Appendix: H6S4-C Minimal Stochastic Classical Pulse Geometry

Parent hypothesis: H6: Classical spacetime emerges from decohered pulse-history structure Status: Completed with final project-rule classification: diagnostic tool Purpose: Select the stochastic classical pulse-geometry route after H6S3 and state the candidate gate without pretending that a law has already been found.

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1. Scope

H6S4-C asks whether the Pulse Model can define the smallest admissible stochastic classical source-response rule after the H6S1, H6S2, and H6S3 gates.

The route keeps geometry classical but allows an objective, causal pulse-geometry noise process if and only if that process is declared before external comparison and passes the ledgers below. It must not reuse arbitrary ensemble branch labels as physical source-response inputs.

This appendix cites:

• H6S1 for the weak-field source/probe discriminator, branch-conditioned pulse-count diagnostics, ordinary instrument-noise separation, no-signaling checks, and conservation guardrails.
• H6S2 for ensemble-decomposition invariance, causal-domain labels, conservation ledgers, coefficient ledgers, and rejection of unsupported remote-basis marginal changes.
• H6S3 for the no-free-branch-variance theorem and the route ledger requiring records, stochastic classical pulse geometry, or non-classical geometry before branch variance is physical.
• review-4 for the decision to try H6S4-C first and to stop adding guardrail-only H6 work unless this route fails.

H6S4-C must end with exactly one final project-rule classification:

• controlled modification
• clean no-go
• diagnostic tool

The skeleton in this file is not yet any of those final results. It is the admission gate for later proof and code.

2. Candidate Gate

The minimum response shape is:

$$R_C(\rho_{\mathrm{local}}, O_{\mathrm{source}}, P_{\mathrm{local}}, D_{\mathrm{causal}}, L_{\mathrm{cons}}, L_{\mathrm{coeff}}, L_{\mathrm{reg}}) \mapsto P(N_C)$$

Here:

• $\rho_{\mathrm{local}}$ is the local source density operator or an explicitly declared local source record.
• $O_{\mathrm{source}}$ is the declared source operator or local source-record observable used by the response.
• $P_{\mathrm{local}}$ is the local pulse-record input available to the probe response.
• $D_{\mathrm{causal}}$ is the declared causal domain.
• $L_{\mathrm{cons}}$ is the conservation ledger.
• $L_{\mathrm{coeff}}$ is the coefficient ledger.
• $L_{\mathrm{reg}}$ is the regulator ledger.
• $P(N_C)$ is a normalized finite probe pulse-count marginal.

The candidate gate is passed only if the response rule is:

• normalized
• finite
• decomposition-invariant when no physical selector exists
• causal
• conservation-accounted
• coefficient-disciplined
• regulator-declared
• separated from ordinary instrument noise
• compared against the required baselines before external comparison

Passing this gate still does not make the rule an accepted law. The final verdict controls promotion.

3. Allowed Inputs

H6S4-C may use:

Input	Rule
Local density operator	Allowed when the response depends only on $\rho_{\mathrm{local}}$ and declared local operators.
Local source records	Allowed only when the record exists in the probe causal past, arrives through a modeled causal channel, or is used in shared-future comparison.
Local pulse records	Allowed as probe-accessible record data, not as an undeclared branch selector.
Source operators	Allowed when the operator, units, domain, and coefficient status are declared before comparison.
Causal domain	Required for every response report.
Conservation ledger	Required for candidate status.
Coefficient ledger	Required for candidate status; fit-after-observation coefficients are rejected.
Regulator ledger	Required when any cutoff, smoothing, finite-size parameter, or correlation scale affects the observable.
Artifact ledger	Required for ordinary instrument noise, environmental decoherence, source-preparation spread, calibration drift, and postselection.

4. Prohibited Inputs

H6S4-C must not use:

• arbitrary ensemble branch labels as physical response inputs
• remote-basis choices as spacelike causes of the probe marginal
• hidden collapse or pointer selection not declared as a physical record route
• fit-after-observation coefficients
• undeclared conservation accounting
• undeclared regulators or smoothing scales
• ordinary instrument noise relabeled as pulse-geometry noise
• environmental decoherence double-counted as stochastic geometry
• external target data to choose the law
• H7, finite-loop, geometry-action, or additional guardrail-only work as a substitute for the H6S4-C route decision

5. Required Ledgers

The later H6S4-C law or no-go theorem must declare these ledgers before code or promotion.

Ledger	Required content	Failure status
Stochastic variable	The noise variable or stochastic process, its domain, and whether it is classical, objective, and local.	Missing law blocks controlled-modification status.
Coupling target	Whether the process couples to $T_{\mu\nu}$, phase density, pulse records, curvature records, potential-operator variance, or another justified local object.	Missing target blocks candidate status.
Causal support	The support of noise correlations and whether the probe lies inside the causal domain.	Spacelike remote-basis marginal dependence is rejected.
Conservation	Energy-momentum or pulse-accounting status, including environment or reservoir terms if needed.	Missing account blocks candidate status.
Coefficients	Names, dimensions, numerical role, and provenance of every amplitude, coupling, or scale.	Fit-after-observation is rejected; exploratory-only cannot support prediction.
Regulators	Finite-size input, cutoff, smoothing, or correlation length, with provenance.	Undeclared regulator blocks candidate status.
GR and QM recovery	Limit in which the response reduces to the density-only or standard semiclassical baseline.	Missing recovery limit blocks law status.
No-signaling	Demonstration that equivalent decompositions of the same $\rho_{\mathrm{local}}$ give the same probe marginal without records.	Remote-basis marginal change is rejected.
Artifact separation	Ordinary instrument noise, environmental decoherence, source spread, drift, leakage, and postselection are separated from pulse-geometry noise.	Double-counting blocks candidate status.
Observable	Fixed excess pulse-count variance, covariance, visibility loss, timing jitter, or a precise statement that no such observable survives.	No fixed observable forces diagnostic-only or clean no-go.

6. Required Baselines

Every later H6S4-C report must compare the proposed route against:

Baseline	Role
Density-only expectation response	Conservative response depending on $\rho_{\mathrm{local}}$ or $\langle T_{\mu\nu}\rangle$, with no free branch variance.
Invalid ensemble branch response	Rejected bad rule used to expose decomposition artifacts.
Pointer-record response	Record-conditioned comparator; allowed only with declared local records, causal support, and ledgers.
Ordinary instrument noise	Probe or apparatus noise ledger kept separate from pulse-geometry noise.
H6S3 no-free-branch-variance theorem	The no-go control: chosen ensemble notation alone cannot create physical branch variance.
Known stochastic semiclassical gravity	Conceptual comparator to check whether H6S4-C is only a renamed known framework.

7. Result-Class Discipline

H6S4-C may be classified as controlled modification only if it supplies a concrete law before external comparison, with all ledgers complete and at least one fixed diagnostic observable.

H6S4-C must be classified as clean no-go if the minimal stochastic classical route cannot pass the required assumptions, collapses into an already-known framework without a distinct Pulse Model contribution, or requires an inadmissible input such as remote-basis dependence, hidden collapse, or fit-after-observation coefficients.

H6S4-C must be classified as diagnostic tool if the work remains a route ledger, baseline comparator, or failure diagnostic without a promotable law.

Only one of these classifications may be the final H6S4-C result.

8. Forbidden Overclaims

This appendix must not claim:

• a Pulse Model stochastic source-response law exists before it is stated
• metric quantization has been proved
• collapse has been derived
• branch-specific metrics are physical by notation alone
• ordinary instrument noise is new pulse-geometry noise
• an exploratory coefficient is a prediction
• a stochastic classical route is novel merely because it uses pulse-count language
• H6 classical-spacetime emergence is solved

9. Appendix Work Boundary

The route-scope skeleton is complete. The next sections state the minimal finite stochastic candidate and keep final status open until executable checks and adversarial review are complete.

10. H6S4-C.2 Minimal Finite Stochastic Candidate

H6S4-C does not choose an arbitrary ensemble branch distribution. The minimal candidate is a finite weak-field stochastic pulse-potential response defined by the spectral distribution of a declared local potential operator.

Let $\hat{\Phi}_C$ be a finite Hermitian pulse-potential operator for the source degrees of freedom inside the declared causal domain of probe clock $C$. Let its spectral values and projectors be $\phi_k$ and $\Pi_k$. The local source state is $\rho_{\mathrm{local}}$.

The objective classical stochastic variable is:

$$\Phi_C^{\mathrm{stoch}}=\phi_k$$

with probabilities:

$$p_k=\mathrm{Tr}(\rho_{\mathrm{local}}\Pi_k)$$

The probe pulse-count values are:

$$N_{C,k}=f_C T_C(1+\phi_k/c^2)$$

The response marginal is:

$$P(N_C=N_{C,k})=p_k$$

Equal pulse-count values are merged into one distribution point. No random sampling is part of the executable rule; the API must return the finite distribution and its deterministic moments.

The density-only baseline uses:

$$\bar{\Phi}_C=\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C)$$

and:

$$N_{\mathrm{exp}}=f_C T_C(1+\bar{\Phi}_C/c^2)$$

The stochastic variable may be written as a zero-mean pulse-count fluctuation around the density-only mean:

$$\delta N_C=f_C T_C(\Phi_C^{\mathrm{stoch}}-\bar{\Phi}_C)/c^2$$

This gives the fixed excess pulse-count variance:

$$V_C^{\mathrm{geom}}=(f_C T_C/c^2)^2[\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C^2)-\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C)^2]$$

The corresponding timing-jitter variance is:

$$\sigma_{\tau,C}^2=(T_C/c^2)^2[\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C^2)-\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C)^2]$$

No visibility-loss law is claimed at this step. H6S5 may map this variance to visibility only if it supplies a separate interferometer or clock-readout observable map.

10.1 Coupling Target

The H6S4-C.2 candidate couples to potential-operator variance in the finite weak-field source/probe arena:

$$\mathrm{Var}_{\rho}(\hat{\Phi}_C)=\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C^2)-\mathrm{Tr}(\rho_{\mathrm{local}}\hat{\Phi}_C)^2$$

It does not directly couple to phase density, pulse records, curvature records, or arbitrary branch labels. Pulse records are probe readouts and ledger inputs, not a hidden branch selector.

In a later covariant theory the analogous object would have to be a conserved stress-energy noise kernel for $T_{\mu\nu}$. H6S4-C.2 does not claim that full covariant law.

10.2 Causal Support

The operator $\hat{\Phi}_C$ is admissible only when its construction is tied to source data in the declared causal domain of probe $C$, or to records compared in a shared causal future. A spacelike remote basis choice may change an ensemble description of $\rho_{\mathrm{local}}$, but it cannot change the spectral probabilities $p_k$ unless it changes $\rho_{\mathrm{local}}$ or a declared local record.

Therefore equivalent decompositions of the same local state give the same stochastic response:

$$R_C(E_Z)=R_C(E_X)$$

whenever:

$$\rho(E_Z)=\rho(E_X)$$

The candidate has no remote-basis marginal dependence.

10.3 Ledger Status

Requirement	H6S4-C.2 status
Stochastic variable	Declared as the spectral classical variable $\Phi_C^{\mathrm{stoch}}$ of the local finite potential operator.
Coupling target	Potential-operator variance in the finite weak-field source/probe arena.
Causal support	Declared local causal-domain operator; no spacelike remote-basis dependence.
Conservation	Conditionally branchwise conserved only in the static finite weak-field arena where each spectral potential value comes from a conserved source record. Outside that arena the conservation status is not-yet-classified and blocks law promotion.
Coefficients	No free stochastic amplitude. The pulse response coefficient is fixed by the weak-field redshift factor $f_C T_C/c^2$. Any extra amplitude is exploratory-only and cannot support prediction status.
Regulator	No new regulator beyond finite operator dimension and any H6S1 softening or finite-size source input. Any softening radius must be declared before comparison and cannot be fitted after observation.
GR recovery	The mean response is the density-only weak-field response $N_{\mathrm{exp}}$.
QM recovery	If $\rho_{\mathrm{local}}$ is sharp in $\hat{\Phi}_C$, or if $\hat{\Phi}_C$ has zero variance on the state, the stochastic excess variance vanishes. No collapse of the source state is claimed.
Observable	Fixed excess pulse-count variance $V_C^{\mathrm{geom}}$ and timing-jitter variance $\sigma_{\tau,C}^2$.

10.4 Comparison With Known Stochastic Semiclassical Gravity

This candidate is conceptually close to stochastic semiclassical gravity: the classical response receives objective noise whose covariance is fixed by quantum source fluctuations. In the finite weak-field setting, $\hat{\Phi}_C$ is the local potential analogue of a stress-energy fluctuation source.

That closeness is a strength and a risk. It keeps the route physically disciplined, but it also means H6S4-C must not claim novelty merely for renaming a known stochastic semiclassical idea in pulse-count language. The possible Pulse Model contribution is narrower: a finite pulse-record response contract, strict no-ensemble-label gate, and a fixed pre-comparison pulse-count variance observable.

If later review finds that this is only known stochastic semiclassical gravity with renamed variables, the final H6S4-C verdict must downgrade to diagnostic tool or clean no-go rather than controlled modification.

10.5 Pre-Code Result Class

The H6S4-C.2 pre-code result is a controlled modification candidate, not an accepted law. It supplies a concrete finite stochastic response rule and a fixed excess variance before code is written.

The candidate must still be blocked or downgraded if executable checks or adversarial review find any of these failures:

• non-normalized or non-finite response distribution
• decomposition dependence for the same $\rho_{\mathrm{local}}$
• hidden use of arbitrary ensemble branch labels
• missing conservation ledger outside the static finite weak-field arena
• fit-after-observation coefficient
• undeclared regulator
• ordinary instrument noise double-counted as pulse-geometry noise
• equivalence to known stochastic semiclassical gravity with no distinct Pulse Model content

11. H6S4-C.4 Baseline And Failure Classification

The executable H6S4-C API compares the candidate against the required baselines rather than treating the stochastic variance as self-justifying.

Baseline	H6S4-C handling rule
Density-only expectation response	The stochastic response must have the same mean as the density-only response. Its only proposed difference is the fixed excess pulse-count variance.
Invalid ensemble branch response	Kept only as a rejected diagnostic. Its variance may expose a branch-label artifact but is never the stochastic law.
Pointer-record response	Kept only as a record-conditioned comparator. Pointer variance requires records and ledgers and is not H6S4-C stochastic geometry.
Ordinary instrument noise	Added only to observed variance as an apparatus ledger. It remains separate from $V_C^{\mathrm{geom}}$.
H6S3 no-free-branch-variance theorem	Equivalent decompositions of the same $\rho_{\mathrm{local}}$ must give the same stochastic response when no physical selector exists.

Route-failure classification follows these rules:

• missing coefficient provenance, conservation ledger, regulator provenance, causal support, or no-signaling guardrail gives blocked
• a diagnostic failure against H6S3 gives clean-no-go-candidate
• complete ledgers plus positive fixed excess variance gives controlled-modification-candidate
• zero fixed excess variance or missing no-free comparison gives diagnostic-only
• accepted_as_law remains false because the final H6S4-C verdict controls promotion

12. H6S4-C.6 Adversarial Physics And Ledger Review

This review stress-tests the finite stochastic pulse-potential candidate before the final verdict. The review distinguishes a coherent finite diagnostic from an accepted stochastic classical source-response law.

Issue	Classification	Review result
Ensemble-decomposition dependence	passed	The response depends on $\rho_{\mathrm{local}}$ and the declared local potential operator, not on an analyst's ensemble labels. Equivalent Z and X decompositions of the same density operator give the same response.
Remote-basis signaling	passed	A spacelike remote basis choice cannot change the probe marginal unless it changes the local density operator or a declared local record. The API comparison keeps the stochastic response invariant under equivalent decompositions.
Hidden collapse or pointer selection	passed	The candidate samples an objective classical potential variable for the response report but does not collapse the source state, select a pointer branch, or import pointer-record variance as the law. Pointer records remain a separate comparator.
Conservation accounting	blocked	Conservation is only conditionally accounted in the static finite weak-field arena where each spectral potential value is tied to a conserved source record. H6S4-C does not supply a general conserved stochastic source or stress-energy noise law. This blocks accepted-law and controlled-modification promotion.
Coefficient fitting	passed	The finite candidate has no free stochastic amplitude. The only pulse response coefficient is fixed by the weak-field redshift factor. Any added amplitude is classified as exploratory-only or fit-after-observation rejected.
Regulator dependence	diagnostic-only	The finite operator and any H6S1 softening are declared, but H6S4-C does not prove regulator-independent continuum behavior. This is acceptable for a finite diagnostic and insufficient for a law.
Ordinary noise double-counting	passed	Ordinary instrument noise is carried as a separate observed-variance ledger and is not included in $V_C^{\mathrm{geom}}$.
Artifact ledgers	diagnostic-only	The API requires an artifact ledger and the tests separate invalid ensemble, pointer-record, and ordinary-noise baselines. Real experimental artifact subtraction is not supplied at H6S4-C, so external prediction remains blocked.
GR and QM recovery	passed	The mean equals the density-only weak-field response, and the stochastic excess variance vanishes when $\rho_{\mathrm{local}}$ is sharp in $\hat{\Phi}_C$ or when the potential variance is zero. No source-state collapse is claimed.
Equivalence to known stochastic semiclassical gravity	diagnostic-only	The candidate is best understood as a finite weak-field pulse-record analogue of stochastic semiclassical gravity, with potential fluctuations replacing a full stress-energy noise kernel. That makes it disciplined but not a distinct accepted new framework.
Overclaims of new physics	passed	The appendix and API keep accepted_as_law false and reserve promotion for the final verdict. The present result must not be described as a new law or external prediction.

The adversarial review blocks a final controlled-modification verdict. The remaining honest final classifications are diagnostic tool or clean no-go.

The review does not force a clean no-go for all H6S4 routes. The finite stochastic response is coherent as a route diagnostic, passes decomposition and no-signaling tests, and produces a fixed variance comparator. The failed assumption is stronger: H6S4-C has not supplied a general conserved stochastic classical source-response law distinct from known stochastic semiclassical gravity.

13. Final Verdict

Final project-rule classification:

diagnostic tool

H6S4-C supplies a finite stochastic pulse-potential diagnostic, not an accepted stochastic classical source-response law.

The missing assumption is a general conserved stochastic classical source-response law distinct from known stochastic semiclassical gravity. The finite candidate is coherent inside the static weak-field spectral-potential arena, but its conservation ledger is not general, its continuum or regulator-independent status is not proved, and its structure is too close to known stochastic semiclassical gravity to claim a distinct controlled modification.

The accepted H6S4-C outputs are:

• a route-scope appendix and final diagnostic verdict
• a deterministic finite spectral-potential response API in src/pulse_model/stochastic_pulse_geometry.py
• focused tests in tests/test_stochastic_pulse_geometry.py
• a fixed diagnostic excess pulse-count variance comparator
• explicit baseline comparisons against density-only, invalid ensemble branch, pointer-record, ordinary-noise, and H6S3 no-free-branch-variance baselines

The rejected H6S4-C overclaims are:

• no accepted stochastic classical pulse-geometry law
• no controlled modification
• no new prediction
• no external comparison authorization
• no solution to the full quantum source-response problem

H6S4-C does not block all H6S4 routes. Pointer-record or collapse-selection routes still need a conservation-accounted selection law, and non-classical geometry or mediator routes still need a scoped sector and recovery limit. H6S5 may use the H6S4-C finite response only as a diagnostic observable comparator, not as a promoted law.