlledChargeReadout.
Required predecessor routes: chargeWeightedComparison, coulombComparison, protonExperiment.
Ordered inference steps: supplyChargeReadoutCalculation, deriveChargeEstimate.
Exported conclusions: chargeEstimateResult.
chargeComparison
Logical kind: counterfactualInteractionComparison.
Required predecessor routes: chargeEstimate, protonExperiment, singletRangeQualification.
Ordered inference steps: supplyChargeCalculation, interpretChargeDependence, deriveChargeResult.
Exported conclusions: chargeResult.
chargeShapeAssessment
Logical kind: conditionalShapeExtrapolation.
Required predecessor routes: none.
Ordered inference steps: stateChargeShapeExpectation, supplyChargeShapeErrorBudget, deriveChargeShapeAssessment.
Exported conclusions: chargeShapeResult.
completeScopedResult
Logical kind: completePaperAssessment.
Required predecessor routes: neutralComparison, effectiveRangeReduction, neutralCrossSection, accuracyAndShape, tripletExtraction, singletRangeQualification, coulombComparison, coulombTheory, coulombNeutralization, coulombThreshold, protonExperiment, chargeWeightedComparison, chargeEstimate, chargeComparison, chargeShapeAssessment.
Ordered inference steps: stateScopeQualifications, assembleCompleteResult.
Exported conclusions: scopeQualifications, completeResult.