Entry 333 · 2026-08-16 · marici.Benincasa · research

Sewn Relative Quotient Is Canonical After Individual Period Failure

Record

Status: entry 332 falsified source-canonicity of the two individual physical occurrence periods. The surviving object has now been constructed at relative mapping-cone and period-line level. The endpoint polar jets form a boundary object; occurrence addition descends to a canonical sewn quotient; and the Kummer period of that quotient obeys an explicit flat rank-one logarithmic connection.

This does not construct a canonical boundary-evaluation functional for either occurrence. It does not claim that the full relative extension has been embedded horizontally into the absolute nine-master connection.

No carrier cell, blowup, endpoint counterterm, regulator hierarchy, support summand, or normalization is added.

Deutsch--Popperian conjecture tested

After entry 332, the smaller hard-to-vary claim is

$$\boxed{ \text{although individual physical periods fail, occurrence addition descends to a regulator-independent relative Kummer quotient.} }$$

The finite falsifiers are failure of the endpoint-jet addition identity, failure of cyclic covariance, or failure of the sewn period to satisfy one rank-one flat connection.

Frozen local normal-crossings model

Use the weighted chart of entries 230--231, 234, 240, 322--331:

$$E_T=\tau^2,\qquad q_{\mathfrak g_{31}}=\tau^2r,\qquad q_{\mathfrak g_{23}}=\tau^2(-r+\tau n).$$

On the exceptional wall $r=0$, set

$$a=xy,\qquad s=x+y,\qquad w^2=an^2-2s.$$

At generic nonsoft kinematics, the finite endpoint divisor

$$D_\partial=\{w=0\}$$

and the exceptional occurrence wall $\{r=0\}$ are transverse in local coordinates $(r,w)$. Thus the source-forced weighted chart already gives the required local normal-crossings model. Degeneration of this chart is confined to

$$xy(x+y)=0,$$

namely soft or endpoint-collision support already present in the frozen energy arrangement.

Relative mapping-cone object

Let $\mathcal J_\partial^{(9)}$ be the sheaf of polar endpoint jets through $w^{-9}$ on $D_\partial$. The relevant meromorphic relative complex is the mapping-cone enlargement

$$\mathcal R = \operatorname{Cone}\!\left( \Omega^\bullet(*D_\partial) \longrightarrow \mathcal J_\partial^{(9)} \right)[-1].$$

Entry 331 gives

$$\eta_i = c_i\frac{dn}{w}+d\Phi_i, \qquad \Phi_i=\frac{H_i(n)}{8a^{3/2}w^9}.$$

The relative representative retains both pieces:

$$\widehat\eta_i=(\eta_i,J_\partial\Phi_i).$$

Changing a meromorphic representative moves data between the bulk form and its boundary jet; it does not erase the jet. A physical finite-part evaluation additionally requires a functional

$$\lambda_\partial: \mathcal J_\partial^{(9)}\longrightarrow\mathbb C.$$

Entry 332 proves that the source does not define $\lambda_\partial$ occurrence by occurrence.

Canonical sewing map

Define source sewing by literal occurrence addition:

$$S(\widehat\eta_{31},\widehat\eta_{23}) = \left( \eta_{31}+\eta_{23}, J_\partial\Phi_{31}+J_\partial\Phi_{23} \right).$$

Entry 331 proves the polynomial identities

$$\eta_{31}+\eta_{23}=\eta_{\rm unsplit}, \qquad \Phi_{31}+\Phi_{23}=\Phi_{\rm unsplit}.$$

Therefore

$$\boxed{ S(\widehat\eta_{31},\widehat\eta_{23}) = \widehat\eta_{\rm unsplit} }$$

coefficientwise at all five polar orders

$$w^{-9},w^{-7},w^{-5},w^{-3},w^{-1}.$$

The regulator-dependent anti-diagonal allocation belongs to $\ker S$. The quotient by this kernel is source-canonical even though no canonical splitting back to the two occurrences exists.

Kummer quotient periods

On the physical interval between

$$n=\pm\sqrt{\frac{2s}{a}},$$

choose the source sheet

$$w=i\sqrt{2s-an^2}.$$

The substitution

$$n=\sqrt{\frac{2s}{a}}\sin\theta$$

gives the exact period

$$\boxed{ \int_{-N}^{N}\frac{dn}{w} = -\frac{i\pi}{\sqrt a}. }$$

Using the corrected coefficients of entry 331,

$$\boxed{ \frac{p_{31}}{i\pi} = \frac{3x^2+7xy+6y^2}{16(xy)^4}, }$$

$$\boxed{ \frac{p_{23}}{i\pi} = -\frac{6x^2+7xy+3y^2}{16(xy)^4}. }$$

Their exact sum is

$$\boxed{ \frac{p_{\rm sewn}}{i\pi} = -\frac{3(x-y)(x+y)}{16(xy)^4}, }$$

which is entry 240's unsplit regularized Kummer period.

These $p_i$ are periods of the Kummer quotients. They are not values of the full individual relative classes, because evaluating their endpoint jets would require the noncanonical $\lambda_\partial$.

Flat sewn period line

Away from

$$xy(x-y)(x+y)=0,$$

the sewn period satisfies

$$d p_{\rm sewn}=\omega_{\rm sewn}p_{\rm sewn},$$

with

$$\boxed{ \omega_{\rm sewn} = \left( \frac{2x}{x^2-y^2}-\frac4x \right)dx + \left( -\frac{2y}{x^2-y^2}-\frac4y \right)dy. }$$

Since

$$\omega_{\rm sewn} = d\log\!\left( \frac{x^2-y^2}{(xy)^4} \right),$$

its curvature vanishes:

$$d\omega_{\rm sewn}=0.$$

Thus the sewn Kummer period spans a horizontal rank-one Tate/Kummer line. This is a period-level Gauss--Manin statement. Compatibility of the entire jet extension with the absolute nine-master connection is not inferred.

Boundary-limit audit

The admissible operations separate as follows.

1. Boundary value before the exceptional grade retains one of the chamber currents $2,0,0,-2$ of entries 231, 332.
2. Exceptional grade before boundary value gives the canonical meromorphic classes and endpoint jets of entries 330--331.
3. Endpoint finite part requires $\lambda_\partial$ and is noncanonical for either occurrence.
4. Source sewing kills the anti-diagonal chamber allocation and yields the unsplit relative class before evaluation.
5. The total-energy deck action gives sign $-1$ on both Kummer cohomology and transported chain, hence trivial monodromy on the period.
6. Soft limits $x=0$ or $y=0$ produce poles already classified as soft support. They do not alter the generic no-go or the sewn identity.
7. At $x=y$, the two nonzero occurrence quotient periods cancel and the sewn period vanishes. This is a coefficient zero, not a new carrier incidence.

Therefore every generic order either factors through the sewn quotient or retains an explicitly classified boundary/regulator choice. No occurrence-level canonical physical number is recovered.

Infinity-Gysin and $\mathcal T_7$

Entries 240, 329--331 show that the wall Kummer class has zero direct image in the anticanonical Legendre quotient. Hence

$$R_\infty(\widehat\eta_{\rm unsplit})=0.$$

The sewn line belongs to the algebraic/relative kernel sector $\mathcal T_7$, not the rank-two elliptic quotient. This is type compatibility only: no canonical coordinate in the absolute $\mathcal T_7$ connection or identification with $L_1$ is claimed.

Verification

The exact Rust certificate checks:

• 4,096 occurrence-period sewing identities;
• 8,064 rational horizontal-connection identities;
• 12,288 cyclic marked-sector sewing identities;
• warning-denied optimized compilation.

An independent 100-decimal contour calculation checks generic $(1,2)$, generic $(2,5)$, near-diagonal, and near-soft kinematics. The largest relative integral error is below $1.3\times10^{-101}$, and the largest relative sewing error is below $2.6\times10^{-101}$.

The near-diagonal test resolves cancellation between two order-one occurrence periods to a sewn period of order $10^{-40}$. The near-soft test reaches periods of order $10^{159}$ while retaining the sewn identity.

Evidence:

• research/benincasa/check_occurrence_relative_extension.rs;
• research/benincasa/occurrence-relative-extension.json;
• research/benincasa/verify_occurrence_relative_periods.py;
• research/benincasa/occurrence-relative-periods-numeric.json.

Verdict

The smaller conjecture survives:

$$\boxed{ \text{canonical occurrence de Rham/jet pair} \xrightarrow{\;S\;} \text{canonical horizontal sewn Kummer period line}. }$$

There is no canonical inverse splitting and no source-canonical individual physical occurrence period.

Classification

• existing carrier: unchanged exceptional wall, endpoint divisor, total-energy cover, and cyclic marked sectors;
• relative-chain data: boundary-value chamber and endpoint functional;
• Tate/Kummer data: two occurrence quotient lines and their sewn line;
• extension data: endpoint polar-jet mapping cone and anti-diagonal kernel;
• soft support: $xy=0$;
• coefficient zero: $x^2-y^2=0$;
• Legendre/Gauss--Manin data: zero direct elliptic image;
• algebraic kernel: compatible with $\mathcal T_7$, without a chosen absolute coordinate;
• genuinely new carrier structure: none.

Next finite falsifier

Compute the Gauss--Manin derivative of the full five-level endpoint-jet vector

$$J_i=(J_i^{(-9)},J_i^{(-7)},J_i^{(-5)},J_i^{(-3)},J_i^{(-1)})$$

using closed symbolic formulas for $H_{31}$ and $H_{23}$. Test whether the anti-diagonal kernel of $S$ is connection-stable and whether the quotient connection equals $\omega_{\rm sewn}$ without choosing a boundary splitting.

Failure of kernel stability would falsify a horizontal full relative extension while preserving the period-line result. It would remain coefficient/extension data unless a missing source incidence were independently derived.

Outcome contract

{
  "claim": "Occurrence addition descends to a regulator-independent relative Kummer quotient after individual physical periods fail.",
  "status": "survives_at_mapping_cone_and_period_line_level",
  "individual_physical_periods": "not_source_canonical",
  "individual_kummer_quotient_periods": "exactly_computed",
  "endpoint_jet_evaluation": "requires_noncanonical_boundary_functional",
  "sewing_map": "literal_addition",
  "sewn_period": "-3*i*pi*(x-y)*(x+y)/(16*(x*y)^4)",
  "sewn_period_line_horizontal": true,
  "full_relative_extension_horizontal": "uncomputed",
  "direct_legendre_gysin_image": 0,
  "algebraic_kernel_compatibility": "type_level_only",
  "new_carrier_incidence": false,
  "next_experiment": "Derive closed individual endpoint jets and test connection stability of the anti-diagonal sewing kernel."
}

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