Projected Curvature: Second-Order Read-Only Analysis

01

Executive conclusion

Two independent finite-window fits — one taken along the 90-degree axis (the upper cap) and one along the 270-degree axis (the cone-apex side) — return projected radii that agree to roughly four parts in a thousand:

This analysis asks where that shared scale near 47 comes from, and answers it in two parts.

rho = 1 / (4 * kappa_sigma) = 47.34582205692646

Hence the overall classification is PARTIALLY_EXPLAINED: a structurally derived base scale plus an isolated, quantified, unexplained upper-cap residual.

Subsequent adversarial read-only analyses sharpened the unresolved side without changing that classification. The 90-degree correction is confined to a narrow, nearly zero-sum cross-ell direction close to (1,-2,1), but its small in-plane drift is not robustly attributable to a fixed active-block pair or even to a face-level demand coordinate across all admissible drift definitions. The strongest surviving follow-up result is therefore geometric rather than causal:

90-degree correction: narrow near-zero-sum cross-ell direction
ell=0 face demand:     extremely stable rank-1 redistribution mode
link between the two:  suggestive in ranks, not robust across definitions

The preregistered read-only stopping rule was reached. The saved primal artifacts identify the geometry but do not determine why the optimizer selects it; resolving that question would require a new experiment that records dual/KKT information. Section 6 gives the complete audit trail.

02

Frozen scope and provenance

This is a sealed-artifact-only, read-only analysis. Every number reported here was derived from 38 previously saved support solves. No new solver run, additional support solve, parameter refinement, ell refinement, or full run was performed — these operations are explicitly prohibited by ANALYSIS_SCOPE.json, and new_solver_runs is 0 in every provenance record in the package.

Authoritative claim boundary, quoted from projected_curvature_result.json and ANALYSIS_SCOPE.json:

The analysis decomposes the saved finite-grid response and identifies coordinate-invariant identities of the frozen linear map. It does not infer continuum-limit curvature, assign KKT dual weights that were not saved, or prove an exact full-feasible-set involution.

PROVENANCE_SHA256.json records the SHA-256 hash of every upstream artifact consulted, including the frozen solve result, the recovered coefficient array, the core implementation module, the gate-2 runner, and the face-microscope package with its own manifest and verification record.

03

Where the number 47 comes from

This is the central derivation. It is short, and it is the reason the shared scale is called structural rather than coincidental.

Let B0 be the ell=0 imaginary-amplitude map on the saved grid. The saved map represents the boundary function sigma(s) exactly to numerical precision:

B0 c_sigma = sigma(s)

Define the projected sigma-mode scale as the second projection functional applied to the representing coefficients:

kappa_sigma = q2_ell0^T c_sigma = 0.005280296954172882

The unitarity-disk expansion together with the exact projection structure then gives, in order:

y2 = (sigma / 2) * x1^2          disk expansion
x1 = 2 * a1                      projection structure
b2 = 2 * kappa_sigma * a1^2      substituting
rho = a1^2 / (2 * |b2|)
    = 1 / (4 * kappa_sigma)
    = 47.34582205692646

The a1^2 cancels. The predicted radius is independent of the path amplitude, which is precisely why the same value can appear on two different axes with different a1. The corresponding universal quadratic coefficient is

2 * kappa_sigma = 0.010560593908345763

Three points about interpretation:

• This is a coordinate-invariant disk/basis/projection scale. It is a property of the frozen map, the disk geometry, and the projection — not of any particular solve, offset, or fitting window.
• It is not a consequence of the large raw z0 coefficient. Section 5 shows that z0 contributes exactly zero projected curvature.
• Raw coefficient norms are not a measure of physical mode importance. All mode-importance statements in this document use physical-amplitude-space or coordinate-invariant quantities. This distinction is material: at 90 degrees the z0 group carries a first-order physical amplitude norm larger than the full path norm (ratio 1.1080417826762037) while contributing nothing at all to curvature.

04

Projected Taylor and curvature decomposition

4.1 Finite-window quadratic fits

Both axes were fitted with a parabola over the four saved projection-fit offsets.

In percentage terms: the 270-degree radius sits -0.00668% from the universal value, and the 90-degree fitted quadratic coefficient sits +0.44108% above the universal coefficient (equivalently, its radius sits -0.43915% below). The observed 90/270 fitted-radius difference is 0.433438%.

4.2 Resolution caveat — read this before quoting the 0.43% figure

The quoted 0.43% agreement is a property of a finite-window parabolic fit, not a demonstrated asymptotic statement. taylor_coefficients.csv records direct Taylor estimates at seven offsets per axis, and the representative direct parametric radii show visible scale sensitivity:

The smallest offsets are visibly contaminated by finite-precision differencing of the saved grid. Consequently:

• Exact asymptotic curvature equality between the two axes is not established.
• No continuum-limit, differentiability, or exact-symmetry conclusion may be drawn from these finite-grid values. The evidence supports a numerical statement about the saved response, and nothing stronger.

4.3 Mode-group decomposition at the representative offset

All group quantities below are evaluated at the representative saved offset 0.010986328125 degrees. Contributions to b1 and b2 are additive across groups.

90 degrees (a1 = -47.770863870657976, a2 = 3693.805124456793, b1 = 9.405568783527052e-05, b2 = -23.852252706408095):

270 degrees (a1 = 47.22984500175838, a2 = 1291.830638283183, b1 = 1.5662900860059252e-05, b2 = 23.5879657134319):

The two axes reach comparable curvature by structurally different routes. At 270 degrees the ell=0 spectral group supplies essentially all of b2 directly, with ell=2 and ell=4 contributing at the 0.01 level. At 90 degrees the individual group contributions are three to four times the net result and carry opposing signs — the subject of the next section.

05

Why z0-only is insufficient, and how the upper-cap rest modes cancel

5.1 The z0-only path has zero projected curvature

Because q2[0] = 0, the z0 direction is annihilated by the second projection functional. A strict z0-only projected path therefore has

b1 = 0,  b2 = 0,  projected curvature = 0,  projected radius = INFINITE

on both axes. This is recorded explicitly in the result JSON as z0_only_projected_radius: "INFINITE" for 90 and 270 alike.

The consequence is structural, and it is the reason Section 3 insists on the distinction between coefficient magnitude and curvature relevance: all finite projected curvature is supplied by rest modes. The dominant raw z0 coefficient supplies none of it, however large its amplitude norm.

5.2 The 90-degree cancellation

At the representative offset 0.010986328125 degrees, the 90-degree b2 contributions are:

ell=0 spectral: +17.47054612371266
ell=2:          -77.65338328346152
ell=4:          +36.33058445334078
-----------------------------------
net:            -23.852252706408095

The gross magnitude of these contributions is roughly 131.45; the net is roughly -23.85. Three cancellation diagnostics are saved:

The third row is the most informative. Evaluating the universal identity b2 = 2 * kappa_sigma * a1^2 at the saved 90-degree a1, and applying the reflection appropriate to that axis, gives a reflected universal b2 prediction of -24.09986072482907. The saved value is -23.852252706408095, so:

Measured against that universal baseline, the individual group deviations are large and opposed:

ell=0 spectral - universal baseline: +41.57040684854173
ell=2:                              -77.65338328346152
ell=4:                              +36.33058445334078

These deviations cancel to 99.8408%, leaving the small residual above. For contrast, at 270 degrees the same accounting shows deviations of +0.007600948861661294, +0.011734827090482791, and +0.011553917043577270 — all small, all the same sign, with a saved baseline-delta cancellation fraction of exactly 0.0. The 270-degree axis simply does not need a cancellation; the 90-degree axis does, and gets one.

5.3 What this does and does not establish

This is strong quantitative compensation toward the common scale: three group contributions, individually several times the net, conspire to land within one percent of a baseline derived independently of them.

It is not a proof. No exact identity over the full feasible set was found that forces this cancellation. The saved solve records contain no conic dual multipliers, and the previously tested full-complement map is infeasible, so the candidate symmetry that would have explained the cancellation was rejected rather than confirmed. The compensation is reported here as a measured fact about the saved response, and the residual +0.24760801842097635 is left explicitly on the unexplained side of the ledger.

06

Follow-up audit of the 90-degree near-null geometry

All analyses in this section reuse the same 38 saved primal solves. They perform no additional support solve, parameter refinement, or optimizer run. Each inferential step was placed in a separate output package, and the later adversarial audits govern the interpretation of the earlier exploratory candidate.

6.1 The cancellation selects a narrow near-zero-sum direction

For each of the five stable differencing widths, define the actual-minus-universal group response

Delta(h) = (Delta_ell0, Delta_ell2, Delta_ell4).

At the representative width, the components are

(+41.57040684854174, -77.65338328346150, +36.33058445334078),

with cancellation fraction 0.9984082220825929. Across the full five-width stable window:

A blind PCA constrained only to the zero-sum plane returns

d_zero = (+0.4337285565, -0.8159549679, +0.3822264114),

which is 2.0870314575 degrees from normalized (1,-2,1) and captures 0.9999623707345359 of the zero-sum projection energy. The 270-degree representative control has three small same-sign deviations and zero cancellation, so this near-null geometry is specific to the 90-degree route within the saved data.

This establishes a narrow empirical direction, not an exact stencil law. Writing the zero-sum component in the orthogonal basis

s = (1,-2,1),  a = (1,0,-1),  n = (1,1,1),

shows that the best aggregate direction can be written approximately as

d_star proportional to (1.0631164788, -2, 0.9368835212).

However, the fitted in-plane coefficient ratio lambda = B/A varies from 0.0511741 to 0.0800424; its maximum relative deviation from the aggregate value is 0.2681688852. The strict fixed-lambda hypothesis therefore fails even though the zero-sum plane and narrow cone are confirmed.

6.2 A provisional active-demand association

The direction was decomposed as

Delta(h) = alpha(h) d_zero + eta(h) d_perp + epsilon(h) n,
r_eta = eta/alpha,  r_epsilon = epsilon/alpha.

An initial preregistered comparison separated the in-plane drift from observable leakage:

Here p_D is a normalized quadratic-demand balance constructed from the odd first response. The fixed pair carries between 95.71% and 95.94% of the active demand over these widths, and the original r_eta association has absolute Spearman correlation 0.90. This justified the provisional label ACTIVE_DEMAND_CANDIDATE, but not a causal or KKT interpretation.

6.3 Adversarial audit rejects the pair-specific mechanism

The candidate was then tested without changing or reselecting the fixed pair.

The canonical pair was never replaced by a better-looking competitor. The correct adversarial decision is

DOES_NOT_SURVIVE
ELL0_FACE_LEVEL_NOT_PAIR_SPECIFIC

The failure does not show that the shared odd-response term mechanically manufactures the association: that universal shared term alone has exact tail 99/120 = 0.825, while the unratioed raw-even eta retains the original sign with tail 8/120 = 0.0666667. It shows instead that the claimed direction-level, pair-specific mechanism is not robust.

6.4 Regression-free face geometry

The four center-boundary ell=0 blocks were next treated as a single normalized demand vector

p(h) = (p_11, p_12, p_24, p_29),  sum_i p_i = 1,

with no regression of drift on face coordinates. Centered PCA gives:

Thus the face-demand variation is genuinely rank-1, but its mode is predominantly s29 against the pooled other three blocks rather than a special s29-s24 exchange. This also explains why all six pairwise demand coordinates performed similarly in the competitor scan.

For each held-out width, the PCA direction and mean were estimated from the other four widths only. The resulting cross-fitted score was compared with four drift definitions using ranks and signs, without fitting a regression line:

The original rank ordering is uncommon under the 120 scale assignments (5/120 = 0.0416667 one-sided), but it fails the preregistered all-five sign condition. The varying positive denominator alpha reverses 6/10 pairwise raw-eta orderings. This documents normalization sensitivity; it does not prove that alpha is a nuisance or that the reversal is an artifact.

The matched four-coordinate 270-degree control is also strongly rank-1 (0.9975978110332967 PC1 fraction), as is the native 29-block ell=0 face (0.9963579652925423). Rank-1 demand variation alone is therefore not a 90-degree-specific signature. What does not reproduce at 270 degrees is the original drift correspondence: matched-coordinate Spearman is -0.60 with only 2/5 positive centered-sign matches.

6.5 Final follow-up classification and stopping point

The follow-up evidence separates a firm geometric result from an unsupported mechanism claim:

RANK1_FACE_GEOMETRY_CONFIRMED_DRIFT_LINK_NOT_ROBUST
FACE_LEVEL_NOT_SUPPORTED

The narrow 90-degree near-zero-sum direction, the small observable leakage, and the rank-1 ell=0 face redistribution are reproducible facts about the saved five-scale window. The identity of the optimizer-level mechanism selecting that direction is not recoverable robustly from the saved primal responses. Under the preregistered stopping rule, the read-only mechanism search ends here. A further causal test would require new solves that save conic dual multipliers, slacks, and active-threshold information, with its decision rule sealed before execution.

The exact permutation fractions above describe only the ordering of five saved scales. They are not population p-values and do not authorize causal, KKT, dual, continuum, or intervention claims.

6.6 Authoritative follow-up artifacts

Every stage has its own scope, provenance record, preflight record, and SHA-256 manifest. Later negative audits narrow the claims of earlier exploratory stages; they do not retroactively alter the saved results.

07

Disk-level and active-constraint findings

active_disk_quadratic_demand.csv records, per active boundary block, the physical disk-level quadratic demand sigma * (x1^2 + y1^2), the corresponding z0-only demand, the second-order supply, and the resulting supply/demand accounting.

Ranking boundary (authoritative, from the result JSON). Ranks use unweighted physical disk quadratic demand sigma*(x1^2+y1^2), not absent KKT dual weights; they are kinematic stress ranks, not unique causal curvature allocations. Nothing in this section may be read as a KKT-weighted causal attribution — conic dual multipliers were never saved.

7.1 The 90-degree cap is effectively low rank

Thirteen boundary blocks are active, but the unweighted demand is concentrated in a handful:

An effective participation count of 1.8067 against 13 active blocks is the quantitative statement of "effectively low rank." The full-over-z0-only ratio of 1.234 is a second, independent sign that the rest modes matter here: the true demand exceeds what the z0 direction alone would generate by roughly 23%.

Demand by ell at 90 degrees: 0 → 28714.95849561105, 2 → 62.28548520077859, 4 → 0.0002070587022949553; the ell=0 share is 0.9978355921834331.

7.2 The 270-degree cone-apex side is broad but overwhelmingly ell=0

Demand by ell at 270 degrees: 0 → 137554.43514200783, 2 → 5.6142181336861233e-08, 4 → 2.6268109124899874e-10.

The apex side is the mirror image of the cap in structure. Far more blocks are active (89 versus 13) and demand is spread across roughly 24 of them, yet the ell=2 and ell=4 apex blocks — while genuinely boundary blocks — carry negligible first-order curvature demand, some ten orders of magnitude below the ell=0 total. This is the disk-level counterpart of the mode-group table in Section 4.3, where the 270-degree ell=2 and ell=4 b2 contributions were likewise negligible, and it is why no cancellation is required on that axis.

08

What is structurally explained versus what remains unexplained

Explained

Also established, and structural rather than numerical: the z0 direction contributes exactly zero projected curvature on both axes because q2[0] = 0, so the entire projected curvature is a rest-mode phenomenon.

Unexplained

The reasoning behind the PARTIALLY_EXPLAINED label, in the words of the result JSON:

• Why not a numerical coincidence. A coordinate-invariant disk/basis identity predicts the shared scale before using either fitted radius and reproduces the lower result to far below one percent; the upper residual is explicitly isolated.
• Why not fully structurally explained. The common 47.3458 scale is exact for the frozen ell=0 sigma mode and explains the lower cap. The upper cap lands within 0.44 percent only after a large cross-ell cancellation. Saved artifacts contain no conic dual multipliers and the previously tested full complement map is infeasible, so no identity was found that forces that cancellation.

The follow-up audits add two statements to this ledger:

Standing claim boundaries

These are limits on interpretation, not open questions to be resolved by rereading the data:

1. No continuum limit. The finite-grid numerical evidence here does not support a continuum-limit, differentiability, or exact-symmetry theorem. Section 4.2 shows the direct Taylor estimates are scale-sensitive.
2. No causal attribution from rankings. The active-block rankings are unweighted kinematic disk-demand ranks. Conic dual multipliers were not saved, so no KKT-weighted causal curvature allocation exists in this package and none may be inferred from it.
3. No mode importance from raw coefficients. Use physical-amplitude-space or coordinate-invariant quantities. The z0 group is the standing counterexample: dominant in raw norm, exactly zero in curvature.
4. No exact symmetry claim for the upper cap. The 90-degree closeness to the common scale is measured, not proven to be forced.
5. No additional solves. This analysis consumed 38 sealed solves and performed zero solver runs. Nothing in this document should be read as implying otherwise.
6. No further mechanism selection from the five saved scales. Pair scans, exact permutations, cross-fitted PCA, and normalization audits have exhausted the preregistered read-only route. Selecting another block contrast or drift normalization from these same five points would reopen post-selection bias rather than resolve the mechanism.

09

Reproducibility and file guide

Verification record

The original handoff records below verify the base projected-curvature package. The five follow-up analyses in Section 6 are separate sealed read-only packages with their own manifests; they are referenced here but are not silently added to the original 12-file handoff manifest.

Every file in this handoff is hash-listed in HANDOFF_MANIFEST_SHA256.txt, and every upstream artifact consumed by the analysis is hash-listed in PROVENANCE_SHA256.json. To confirm integrity:

sha256sum -c HANDOFF_MANIFEST_SHA256.txt

The upstream package's own record is preserved verbatim in MANIFEST_SHA256_SOURCE.txt and PACKAGE_VERIFICATION_SOURCE.txt. Note that the source manifest lists two files not carried into this handoff — analyze_projected_curvature.py and read_only_preflight.json — along with the source package's own README.md, which appears here under the name README_SOURCE.md with an unchanged hash.

File guide

The 38 individual solve JSON files and the analysis source code are intentionally omitted from the original handoff. All README-relevant base values are present in its JSON and CSV artifacts; the later near-null and face-geometry values are governed by the separately manifested result files listed in Section 6.6.