@@ -4,6 +4,50 @@ All notable changes to this project will be documented in this file.
 
 ## [Unreleased]
 
+### Added — 2026-09-11 Binary64 REMESH boundary and P2 causal binding
+
+- Centralized the production delayed-REMESH scalar evaluation in
+  `_delayed_remesh_kernel.py` and added an exact pairwise observer. It records
+  the Jensen denominator, ideal/raw/bounded separations and signed rounding,
+  clipping and total defects without promoting one pair to a uniform class.
+  A normal-valued `alpha=1/2` witness requires exactly `eta=2^210-1/4`, so
+  strict robust-envelope contraction requires `q<4/(2^212+3)`; a bounded hard-
+  clipped interval alone is insufficient for the existing `q=9/16` policy.
+- Added the sealed `alpha=1` hard-clip REMESH class. On fixed ordered support,
+  fixed positive metric and sufficient finite represented history inside one
+  interval, the runtime recurrence copies the global delayed row numerically,
+  has uniform relative defect `eta=0`, and preserves the class under REMESH.
+  Signed-zero bits, schedules, repeated event execution and future runtime
+  behavior remain outside this certificate.
+- Added the restricted global binary64 P2 half-Reception composition. With two
+  mutual singleton neighbors, an immutable all-target EPI snapshot, the exact
+  configured mix `0.5` and one common hard clamp, both proposals are
+  numerically equal for every finite represented pair in the source interval.
+  The numeric kernel gain
+  is therefore `q=0`; combined with `alpha=1`, active-history spatial
+  disagreement is exactly zero after `tau_global+1` cycles. The certificate
+  covers arbitrary finite repetition of these numeric EPI kernels while
+  withholding complete Reception-stage, grammar, live graph and solver claims.
+- Added a finite causal adapter for one executed P2 half-Reception stage. It
+  binds a sealed event result, two-phase EN evidence, ordered graph support,
+  runtime singleton neighbors, exact half mix, hard interval, metric ray and
+  captured endpoints to the global `q=0` kernel by bit-exact replay. REMESH
+  graph binding, auxiliary Reception state and repeated/future runtime claims
+  remain outside its scope.
+- Added a sealed finite causal P2 Reception/REMESH sequence certificate. It
+  binds every selected executor-owned EN stage to the global `q=0` kernel and
+  every same-cycle `alpha=1` hard-clipped REMESH to its represented global-delay
+  row with `eta=0`. For an observed sequence with
+  `N >= tau_global+1`, it verifies the active history suffix and exact
+  post-horizon spatial-disagreement extinction. Stale retained rows before that
+  suffix, future or unobserved repetition, auxiliary state, solver properties
+  and full TNFR stability remain outside its scope.
+- Added examples 173–176, including
+  `176_runtime_p2_reception_remesh_sequence.py`, exact public stubs, facade
+  exports and adversarial tests covering normal/subnormal extremes, signed
+  zero, inactive and active clipping, hostile inputs, private reseals and the
+  exact energy-gain-four witness for the canonical default EN factor.
+
 ### Added — 2026-09-11 Relative-defect REMESH/schedule stability
 
 - Added `remesh_schedule_relative_defect_stability.py` and its public stub. For
@@ -64,9 +108,12 @@ All notable changes to this project will be documented in this file.
   and `169_event_remesh_causal_runtime.py`. The outer executor runs ordered
   `EventRemeshCycleExecutionSpec` values on one graph in one transaction and
   seals ordinal/spec/result receipts, including exact schedule identity. Its
-  result retains the ordinary offline cycle observation and compatible runtime
-  schedule/history telescope, thereby certifying same-invocation causal order,
-  common graph identity and finite graph-owned atomicity. It does not compose
+  result always retains the ordinary offline cycle observation and, by default,
+  the compatible runtime schedule/history telescope. The explicit
+  `require_runtime_telescope=False` branch always stores `None`, so causal words
+  without a common affine metric still retain same-invocation order, common
+  graph identity and finite graph-owned atomicity; telescope-specific claims
+  remain false. It does not compose
   schedule and REMESH gains or prove a uniform positive normalized block
   margin over a declared forward-invariant runtime class, intrablock runtime prefix control,
   solver accuracy/order, mesh convergence, repetition or future stability.
 
@@ -467,12 +467,16 @@ executes a declared finite sequence of `EventRemeshCycleExecutionSpec` values
 on one graph inside one outer graph transaction. Each
 `CausalEventRemeshCycleReceipt` binds its ordinal, the exact submitted spec and
 the resulting cycle, including identity between the executed schedule and the
-spec schedule. The sealed `ExecutedEventRemeshCycleSequence` retains those
-receipts, the derived `ObservedEventRemeshCycleSequence`, and the compatible
-`RuntimeRemeshScheduleSequenceObservation`. This wrapper establishes
-same-invocation causal order, one graph identity and graph-owned atomicity for
-that finite execution. The two nested observers remain offline artifacts and
-retain their own negative provenance properties when used independently.
+spec schedule. The sealed `ExecutedEventRemeshCycleSequence` always retains
+those receipts and the derived `ObservedEventRemeshCycleSequence`. By default,
+`require_runtime_telescope=True` also requires and retains the compatible
+`RuntimeRemeshScheduleSequenceObservation`. Setting it to `False` always stores
+`runtime_telescope=None`; this permits an otherwise valid causal word that lacks
+one common affine schedule metric, and telescope-specific certificate
+properties remain false. Same-invocation causal order, one graph identity and graph-owned
+atomicity still hold for that finite execution. Downstream telescope and block
+observers require the default branch. The nested offline observers retain their
+own negative provenance properties when used independently.
 
 Causal provenance does not turn the finite telescope into a global executable
 gain or a theorem for repetition. Schedule and REMESH gains remain separate;
@@ -656,7 +660,8 @@ family or its forward invariance.
 
 [`observe_executed_event_remesh_relative_defect_block(...)`](src/tnfr/physics/runtime_remesh_schedule_relative_defect.py)
 checks that hypothesis on one nonempty contiguous block of an intact causal
-execution. At each boundary it reconstructs `J`, the ideal and bounded head
+execution that retained its compatible runtime telescope. At each boundary it
+reconstructs `J`, the ideal and bounded head
 energies, the signed defect and its exact slack; requires the represented
 schedule gain to be no larger than the declared `q`; and verifies the complete
 history-energy vector against `diag(q_eff,1,...,1)P`. It then checks the finite
@@ -668,6 +673,76 @@ future binary64 stability remain open.
 presents both an exact-zero defect block and a positive binary64 defect
 accepted at its exact minimum `eta`.
 
+[`observe_binary64_remesh_pair_relative_defect(...)`](src/tnfr/physics/binary64_remesh_relative_defect.py)
+then resolves the represented-number boundary pairwise. For two temporal input
+tuples it replays the shared production REMESH kernel and records the exact
+Jensen denominator, ideal/raw/bounded squared separations and signed rounding,
+clipping and total defects. One normal-valued `alpha=1/2` pair requires
+`eta=2**210-1/4`, so strict contraction needs
+`q<4/(2**212+3)`; a bounded hard-clipped binary64 interval alone cannot support
+the existing `q=9/16` witness policy. This observation is local to one pair and
+does not certify a uniform class.
+
+[`certify_alpha_one_hard_clip_remesh_class(...)`](src/tnfr/physics/binary64_remesh_relative_defect.py)
+supplies the first uniform represented boundary. On fixed ordered support, one
+fixed positive metric, fixed positive delays and sufficient finite history
+inside a common hard interval, `alpha=1` copies the global delayed row
+numerically. Its relative defect is uniformly `eta=0`, and REMESH alone
+preserves the class. Signed-zero bits, a schedule family, repeated event
+execution and future runtime behavior remain outside the certificate. Example
+[`173_binary64_remesh_relative_defect.py`](examples/02_physics_regimes/173_binary64_remesh_relative_defect.py)
+shows both the obstruction and this boundary class.
+
+[`certify_p2_half_reception_remesh_stability(...)`](src/tnfr/physics/binary64_p2_reception_stability.py)
+closes a restricted global numeric EPI-kernel composition. On two mutual singleton
+neighbors, an immutable all-target EPI snapshot, the exact configured
+`EN_mix=0.5` and one common hard clamp, the two Reception proposals are the same
+reversed sum of half-scaled operands. They are numerically equal for every
+finite represented input pair in the source interval, including underflow
+cases, so the global
+centered-energy gain is `q=0`. Composed with the `alpha=1`, `eta=0` REMESH
+class, the active post-schedule history has zero spatial disagreement after
+`tau_global+1` restricted cycles. This proves arbitrary finite repetition of
+the numeric EPI kernels. It does not certify the complete Reception stage,
+grammar admission, live graph/event execution, a global affine binary64 map or
+solver behavior. Example
+[`174_binary64_p2_reception_remesh_stability.py`](examples/02_physics_regimes/174_binary64_p2_reception_remesh_stability.py)
+reports the exact composition and its signed-zero boundary.
+
+[`certify_executed_p2_half_reception_stage(...)`](src/tnfr/physics/runtime_p2_reception_stage.py)
+crosses the next finite causal boundary. It selects one Reception event from an
+intact `OperatorEventExecutionResult` and requires executor-bound two-phase
+stage evidence, the exact ordered P2 targets and mutual singleton runtime
+neighbors, binary64 mix `0.5`, the source hard interval and metric ray, captured
+left/right endpoints, and a bit-exact production-kernel replay. The resulting
+certificate applies the global `q=0` theorem to that one executed EPI stage and
+records its finite grammar admission and whole-schedule graph atomicity. It
+does not bind REMESH history or configuration to that graph, audit every
+Reception auxiliary write, retain current live-graph identity, or prove future
+or repeated runtime stability. Example
+[`175_runtime_p2_reception_stage.py`](examples/02_physics_regimes/175_runtime_p2_reception_stage.py)
+reports this executor-bound endpoint result and its negative scope.
+
+[`certify_executed_p2_half_reception_remesh_sequence(...)`](src/tnfr/physics/runtime_p2_reception_remesh_sequence.py)
+binds that stage result and the `alpha=1` boundary to one completed
+`ExecutedEventRemeshCycleSequence`. For every observed cycle it identifies a
+selected executor-owned EN stage with the complete schedule EPI endpoints and
+the global `q=0` kernel, then identifies the same cycle's applied hard-clipped
+REMESH with its exact represented global-delay row and `eta=0`. Support, metric,
+interval, delays, history capacity and runtime alpha-source provenance remain
+fixed. If `N >= L = tau_global+1`, the active suffix of exactly `L` chronological
+history rows remains inside the source interval, and the recorded post-horizon
+and final post-REMESH fields have zero spatial disagreement. When
+`tau_local > tau_global`, that suffix is shorter than the full source-class
+history requirement; the local-delay rows are nevertheless inactive because
+`alpha=1`. Older retained history rows remain outside this claim. The sealed
+certificate concerns
+only that finite same-invocation trace; it does not certify future or unobserved
+repetition, auxiliary Reception state, current live-graph state, solver
+properties or full TNFR stability. Example
+[`176_runtime_p2_reception_remesh_sequence.py`](examples/02_physics_regimes/176_runtime_p2_reception_remesh_sequence.py)
+records the finite causal extinction boundary.
+
 [`observe_runtime_remesh_history_bridge(...)`](src/tnfr/physics/runtime_remesh_history_stability.py)
 binds one applied executor-sealed cycle to that exact companion. It converts
 the retained runtime history to newest-first order, replays the nested